So, What IS the Point? 2
The voice of the
caterpillar 3
Philosophy and
Science 6
Primitives: a
toast to bottom‑up 10
Science,
Sociology, Substance 17
Science, Dogma, and Scientists 18
Occam, Ockham and all that 19
Magic. 21
Cliché: Why is
There Anything At All? 22
Is There Really Anything but Solipsism? 23
Why "Why?"? Well, if Anything:
"Because why". 25
Operation à la mode 29
The Stubbornness of Underdetermination. 30
Semiotics,
Language, Meaning, Comprehension 33
Fundamentals:
Choose your turtle 40
Fundamentals:
Axioms and Assumptions 41
Brass Tacks 47
Preconceptions,
Mathematical and Other 47
Pure
Contention 47
Science, evidence, and near‑proof 56
Guess, Grope, Gauge, Accommodate 59
De‑, In‑ and Abduction 60
Common sense and logic. 71
Conjectural lemmata 72
Gedankenexperimente: Thought experiments 73
Infinity, Finity,
and Cosmology. 75
Entities and atomism and not much confidence. 75
Atoms in spaces. 77
What does it take to make a dimension? 81
What Then, Is Physical Algebra? 84
Cause, causality, and implication 86
The Emergence of Emergence 101
Nothing with no time, and no time with
nothing 103
Media, Messages, Observations, & a Hint
of Semiotics 104
Entities in emergence and reductionism 105
Putting together timelines to please Theseus 111
Prediction of the Knowable and Unknowable 113
Emergence and epiphenomena 117
Levels of emergence 118
Generic and Specific Emergence 120
Emergence, Scope, Scale, and Type 123
Tomic Entities, their Origins and Fates. 127
So What? 129
Generalisation, Reductionism, Reductionistic fallacies 129
Existence, Assuming it Exists. 132
Indestructible
Information 137
Existence of Entities in Formal systems. 142
Existence of Entities in Material systems. 149
Euclid’s and Others’ Excesses 152
Nothing Determined 157
Determinism, Information, Time's Arrow 160
Fanatics
. . . . may defend
. . . . a point of view
so strongly
. . . . as to prove
. . . . it can't be true.
. . . . . . . . Piet Hein
A large part of the content of this
essay deals with philosophical topics that have been chewed to rags again and
again for millennia rather than centuries. I am not even a philosopher, so what
qualifies me to rouse the sleeping dogs yet again?
That I cannot say in detail, but many of the
standard philosophical questions I discuss, mainly questions relevant to science,
still puzzle and confuse most people, including me; and yet those puzzles seem
to lose their substance, sometimes even their interest, when regarded in terms
of information and techniques for handling information. And some of my points
seem to me to have attracted too little serious attention.
The effects are so strong that I increasingly
find it difficult to read some fairly new works, 20th or even 21st
century, for sheer irritation at the persistence with which they miss the
essence of several pivotal problems. I’m not saying there are no good new
works; some of the later works, and even some of the early works, are excellent
and inspiring, but the frustration persists down the works of the centuries.
Is this just ignorant arrogance on my part?
Possibly, but if so then my views should be easy to refute, and to refute with
something better than pointing out (often correctly) that I plainly had not
read Descartes and Kant and Nietzsche and Hegel and Marx and Tolstoy, and who
was that other fellow ...? And particularly had not read them in the original
editions, or languages. So I bare my breast: make the most of the opportunity.
In particular, I find a few of
the works of scientists on the philosophy of science superior to the nearly all
of the works of philosophers on the philosophy of science. A particularly fine
example is Arthur Eddington's "Philosophy Of Physical Science", in
which he showed how much more valuable it can be for someone writing on the
philosophy of a subject to know the subject, than to know philosophy. After
all, plenty philosophers writing on philosophy show how it is possible for a
philosopher writing on philosophy, not to know the subject either.
And current popularity of sources need not be
cogent: for example I am not keen on Popper either, even though he still is
fashionable among undergraduates and even graduates who have not thought
seriously and independently about falsification. As I see his work, it looks
self‑indulgently shallow and erratic; not that I claim to be the only one with
that opinion.
As for what I offer here, I offer it with
little apology; I hope anyone who reads it will do so with enjoyment, even if
only sardonically, and that some readers will profit from it immediately, but
that they will profit even more in due course.
One matter I do apologise for, is the
structure of the essay or at least its lack of structure; I would have liked to
present everything in a simple, logical sequence, but I am increasingly
persuaded that there ain't no sich thing. The world consists of bits, items, that
entangle each other in all sorts of ways at once, so I have to wander as I
wonder. I beg no forgiveness, but I do at least assure you that it is not by
choice that I am incoherent, much less out of malice.
I know almost
nothing about almost nothing,
and absolutely nothing about everything else.
However, I do not vaunt this accomplishment
as unique, or even unusual,.
So, who am I?
As Alice did in her day, I in my turn recuse
myself from answering that question; in my case because I assume no authority
and assert no matters of opinion as matters of fact, so it seems to me that the
question of who I am is hardly relevant.
As for authority, I am not immune to
authority, and life is too short for rejecting all authority. And for my part,
I claim no authority but my own opinion (an ill-favoured thing, but mine own)
and my own opinion remains my own opinion.
All the same, authority, however
respectable and respected, constitutes, as such, neither proof nor logic; as a
healthy principle, remember how Horace put it in classical Roman times, and the
Royal Society in the 17th century reaffirmed: “Nullius in verba”. I am partial
to a bit of that myself, and I commend the principle to readers.
For many reasons however, that
principle itself is not an absolute; people speak of believing only what you
can see, and rejecting whatever is stated as fact without support and all that
hard-nosed, commonsense, intellectually independent Good Stuff, but resources,
particularly time and one's own intellect, are limited. Consider: one cannot
personally verify the truths in one's own field of knowledge, let alone all
human knowledge. Even mastering the sense or establishing the factuality of
every statement made inside a university lecture theatre is beyond anyone. We
cannot delay Biology 101 while each student personally verifies every
individual assertion presented in class. Nor can we delay Biblical Hermeneutics
101 while each student personally decides whether to accept the book of Job as
literally true or as allegorical, in conformity with the associated curriculum.
The best support I can offer for my own views
is observation, deduction, conjecture, and speculation, with perhaps too much
rationalisation. Of course the fact that I do so from inside the world I
inhabit, entails certain limitations. I do not claim to be any better than a
product of my times, nor to be original. As Kipling put it:
Raise ye the stone or cleave the wood
to make a path more fair or flat;
Lo, it is black already with blood some Son of Martha spilled for that!
Not as a ladder from earth to Heaven, not as a witness to any creed,
But simple service simply given to his own kind in their common need.
Mind you, I do try to credit my sources when
I am conscious of them. Bierce said, in the introduction to his Devil's
Dictionary: "This explanation is made, not with any pride of priority
in trifles, but in simple denial of possible charges of plagiarism, which is no
trifle." People who reject my versions of observation or those of
anyone else, may do so without incurring my resentment, but without necessarily
securing any commitment of mine to accept or even respect their preferred
opinions.
If I so much as offer food for thought, and
raise questions I cannot answer, but that others might build upon, it would
please me to think that I stimulate anyone to think to some purpose, possibly
to attack material problems, or possibly for fun. Or both.
From time to time while writing this, I
understandably have been partly encouraged and partly disappointed, though not
at all surprised, to find that some aspects of concepts related to those that I
have formulated, and still am developing, were emerging in mainstream mathematics
and physics and science and life in general long before I started; so certainly
some aspects of my views are far from new.
As I said: that is no surprise. However, I
had encountered the concepts by idiosyncratic routes of my own, so interested
persons might find some to be worth another look in the hope of fresh
perspectives, even if some of those concepts are not at all new.
That is why I now write this essay.
As for style, format, and similar items,
readers might like them or write their own essays to their own tastes.
For example, some readers dislike epigraphs
and quotations.
Tough.
I like mine.
Philosophers say a great deal about what is absolutely
necessary for science,
and it is always, so far as one can see, rather naïve, and probably
wrong.
Richard Feynman
For many centuries philosophy was confused
with science, largely disastrously. In the period during which science and
philosophy were such small fields of study that many polymaths could master all
the known works, there were so many holes in both classes of study that both
classes suffered: science suffered from the preconceptions of the philosophers
of the day, and philosophy suffered from misconceptions of the scientists of
the day.
Then the eruption began to break through.
That volcano had been rumbling for a long time, but for several centuries,
starting perhaps about the time of Galileo, developments in the study of
reality, increasingly had become so coherent and compelling that science was
seen as increasingly threatening to the dogmata of the authorities; those
authorities became really nasty about those threats; it was a long time before
the thugs began to realise that no matter how fast you burnt heretics, there
was a limit to how deeply you could bury the implications of reality,
especially if you yourself didn't understand the implications of the reality.
Or for that matter, the implications of the
dogmata.
Or even the implications of the actual
beliefs and conceptions of the believers, irrespective of the dogmata.
Many of those authorities still have not
realised that if they were to write down their dogmatic claims, and all copies
of their texts were lost or forgotten, those dogmata would never resurface in
the same form again, except as variations on the same weary rhetorical
fallacies that bullies use to justify their bullying. In contrast, the facts of
life and the fruits of valid reasoning, as revealed by the activities of
researchers, if similarly lost, would keep resurfacing for as long as new
children are born asking inconvenient questions and sometimes following them
up. Especially if those children grow into scientists seeking cogent answers,
convenient or not.
Those dogmatists failed to understand that
the essence of their problem was that they were trying to persuade water to
flow uphill. Commonly they still fail in that understanding. Those who fail
most catastrophically are the wishful thinkers who resent their own inability
to grasp scientific advances, and the parasites who sell the dogma to wishful
thinkers, whether the dupes have scientific ambitions or not. Occasional
specimens who do in fact have the mental equipment, may react so bitterly
against their own fears or resentment of reality or authority, as to deny the
undeniable.
Not that that bothered Jane and Joe Average,
who regard it as a virtue to follow the herd in ignorant, irresponsible loyalty
to the cruel, greedy, and ridiculous.
All such classes of wishful thinkers are the
lapdogs of those Authorities who have axes to grind, and who will stop at
nothing to support their social and intellectual parasitism: their dupes or
victims are those who just want to accept, without effort, comprehension, or
doubt, whatever their exploiters assure them, irrespective of any basis in sense
or fact or honesty.
So much for universal education once the
politicians get their claws into the rabble.
"I saw it on TV" or "My friend
saw it on the Net" or "The experts all say that (check one of: cooked
food, raw food, red meat, boiled water, dairy products, or gluten) will give
you straight hair and curly teeth…"
If there is a limit to such slavishly
mindless idiocy, I have not seen it yet.
At some time, arguably during the nineteenth
century, science grew too large and too obtrusive for the Authorities to
ignore, or for most polymaths to master — not that some of them
didn't delude their fans into thinking that they had mastered it all the same.
By about the mid‑20th century, after the flood of discoveries and
reasoning had grown beyond the capacities of any one person, that air of smug
omniscience faded into gnashing of teeth and bewailing of specialisation and
silo mentalities and interdisciplinarity.
In other words some of the would-be
Authorities could no longer support their pretension of personal access to
omniscience, and they hated to admit it. Of course, some still refuse to admit
their ignorance; or they even represent it as a virtue, but those impress only
the dupes who are unsalvageable anyway.
But none the less, those dupes that commonly
constitute the voting majority or thug secret police.
For my part, if you read much of this
document and think that I think I know a lot, let alone know everything, or
have access to knowledge of everything, then don't bother to ask for your money
back: you didn't read carefully enough to earn it.
The philosophy of science in days of old
lagged simply because most philosophers weren’t deeply or functionally
interested in science itself (to judge from the topics that they published or
things they said) or weren’t scientifically trained to sufficient depth to
enable them to support or broaden and deepen their views; or they were writing
on the basis of assumptions that the scientific competence of the day could not
adequately deal with. There still are many such writers. Some are scientists
who had not turned to philosophy until they had passed their philosopause, and
so their philosophy tended to be flaky, even if perchance their science once
had been competent.
Sound science and sound philosophy do not guarantee
each other, but unsound philosophy does augur poorly for robust science.
On the other hand, many modern professional
philosophers who did try to write on the philosophy of science tended to be
writing on the basis of their own readings of the science of two or three
generations earlier; commonly the results were so pathetic that by about the
mid 20th century there was increasing backlash from working
scientists, who could tell that for a long time the philosophers in question
hadn’t known what they were talking about.
Unfortunately, scientists who, in their
reaction, impatiently reject the philosophy of their own fields of expertise
wholesale, thereby cripple their ability to advance their own fields of study
and practice. They thereby invite stagnation, confusion, and even
retrogression.
The field of philosophy of science in which
the practitioners have produced the worst of such superficial nonsense in my
opinion, is metaphysics, not that I am prepared to defend that opinion. To
describe the basic concepts underlying reality, metaphysical work is necessary
in certain contexts, such as in establishing basic concepts of reasoning and
knowledge, but all the same, no field of human thought is more in need of
continuous sceptical criticism.
Except religion perhaps.
Because metaphysics is so open to arbitrary
mystical ideas from writers who see their own intuitions as cogent, necessary
truths, metaphysical work is prone to rampant pathological growths if not
checked regularly for consistency with empirical evidence.
And "evidence"? What is that?
Evidence
is any information that might rationally affect anyone's assessment of the
relative strength of relevant rival hypotheses.
In classical or pre‑classical times, some
schools of philosophy rejected all inconvenient demands for consistency between
logical and physical evidence, on their assumption that reason was infallible,
whereas our senses were subject to error. Well, however fallible our senses,
reason certainly is fallible, as we can tell from the radical, and often bitter
disagreements between reasoners, but anyone who nonetheless insists on the
infallibility of reason, falls foul of the implications of reason itself, as
follows: one important formal logical operation is that of implication,
and in formal logic a truth cannot imply anything but truth, so reason tells us
that any proposition that predicts an observation that contradicts our actual
observation, cannot be true.
That is not the whole story, of course;
real-life, commonsense implication is a treacherous beast at best, but still,
the challenge of reason by reason remains powerful.
Typically, mystical or metaphysical ideas
that predict wrongly, accordingly are neither cogent, necessary, nor even true
in the sense that their predictions, if any, are novel and successful. This is
adequate demonstration that the philosophers in question indeed hadn’t known
what they were talking about. Furthermore, like so many religious
fundamentalists, their "cogent, necessary truths" might have been
more interesting, if not necessarily conclusive, if their respective apologists
could persuade each other.
Commonly they do not. Quot homines, tot
sententiae. And the very fact that rival arguments of reason could
disagree, implied the fallibility of reason, of observation, of interpretation
and of assertion. Disagreement means that at most one of the disputants could
be at least partly right on each point; it also is perfectly possible for them
all to be radically wrong about whatever they see as truths that they would be
willing to kill for. And not just wrong in minor degree or detail, but
radically wrong in principle as well as gross matters of fact.
It happens ...
Conversely, whenever the failures of reason
can be ignored, that very fact implies that the reasoning itself has no
real-life relevance, and therefore should be dismissed or ignored. What
metaphysics, independent of science and reality, might be good for, apart from
formal philosophy, is not of much concern in this discussion.
Intrinsically, this essay is largely a
discussion of applied philosophy — not of purely formal
argument.
The backlash from the working scientists
stemmed largely from the late 19th century and onward, when the
frontiers of science started encroaching on fields in which the results looked
like nothing better than nonsense to the layman, and were fraught with traps
for philosophers who had failed to keep informed on so much as the bald facts
of the disciplines, never mind their implications. In this respect they
themselves qualified as laity in relevant aspects of science.
A mild example of how far philosophers'
brusque, down-to-Earth pronouncements on science can be downright wrong:
...philosophers
have said before that one of the fundamental requisites of science
is that whenever you set up the same conditions, the same thing must happen.
This is simply not true, it is not a fundamental condition of science.
The fact is that the same thing does not happen,
that we can find only an average, statistically, as to what happens.
Nevertheless, science has not completely collapsed.
Richard Feynman
In the early 20th century the philosophic implications of
scientific advances began to develop a lot of traps for the scientists too.
There still are rival camps bandying more insults than insights ...
And what was worse, some such philosophers
simply did not realise why it mattered. In the laboratories and the
field on the other hand, there was a growing tendency to comprehensive
rejection of philosophy of science, especially from the “shut up and calculate”
school of thought.
That might seem to be pretty attenuated cause
for concern, but philosophy is supposed to be the discipline that deals with
thought about thought, and science is a field that is so dependent on a high
standard of thought, that there is hardly a discipline more important to
educated scientists than philosophy of science. If that is beyond him, how can
we see him as more than half a scientist, if that? Half-sense is rarely better
than non-sense. A practitioner in a scientific field, one who eschews thought,
might well be a genuinely valuable hack worker — never let it be said
that I disrespect hackwork — but it is unusual for hack workers to
produce fundamentally novel intellectual or material breakthroughs.
Genuinely valuable data and technology they
often deliver, but, however important, those are something other than
intellectual breakthroughs. It is in such aspects that we see the differences
between the roles of theoreticians and experimentalists. Neither category is adequate
on its own.
I do not suggest that the experimentalist is
necessarily a hack worker mind you! Brilliance in creativity occurs in all
walks of life and all sorts of functions. And theoreticians can be quite as
pedestrian and as intellectually adequate as any jingle writer.
In some circles, both philosophers and
scientists, there has long been a tendency to contemn the history and
philosophy of science as trivial or useless, but lately there has been some
improvement. History and philosophy of science currently are burgeoning as
fields of study, and publications reflect the tendency. I’d hate to have to
supply supporting figures though, especially figures for how valuable specific
works in the philosophy of science might be.
Still, in the last few decades my readings
largely have been disappointing. Material I find about communications and
mental processes and reductionism and emergence and many more, solemnly
masticates themes that commonly seem to me to be repetitive, redundant,
irrelevant, out of date, or downright mystical. They seem to ignore the nature
and relevance of the role of information in the processes and principles
and concepts under consideration.
Maybe I just have been unlucky in my reading;
I hope so. But has my bad luck been bad enough to explain my misapprehensions?
Now, a quick word on one
of my heroes, Richard Feynman, and his views on philosophers' views on science.
In that context I did not get my views from him, hero or not. He is alleged to
have asserted, among other uncomplimentary pronouncements: "Philosophy of
science is about as useful to scientists as ornithology is to birds". It
does not much matter how accurate that attribution might be; it is in line with
some of his other remarks. But we can discount his invective as irrelevant,
because Feynman was not above making unphilosophical assertions about science
and unscientific assertions about philosophy. And he himself commonly made
philosophical assertions about science; often penetratingly.
Far more relevant would be his views on the
reliability of the views of experts, in particular:
"Outside
of their particular area of expertise
scientists are just as dumb as the next person".
Quite.
I agree, but with the reservation that I
regard the very term "scientist" as sloppy terminology: as I
see it, the relevant concept is not the role to which one might apply the term
“scientist”, but scientific behaviour as an activity. And someone who by
calling or profession is committed to scientific behaviour, is no more to be
relied upon not to deviate from it, by either error or intention, than an
ecclesiastic can be relied upon not to deviate from virtue.
But, as a layman in philosophy, speaking
about philosophy, Feynman didn't do too badly on average: perhaps somewhat
better than laity speaking about science. As you may see from some of my
quotes, quite a lot of his lines are straightforward philosophy of science,
though he did not claim them to be anything of the sort; not in any claims that
I read, anyway.
Everything should be built top-down, except the first
time.
Alan J. Perlis
One of the basic concepts in science and
philosophy, very near to something ineffable and undefinable, is the idea of a thing:
an entity. I am not sure whether to regard that concept of
"entity" as a primitive, either at all, or possibly as a class
of several primitives that deserve separate comprehension. Whether the concept
of entity is a primitive or not however, I usually say "entity" when
I speak loosely of something's "thingness" and say "thing"
when I refer to something — some entity — without necessarily
considering any particular aspect of its attributes.
With apologies to every reader familiar with
the concept of primitive concepts, my excuse for explaining the term is
that I have on occasion been abused by persons who thought that "primitive
concepts" or "primitives" had something to do with
savages, and that I was sneering at such savages as being inferior, mentally or
otherwise. Or they thought that to speak of something as "primitive"
was to disparage it as being appropriate only to savages.
So bear with me here: a primitive concept in
a particular context is one that we take as basic, as something
given, something that cannot or need not be broken down further into
anything simpler, that cannot or need not be explained in terms of more
fundamental concepts, or that in context we have no need to simplify further
for the purposes of our discussion.
Typically the importance of a primitive is
that we can use it as the basis or part of the basis for more complex concepts
or relationships, much as we use axioms in mathematics as the bases from which
to derive theorems of arbitrary complexity.
But the term is relative. I do not assert (I
just do not know) that in our universe there exists any such thing as an
absolute primitive, but primitivity is a convenient
concept — possibly even a necessary concept — in a universe such
as ours, in which there is no physical capacity for infinities, so I do assume some
absolute primitives, whatever such things might be; indeed, I make that
assumption as a matter of convenience.
And sure enough, the concept of primitives
crops up frequently, including in this essay. Just do not assume when you see
such a reference, that I am under the illusion that I am formally proving
anything; take it as a concept on which one can base assumptions for the sake
of discussion, analogous to the Euclidean assumptions of points, lines, planes,
and the like, none of which has any physical reality.
Now: as I see the idea of an entity,
or use the term, it is whatever you could think in terms of. Whether it
is a well‑demarcated physical object such as a crystal, or it is a poorly‑demarcated
object, physical or otherwise, such as a ball of fluff or a crowd or a concept
or a river or a species or an event or the state of something (say fluidity, or
spin, or colour, or anger) or whether it is an imaginary object such as a
unicorn, or an abstract object such as a number — anything you could give
a name to if you wished to discuss it — you could, in a suitable context,
regard it as an entity. Sometimes there are tricky examples, such as objects
that cannot be distinguished in terms of quantum mechanics: when two electrons
from separate sources have collided and have recoiled along separate pathways,
then if it is at all meaningful to say which of the two outgoing electrons
corresponds to which incoming electron, we do not know how to say which is
which, either for lack of means of measurement, or, more fundamentally, for
lack of physical meaning to the idea of their having respective identities at
all.
But it often is unclear whether such marginal
examples are relevant in practice.
An entity could be atomic, a concept I
discuss later, meaning that that entity cannot be split into simpler units,
either at all, or without changing its nature. So far, so simple, but at this
point I introduce a neologism: I started writing about splittable items as
being “non‑atomic”, but the clumsiness of a word whose meaning amounts to “non‑non‑splittable”
became irksome, so I have changed all those references to “tomic”.
I could not at first find that back‑formation
in use anywhere else, but the term seemed to me to be convenient in this sense,
so I present it here. My apologies to anyone who hates the word, but my need
was the greater I think, and I am the current speaker so, when you are the
author, you may choose your own terms: tomic, atomic, or a-atomic; but till
then, suffer!
Since writing that, I have indeed found
"tomic" and related terms used in the study of poetic scansion,
referring to pauses between words, but even that does not seem to be in wide
usage, and threatens no conflict with its semantics in our current entitic
context, so let's proceed, tomically or otherwise.
Whether the entitic nature of any particular
entity is in any way intrinsic to that perceived entity, or whether it is an
effect of that entity's existence in the world, or its relationship to the
world, or whether it is a vacuous mental delusion, an intellectual crutch in
dealing with the world, I do not address here; but I do not know how to do
without that crutch or convenience, so ...
Examples of entities, tomic or atomic,
not all of them distinct, might be, say:
- individual elementary atomic
units (atomic in the sense of being primitive, not being tomic,
neither physically nor logically divisible into simpler or smaller units
or categories) or
- any recognisable status,
action, or quality, such as a smell, a colour, momentum, emotion, or
- sets or structures of elementary units or entities united by particular
relationships (such as adhesion, repulsion, location, or resemblance),
each set regarded as an entity, for example a crowd, or a pile, or an
aggregate, or
- more generally, any relationship
that unites entities into a less primitive entity, such as a membership
of a constellation, a flock, or a team, or
- relationships between entities, such as processes, events, ideas, reputations, recipes, or
concepts that exist in the form of relationships between neurons, and
possibly between neurons and perceived objects.
- clearly or vaguely defined or
delimited, as we see in complex structures or in clouds or impressions.
- More generally still, in some
senses the reality of an entity could include its relationship to every
other entity in an observable universe, but for practical purposes,
this might be too obsessive to take seriously at this point in our
discussion. It would however, involve enormous complications: for example,
imagine four bodies roughly equally far from each other, as seen from a
central fifth body, each of the four barely within the limits of the
central fifth body's observable universe, as defined by its red-shift
boundary; then it would take billions of years for any event at any one
body to have any effect on the other four. I do not deny that the quintet
would comprise an entity in various contexts, either abstract or material,
but I am unsure of the implications of the concept; suppose each of the
five to comprise a civilisation on its own planet — then suppose each
sends a message to each of the other four; For the rest of them the
central recipient of the messages would have to play relay station, and
for each message, the recipient might no longer exist when it arrives, and
the message dispatcher would have no way of knowing whether the source
civilisation still exists until the reply arrives, much less exists in the
same form as when the message was despatched. Less dramatically, but more
difficult to resolve, would be the question of whether any of the outer
four would be within the other's observable universe. Such communication
would at the very least lend a new dimension to the exhortation to: "Cast
thy bread upon the waters: for thou shalt find it after many days."
Very many days indeed ...
Now: in dealing with the world, and managing
one's own patch of it, the concept of entities seems to my limited imagination
to be unavoidable in various ways.
For instance, it has become a cliché in naïve
computing or design circles, that there exists a right way to design complex
structures (entities, if you like) such as programs and bridges, and that such
a right way necessarily must be "top‑down". I, for one, have used the
top‑down concept repeatedly throughout my career, and would go on doing so indefinitely.
But not invariably.
Roughly speaking, top‑down design amounts to
conceiving the desired outcome first, only thereafter conceiving the major
components and their nature and operation, and then their components in turn,
stopping only when you have everything you need for the desired end‑product.
This might sound silly to the uninitiated, but it really is a very powerful
principle, and its sophisticated application can be effective in ways that
amaze or confound a sceptic. Beginners tend to be puzzled or irritated at the
disciplines of top‑down work, but beginners commonly are easily puzzled or
irritated anyway, and once they eventually begin to become comfortable with top‑down,
it helps them through the struggle.
However, there is a trap. It is true that top‑down
commonly is more powerful and generally faster and better suited to teamwork
and modularity and accommodation to future developments, especially in dealing
with complex problems — but it relies on the existence of components that
are familiar, well‑understood, and commonly recognised, components that one may
use in the design or construction. Such components might be primitives, or they
might be suitable perspectives and tools, whether primitive, simple, or
complex; as entities they must be recognisable in the same form, and available
in such a form, to all the relevant parties, and that is more than we generally
can rely on. In working on a new material, if we rely on our familiar nails,
hammers, screws and screwdrivers without examining the implications and
options, the results might turn prove embarrassing, expensive, or disastrous,
say if the workpiece is an expensive glass, a dangerous explosive, or a new
environment.
Even in programming, top-down can lead naïve
practitioners into unsuitable algorithms with unobvious inefficiencies, special
case logical traps, legal liabilities ...
So, when the necessary components are
unavailable or poorly understood, one commonly needs to begin by establishing
the necessary primitives. Items that later prove to be of value to top‑down
designs, often have been discovered independently of the application, without
any idea that they might be useful beyond our immediate needs. Some we stumble
upon; some originated as whims, speculations or toys; examples include
computers, saws, money, flour, telephones, vaccines, rockets, radio, wheels,
and antibiotics.
Such things most people see as primitives;
forgetting that they once were new, and still are anything but primitive.
The top‑down bigot, before assimilating the
relevance of both aspects in any intellectual or practical field, on
encountering a refractory problem, is likely to degenerate into the mindset of:
"If it still don't work, I gets a bigger 'ammer!" That is no more
competent than the troop of monkeys climbing the biggest tree in sight, saying
"Look! We are going to the moon!" That is a valid example of the top‑down
approach: they see the moon alright, as well as the objective, and they see the
direction to travel, but they have a lot of bottom‑up work to complete before
their project is better than futile. And their final solution had better not
rely on that tree.
Thinking of the intrinsic hazards of top‑down
work, I am obliquely reminded of the brilliant Harris cartoon I saw decades ago
in the Scientific American:
On the other hand, with all its associated
temptations to reinvention and confusion, bottom‑up discovery and design is
ubiquitous in creativity. The more bottom‑up concepts and tools people master,
the more powerful their options for top‑down design. After the bottom‑up
creation, the new resources should be documented and made available for future
use. We shouldn't be reduced to re‑inventing our familiar nails, hammers,
screws, and screwdrivers for every bit of carpentry. Accordingly, for example
in mechanical engineering, whole books have been published, showing thousands
of previously invented mechanical movements. Designers may refer to them for
either instruction or inspiration when they recognise a need in a particular
project.
Such remarks might sound terribly prissy,
self-justificatory, and academic, but consider an example of the practical
hazards of common-sense top-down approaches in innocence: some decades ago a
buried electric cable was laid across country in rural India. Suddenly
the cable failed, and the power supply with it. Locating the fault in the cable
proved to be unusually difficult and local residents were mystified and the
technicians frustrated. Eventually did find the break in a field where the
buried cable had been snapped by a plough. The ploughman had realised that the
break was unacceptable, so he fixed it: he knotted the two broken ends together
and re-buried the knotted part. After all, that was simple common sense;
everyone knows that when a cord breaks, you fix it by knotting the ends
together: see a problem, solve the problem.
And the remedy worked too! No one came to
shout at the ploughman, so patently his fix had been satisfactory; no bottom-up
thinking, no problem! Top-down wins again!
Such blunders are not limited to rural
naïveté; so many years ago that cars still had a choke lever on the dashboard,
a woman took her new car back to the garage and complained that its fuel
consumption was ridiculous. The puzzled mechanic began an examination, and
noticed suddenly that the choke lever was pulled all the way out; he asked why.
"Oh that," said the woman, "I never use it, so I just
keep it pulled out to hang my handbag on."
In top-down terms, that was perfectly
reasonable; simple decision deferral: bag gotta hang, hang it on something
obviously otherwise useless, but adaptable for hanging things. That is what
such hooks are for, right? After all, why else would the lever be that
shape?
Not only car maintenance and cables, but
computer hardware and software war stories, abound with analogous cases.
Such examples illustrate whole classes of
prerequisites for either bottom‑up or top‑down design. Too narrow a view can be
fatal in various ways. The bottom‑up discovery of a way in which a couple of
rocks can be persuaded either to conduct an electric current, or to interrupt
it, might not immediately suggest that multi‑billion‑switch, super‑fast
computers, small enough and cheap enough to be wasted on domestic devices,
could be based on that principle; conversely, top‑down designers might fail to
understand why climbing trees could not be the way to get to the moon, or why
knotting and burying broken cables could be anything to make a fuss about.
Similar considerations apply to the
conception and design of perpetual motion machines and homeopathic remedies.
Flying machines were conceived top‑down for
millennia, but were rejected as impossible before the necessary developments in
aerodynamics, aerostatics, mechanics and combustion engines were created,
largely laboriously bottom‑up from the point of view of achieving flight.
And so it is with engineering, science and
philosophy. In real life a distressingly large proportion of progress
— and in particular, of wasted progress, amounts to standing on the toes
of predecessors instead of standing on their shoulders. And intrinsically,
climbing onto shoulders is largely bottom‑up.
One needs to balance outlook and context, and
be ready to explore and explore. John Donne had the right of it, in saying that
he, that will reach Truth, about must and about must go ...
Other concepts that might be considered in
various contexts as primitive or nearly so, are variously defined and widely
disputed, and my discussion of them here I suck out of my own thumb, and it is
not to be taken as gospel. They include:
- Information, where it means something like: whatever states distinguish
the relative acceptability or adequacy of alternative hypotheses. It also
might be seen as: whatever relationships between entities, affect the
relative probability or outcome of alternative physical events.
- Randomness is hard to define, and different people define it in different
ways for different purposes, but for my purposes I define it here as lack
of information as I have just described information. It takes at least
two forms:
- Where sufficient
information to determine a state may exist (did that coin fall heads or
tails? Is the cat in that box alive?) but is not available to the subject
or observer, or:
- Where
information does not exist at all to determine a given question, not to
any observer in any sense, and not to "nature" itself
(when will that unstable nucleus decay?)
- Probability is the degree to which one might regard any particular
hypothesis concerning existing states as being stronger or weaker than
another, in the light of the available or existing information.
This implies that different observers, or the same observer at a different
time or other different coordinates, might rationally assign different
probabilities to the same set of conceivable events.
As an adolescent I aspired to lasting fame, I craved
factual certainty, and
I thirsted for a meaningful vision of human life - so I became a scientist.
This is like becoming an archbishop so you can meet girls.
Matt Cartmill, anthropologist
It has been a long time since I craved
factual certainty, partly because I am deeply sceptical of the idea that
anything of the type is accessible at all. I am equally sceptical of whether
the concept of factual certainty itself has any substantial meaning in our
world in general, or can have meaning in brains of our human type in
particular. I am confident however, though without formal proof, that I exist
in a world that does exist and that does briefly include me, in whatever sense
that such concepts might make sense at all; I am in fact sufficiently confident
in writing this essay, to relegate to the indefinite future, impotent
speculations on unobservable worlds beyond my immediate topic.
Furthermore, I am confident that this
universe of my perception intrinsically comprises certain objects and certain
interactions of objects, that behave in such ways as to achieve certain classes
of consistency — what we might call logical behaviour. Reasons for this
view, I discuss later, but only superficially.
Any list of the various views of the
philosophy of science, such as positivism, empiricism, instrumentalism,
materialism, or falsification, in their various forms and combinations, would
exceed anything that I could afford to deal with here. Accordingly I have not
even classified my own view formally, and am not even much interested in trying
to do so. Life is too short.
However, one of my pet irritations is when
people confuse science in terms of its subject matter, with the alleged
sociology, psychology, and related views of science and scientists. I do not
claim that every such topic is without interest or importance, but it is not of
primary interest here; I pay little attention to the works, much less the
conclusions, of the likes of Kuhn, Derrida, or Feyerabend, irrespective of some
thoughts they propound among the drivel
My interest is in the nature, rationale, and material significance of
scientific activity itself.
And the fun.
Insofar as they may be coherent, my philosophical
views of science are along the lines of those called realist philosophy, though
I am not necessarily conventional according to any recognised school of
scientific realism (variations are many). By my own version of realism I mean
in essence that I see the observable universe, including myself, as existing —
and existing in a form reasonably consistent with the impressions we can gather
from empirical evidence (evidence of our senses, as some like to call it) and logic (the evidence of our
sense, insofar as we can deal with
it). I do not insist that this view is correct, but anyone wishing to
convince me of anything to the contrary, will need impressive powers of
persuasion and argument.
To save readers speculations on my
underlying. intentions and illusions, I include a few remarks on important
assumptions about science. I do not justify them here, because all my thoughts
dangle together, and I can't cover everything in just one essay. Nor in several
others I have written elsewhere, for that matter.
Too bad.
Or maybe not so bad: here goes.
Philosophy is questions that may never be answered.
Religion is answers that may never be questioned."
quoted by Daniel Dennett
Definitions of
"science" are varied, hackneyed, and largely uncomprehending; and the
definition of "scientist" is, if anything, worse. One correspondent
(impressively qualified at that) in disagreement with my view of science,
quoted points of some august body's definition (Royal Society? Can't remember —
doesn't make no neverminds). And yet that definition was transparent nonsense:
it dealt with examples of good practice in science — proper experimental
design, controls etc.
All good stuff in
itself of course, but naïve in the extreme and other places; it didn’t even
address the question: that of the definition of science.
More realistically,
the following is closer to the point, derived from text I produced elsewhere:
Science in the
sense that we are discussing, is the opposite of religion in that, far from recourse
to dogma as the ultimate basis of authority for defining the fundamental
assumptions, much less recognising dogma as a basis for justification of choice
of action, science intrinsically has no conceptual scope at all for
ideological dogma.
Science does not
even deny dogma, any more than religion denies noise.
Science, or to
be more precise, scientific practice,
is in essence the application of a range of processes for finding and using
information in constructing, identifying, urging, or selecting, the strongest
candidate hypotheses to answer any reasonably meaningful question, whether
deductive, inductive, or abductive, and whether in contexts that are formal,
material, or applied. No appeal to dogma, in fact, no appeal to any assertion
at all, whether empirical or philosophical, transient or eternal, has cogency
in science in this sense, because the only means available for convincing
persons who refuse to accept your arguments, is by letting them convince
themselves in the light of available evidence, including any evidence that they
unearth for themselves. And conversely, your adversaries' options for
convincing you of their views, are equally constrained in turn. There is no
guarantee that either party is more or less right.
Informality of
wording apart, that is an approach to a definition of "science" in
the sense of "scientific behaviour".
All that stuff
about controls and Bayesian theory and Ockham’s razor, and predictive power and
more, is well and good, but in itself it isn't science, just lists of
components of good, effective, practice and principle. None of it promises
correct or even ultimately predictive conclusion or formal proof — science
isn't about formal proof any more than about authority, but rather about
selecting the currently best-supported working hypothesis, while perpetually
considering either improvements, or total replacements to current concepts and speculations.
As for what a scientist is, I do not
regard the question as very useful; if I were to meet someone wearing a hat with
big red letters saying “scientist” it would no more impress me than a hat that
says “Lion Tamer” (or MAGA, for that matter).
“Scientist” might appear in your job
description in some contexts, but the impression it conveys to Janet or John
Average would not be very informative.
Perhaps the term “scientist” could
better be justified as designating one’s vocation rather than anything else.
Essentia non sunt multiplicanda praeter necessitatem.
(Essential assumptions are not to be multiplied beyond
necessity.)
William of Occam
(attrib)
Because it is not the main point of the
essay I will say little about Ockham’s razor; it is anyway a topic more
honoured in its incomprehension than its application. Some claim that it is a
social construct of no substance, and I am not inclined to waste my keystrokes
on their naïveté. Others speak of it as in effect so much of a holy writ that
to invoke it is sufficient to eliminate from serious consideration, any
proposal that they dislike. That too is not worth serious attention. Yet others
regard it as the basis of science in every respect. They at least can raise
arguments of some substance, but I reject their assertion as too simplistic to
be sound in general.
His term “essentia” is slightly confusing. I have translated it as: “essences”,
or “essential assumptions”, but you might find it helpful to think in terms of
“basic ideas” or something similar; or a more compact colloquial expression, such
as “KISS”: “Keep It Simple, Stupid!”
For my part I regard Ockham’s principle as
being healthy in the current philosophy of science, and certainly an insight of
brilliance in its day. Its merit remains relevant in our time, but like many a
valuable precept, it requires good sense in its application. In this
requirement it resembles all worthwhile principles in life in general, and in
science in particular.
And what does it apply to? Anyone (in our
sense in particular, anyone with any interest in scientific principles) should
bear Ockham in mind whenever seeking to understand a concept or phenomenon in
terms of its associated phenomena. Are you really sure that you need to insist
on this point or that, as essential to your thesis? Even if it is valid, even
if it is correct, then if it can be considered separately, that is what should
be done. There is more to the razor than just eliminating what is false, or
even doubtful. Considering concepts in isolation is as important as considering
their roles in combination. Combinatorial problems can be as misleading and
expensive as outright errors, they can change the behaviour of individual
items, and they may mask outright errors.
Still, even if we have isolated our issues
competently, there is no single way to achieve perfect understanding, either
comprehensively or uniquely simply; there always will be more to the subject of
study than we can describe and comprehend, and more scope to speculate on its
nature than we can in principle dismiss in terms of necessary logic. William of
Ockham advised in effect: to address the difficulty by cutting it down to
manageable proportions by discarding all assumptions that you can do without for the present.
Now, the best way to do this is, in my
opinion, to guess at what at first seems most obvious, but bear in mind that you
will be oversimplifying it in some ways, and overcomplicating it in others. No
problem, don’t panic, this is what you are in the game for if you are any kind
of scientist.
Suppose, one wishes to see whether a thin
sheet of material such as paper, used as a bridge, can bear the weight of say,
a dry utensil, such as a knife or spoon; it can barely bear its own. But if you
fold it concertina-wise it can bear hundreds of times its own weight.
This illustrates the difficulty of applying
the principle of the razor: there is more to it than the elimination of
essential assumptions — there is the related concept of the simplicity of
the essential assumptions: the modern version could be paraphrased as: “Make
everything as simple as possible, but no simpler” Attributed as is much else, to Einstein.
But simplicity is anything but simple: you
can measure simplicity in terms of the number of components, which might mean
that you need to increase the number or complexity of the principles invoked.
Alternatively you might be able to improve the conceptual complexity by
increasing the number of components.
Suppose someone is reporting the tosses of
a notionally fair coin or die. You keep tossing it and the output looks pretty
random, so random that you get suspicious; could it be that there is an
intelligence in the coin that keeps its output apparently random? Or an
intelligence that is passing out an encoded message? If so, could the first
fifty million heads-or-tails really be an encoding for the full works of Shakespeare?
Yes, it certainly could, that is elementary
information theory. But is it a helpful assumption? Certainly not. That sort of
assumption is just what Ockham warned us against.
A less subtle challenge would be to guess
the shape and numbering of the tossed object. Given numbers from 1 to 6 would
suggest what? A cubic die? But there are many shapes and numberings that could
give you that; a mindless adherence to Ockham would immediately leave you with
the problem of defining simplicity: what is the simplest shape for a die that
fairly yields numbers from 1 to 6? And what is the simplest numbering pattern?
And how many fair numbering patterns are there for any given value of 6n? And
what shape that could in principle give you values from say 1 to 12 or 14 if
tossed often enough, but arbitrarily rarely; how often would you have to toss
them before Ockham and Bayes could warn you that the assumptions you currently
hold, multiply your essential assumptions insufficiently.
For example, it is possible to get a fair
six-valued die with any of 6n-sided polyhedra, And most people would guess that
the simplest fair hexahedron would be a cube, in which n=1. And they would be
wrong; there are simpler fair hexahedra in which n still equals 1. A fair
triangular hexahedron has fewer apices than a cube, and fewer edges. And what
about a fair sphere, where n arguably is zero?
(And yes, I possess fair spherical dice
that give unambiguous readouts!)
And where n increases in value, the
polyhedra that meet the fairness requirement, increases rapidly.
Ockham was a great of his time, but his
admonition needs to be applied with care and with thought, then and now.
This is clear when one examines the history
of science. One begins with an idea (commonly abductive, as I mention later) and
if it does not fit newly emerging observations, you may simplify it, or
compound it, or discard it entirely and replace it with something new, either
simpler or more complex.
Ockham spoke of multiplying assumptions unnecessarily; he offered no criticism
of multiplying essential assumptions necessarily.
Practically any branch of science could be
adduced to illustrate these principles, but astronomy-and-cosmology would be
the most visually dramatic: Heavens rotating round the Earth, Earth-centric
sun, Solar-centric planets, stars as solar-(stellar?)-systems, nebulae as
clouds, galaxies as milky ways of many types,.
Ockham’s razor would have no special role
in many of such surges in progress, in which many of the steps led to barely
thinkable changes of viewpoint.
It certainly is true that in any particular
view of a field, Ockham’s principle might be useful, even vital, but it is no
substitute for basic principles of science, such as re-thinking theories when
they begin to conflict with new observation or logic.
In short, Ockham’s principle is not all
there is to science, especially exploratory science; his principle is primarily
of parsimony, which is good, but pioneering observation, insight, abduction,
are important as well, and so is explanatory richness. When proposed theories
have been formulated, there is plenty of time to consider reformulating them
more parsimoniously, or even rejecting them outright.
It is called research.
If all our common‑sense notions about the universe
were correct, then science
would have solved the secrets of the universe thousands of years ago.
The purpose of science is to peel back the layer of the appearance
of the objects to reveal their underlying nature. In fact,
if appearance and essence were the same thing,
there would be no need for science.
Michio Kaku
I use the term “magic” several times
in this discussion, italicised to avoid confusion with more familiar contexts,
so let’s clarify what I mean by it.
I do not mean what Arthur C.
Clarke meant when he said that any sufficiently advanced technology was
indistinguishable from magic. He did have a point of course, but that is not
the point I deal with here.
From time to time in this essay, I propose
more or less Socratic thought experiments in which I exclude some
practical considerations, or internal contradictions, or I introduce
unjustified assumptions that are incompatible with real life, or where I invoke
impossible powers, always purely for the sake of illustration.
For example I might speak of doing things
with something too large to fit into our observable universe, or of balancing a
vertical tower of a few dozen loose snooker balls, each on the one below. It
might be something mathematically describable on suitable assumptions, or it
might not, but it generally would be something that is not practically
possible — such as violation of thermodynamics.
To support anything of the type in the real
world would need magic.
So, if I speak of “magic”, that is all
there is to it: nothing to do with superstition or witchcraft or anything
occult or mythology: it is pure analogy or abstraction for convenience in
illustrating a principle — not proof, please note, just hand‑waving to
avoid getting bogged down in unconstructive quibbling.
Over your head
The rigid, pure, persistent ray
Pierces the darkness like a blade,
Wherein is no thing seen
Save that the dust-motes in their millions
Eddy and play
In carols and cotillions,
Until it breaks upon the screen,
And then
Appear the shapes of driving clouds
And desperate men
Sailors in the shrouds
Of labouring ships,
Sails shaking,
Seas breaking,
Men and the sea at grips;
The empty, lifeless band of light
On unimaginable waves
Carries the terrors of the stormy night,
Dragged from their graves,
And makes to live again
The struggling men
In your sight.
Just so our earth,
"With all its striving and its stresses,
Its tears,
Its mirth,
Its loves and hates.
Riven souls, relentless fates,
Cities proud and haunted wildernesses,
Is not, as men have guessed,
Some god’s uneasy dream,
Or selfish jest,
But just the interruption of a beam.
Arnold
Wall The Cinema
Yes, why is there something rather
than nothing (assuming anything at all)?
Is There Really
Anything but Solipsism?
It is necessary for the very existence of science
that minds exist which
do not allow that nature must satisfy some preconceived conditions.
Richard
Feynman
Let's first deal with the idea of solipsism: that nothing exists but
myself, and even then your existence or mine is only a personal delusion:
essentially a nothing that vainly fancies that it exists.
Well, I can fancy myself denying or imagining
the existence of gods, unicorns, and Rumpelstilzchen, in some senses, but how I
can imagine my own existence unless I exist to do the imagining, defeats me. I
am not a great fan of Descartes' cogito ergo sum, but it is not without
point. To imagine is by definition to do
something (imagining!) and in my opinion anything that at least does anything even if it does nothing
more active than to lie as a doorstop and exert a gravitational effect, thereby
exists; that practically amounts to the definition of existence
As definitions go, it has elements of
circularity, but so does cogito ergo sum, so I move in exalted circles.
But solipsism leaves me with major unresolved
questions. For example: "cogito ergo sum" itself: "I think, therefore I am"
might establish my existence, but only if we concede similar assumptions about
other physical realities; for example it would not deny: "You claim to
think therefore you exist", and various similar aphorisms or speculations
that various entities contribute to events. For that to be possible would
immediately and intrinsically negate solipsism as strongly as solipsism would
negate non-existence. It would mean that if that implication was valid for me
and my existence, it would be equally valid for other agents and operations.
Not to mention the argument of my perception of them as constituting existence.
Still not proof, but then "I
think therefore I am" also is not proof; as Perliss, for example, pointed
out:
"You
can’t proceed from the informal to the formal by formal means".
To me it equally is not clear that one could
proceed from the formal to the informal by formal means; one certainly cannot
prove the completeness of a formal system from within that same formal system.
So immediately I reject solipsism: it is
unpersuasive at best, vacuous, and most certainly based on an arbitrary
assumption, namely that thinking implies existence; even if that were in fact a
cogent assumption, it would not follow that it is the only ground one could
find to support the concept of existence.
To begin with, I introduce the concept of an algebra
of physics. There will be more about that later, but to avoid confusion
at this point I define algebra here without explanation and
without originality, as follows: an algebra, whether formal or material,
consists in a set of objects or object types, plus a set
of operations
on objects in that set. We most familiarly think of numbers as the objects that
algebra operates on, but that is not a logical requirement; there is an
indefinite range of algebras, operating on an indefinite range of object types
or categories of types, and numbers are just a limited subcategory.
Now suppose we assume that a universe exists,
how could it be meaningful to assume that a universe with no algebra — no
patterns of implications of events — could exist? Having no algebra would
make it very difficult to define an event, and to eliminate the concept of an
event would practically define non-existence.
And an assumption such as "I think
therefore I am" is an operation of implication, and accordingly is a
component of an algebra of physics. Imagination in turn is a class of thought,
and as such is crudely physical, comprised of operations on information in a
brain — all of such things being fundamentally physical.
And I claim that to deny that what is
physical exists, is a contradiction in terms, because denial comprises physical
objects, operations, and events. We may neglect the denial as being
self-nullifying.
And how do I justify my claim that
imagination demands and comprises information? Because there are many different
alternatives to any item of imagination, You might imagine some thing, and I
some item that conflicts with that thing, and again, I might imagine first one
thing, and then another, different thing, and information is what it takes to
determine any difference between things, whether the differences are intrinsic
in nature or are external coordinates.
For example I might imagine a round bottle
and a square block of ice (two things that intrinsically differ materially) or
imagine a near bottle and a far bottle (differing only in coordinates) —
telling such imagined objects apart is a matter of information.
Those who assume the delusionary nature of
this world, commonly assert that the alternative assumption — the
assumption of the actual existence of an enormously complex material
universe — is invalid because it falls foul of Occam’s razor; but that
view is open to disqualification from two objections: firstly Occam’s razor
itself actually is more of an arbitrary assumption, a rule of thumb, a sensible
practical principle, rather than a cogent disqualification of every new idea;
in practice many an intellectual or technical advance does demand the
introduction of new concepts. This may be come about either by splitting an
existing concept into distinct concepts, or by introducing a radically new
additional concept.
Truth is not to be determined by counting
concepts.
Obedience to Occam’s exhortation is no more
than a healthy mental habit: that of parsimony of concepts; once make an idol
of it, and you find that, as with other idols, there is no breath at all in the
midst of it.
Such creation or recognition of a new
concept, Ockham never denied, but many who invoke his words seem to think that
they automatically are justified in kicking up a fuss any time that they fancy
that anyone is introducing a new concept. And, since the world is large and
knowledge and language are small, new concepts continually crop up and need
identification and assimilation.
But even without rejecting or constraining
Ockham, the assumption that we exist, and that each aspect of the world exists,
more or less as we perceive it, is less of an assumption than that every
thing that we appear to perceive is not there, but just an ad hoc illusion.
That illusion comprises information. Someone or some thing that creates or
comprises the information is just as unlikely as the information itself, and it
would introduce a go‑between just as complicated as the physical thing plus the
go‑between.
Therefore the parsimonious assumption is that
the world does exist, rather than that I personally imagined it. I still might be
wrong, but if so, I propose that my view falls down mainly in my inability to
identify what, later in this essay, I shall refer to as the bottom turtle,
the truly basic assumptions that I am unequipped to identify or characterise.
And as far as I can tell, our world is a
world of implication and consequence — but it does not follow that it is a
world of complete or perfect information. More about that later.
Trying to answer rhetorical questions instead of
being cowed by them
is a good habit to cultivate.
Daniel C.
Dennett
What I write here will not delve deeply into
the question of why things exist, firstly because the way that question
is asked usually amounts to a non‑question. In fact it usually turns out to be
an elaborate exercise in question begging.
The question itself implies preconceptions of
many sorts, such as the meaning of: "is", or "existence",
or "concept", or "event", and first and perhaps most of
all, to speak of "why".
The very idea of existence is troublesome; I
take it for granted here, because the question itself takes it for granted. In
a later section of this essay I discuss existence, and establish it as a useful
concept in context, but it is too long for this section, so let it wait.
As for "why" ...!
For years I have regarded
"why" — and the associated "because" — as being
among the most treacherous words in human languages — in various forms
their various senses and meanings differ radically, and people confuse them,
often so badly as to make no sense at all.
In one sense the sorts of things people mean
when they say "why" or "because" commonly have to do with
the way things happen: "this might be expected to cause that, or imply
that, or this did cause or imply that, or this frequently or always will cause
that, or be followed by that". Things tend to happen according to what
we might call the logic or algebra of the universe, and, in particular senses,
words like "why" and "because" deal with that assumption
that things happen because the reigning rules of the world imply them, whether
in the past, present, or future. In fact in some languages the word
"why" literally means something like "how come" or
"for what"?
In such terms, "why" and
"because" (or, if you prefer, "wherefore" and
"therefore" or "caused by" or something similar) have to do
with the logical operations: implication and inference, or if you
prefer, consequence, or deduction. And those are among the most
fundamental logical operations, both in formal logic and in dealing variously
with what we see as reality. In logic or mathematics as opposed to physics,
"why" and "because" refer to how axioms lead to the
theorems or conclusions that follow from them — how they imply them —
and from which axioms, assumptions, and arguments a conclusion might have
followed.
And yet, that difference is artificial: in
physical science, we empirically find that the world wags in a particular way:
that events lead to other events when entities interact, and that the
interaction is not in all cases purely arbitrary, but partly predictable, at
least in principle and according to rule. Based on that finding, we might be
willing to say why a billiard ball might or might not fall into a
particular pocket, or if it already is in the pocket, in how many ways it might
have got there, and in which other pockets it might have landed instead, or why
we do not believe that the ball appeared as a pigeon that landed on the cloth,
turned into a billiard ball and rolled into the hole.
From that
point of view, we argue that the universe behaves according to certain
rules — not a book of rules that anyone promulgated, but rules that
reflect the way things consistently happen and cause other things to happen.
In short, such is the way that events and
states follow other events and states. In our universe the algebra does not
facilitate the conversion of pigeons landing on billiard tables and turning
into billiard balls. It may not, it usually does not, predict that exactly the
same thing will follow the same causes in exactly the same way every time, but
the ways events imply other events commonly are fairly consistent, and
sometimes are highly precise.
Why do we argue that way? Well, for one
thing, it is hard to imagine a universe without constraints on what sort of
events follow particular precursors. In fact I suspect that, given the way any
universe might work, the concept of an empty universe, or the non-existence of
any universe anywhere, or the existence of any universe that has no particular
behaviour patterns, somehow leads to some internal inconsistency. Don't ask me
to justify that suspicion. And don't ask me to imagine what sort of fundamental
primitives could be the basis of an internally inconsistent universe: I boggle.
Anyway, later I shall discuss the concept of
an algebra of physics in more
detail.
The rules of physics are at least as
constraining in most formal disciplines as they are in empirical studies. For
example if we are told that the result of an unspecified calculation (boringly)
is 211 in decimal notation, then we can find all sorts of calculations that could
have given that answer.
We also, arguably more importantly, could
find all sorts of calculations that we know could not have given that
answer: say, multiplication or addition of any two prime numbers. (Bear in
mind: no matter what logic you prefer, in current number theory, the number 1
is defined as not being a prime! This is because each prime has
precisely two distinct divisors, whereas 1 has only one divisor: 1. And that is
not someone's idle whim; it has important consequences. I do not discuss it
here, but if you doubt me, read up on the Fundamental Theorem of Arithmetic).
For a more crudely physical example, if we
are told that two isolated bodies in free fall in space, in mutually stationary
circular orbit around each other, will remain in that orbit until they are
disturbed from outside. Again, that will not tell us how they came to be
in that orbit, but we can identify all sorts of things that could not
have given rise to the situation.
So far so tedious, but important.
People often speak of those ways in which
things happen, or apparently tend to happen, as “laws” of “science” and similar
names. Such terms are harmless, as long as one does not confuse the patterns of
events with laws in the sense of human laws. The two concepts have little to do
with each other. One might as well speak of "laws of arithmetic"
being something to do with human legal systems. In fact, if that use of the
word "laws" means anything at all, it really means something more
like: "the way things happen when one applies the various operations
defined in that arithmetic".
The "laws" also have not much to do
with science: science does not make rules, though scientific work might
lead to the discovery of rules, patterns of events, and, at some level,
to developing some idea of why and how some of those rules apply.
As it happens, there is yet another aspect to
our real world, something that many people do not realise; in fact many would
categorically reject the idea, although it is inescapable and very important:
Sometimes
there is no “why” nor any "because".
In other words, as I shall point out, some
things do happen randomly in some
respects — truly, inescapably, randomly.
As I use the term “randomly” in this
connection, that means not just that we ourselves happen to lack the
necessary information to guess why
one particular thing should happen rather than another; it also means that no
information in the universe exists, that determines that that particular
thing should happen rather than some other; at least not until after the event
has happened, and often not even then.
Such missing information physically does not
exist — not for you, not for me, not for Schrödinger's cat, not at all.
Also, it means that if there is some such information, but not
enough to determine the outcome absolutely, then such non‑definitive
information favours some notionally possible outcomes rather than some
others. This means that such information makes those outcomes more probable:
“more probable” means that such information as does exist makes the more
probable outcomes occur more frequently if that class of event is repeated
indefinitely.
How far the principle of non‑existent
information applies to the tossing of a fair coin or a die, such that it comes
to rest more often on a face than on its edge, I cannot say, but it certainly
would apply to the fact that there is a greater probability that any particular
isolated atom of an isotope of uranium will spontaneously undergo alpha decay
rather than that it will split into say, barium plus krypton plus a job lot of
neutrons.
With due respect to Einstein, and due credit
to Hawking: God most assuredly does play dice, and does so on a scale
that numbs the mind. Of that more later, but for now we can ignore the point,
though we shall have to return to it.
As I see it, those two types of questions,
logical and physical, do not differ in essence, because I am of the opinion
that mathematics is a branch of physics, and not vice versa.
(Should I prove to have been wrong in this, the two types, logical and
physical, still need not differ fundamentally, but let that go for now.) I
accept and affirm that information is in at least
some real, literal senses physical.
Mathematics, I repeat, is
a branch of physics: without physics there can be no information, no
implication (as opposed to determinism), no relationships, and therefore no
logical operations. To eliminate all such things would at a stroke eliminate
all forms of mathematics, whether applied or purely formal, whether meaningful
or just meaningless noise. Relationships and logical consequences just could
not apply, let alone exist in any sense, without at least the physics of
information, and conversely, given physics, or any conceptually related system
in any imaginary universe, the possible mathematics pops out as a consequence,
an abstraction, of the way things are.
Accordingly, even the
most purely formal mathematics is real and material, just as the physical
observations are real: it follows that there are no points or lines; and
identical bosons can share each other’s location, whereas identical fermions
cannot.
These things follow from
the nature of the objects in our universe, plus the operations that they can
undergo.
A universe in which no
events could happen, even in principle, would be hard to define, let alone
imagine, and one in which events do not affect or determine each other, whether
rigidly or not, would be no better. On the other hand, a universe in which events
arise as plesiomorphically consistent interactions between entities, would
necessarily imply causal behaviour: formal operations on objects, with
consequences in line with their probable outcomes. Those operations and objects
would simply be aspects of the physical algebra of that universe.
By way of naïve example,
to imagine a non-trivial universe without those implications, would make as
much sense as expecting water not to splash when poured; whatever happens in a
universe is part of the consequence of the algebra of that universe. And the
formal and material nature of the algebra and behaviour of the universe are
implicitly inseparable; each is part of the same thing. To ask that three plus
three not formally make six, and vice versa, is to ask that six eggs not make
half a dozen eggs.
Go ahead and try to
imagine such a universe; you will find yourself in difficulties because you are
trying to do it with a brain and an algebra constructed and operating according
to the algebra of our current universe.
Of course, some
metaphysicists might deny that any physical "realities"
"exist", but if that were correct, then metaphysicists could not
exist either, so what could the maunderings of nonexistent metaphysicists
matter?
Furthermore no
mathematical or logical state or assertion can exist meaningfully and no
operation, formal or otherwise, can be performed on any entity, without
constraints imposed by information — information
that in turn cannot exist without mass/energy/space/time and all the things
that, in one form or another, in one way or another, in one combination or
another, make up our basic existence as far as we can tell, and do so according
to that algebra.
Consider: in principle any formal statement or proposition can be manifested mechanically
or materially (for example, by writing it in ink, shouting it into a void,
chiselling it into tablets of stone, typing it into a mechanical calculator,
constructing a computer to model such a statement, or impressing it into a
brain) and, in principle, any such mechanical representation in turn can be
formally described at least at one further level of abstraction. Also
therefore, in principle, any mechanical system can be mechanically abstracted
or duplicated at least at one more level by some other mechanical system. In
each case, some imperfect representation, some plesiomorphism, is involved. (I
discuss the term plesiomorphism in a coming section; pending that, it
will do to think of it as similar to "isomorphism".)
Conversely, nothing of any of those types can occur or persist or be stored or
transmitted without physical media.
Information generally is exponentially smaller than the material system of
which it constrains the states, but without the material and its states, it is
nothing. No material, no information, and without either: neither and nothing!
Purely formal operations, such as binary or
other Boolean implication a→b, meaning a<b, should not be confused with
logical consequence. Formal operation deals with truth values, not with the
states, events, and objects that might have encoded or transmitted or embodied
the relevant truth values or their processing, and the fact that two values are
consistent with an implication-relationship has nothing to do with cause or
consequence.
To deal with the relationships and nature of
the objects on which the operations of an algebra may be performed, we need an
extra concept; that of modal operations, modal logics,
modal algebras, that can take the natures of the objects into account. The concept
is enormously important, but too large for me to do more than to recognise in
the context of this essay.
Without due consideration of modal concepts
equality of measures, such as cardinal numbers or truth values, does not mean
identity of the properties of the populations subjected to the operations of
comparison. That would be analogous to arguing that four apples are identical
to the number 4, so that four bricks would be as good as four apples for making
apple pie.
Similarly, implies-and-is-implied-by
relationships between truth values does mean quantitative equality of the truth
values, but does not mean identity of the values derived (the number 1
applied to drops of water, does not behave like 1 applied to
probabilities or ball‑bearings.) and 1 ball‑bearing is not identical to 1
thousand events or 1 event or 1 probability.
Such modal considerations can be very
profound or very obvious, but in either case are very important and not safe to
neglect. They generally include assumptions that are context-dependent, such as
being related to time, ethics, mathematics and many more.
As for what information fundamentally is: I regard information in any
given form or context, as being any physical state that distinguishes
alternatively possible, perhaps hypothetical, states or other entities from
each other.
Pardon the vagueness of my terminology, but I
am unsure that I have the vocabulary to put it more plainly. In fact I am not
sure that humanity has yet defined such a vocabulary, let alone generally accepted
and comprehended one.
A scientist seeks the truth, wherever that may lead.
A believer already knows the truth,
and cannot be swayed no matter how compelling the evidence.
Anonymous
Wherever we cannot definitively and uniquely
assign the origins and causes of observed states or events, we say that origin
or causes are underdetermined, meaning that more information would be
necessary to distinguish some alternatively possible causes or origins, or
to associate observed states with whatever had led to them.
Suppose for example that I saw a coin
tumbling down, a coin that I accept to be fair, and I watch it in due course
settling flat on what I take to be a fair plane surface. Suppose that from my
position I could tell the location of the coin, and that it landed flat, but
that I could not see sufficient detail to decide whether it shows Heads or
Tails, as seen from my side of the plane. If so, I still would need enough
extra information to tell the toss.
In principle, one extra bit (binary
digit) of information could suffice. I already have a great deal of
information, such as that the coin did not settle on its edge, nor shatter on
landing, nor land somewhere else, but my information still is not complete, and
I propose that I never could have complete
information about any material
situation. All the same, such an extra bit of information already is enough
to limit the item of usual interest either to Heads or to Tails. Had we
tossed an octahedral die instead of a coin, three bits (equivalent to one octal
digit) would have been necessary to say which face was up. And a fair
hendecimal die would require one digit to the base 11 (nearly 3.4595 bits), to
express which of its 11 faces had come up.
(Challenge: design such a die.)
A more fundamental example of
underdetermination is: you find an ordinary six‑sided die lying with one face
up — how many alternative ways could it have got into that situation?
Was it tossed? Dropped accidentally?
Carefully placed? Shaken in a cup and thrown? How many times did it
bounce, and in which directions? Was it created supernaturally and left there
in that position a moment before we looked? Formed from a scrap of meteoric
material that happened to come down in that shape?
Not all those explanations and speculations
are equally likely, but in principle all those and more are possible. And there
are indefinitely many subtler variations on each possibility; suppose for
example that you had good evidence that the die had been tossed from a
cup — would that determine the colour of the cup or how many times the cup
had been shaken or with the left hand or the right? And how relevant would each
possible variation be, to which face was on top?
Those variables might or might not affect the
position of the die, or the number appearing on top, but every one might be
important in some other connection: elsewhere in this essay I illustrate that
there is no logical limit to how small an event might have indefinitely large
consequences in our universe.
From at least as early as the time of Newton,
and as late as the time of Laplace, the dominant view of physicists and
philosophers of science (irrespective of any religious or fatalistic view) was
roughly that every physical situation was rigidly determined by what had gone
before, and equally rigidly in turn had determined its consequent events. That
view we call determinism. It still is popular among people who have not studied
the realities of the matter deeply enough. However, in principle it could have
been faulted even at the height of its popularity, and when quantum theory
became established in the early twentieth century, determinism was essentially
dismissed as a principle in physical reality. This did not invalidate the
concept of causality, but that is something subtly different, though
importantly different.
Anyway, such concepts concern underdetermination:
something occurs or is observed with greater confidence than the confidence
with which one can assign a specific cause or explanation. Note however, that
underdetermination as a principle is not limited to past causes, but applies to
future effects as well; as I point out in this essay, both quantum mechanics
and classical physics imply underdetermination of predictions, as well as of past
causes of events.
I suspect, but am uncertain, that
underdetermination of future and underdetermination of past events arises from
similar principles. However, that does not imply, as Laplace
suggested, that if we somehow magically turned time, or the course of
events, back to front, like running a cinema film show back to front, we would
see everything running exactly in reverse; all sorts of things are unfriendly
to that idea, and our universe is in many ways unlike a film
show of deterministically successive frames that can run backwards as well as
forwards.
For example imagine setting a vacuum cleaner
to blow, and using the blower to clear a sprinkling of sand off an area of
floor. Once you have blown a strip of sandy floor clear of sand, you stop, connect
the nozzle to the suction end, and see whether the suction will bring the sand
back to where it had been before.
It doesn’t even begin to work, does
it? The distant sand doesn’t stir, and the closer sand beside the clear strip,
vanishes up the nozzle.
This is one aspect of some very important
principles that Ilya Prigogine clarified: principles that prevent time from
running in reverse. Nor, if time were to run in reverse, whatever that might
mean, would the world be indistinguishable from time running forwards; there
would be no sudden ability to use sucking to reverse what blowing had done, or
for a toppled needle to stand erect where it had been balanced on its point, or
for the shards of a broken window to come together if we threw the ball again
backwards.
There are whole categories of effects that,
even if they do not absolutely forbid the undoing of events by running all the
particles involved in reverse, would require magic to do so. The vacuum
cleaner is a good example: in blowing the sand, the air blows in something like
a narrow stream, so that the force of its blast decreases not much more than
linearly for a fair distance, so, held steady for long enough, it can blow the
sand quite far. However, when it sucks, its suction swallows air from almost a
complete sphere of air, but only near to the nozzle, so that the strength of
its suck decreases roughly with the square of the distance from the nozzle. For
practical purposes one could forget about reversing the effect of blowing, by
changing to suction instead.
And pointing out that in either situation,
the elementary particles involved would be obeying the same rules irrespective
of their direction, cuts no mustard. Their immediate effect on each other is
mainly short range — almost unimaginably short range when we think of
a proton bouncing off a neutron, whereas a blast of air molecules or the
turbulence of a flowing fluid takes effect over millions of millions of times
greater distances.
There is a good deal more to that topic, but
that will do for now. As I see it, in such terms information, or the lack of
information, in its role in determining or underdetermining events, is about as
close to fundamental as anything can be in real life; always assuming that real
life really is real in some relevant sense.
Which in my real opinion it really is —
in many ways at least.
Whether there is anything still more
fundamental that determines the nature and history of our physical, empirically
observed, universe, whether it all comes down to quantum entanglement or any
similar principle, I have no idea and I cannot imagine how to discuss such a
matter in non‑circular terms.
But be that as it may, the concept of what
sort of answer to give to a "why" question is not always the same as
the concept of “law of nature”.
"Why" also refers to questions of
justification of actions, of how we base them on motivation and justification
in terms of personal values, of opinions, of rationale. "Why did he do
it?" or "Why should I do it?"
More trivially it can amount to
temporisation, such as in: "Why, I think I can ...".
And more, according to taste. But the main
point is that we must distinguish between those meanings, not just assume that
the idea of "why" is obvious; otherwise we never have any coherent
meaning for "why" questions or "because" answers. It
follows that the question of why anything exists at all, reduces to
confusion because the asker rarely has worked out what the question means, nor
what it could mean, if anything, nor what sort of answer could settle such a
question. And if he has worked it out, then he needs to express the question in
answerable terms. As it stands it is not answerable.
In the 1950s Robert Sheckley wrote a
brilliant short story called “Ask a Foolish Question”, a story that in my
opinion every would-be philosopher of science should read with care. It is
available online. I recommend the story to anyone who doubts my reservations on
“Why is there Something Rather Than Nothing”.
Elsewhere I discuss such matters from other
points of view, but for now I do little more than to note the point that it is
conceivable that the concept of nothing, or a universe of nothing, a null‑universe,
if you like, might prove to entail some internal inconsistency, so that the
very idea of there being nothing, in that nothing exists, instead of something,
would be meaningless.
But such fields are treacherous at best.
Martin Gardener quoted Bas van Fraassen’s pretty quip: "The fool hath said
in his heart that there is no null set. But if that were so, then the set of
all such sets would be empty, and hence, it would be the null set. Q.E.D."
And, as Kipling said: "there was a great
deal more in that than you would think."
Or possibly less.
Accordingly I ignore such questions as a
rule, but some of their forms do arise in the following discussion, so I try at
least to dispose of them first even if I cannot answer them meaningfully. And
the first step is to establish the position from where we start. At times I
myself use the word "why", but then I do try to make it clear what
the sense is.
But for the present, I just accept that there
something other than nothing actually
exists. A sort of lazy man’s axiom, or, more properly, assumption. That
question is not the intended topic of this essay.
But thinking about it, at least a little, can
save a lot of head bumping.
So, if my assumption, lazy or not, is wrong,
make the most of it.
That assumption entails some strong
suspicions, even if it does not formally provide an answer to the question. In
particular, it suggests that if something does actually exist, seemingly at
least as part of a universe, then some subsets or component structures seem to
exist within that universe. Without components, how could you have a universe
with any content that is not the whole universe? And it suggests that for any
component to exist as experienced by other components (components such as
ourselves, or atoms, or stars) then their existence must mean that components
in their various combinations cause events by limiting the forms of their
actions or interactions.
For example, existing entities could
interact by such principles as certain classes of entities not occupying all
the same coordinates at once, or they could interact by attracting each other
gravitationally; and when they do interact, there are outcomes that differ from
what the outcomes would have been if there had been no interactions. That is
one view of what existence means, if anything at all.
And such a meaning has crucial implications
for the concepts of entities, events and causes.
Of which I might say more when we encounter
them, from time to time.
Semiotics is in principle the discipline studying
everything that can be used in order to lie.
If something cannot be used to tell a lie, it conversely cannot be used to tell
the truth:
it cannot in fact be used ‘to tell’ at all.
I think that the definition of a ‘theory of the lie’ should be taken as
a pretty comprehensive program for a general semiotics.
. . . . . . . . Umberto Eco
If I had any sense I might have omitted this
section, but it might help in justification of why I did not omit the whole
document.
Semiotics is one of those simple terms that
covers such wide fields of concepts that we cannot define them simply. The
subject at one time was regarded as too recondite to be of interest, until in
recent decades it was adopted by pretentious authors, critics of arts and
politics, who collectively diluted it nearly to meaninglessness. I do not
pretend even to define semiotics properly, but hope to put a few important
items into perspective. For anyone unfamiliar with the field to get a proper
understanding, I recommend books by genuine semioticians, such as Umberto Eco,
who wrote the readable "A Theory of Semiotics". There also are
valuable Internet articles in Wikipedia and the Stanford Encyclopaedia of
Philosophy.
Here however, I hardly more than illustrate a
few concepts relevant to this text. Semiotics at its most essential, has to do
with information, and information is fundamental to my discussion.
Semiotics in particular, deals with
communication, the signals, signs, words, tokens or pictures, the ways they
function or are used, and the ways in which they affect the users or subjects
that play roles in communication.
That is a broad field, with more topics than
most people realise, and here I deal mainly with three classes of subject that
readers might do well to bear in mind in making sense of this document. Many
books have been published on each of them:
·
Semantics: deals with the relationships between signs and their meanings.
When people argue about the implications of attaching different meanings to the
same word, or the same meaning to different words, the problems that arise are
largely semantic. It accordingly is important to be sure that in any discussion
the participants share the same semantics. For example, if someone who is
thinking of his smallholding as a farm, gets into a farming discussion with a
rancher, and neither realises their differences, things may go badly wrong.
This is such a common class of problem, that innocents, especially political
bigots, who have trapped themselves in logical blunders, blame
"semantics", thinking that the word is simply a fancy way of saying
"quibbles"; that itself is a semantic error, an error that reveals
the perpetrator’s lack of education.
- Syntax:
deals with the relationship between signs in the same message. This takes
many forms, both in similar messages of the same form and in different
forms. In a given language there might be grammatical differences for the
same word in case, voice, and the like, or in the sequence of words.
Compare:
- "Him she
doesn't like." with "She doesn't like him." They have
largely the same meaning, though the subtexts may differ.
- "Bob likes
Alice." with " Alice
likes Bob." They do not necessarily contradict each other, but do
not mean the same thing at all.
- “Stand)?doggerel
floats am” does not clearly mean anything because it is not obvious that
any part of the message has any functional relationship with any other,
unless you accept that the words are correctly spelt.
Such games can be elaborated indefinitely,
but syntax in various media, such as in different languages, whether written,
spoken, gestured, is important not only in understanding, but in efficiency and
reliability. It overlaps semantics in more ways than are immediately obvious.
Syntax is intrinsic to mathematical
notations, as much as in verbal languages; for example, in common infix
notation, the expression:
. (a+b)×(c-d)
has about the same meaning as the postfix notation:
. ab+cd-×
The difference is essentially in the syntax.
As an example of
the role of syntax in semiotics, consider the kerfuffle that appeared in the magazine “Popular Mechanics” for 2019, July 31st. It
included a problem that they said drove their “entire staff insane”. I saw it
only recently, and it is either very pretty or very ugly, depending on how you
see it. The difficulty arose because at least the majority of those challenged
did not recognise the fact that it was not a problem in mathematics, but of
notation, and therefore of syntax. In essence the problem was to evaluate the
expression:
8÷2(2+2).
If you have never seen the problem before, you might like to evaluate it
yourself before reading on. Apparently they got answers ranging from one to
sixteen.
Stop now to think it out if you are interested enough; this paragraph is in
effect a
s
p
o
i
l
e
r
p
r
e
v
e
n
t
a
t
i
v
e.
- If you have not yet
peeped and wish to think it over, now is the time, else carry on.
The fact is that the problem is one of semiotics rather than mathematics,
and in particular, a problem of notation, that is in this case to say: of syntax.
The syntax in question determines the sequence of operations and thereby
the answer. But the syntax depends on the context: specifically the
notation, the convention adopted by the interpreter. And that notation is
not logically defined; in fact, in most contexts the input string is
simply wrong, and as such, meaningless. It certainly is meaningless as
reverse Polish notation, and would be bounced by any calculator I have
seen and any computer language I have used, though it is in principle
possible to write a forgiving compiler or interpreter that would interpret
it consistently in the same sense as school arithmetic.
Three of the most obvious answers, each of which could be based on a
consistent syntax, would be 1, 10, and 16. Most compilers would simply
baulk at the expression as given, and some would give different answers
anyway.
So, who would be right?
Trivial.
The right notation and interpretation would depend on the semantics. To
argue that point would be like arguing whether someone speaking German,
rather than someone speaking Polish, was speaking properly — when
addressing someone whose language happened to be English.
Most computer languages use infix notation with explicit operations; so,
to get them to accept, and correctly interpret, your input, you would have
to correct it to either - 8÷2*2+2 or
- 8÷22+2 or
- 8÷2*(2+2) or
- 8÷(2*(2+2))
- And each would give you consistent
respective answers, though not all would give you the same answer to the
same instruction. None is wrong unless it is not in the required notation,
and none is right unless it is in whichever is the notation, in semiotic
terms the syntax, required by the relevant system.
- Pragmatics: deals with the relationship between the message and its
effects on the participants in the communication. Pragmatics might be
affected by choice of words, of voice, of topic; think of ideas such as
"damning with faint praise", of telling the wrong joke in a
given company or on a given occasion. Think of tact; as Ernest Bramah
pointed out: "Although there exist many thousand subjects for elegant
conversation, there are persons who cannot meet a cripple without talking
about feet."
That would be a palpable blunder in pragmatics.
In these connections language is the
system of signs and messages you use; it can be in many forms, such as spoken
words, signalled words, coded words, gestured words, technical words, jargon
words, and, always in suitable contexts, sentences, strings and structures.
In such senses language and notation can
include symbolic statements such as:
"(a,b,c,i,j Î ℕ)&(a=ij &
b=i2-j2 & c= i2 + j2)↔(a2+b2=c2)
That is equivalent to a verbal description of
what determines a Pythagorean triple.
The relationship between an algebra and a
language is very close, as you may see later in this document.
Language is fundamental to most of what we
commonly call communication, including what we might call selective
communication, in which we intend the exclusion of some communicators
from some of the messages. Examples of such exclusion include enemy
communicators, "outsiders", and "not before the children".
Think of the two women who had come to visit
a friend, who left them in the front room with her daughter while she went to
prepare tea. The daughter of six or eight years old was no beauty, and one of
the visitors spelt out to the other: "Not very p-r-e-t-t-y!"
"No," said the little girl,
"but very i-n-t-e-l-l-i-g-e-n-t."
I leave you to think of what such things have
to do with pragmatics, and how they depend on meaning and comprehension.
And as for the meanings of "meaning"
itself, they still are open to discussion, and there are whole books on the
topic — and even on "The Meaning of Meaning". In this
document I generally use the idea mainly in the semantic sense of the
relationships between a symbol or statement and the entity that it refers to.
There are other usages that are of little use to most people, but in case you
wish to go into more detail, there are good articles online; a good place to
start is in Wikipedia, under: "Meaning (philosophy)".
Comprehension
may be seen in suitable contexts, as the relationship between the receiver of
information and how that information affects the receiver.
So what is there about that entire topic,
that is relevant to us here?
Mostly, that every single item in it had to
do with information, receiving it, formatting it, processing it, acting
on it, and propagating it.
It might seem all terribly superficial, but
don't bother to tell me about it before you have assimilated and comprehended
the following parable; its origins are obscure, but it has been attributed
apocryphally to Einstein:
A blind man and his friend were walking on
a hot day, when the friend said:
"I wish I had a nice glass of cold
milk!"
"Hm? What is that? 'Glass', I know; 'cold' I know, but
'milk'?"
"Milk? Surely you know milk? A white
fluid!"
"'Fluid' I understand yes, but what
is 'white'?"
"White? It is an attribute of the
feathers of a swan."
"'Feathers' I know from pillows, but
what is a 'swan'?"
"A swan is a bird with a long, curved
neck."
"A 'neck' I know; I can feel my own,
but what is 'curved'?"
"Curved? 'Curved' is like 'bent';
give me your hand, stretch it out, feel that: your arm is straight! Now I bend
it; feel that: it is curved, like a swan's neck."
"Aaah! Now I know what 'milk'
is!"
I have read it described, that when that
story was related at a particularly high-powered conference on conveying
mathematical ideas, the audience sat silent for some time, till one of the
biggest names present erupted with: "But what the expurgated imprecation
does that mean?"
I cannot but sympathise, and recommend that
readers stop here and think for a while about whether it means anything worth
thinking about.
A different, but related, aspect of
perception and interpretation, emerges from the oriental parable of the blind
men and the elephant: one of them felt the tail and said that an elephant was
like a brush; one felt the leg and said no, an elephant was like a tree; one
felt the belly and said it was like a roof; and one felt the trunk and said an
elephant was like a snake.
I first got those ideas from a school friend
who had read them somewhere, at which time I glossed over the implications.
Since then I have come to regret that I had not thought it over more seriously
at the time.
What bothered me was not so much querying the
partial understandings of the men examining aspects of the elephant, but trying
to imagine what sort of understanding of milk the blind man could
imagine from such vague and indirect analogies. Close your eyes and envisage
milk if you can, in terms of necks, birds, elbows and feathers.
That reservation remains with me today,
decades later, reminding me of more famous parables. Consider for example:
"
...without a parable spake he not unto them:. That it might be fulfilled which
was spoken by the prophet, saying, I will open my mouth in parables; I will
utter things which have been kept secret from the foundation of the world."
Why there should be virtue in avoiding clear
speech when clear speech is possible, I cannot guess, but when no clear
demonstration is available, we must make do with what we can. Consider our
understanding of the world around us: some people say that they have too much
common sense to believe anything but what they see with their own eyes —
but, for the following reasons, they are no better off fundamentally, than the
blind friend with the abstract, attenuated conception of milk in terms of
feathers and curves.
The logical parallel is uncomfortably
disconcerting.
Firstly such people clearly don't understand
that vision itself is a complicated pathway, incompletely understood even now,
where the available light, the refraction of the medium, the shapes and
constitutions of several media in the eye itself, affect the image on the
retina, the way the retina registers the image, and begins the process of image
processing, the way neurons on the way to the brain cross over, and pass on the
data to the proper parts of the brain, and the way that all those stages can be
fooled into producing optic illusions.
And that is just the start. When we look at
ever smaller items, we soon are unable to make out anything without lenses, and
after that, microscopes. And by now, we no longer can trust light itself, but
must massage it drastically to see more, having recourse to UV and X-rays, and
after that to electrons.
Not far beyond that, and we have resort to
accelerators and advanced mathematics to understand sub-atomic realities.
So, how true to any underlying realities are
our perceptions, let alone our comprehension, of apples in our hands, cells in
our bodies, of molecules, of atoms, electrons, protons, quarks, or neutrinos,
when we have to see or even conceive them in terms of light passing through
lenses in our eyes, of stresses in our tissues,. of impulses in our neurons, of
wires and springs in our instruments, or marks on our rulers?
Long before we go as far as that, we are
beyond our blind friend's attenuated view of the meaning of milk. It is not for
nothing that Richard Feynman, said: "If you think you understand
quantum mechanics, you don't understand quantum mechanics".
And if you think you understand the world in
terms of what you see with your own eyes, you don't understand the world or
your own vision, any more than a believer in the flatness of the Earth does.
Or perhaps any more than the blind friend
understands milk.
As I see it, it amounts to a more
sophisticated, less laboured, version of Plato's cave, and I prefer it greatly.
We need not despair; it is better, or at
least more worthy, more effective, to work on the hypotheses that we derive
from the information available to us or our perceptions; they are not random
guesses as some ignorant anti-scientists claim — each new hypothesis must
in the first place correspond closely enough to what we already have seen to happen;
it then must enable us to make better sense and better predictions about what
we still are trying to explain or discover.
At this point it may be good to remember that
Edward Teller quote: "What is called understanding is often no more than a
state where one has become familiar with what one does not understand" ...
And if you think we are in a bad way to
understand our world, see how well the lion, the antelope, and the grass
manage, with no frustrating thoughts of understanding swans and milk, but an
impressive accommodation to the world as they see it.
But the more we progress, the narrower our
scope for dealing with error. If you doubt this, compare your lifestyle to
lifestyles of affluent people two centuries ago; then compare those with the lifestyles
of one or two millennia ago. The latter two differ less than yours differs from
either.
But be cautious in your comparisons. We all
are limited in taking things for granted. A colleague in our computer team
related a conversation that arose from the news of the discovery of the wreck
of the Titanic in 1985. Someone created a pause in the discussion by asking in
horror: "Why didn't they send out Boeings to rescue the people?"
After a dumbfounded silence, someone said: "That was in 1912!"
The response was: "So what is your
point?"
Making sense of historical contexts can be
challenging; failing to do so can be disastrous as well as ignominious.
As for our comprehension of our world,
perhaps someday we will get to the point where we understand more of what at
present we see as mysteries, or as working hypotheses. Examples include aspects
of subjective consciousness, or of quantum mechanics and
relativity — but perhaps we always will have to accept the swan's
neck and feathers as representing the true nature of milk.
And if so, we must take whatever parable we
have, whatever working model or hypothesis, however blind, as a valid analogy
to fact. One does what one can with such information as one can get, whether
about milk or about elephants, by such channels as we have access to. And one
mark of a civilised education is that one's views change as one's information
changes.
And all considered, it works amazingly well,
increasingly well, lifetime after lifetime. We call that progress, though it is
not clear what progress amounts to when education lags too far behind. Do not
rely too unthinkingly on anyone sending Boeings to rescue you.
"I've got a better theory," said the little old lady,
"We live on a crust of earth on the back of a giant turtle."
"If your theory is correct, madam, what does this turtle stand on?"
"The first turtle stands on the back of a second, far larger,
turtle."
"But what does this second turtle stand on?"
The little old lady crowed triumphantly, "It's no use, Mr.
James —
it's turtles all the way down!".
after John
Robert Ross: Constraints on Variables
in Syntax
The reason that I try to deal with questions
concerning the why and how of our existence, is that there is a logical and
practical difficulty that I cannot dismiss: in accounting for the nature,
origin, and history of our universe or universes, no one has shown me how far
down we need to extend our stack of turtles.
The question of ultimate causes and ultimate
explanations amounts to part of what is necessary for establishing key primitive
concepts, as I already have defined the idea of primitives. Religion offers
no help, for obvious reasons: most religious answers amount to selecting a
particular turtle in the stack (call him "Good Ol' Dick", if you
like) and asserting that he doesn't need anything to rest on. All the
turtles above him do though, because the very idea of a turtle with nothing to
rest on is absurd.
Alternatively one could follow the stack all
the way to solipsism, as I have described it: the assertion that your own mind
is the only reality and that the apparent world is no more your mind's dream:
no turtles required. In effect, you accept your mind as the only turtle, and
the stack as a dream; probably a fevered dream at that. No more primitive
concept would be necessary, I think.
But, as I see it, such solipsism comes at too
high a price and offers too little predictive or explanatory substance to be
worth considering.
Personally I stop far short of solipsism. I
do not try to find a final turtle, nor to assert any infinite regress of
turtles all the way down. Instead I assume without proof but as an
empirical basis for discussion, that our current state of sceptical,
hypothetical, experimental, inductive, abductive, deductive science is a good
practical start to learning indefinite amounts about the universe in which we
find ourselves — or in which it seems to me that we find ourselves.
I say more about de-, in-, and abduction
shortly; if the terms are unfamiliar, just think "common sense", and
you will be pretty close. Or if you like to put it that way, I deal with
guesses, evidence, and reason, so far as I can: guess, grope, gauge, and
accommodate — or perhaps rationalise.
In short, I do not deal with the origins of
origins, but with the most suggestive and persuasive aspects of empirical
appearances — what I seem to see about me.
Which leaves a lot of scope for error,
whether that error matters or not.
Such abductive guessing and groping may yield
insights into basic questions, or may not, but it is better than floundering
indefinitely for lack of the courage to think for yourself. Even wrong
assumptions give us something to start from: a hypothesis that we can adjust as
we learn more. Abductive approaches also are appropriate to this essay, which I
try to make largely pragmatic. Not that I take anybody's theory of formal
pragmatism for granted, but I do reject the impotence arising either from
mysticism, or from demands that we start out from a formal demonstration of the
empirical basis and nature of the universe.
The
sciences do not try to explain, they hardly even try to interpret, they mainly
make models.
By a model is meant a mathematical construct which, with the addition of
certain verbal interpretations, describes observed phenomena. The justification
of
such a mathematical construct is solely and precisely that it is expected to
work.
John von Neumann
A popular ideal, common especially to many
schools of philosophy that try to be at once materially sound and logically
unassailable, is to establish a structure of the same form as axiomatic
disciplines — formal mathematics in particular: pick your axioms, and base
the entire structure of your discipline on that. As I point out elsewhere in
this text, that makes some sense in formal disciplines: in those one has
latitude to choose axioms almost as one pleases.
When dealing with material reality on
the other hand, the so‑called "axioms" really are assumptions
about the presumed primitives. As such, no matter how logical or
ingenious or obvious they seem, their validity is no greater than the degree to
which they can be shown to match the nature and behaviour of the presumed primitives.
Accordingly, it is not valid to work on the
basis of assumptions in the same way as working on the basis of axioms.
In this essay I think of "formal
axioms" as the arbitrary, unproven, possibly internally meaningless, bases
of formal systems in which the theorems are ultimately derived from their
respective axioms; the formal part might not appear within the axioms
themselves, or might be degenerate in the form that they do appear. What
unavoidably must have form, is the process of deriving theorems from
those axioms or previously established theorems.
In contrast, but in analogy to, formal
axiomatic systems, "material assumptions" or "material
primitives", right or wrong, are the assumptions on which we base our
reasoning about physical "laws", "behaviour", or conclusions
of "fact" in the universe as we seem to see it. Right or wrong, they
have meaning in terms of what we seem to observe.
That is to say that they refer to items in
the algebra of whatever universe they refer to, its objects or object types,
plus the operations on objects in that algebra.
That is a big, big, contrast, so pardon me
for being captious about people who vaunt their "axioms" when what
they really mean is "assumptions".
So what is there to see in our empirical
world, and what is there to do about it? If it looks like a toad, say I,
waddles like a toad, and croaks like a toad, and I have no hangover, then perhaps
it exists as a toad. That is not proof, but it might be of value as a
working hypothesis, pending anything better. My guess might be wrong, but to convince
me, any rival diagnosis would need support at least a little stronger or
more persuasive than my own impression and assumption. My most sensible
choice is to judge from my notional toad's apparent waddling and croaking and
swallowing of worms. After that I act according to my ability to predict and
explain, or at least speculate on, its doing whatever toads usually seem to do.
And of course I might be wrong in any of many
senses, and in many details and principles, but the sense I rely on is common
sense; if anyone has an alternative suggestion he would be welcome to propound
it, but he had better command impressive powers of persuasion. To be sure, when
I do find the persuasion compelling, I am willing to amend my own assumptions
accordingly; but not till then.
Some dominant schools of philosophy in
classical Greece
had the opposite idea: they insisted that the world was in some sense illusory,
so that abstract logic trumped empirical evidence. That assertion might
have had some persuasive power, if only their logical conclusions were
consistent, but in fact, their various philosophers contradicted each other
wholesale, and sometimes bloodily.
As Edmund Burke remarked: "The nature
of things is, I admit, a sturdy adversary ..." And where formal conclusions
conflict with empirical evidence, the evidence. sturdily outfaces the logic of
self-assessed philosophers. Formal or not, if an assertion, based on proof, no
matter how persuasively, leads to assertions that conflict with empirical
outcomes, then something is wrong with assertions or observations: if it is
shown that the fault is with the evidence, that does not prove that the logic
must be right. Both could be at fault, and in history, both commonly were.
Consider the history of flat-Earth theories.
And the same goes for the rest of the visible
part of the universe. Anyone denying the existence or the nature of the
observable universe, must present rival support at least a little more
persuasive than the empirical evidence — the impression we get from what
we can see or otherwise examine.
In philosophy, agnosticism has its merits in
suitable contexts, but agnosticism offers no intrinsically better justification
for rejecting a position than for supporting one. Rationally, philosophical
agnostics can demand no more than that a favoured theory either supplies
cogent, and cogently supported, argument, or that alternative proposals be
considered equally seriously, if not necessarily given equal weight.
In other words, to argue that my inability to
prove my point of view compels me to accept your equally unproven point of
view, is asking too, too much.
It is very difficult (I, for one, have never
seen it done, nor nearly done) to come up with any coherent and cogent
assumption of basic truth about our universe. Cogito ergo sum is no
better than a militant gesture, and Cogito cogitare ergo cogito me esse,
however witty, is not much more helpful.
Now, having no basic factual assumption that
we can rely on as unconditionally true, may sound really terrible, but matters
could be worse. We still can look about us and seem to seem to see what we do
seem to seem to see.
Remember the conception of milk in terms of
the neck of the swan! How much better are our conceptions of our world in terms
of our sense organs and brains?
On an analogous principle we can base
hypotheses and rationalisations. We have no need to despair in the face of
Thomas Nagel's question: "What Is It Like to Be a Bat?" It is not a
question that I ever have heard of a bat despairing over. The things that we
experience in our various lives, whether we be bat or mole or hawk, are parts
of the same universe. This suggests that there is likely to be at least some
functional sameness to our functional perceptions and to our dreams, if any: a
sameness of how the universe imposes information on existing entities.
Such a sameness I call a plesiomorphism, by
which I mean that there is, for our purposes, sufficient resemblance between
notional entities, for us to regard mutually plesiomorphic entities or their
aspects, as being — near enough for our purposes — equal. I
distinguish between this and isomorphism, where isomorphism ideally would imply
exact matching. For example, we might refer to a plesiomorphism between aspects
of the reality we find ourselves in, and our calculations, or functional
perceptions, or our dreams.
For my part, I see functional transmission of
information between any aspect of reality, and any given entity, as
fundamentally equivalent, whether the recipient is sentient, has a mind, or
not. Whether it is an anemone, a rock, a
bat, or a mystic, transfer of information is the basis of physical cause and
consequence. And it is plesiomorphic, not exact.
Apart from anything else, Quantum Mechanics
does not permit the transfer to be exact.
But neither would the constraints of the
nature of physical information in classical physics permit it to be exact.
Because we cannot be compellingly sure
of any underlying Truth about which we speculate, or even sure of any
meaning, we have scope for multiple possible hypotheses about anything,
including our own existence. Wittgenstein for example remarked on never having
seen his own brain.
Also, by our nature we are not in a position
to comprehend everything at once, so we have to start somewhere if we are to
start at all, and because there is no simple limit to arbitrary formal
speculations on which we might base our world view, at least one practical
option is to base it on our subjective empirical perceptions: what we seem to
see, hear, touch, or otherwise perceive.
That sounds discouraging, reminiscent of the
swan-to-milk Platonism; and yet, as I see it, that is better than the smugness
of the classic Greek philosophers who dismissed empiric evidence as inferior to
what they saw as rationality.
So if we have the time and tools and
interest, we next can compare the most acceptable hypotheses and
rationalisations, and select those that seem likeliest, most fertile, most
rewarding, most consistent, and with the greatest capacity for progress to successive
findings and rationale.
The choice we rate most highly at any
stage becomes our working hypothesis. We then can make predictions to test
the limits of our working hypotheses, and check the outcomes with observations
that might affect our ranking of the relative strengths of some of our
hypotheses. If the evidence changes, then we change our preferred hypotheses.
As long as none is satisfactory, we adjust them, or try to think of totally new
hypotheses, then back to the coalface for new reasoning or observations.
If no suitable observations are available, we
actively try to design and execute experiments to yield helpful observations.
If the desired experiments are beyond the resources at our disposal, we may be
reduced to thought experiments: in effect we inspect our ideas to see how much
sense their implications seem to make.
However, especially in an underdetermined
universe, we never can tell whether our hypotheses about anything include
any fully correct hypothesis, so the best we can do is to do our best to
make do with our best and to better it as well as we can whenever we can.
Up to the present, unless your definition of
science includes formal disciplines such as mathematics, practitioners
of the disciplines of empirical science as we know it generally do
not prove anything formally: they uncompromisingly
work at establishing progressively more successful hypotheses about how various
things work, or seem to work, or seem to seem to work (wake up down there,
turtle number seven!)
According to their conclusions the
practitioners of science (scientists, we hope!) establish explanations of what
they have observed. If the explanations do not support their preferred
hypotheses, they must adopt rival hypotheses to supplant them. If the
explanations support no as‑yet‑considered hypotheses, then it is time to
formulate new hypotheses. If the new hypotheses are worth anything, they must
imply new predictions of observed or observable phenomena. Then, as far as they
can, the scientists plan observations that could be expected to show which
hypotheses have the greatest predictive strength and explanatory power.
Or something.
And the gold standard for evaluating any
hypothesis is how well, how generally, and how powerfully, it predicts items
that as yet are unknown, and how well it offers explanations of existing
observations and suggests new hypotheses or even radically new topics or world
views. A new Weltanschauung if you like.
This tedious reliance on ignominious
blundering may sound very unimpressive, not to say unattractive, but in our
last few centuries such scientific activity has achieved more than all of
humanity had achieved in the last two, or six, or fifteen millennia, or
longer — depending on who is counting. So, till something better emerges, science
as she currently is spoke, deserves the respect due to success.
I freely and cheerfully, if somewhat
wistfully, accept that there are things about the universe that I never will
begin to understand and am not equipped to understand, and some things that
possibly no one is equipped to understand or ever will be equipped to
understand, but so what? Just a few centuries ago there were all sorts of
things that we had taken for granted, accepted as primitives: they had been so
accepted since before living memory, things that humanity not only did not
know, but did not know that they did not know, and would have derided as
nonsensical if they had been told of them. In fact, billions of people still
deride everything they don’t understand, and rely instead on nonsensical fairy
tales propounded by frauds whose scams they think they understand, and
mythologies that have neither predictive nor explanatory power, either in
planning or understanding anything.
Until science began to mature, engines
wouldn’t drive vehicles, sun and wind and horses wouldn’t light a room at
night, planes wouldn’t fly, vaccines wouldn’t prevent diseases, and antibiotics
wouldn’t cure them.
No one who drives a car on a concrete or
macadam road, flies in a plane, reads affordable books, crosses the ocean on a
liner without making a will, washes with soap, wears a wristwatch, plays
recorded music, watches television, writes on paper or wipes himself with it,
benefits from hurricane forecasts and dental implants, girds at yellow fever, beriberi,
and scurvy, or looks through a glass window — no such person is in any
position to sneer at science. Tens of thousands of years of mysticism and
ignorance and respect for seniority or tradition failed to give us the things
that recent centuries have made so ordinary that Joe Public can afford them, or
at least afford to use them, as a rule unthinkingly — in the affluent
world anyway.
Things such as Boeings ...
To be sure, carpers
criticise science, pointing out problems of health, wealth, and happiness
arising from application, mis-application, and exploitation of science; they
point out consequences of pollution, lifestyle, and destruction of resources,
but the flaw in their rationality is not the nature of the problems, but in the
fact that:
the
science provided the power and the users provided the problems.
If they then refuse to use science to avoid
or amend problems, the fault lies with them, not the science. One cannot
rationally blame steel for being beaten into swords, instead of ploughshares,
nor hammers for being used for murders instead of carpentry or blacksmithing.
The products of science in our intellects and
industry, include many that we have been so familiar with for so long, and have
put to practical and intellectual use in so many ways, that. most of us hardly
ever notice them, do not even realise that they are real or necessary, nor even
understand them at all. In fact, many of these familiar wonders are
replacements of previous major advances, marvels that in their turn have been superseded
and largely forgotten. And the latest miracle will not generally be the last,
if ever there is to be a last; we have to learn yet more radical things before
we can start counting turtles.
We will have to continue our search for ever
more primitive primitives: if you like, to search for the bottom turtle.
Counting turtles comes later, if ever.
I hope in this essay to deal with some
examples.
You cannot question an assumption you do not know you
have made
Richard
Buckminster Fuller
My desire and wish is that the things I
start with should be so obvious that you wonder why I spend my time stating
them. This is what I aim at because the point of philosophy is to start with
something so simple as not to seem worth stating, and to end with something so
paradoxical that no one will believe it.
Bertrand Russell, Philosophy of Logical Atomism
Nothing in this work is a claim to reveal
anything new in mathematics nor, for that matter, in logic or physics, but
there is a surprisingly large population of persons confident in their
mathematical competence, who take either unexamined or ancient assumptions for
granted. Some of these assumptions need clarification before I continue,
because the context is critically important to the topics I examine.
University President asks:
"Why is it that you physicists always require so much expensive equipment?
Now the Maths Department requires money only for paper, pencils, and erasers.
And the Philosophy Department is better still. It doesn't even ask for
erasers."
Related by Isaac Asimov
The first assumption I examine is an item of
semantics, which is a field in the discipline of semiotics, as I already have
explained. However, semantics commonly is a troublesome item in discussions and
conceptions, and I am interested in the topic of semiotics, so put up with it
or pass on by. I regard semantics as important in itself, and critically
important in topics in which fine distinctions make all the difference.
This first such topic is the distinction
between "pure" and "applied" mathematics and equally in the
distinction between other "pure" and "applied” formal
disciplines, such as branches of logic and some branches of philosophy.
Personally I prefer the term
"formal" to "pure". This may seem a pointless niggle, but
the evaluative overtones of "pure" introduce judgemental cross
purposes into many a discussion. In context, "pure" doesn't mean
anything anyway.
One thing that sustains the confusion about
this hardy perennial topic, is the sheer confidence of many who deal with the
assorted disagreements by slowly and loudly repeating their personal views to
each other as proof of the obvious.
Well, good luck to them. Now then:
First the taxonomy — the classification
if you like: the principles of defining, naming, and identifying classes of
things and allocating particular things, particular entities if you like, to
particular classes.
No matter what the application, useful
classification depends on intrinsic, relevant differences between
entities. If such differences cannot be clarified, there is no point to arguing
about distinctions. If ducks had big flapping ears and elephants had feathers,
ducks and elephants would be that much harder to tell apart. As it happens,
those two attributes or parameters: big ears and feathers, are innate and
intrinsic to elephants and ducks, respectively, so they suffice to tell them
apart without recourse to other differences: "In the dark I feel
feathers, so this is unlikely to be an elephant; no need to panic ..."
But, among their other attributes, the tameness
or wildness of ducks and elephants will not suffice for telling them
apart; those attributes are neither intrinsic nor innate nor diagnostic: one
gets tame ducks and wild, as well as tame elephants and wild, and plenty of
other creatures that might be tame or wild, and an initially wild duck or
elephant might be tamed after some time. And whether a creature happens to be
tame or wild, that will not be because it is or is not a duck.
Now, in practice nearly all the arguments
about classifying formal disciplines on the one hand, as distinct from applied
disciplines on the other, are about subjective, contingent differences.
Some people assert that there are two or more totally separate classes of such
disciplines, others recognise just one, asserting that there are no substantial
differences at all between pure and applied disciplines.
One extreme of classification is that of G.
H. Hardy, who denied that formal and applied maths had anything whatsoever to
do with each other; Martin Gardner on the other hand claimed that there was no
divide whatsoever.
When great minds differ on abstract issues,
it might be that they are at cross purposes. One reason for cross purposes could
be that they hadn't worked out the relevant functional semantics, in which case
both might well be wrong, whether they were genius or not. Without
functional semantics even great minds cannot command functional
distinctions — if any distinctions at all.
In this essay, the first clue to a functional
distinction lies in how the two classes of disciplines resemble
each other. And resemble they do. For example: Whether "abstract"
("pure" or "formal") or "applied", both
disciplines share largely the same mental mechanisms, the same laws of
inference, the same basis in axiomatic structures.
From this point of view, both formal and
applied disciplines can be seen as examples of what I call Implicatory
Activity, because they assume that derived, or discovered conclusions are
constrained (though not necessarily determined) by the fact that they must be
implied by the axioms or assumptions.
Please note: this does not imply that the
axioms or assumptions of the formal and applied workers need be the same, nor
for that matter necessarily different, just that the concept of deriving
assertions from basic axioms or assumptions concerning the outcomes of accepted
operations on entities (such as that A implies B) is basic to both
classes of activity.
If they conflict, then, in the case of
applied work, either the conclusions or the assumptions must be adjusted or
discarded. Analogously if the work is formal, a new branch of the discipline
must be based on new, elaborated, or modified axioms that lend themselves to
the desired discoveries. Classic examples might include the recognition of
negative or imaginary numbers, complex numbers, transfinite numbers, or
non-Euclidean geometries.
And we may ignore the nonsensical idea that,
because all the steps in deriving any theorem are essentially tautological,
therefore within an axiomatic structure no derivation can be achieved in
essence. In practice the fact is that the very nature of an algebra whether
mathematical or formal, is that it consists in a set of objects or object types, plus a set of operations on those
objects. And a derivation within such an algebra is a sequence of
operations upon information. And operations upon information are
intrinsically material, physical — irrespective of how trivial or complex
or formal they might be.
If you doubt that, try proving the likes of
the Pythagorean theorem, or the four-colour conjecture, or Fermat's last
conjecture, without operations that involve entropy. Formal operations are
physical, and without operations, there is no outcome, whether the outcome is
to be a proof or not.
And try to make sense of Gödel’s
impossibility theorems in terms of physics.
Some people, Hardy for one, argue in essence
that the pure and applied disciplines have nothing in common because they
differ in their objectives: formal disciplines deal with the formal
proof of abstract theorems, irrespective of whether those theorems have any
meaning, while applied disciplines deal with valid derivation of assertions
about objects of study: this implies that such assertions have meaning.
However, that argument about the objectives
is not cogent; those objectives are intrinsic, not to the disciplines
that we wish to distinguish, but to the practitioners. Accordingly,
though those objectives, viewed as attributes, might affect the taxonomy of the
practitioners, they do not affect the taxonomy of the disciplines, and it is
the disciplines that we are trying to distinguish, rather than the
practitioners.
To make that clearer by means of example,
that operative difference is about as cogent as arguing that a hammer has
nothing in common with a paperweight. Because of their extrinsic
attributes, those that we apply as labels, hammers and paperweights certainly
would not appear in the same category in the yellow pages or in a mail order
catalogue, but to the user, their intrinsic attributes might well put
them in the same category in a gale on a building site with the site plans
threatening to blow away: a hefty hammer might then make a very good
paperweight.
In contradicting Hardy, Gardner argued that "recreation" is
itself an application. In this his logic was no better than Hardy's, but anyway
he concluded in effect that, since absolutely any formal discipline could be
applied as recreation, all maths is by definition applied.
However, in systematics such an argument is
futile because objectives such as recreation or purely formal mathematical
activity plainly are intrinsic to the practitioners; they are not
intrinsic to the disciplines. Permit me to stretch an earlier
analogy: a tool such as my hammer can serve effectively as a stationary
paperweight by virtue of its intrinsic heaviness; but its belonging to me is
not intrinsic — so if someone stole it, it would cease to be a paperweight
for me, but in its new extrinsic role as a possession of the thief, it would be
as good a paperweight or hammer, or as valuable an asset to pawn, as if it had
never been mine.
The point of that analogy was to illustrate
that objectives in performing the mathematics do not affect the operations
you perform in working the mathematics, whether in performing a proof or
derivation, or in calculating a conclusion, whether in recreation, in theory,
or in engineering.
In formal mathematics the typical
activities and their objectives are to design axioms, prove theorems, and so
on. If instead you also use the same mathematics to deal with the description
of the nature or activity of some object other than yourself and your
mathematics, it need not follow that you use different mathematics. Whether
applied or not, it is not the mathematics that differ intrinsically, but the
practitioners or the problems.
So far, no hard distinction.
Where the intrinsic differences begin
is that in "pure", "abstract", "formal",
disciplines one may choose axiomatic structures at pleasure, as long as the
axioms are internally consistent (or paraconsistent) and agreed upon. In
practice one could go further; one also might demand that the axioms be
mutually independent, and be intellectually fertile — in other words that
the axioms are not negligible. It might be nice if they also were complete,
parsimonious, elegant, mutually relevant and so on, but we must not be greedy.
Traditionally axioms were chosen as “plainly, intrinsically, true”. (In using
the inverted commas I do not represent those words as a literal quote, but
demarcate them as a concept.)
For good reasons however, the truth of formal
axioms is no longer generally accepted as relevant, nor even necessarily
meaningful, let alone necessary.
In particular, in a purely formal discipline
the very concept of "truth" is doubtful; the closest we can get to
“truth” is to show that some particular conclusion follows from certain axioms,
not that that conclusion is true or false or even meaningful in any other way.
Ideally this means that to prove a proposition X we must be able to show that
the statement of X amounts to the restatement of one or more of our axioms in
some particular sequence, form, or context. In theory that is what formal proof
amounts to; in practice of course, such a viewpoint tends to be too puristic,
even for formal mathematics.
Anyway, within those limits, if I present
you, as a "pure mathematician" or "formal logician", with
an axiomatic structure, and you find my axioms inelegant or redundant, or
offensive, or uninteresting: bad luck!
Your view may amount to valid criticism of my taste or mental
limitations, but that is not the same as refuting my axioms, nor any theorems I
validly derive from them. To refute an axiomatic structure on the basis of
claiming that the axioms in isolation make no sense, is something that arises
in applied mathematics, not purely formal mathematics.
To apply a formal discipline
(typically a branch of mathematics or an axiomatic structure in mathematics)
generally means that one uses that discipline to model some part of the
behaviour of some distinguishable set of objects, typically items or a
structure or process that might not necessarily be part of that same formal
discipline itself. So, one might apply maths in studying the distribution of
primes (applying maths to formal maths) or one might apply mathematics to
studying the distribution of trees and pests (applying maths to ecology). One
might apply formal logic in studying ethics. (applying formal logic to philosophy)
or apply ethical theory to the study of business (applying possibly formal
philosophy to human affairs, economics etc).
But such modelling demands that the logical
structure of the part of the formal discipline involved, say maths or
logic, is isomorphic or at least plesiomorphic to the relevant
behaviour or nature of the subject, say mechanics or epidemiology, or indeed,
mathematics.
Anyone trying to impose such a concept as
axiomatic to an application of a formal subject, would have to explain very
carefully whose axioms they are, and what relationship they have to of the
physical universe. Until we understand, and can demonstrate an underlying
algebra of physics, we cannot meaningfully speak of any fundamental axioms of
physics, only axioms of physicists. And in dealing with empirical realities,
assumptions of context-free, unconditional truth can hardly be realistic.
To call them axioms rather than assumptions
is not cogent. The plesiomorphism of the application must be sufficiently close
for us to describe or measure or predict relevant aspects of the object with
sufficient reliability and precision to make our efforts worth while in the
context in question. Precision need not in all cases be absolute, but it must
be adequate in terms of our assumptions.
Isomorphism in
this sense (the word is used in several senses in which some practitioners
seem to assume that their own parochial definition is definitive and exclusive)
means that in the mathematics or other formal discipline that we apply to the
subject matter, there is a logical structure that matches the logical structure
of the relevant part of the subject. So, if we apply a process correctly
according to our axiomatic structure, we expect to get an answer that, sufficiently
nearly correctly to meet our objectives, describes the entity or event we
are calculating.
As I have mentioned, I have coined plesiomorphism
to refer to application where sufficiently nearly correctly means that
you don’t expect to be absolutely correct, whereas isomorphism literally
means that you expect to be notionally absolutely correct in every
relevant respect.
In either case, it also means that the
resemblance between the abstract logical structure, and the practical, applied
structure is sufficiently close, though not necessarily absolute.
For example, if I use the common arithmetic
of integers to calculate the number of apples in a regular row, counting should
yield a precise number of apples because the relationship between the
apples and the row usually is simple and matches the relationship between
integers and the cardinal numbers of elements in a set; on the other hand,
counting will not yield me an exact mass or volume of apples, because
apples vary in mass and volume, both from second to second (as the apples
respire or their moisture evaporates, or as moisture condenses on them) and
from apple to apple, because apples are no all identical, so that the mass and
volume deduced are not precise, though they might be satisfactory in
practice.
This does not invalidate counting as a basis
for the estimate of mass for routine purposes. Being informed that ten apples
on a tray will weigh a kilogramme will not generally be correct, but it will
commonly have a high probability of correctness within an order of magnitude,
distinguishing the weight of ten apples, from the weight of 100 apples, or of
ten strawberries or elephants.
Similarly most other practical physical
applications of mathematics are inexact. Consider measuring length with a
ruler, calculating a numeric value as represented by the length of its
representation on a slide rule, estimating light intensity with a photo cell,
deducing precipitation from the reading of a rain gauge, predicting the path of
a meteoroid from sightings through telescopes, etc. All of these intrinsically
differ from most formal mathematical considerations, even when exactly the same
operations are performed on the same variables.
The subject might be material; we might use
calculus and Newtonian laws to predict say, the flight of projectiles or
volumes or surfaces of containers, but we need Einsteinian theory for yet more
highly precise prediction of long‑range space trajectories.
On the other hand, the subject might be
formal: we might use probabilistic arguments in dealing with the occurrence of
prime pairs or Goldbach numbers or the distribution of digits in the decimal
expansion of pi.
The isomorphism between model and subject
might be precise, as in counting discrete events, or it might be plesiomorphic,
that is to say, rough, but precise enough to be useful in relevant contexts,
for example in simplifications such as ignoring air resistance in dropping a
cannonball from a tower, or it might be contingent, such as in using part of a
mathematical curve that conveniently matches a totally different function over
a limited relevant range. In formal Euclidean mathematics, you cannot calculate
the value of the diagonal unit square, but in application to the Euclidean
construction it becomes a problem in physical measurement, and quite simple
down to nearly molecular precision.
An example of plesiomorphism between maths
and measurement.
Except in one or two respects everything that
we said about the purely formal discipline applies to the applied discipline;
commonly it even might be the same discipline.
One exceptional respect is that in
applied mathematics the choice of axioms is no longer free: there now is an
added requirement.
It is a requirement so fundamental that one
could argue in favour of at least limiting one's use of the term
"axiom" in applied mathematics: it might be better to speak of assumptions.
Those assumptions, no matter how ingenious, how old, or how new, must be sufficiently
compatible with the structure, the behaviour, of the subject that
your system is intended to model, describe, or predict.
Otherwise they are not usefully applicable;
your plesiomorphism is inadequate.
In short, in applied fields we must add the
concepts of sufficient truth of axioms or assumptions and
sufficient truth of deductions or theorems. I am inclined to prefer the
term "assumption" rather than "axiom" in applied
mathematics or other applied disciplines, and "assumptive" rather
than "axiomatic", whenever there is. a material constraint on how the
assumption is to be formulated or to be applied to the subject matter.
Such an assumptive structure, whether meeting
all the demands of formal mathematics or not, if making inappropriate
assumptions by failing the proper isomorphism or plesiomorphism, would be wrong.
One cannot for practical purposes substitute
say, Cantorian set theory for partial differential equations in dealing with
orbital mechanics, or for everyday arithmetic in bookkeeping; such examples
violate the principle that the intrinsic attributes of the assumptions in
applied mathematics must have the necessary isomorphisms to the objects and
operations they refer to in their respective applications.
For example, addition of infinities is not at
all the same as addition of real numbers.
Less essentially, more than one logical structure
might be applicable to the same problem, though very often some such structures
will be more profitable than others. For an artificial example, there is no
fundamental reason why one might not use complex numbers, or even octonions, to
count apples, but for reasons of convenience it is not common practice.
Again, if the only reason for the counting is
to compare two sets of apples, even numeric counting might be overkill;
matching the apples in one set with the apples in the other set, might be adequate,
so why introduce all the axioms or assumptions appropriate to the arithmetic
of. integers?
In short, we have added another, weaker
distinction: feasibility or cost; parsimony, if you like. In formal
mathematics we do not insist in all connections that a calculation need be practical
or even physically possible; physical possibility might not be of
mathematical interest at all. In applied mathematics on the other hand, it is
necessary for calculations to be practical as well as for plesiomorphisms to be
adequate.
It is possible to choose consistent and
meaningful assumptions or related specifications in applied mathematics,
specifications that definitely are wrong for the application. For
example, to calculate the necessary working strength of a rope for slowly
raising or suspending static loads, it usually is sufficient to add the weights
of the object in any one load.
In contrast, I once read that some arts
students wishing to set some record or other, allegedly used elementary
arithmetic to calculate the necessary strength of a rope required to support
the static weight of a group of students. They allowed an arbitrary safety
factor, as commonly is required in applied mathematics, then used that rope to
support a swinging mass of students.
The ignorant students did not realise that
there are differences between the logical structures of statics and dynamics
...
The rope broke, causing serious
injuries — some fatal if I remember correctly.
Wrong assumptions (or axioms?)
Again, when calculating the strength of steel
cables for lowering cages into very deep mines, it is common to neglect the
weight of the cage and its contents, and instead calculate the weight of the
cables.
Formally this is wrong.
Plesiomorphically it is quite acceptable in
applied maths in engineering.
Furthermore, in applied disciplines, the
concept of precision is likely to be relevant: precision must match or
exceed the required precision of observation and prediction, but also must
not exceed the required precision by too much, because that may be
expensive or imply inaccurate gratuitous assumptions of practical realities. To
calculate a human’s height to the nearest micrometre would suggest to a reader
that such a measurement were possible, whereas even calculation to the nearest
millimetre would hardly ever be practical or useful, or anything better than
delusory.
For example, after sleeping overnight in bed,
we are several millimetres taller than when we go to bed at night after a full
day of working erect.
In formal disciplines the concept of
precision might not even arise — precision could be absolute in theory:
the same arithmetic rules reign in the googolth decimal place as in the first.
On the other hand, paradoxically, in some
subjects the formal mathematician might scorn to contemplate precision at all;
Otto Frisch related that Stanislav Ulam complained that as a formal
mathematician he was used to working with abstract symbols, but had sunk so low
that his most recent paper in the fission bomb project, not only contained numbers,
but that some even had decimal points!
That was an example of applied
mathematics as seen by the formal mathematician. In contrast the formal
mathematician does formal work according to form (otherwise of course it
is not formal).
How does science fit into this? That is not
easy to resolve, because the definition of science is largely arbitrary.
Mathematics used as a tool in science fairly clearly would be a category of
applied maths, and notionally has about as much to do with formal maths as the
arithmetic of the shop assistant counting apples into a bushel has to do with
formal maths.
Commonly (though not universally) we do not
count purely formal disciplines such as mathematics as science, because they do
not necessarily have much to do with anything outside themselves. And
procedures within the discipline are in effect the juggling of axioms and the
theorems derived from them.
Personally, for several reasons I reject the
idea that this distinguishes mathematics from "science"; for one thing,
mathematics of absolutely any kind deals with information and amounts to
physical manipulation of information, for instance by showing in effect that a
given theorem comprises at least some of the same information as one or more
other theorems or axioms, or their necessary implications; and those too
comprise information.
And information I unapologetically regard as
being part of the subject matter of physics.
So I see addition of 3+1=4 as a physical
operation.
But suit yourself about that point — it
hardly matters in this context.
There is at least one other aspect of the
comparison between many forms of formal work, as opposed to applied work:
fundamentally their objectives are almost opposites: formal work, whether
exploratory or striving towards an objective, derives everything from the
axioms that are unreservedly accepted as unassailable; if that does not lead to
anything sufficiently constructive, then one necessarily adds or changes
axioms, and this amounts to moving to a different axiomatic basis, not to
correcting a wrong axiomatic structure.
For example, Cantorian infinity theory
accepts certain axioms that conflict with basic arithmetic theory; the
principle that aleph-m plus aleph-(m+n) (where n>m) generally equals
aleph-(m+n), is not generally valid in finite number theory. And the idea that
there might be a smallest infinity contrasts sharply with traditional number
theory, in which. there is an x-1 for every value of x, in which sense there is
no smallest number. And the question of whether there is any infinity between
two alephs was never settled before new axiomatic structures were proposed.
Note that this does not amount to
showing that the earlier axiomatic structures were wrong, just that they
had not been shown to be suited to the problems hitherto under consideration.
In applied or material disciplines the
opposite is the case. Assumptions are as a rule proposed to be conveniently
close to truth, at least until continual attacks show them to be unacceptably
false, in which case it is necessary to modify them in whichever aspects they
have been falsified. Often this research takes the form of comparing rival
assumptions to find which ones stand up best to falsification.
And in theoretical branches of science,
attempting such falsification is perennially under way. In applied science this
is largely true as well, but as with any other applied activity (technology, if
you like) earlier, notionally discredited, assumptions might widely be used as
plesiomorphic tools of convenience: local maps may assume a flat Earth;
Newtonian orbital mechanics are good enough for terrestrial navigation and
major eclipses; elementary valency theory is good enough for routine chemistry;
and so on.
But for exploratory science, seeking the
essence of aspects of reality, the relentless attempts at falsifying
assumptions is a major aspect of the vocation. The contrast with the likes of
mathematics or logic, is stark.
If such things do not interest you, it now is
safe to open your eyes and read on. This section largely covers what I have to
say about formal, as opposed to applied, mathematics and reasoning.
Or does it? Metaphorically I bite my tongue.
The more important fundamental laws and facts of
physical science
have all been discovered, and these are now so firmly established that
the possibility of their ever being supplanted in consequence of new
discoveries
is exceedingly remote.
Many instances might be cited, but these will suffice to justify the statement
that
"our future discoveries must be looked for in the sixth place of
decimals".
Albert Abraham Michelson. 1903
In keeping with foregoing discussion, there
are two classes of science: formal and empiric (also called
"analytic" and "synthetic").
Empirical science deals with the world we
seem to see ourselves in. In empirical science we have no unconditional axioms
about that world — we can do no more than propose theories based on
assumptions about our observations and the perceived behaviour of the world.
For instance we generally assume that:
- the world operates on principles
consistent enough for us to generalise meaningfully when appropriate
- such information as we can derive
about the world from our sensory perceptions forms a practical basis for a
mental image, a model that has relevant and practical plesiomorphisms to
some sort of presumed underlying reality that has a meaningful
relationship to that which is apparent to us
- the theory of probability may for
practical purposes be assumed to be isomorphic or plesiomorphic with
relevant aspects of the behaviour of the perceived universe. This is the
basis of the ubiquitous applicability of statistics as a practical and
philosophical tool in science.
The current discussion is mainly about
empirical science — formal disciplines have little to do with belief,
since one can construct as many independent assumptive structures as one likes,
and can design them to be compatible with practically any coherent belief one
likes. These structures would not differ from each other in their
"correctness" but only in their interest or usefulness and
applicability. Whether such formal disciplines are relevant to anything
material, is another matter.
In spite of the popularity of the phrase:
"scientific proof", empirical science has little to do with formal
proof; because the inherent uncertainty, empirical predictions and observations
cannot formally prove anything, but they do permit us to compare the
defensibility of rival hypotheses that imply observable phenomena. Observations
that constitute confirming instances of predictions, can serve as a basis for
working hypotheses: they are a weak form of support, abductive or inductive,
that can be assessed in terms of statistical theory.
This is all on the assumption that the
hypothesis has been suitably expressed for the procedure to be meaningful.
Experiment design is a treacherous field because it is subject to the principle
of GIGO: "garbage in: garbage out'. Even modern scientific practice
accommodates a great deal of garbage in, and thereby puts out a great deal of
wasted research, outright delusion, or even bad faith.
The fundamental reason that much of such work
is wasted, or at best expended for little reward, is that it is based on
misconceptions or misformulations or simplistic guesswork; a fair number of
peer‑reviewed works get published in spite of being based on just such
research; after having missed a hidden conceptual flaw the researcher may
perform the rest of the work coherently and competently, and then it might be hard
for a reviewer to spot the relevant flaw. Even having spotted it, it might be a
struggle to justify the view that the paper is ill‑founded. A major source of
such disasters is not poor work, so much as experiments based on preconceptions
or poorly constructed or inapplicable questions. Even flawless work on
meaningless questions produces meaningless answers, and preconceptions often
mask or rationalise the futility.
Whether experiments in good faith and good
practice in science have been well designed or not, if the observations are too
poorly consistent with the predictions, we discard the hypothesis, modify it,
or try again with a totally new hypothesis. We never prove it. We
never forbid anyone to doubt our work or re‑test the hypothesis
or propose alternatives or extensions. We never demand that
anyone accept a hypothesis. We only refuse, when anyone proposes
an alternative, to accept such an alternative before we in turn have convinced
ourselves of its merits.
It does not matter whether this is necessarily
because "we" as "scientists" are so virtuous,
so liberal minded, that we would never dream of imposing our diffident
opinions; we know too well that if we did try to impose them it would have
little effect. That is how the process of science works.
Science depends on conviction.
Conviction by compulsion certainly has worked
very frequently and widely in history and in contemporary education, religion
and politics, but as conviction goes, conviction by compulsion is transient.
After a century, or a professional lifetime, or sometimes within a year, future
generations, rightly or wrongly, will come to hold it to scorn.
It does not follow that because a hypothesis
is untestable by any observation accessible to me, it cannot be investigated
and falsified by some other subset of the scientific community, perhaps even by
a single member. Members of such a subset may be perfectly scientific in their
work, no matter how scientific or unscientific my work had been. Nothing in the
nature of science guarantees that every proposition that is meaningful in terms
of falsifiability to one worker, must be equally meaningful to every other.
There might be differences in skills, in equipment, in resources, in chance
observations. How is one to react to a scientific claim that one is not in a
position to test personally? Is every such claim meaningless by definition, to
everyone but the comprehending observer in person?
Not necessarily. It depends on our personal
world view and intellectual taste, how high a level of confidence we demand
before we are willing accept a given assertion as a working hypothesis. The
principles of science neither demand that we believe, nor that we disbelieve.
The world is too large for everyone to investigate all of it personally in detail.
And as I already have pointed out: we cannot delay elementary classes while
each student personally verifies every individual assertion.
In discriminating between rival hypotheses,
we need not consider only formal falsifiability by personal experiment; it is
reasonable and in practice it also is necessary, to give appropriate weight to
weaker evidence, such as:
- a claim's consistency with our
experience and opinions
- the word of other observers
- the opinions of persons whose
skills we respect
- its consistency with coherent and
logical bodies of theory
- other criteria than direct
evidence, such as parsimony and explanatory richness.
None of these is proof either, but they are
useful in practice and historically have been of enormous power and value.
Weak or indirect evidence still is
evidence — evidence, I repeat, is every item of information that has
weight in rationally influencing one's choice of particular hypotheses as being
the most persuasive — or completely untenable. Strong evidence carries the
most weight; weaker evidence carries correspondingly less. There is no
generally cogent basis for assessing the weight to assign to any item of
evidence for any particular item; its strength keeps changing according to
context and in the light of new evidence, and in any case one's appraisal of
context and weight necessarily are largely arbitrary.
Except in religion there is theoretically no
such thing as absolute evidence, only a range of cogency that extends from an
interesting speculation at one extreme, to repeated, independent, precise,
practical observation, predictable, quantitative, and explicable, at the other.
There is yet another problem with the concept
of formal proof in empirical science: because of the principle of
underdetermination, we never can show formally that we have listed all possible
meaningful hypotheses about something that is observable and falsifiable in
principle. It accordingly is not so much as possible to prove that the correct
hypothesis (the "god's‑eye‑view", or some simplification or
representation thereof) either is the one that our observations support best,
or even that any part of it is one of the alternatives that have been
considered.
We cannot even be sure in principle that our
conception of the phenomenon is framed in terms that can meaningfully be
related to the "god's‑eye‑view," the G‑E‑V.
To illustrate this very important point,
consider someone at the level of technological sophistication of the typical
hunter‑gatherer, who for instance had never seen or heard of magnetism or
electric sparks or currents, and had no conception of electricity or magnetism:
such a person would have great difficulty at several levels, formulating a
meaningful theory about how a battery-operated fan works. Or imagine a remote
islander who happens to have no knowledge of modern technology: he encounters a
battery and a radio transmitter. He finds that if he puts the battery into a
likely-looking slot, some lights go on. He soon recognises this as an emergent
effect. and not one that he could have predicted. A radio technician could have
predicted it, but the islander, no matter how intelligent, is not that kind of
technician; he understandably assumes that producing the visible light is the
function of the transmitter‑plus‑battery.
What he cannot see, and is not trained even
to imagine, is that the light he sees is not the assembly's primary function,
which is the invisible radiation of radio signals. Nor would he guess
that a suitably matching distant receiver of the radio signals could say,
reproduce sounds detected by the transmitter’s microphone, steer a drone, or
set off a bomb.
We in turn have no idea at present, how many
levels and dimensions of sophistication we stand below the TOE of the G‑E‑V.
To be sure, we have some persuasive views
about our the current standard of our scientific world view, but so did
Archimedes, Galileo, Newton,
and any number of 19th century geniuses. Until we have some better
perspective on our own level of sophistication, it is not for us to sneer at
that hunter-gatherer.
He's not of none, nor worst, that seeks the best:
To adore, or scorn an image, or protest,
May all be bad. Doubt wisely, in strange way
To stand inquiring right, is not to stray;
To sleep or run wrong, is. On a huge hill,
Cragged and steep, Truth stands; and he, that will
Reach her, about must and about must go,
And what the hill's suddenness resists, win so.
John Donne Satire III
A major problem I encountered in composing
this essay, was trying to sequence the topics. I find it hard to tell when to
approach such material bottom‑up, and when top‑down.
This is consistent with what I regard as the
most valuable lines John Donne ever wrote. The heuristic nature of scientific
progress may demand alternating attacks, first one way, say top‑down, then a
different one, very likely bottom‑up.
Or transversely?
To insist instead on
imposing your preconceptions all the way through, usually top‑down, or just
confused, tends to harden the mental view and rationalise or complicate ideas
where rigidity would be a blunder at best.
What is called understanding is often no more than a
state where
one has become familiar with what one does not understand.
Edward Teller
Deduction, induction, and abduction are
loosely-defined, loosely‑used terms for some of our commonest forms of
reasoning, especially reasoning in science. Various authors in various
languages have defined their concepts variously and inconsistently for
millennia rather than centuries. I do not undertake to deal with them
coherently, partly because of the sheer volume of the existing published
material, and partly because there is no item in the entire topic on which
various authors have not contradicted each other or themselves, either in
definition or in practice, in logic or semantics.
None of the more supportable versions of
their views is purely right or wrong in itself, and they are not as cleanly
distinguished as some of the smuggest authors suggest, but whole lists of fallacies
violate their various principles one way or another. This chapter is a
commonsense (or at any rate, informal) exploration of a few modes of thought; I
do not claim to offer anything definitive myself.
So don't waste energy on pedantic criticism;
you might like or reject these views, but my intention is not formal
instruction: only to supply a basis for thought, or gaining a perspective of
some of the views I try to express in this essay.
Deduction
As a method of sending a missile to the higher, and even
to the highest parts of the earth's atmospheric envelope, Professor Goddard's
rocket is a practicable and therefore promising device.
It is when one considers the multiple-charge rocket as a traveler to the moon
that one begins to doubt. for after the rocket quits our air and really starts
on its journey, its flight would be neither accelerated nor maintained by the
explosion of the charges it then might have left.
Professor Goddard, with his "chair" in Clark College
and countenancing of the Smithsonian Institution, does not know the relation of
action to re-action, and of the need to have something better than a vacuum
against which to react .
Of course he only seems
to lack the knowledge ladled out daily in high schools.
New York Times Editorial, 1920
Let's first deal with deduction;
it arguably is the least contentiously described. Superficially it seems to be
the tightest form of reasoning, because it is the basis for formal
proof. Popper's major works largely were directed at finding means for basing
fundamental reasoning in research on deduction rather than induction. To my
mind he failed dismally, largely for insufficient recognition of the difference
between formal and applied reasoning, and their respective relevance. I also
saw precious little sign of his appreciation of the significance of
underdetermination.
As I see it Popper's falsification principle
was a blunder; I suspect that in practice one might argue that deduction is not
the most, but the least, valuable form of reasoning in science.
Mind you, deduction as a mode of deriving a
conclusion, whether tentative or firm, is neither dispensable nor even
unimportant, but much of what is intended or purported to be deduction is
neither formally nor functionally deduction at all.
Commonly people take the word deduction to
mean any logical line of thought that leads to solution of a puzzling problem,
but that is not at all the precise technical meaning. Consider:
The basis of deductive reasoning is binary
logical implication, in which:
If A implies B then:
if A is true, B is true;
if A is false then B could be either true or false.
For example:
If the clock is set correctly and the
time is three o' clock,
the clock will strike three.
If not, it variously might strike three
or any other number of times, or not at all, and it might do so or not, whether
the time is three o'clock or not.
Probabilities might be relevant too, meaning that the logic need not be fully
binary, but let that pass for now.
The name for this mode of reasoning is modus
ponens, but I mention that only in case someone tries to impress you with
the Latin.
The other most prominent mode of deduction is
called modus tollens, and in research it is arguably more important than
modus ponens:
If A implies B then:
If B is false, then:
either A is false, or A does not imply B.
This reasoning is the basis of the mode of
research logic called falsification:
For example:
Diamond can scratch glass, so I can test
whether a crystal is a diamond, by trying to scratch glass with it. If it fails
I can be sure it is not a diamond (at least if I can be sure that the
glass is really glass; remember to keep underdetermination in mind!) We
say that I have falsified the hypothesis that the crystal is diamond.
However, if the crystal does scratch
the glass, that test can at most weakly verify or support the hypothesis;
it cannot formally prove by deduction that the crystal really is
diamond: for one thing, some other crystals, such as corundum and carborundum,
can scratch glass as well. Our test was useless in some contexts, but not in
all; we have at least eliminated most possible alternatives, such as that the
"glass" is diamond, and the crystal is sugar.
That is about as much as I can offer here,
because the subject is too big. Still, that little hint might help you avoid
some of the rawest failures in common sense. And in science.
What is special about deduction,
properly applied, is that although, as a mode of reasoning, it cannot prove
everything, nonetheless whatever it can prove, really is proof in formal subjects
such as logic and mathematics.
But again, and always, and especially in
applied logic, as opposed to formal, beware the treachery of underdetermination.
Such deductive power looks very tempting in
science, and many thousands (millions?) of junior students have fallen for it,
especially in the form of falsification, but deductive logic is no silver
bullet. At first sight deduction seems almost infallible, but one must rely on
having some standard assertions known to be true: some relevant
"facts" from which to derive one's conclusions. The history of
science is rife with confident conclusions derived from faulty observations,
faulty assumptions, personal delusions, or received wisdom, and yet totally
wrong in spite of having been taught by generations of authorities.
Formally we have no such thing as an empirical fact at all, and if anyone claims that you can derive formal
truth from empirical data, smile politely and change the
subject.
Deduction also is as close as we can get to
firm proof in empirical science, that is to say as a rule: applied material
studies. That sounds marvellous of course, but in fact, beyond formal studies,
science has very little to do with formal proof at all. More frequently
we work on construction of hypotheses and comparison of rival hypotheses to see
which are most powerfully predictive, or even which ones, known to be formally
false or falsified, are most useful or convenient as fictions or
plesiomorphisms.
Such fictions commonly occur as
plesiomorphisms; for example, although we know this to be incorrect, we
commonly say that a thrown ball follows a parabolic trajectory; that is close
enough for most everyday purposes such as cricket and shooting, and is a lot
easier to calculate than ellipses and air resistance.
So, in practice, deduction in applied fields
is a slippery tool, and imprecise at best. As I shall show, induction and
abduction are more frequently useful in research, once we get past obviosities
such as:
A camel that easily can bear a total burden
of 100 kg, easily can bear a smaller burden, and a 10 kg burden is a smaller
burden than 100 kg.
This camel easily bears 100 kg;
Therefore this camel easily could bear 10
kg.
We don't usually notice such obviosities,
because we take them for granted, but we rely on them continually, so perhaps
we should be more aware of them; if we uncritically drop our guard, fallacies
creep in.
Such insights are not new. They are the basis
of many puzzles. Ambrose Bierce satirised them more than a century ago, as follows:
The basic of logic is the syllogism,
consisting of a major and a minor premise and a conclusion — thus:
. Major Premise:. Sixty men can do a
piece of work sixty times as quickly as one man.
. Minor Premise:. One man can dig a
posthole in sixty seconds; therefore —
. Conclusion:. Sixty men can dig a
posthole in one second.
This may be called the syllogism
arithmetical, in which, by combining logic and mathematics, we obtain a double
certainty and are twice blessed.
A more familiar modern example, is that you
cannot produce a baby in one month by impregnating nine women at once.
Deductions from considerations such as
whether or when to seek a vaccination, seem to be beyond the capacity of many
people.
Another problem, even more serious in my
view, is that formal deduction, in spite of its obvious strengths, is very poor
at discovery, at generating new insights, exploring new ideas and
hypotheses, or seeking solutions to problems. We see more about this in
considering thought experiments.
Induction
It is a well established and repeated observation in
the practice of science, that
the greatest scientist is not necessarily the one who finds the best answers,
but very likely may be the one who frames the best questions.
Anonymous
Now let us consider induction.
A lot of our terminology in science,
mathematics, and philosophy is inconsistent, often for historical reasons. This
is logically trivial, but can be troublesome and confusing. The term
"induction" is one such, and induction comes largely in two flavours: mathematical
induction, and empirical induction (or inductive reasoning),
though the actual terms vary wildly in their usage.
Both forms of “induction” are valuable in
practice — insofar as one manages use them intelligently and
appropriately: Joe Average commonly does not even know about mathematical
induction, and has no clue about how to use empirical induction validly.
The mathematical version of induction is
pretty tight reasoning; in spite of the name, it really is deductive in
nature. The use and form of the method varies slightly according to
convention, but I do not urge any particular convention; suit yourself about
the details. Fundamentally it applies wherever:
- One identifies a set of objects
and can show that there is a procedure (in this sense, an algorithm) for
enumeration of the set, such that the enumeration is certain to include
every member of the set exactly once,
and: - One can show that a particular
assertion is true of at least some first member of such a set under such a
form of enumeration,
and: - One can show that if the assertion
is true for any member of the set, then it will be true for the next
member in the enumeration, for as long as there is an as‑yet‑unenumerated
member,
Then the assertion will be true for every
member of the set.
For example, suppose that we can show that if
we keep adding whole numbers (integers), starting from zero, and going up one
at a time (0+1+2+3…n+(n+1)) then each time we add the next integer then the sum
we get, will be half the product of the last integer times the next integer.
This is easy to show, as Gauss demonstrated while he was still a child at
school.
If we then can show also that it is true for
any one example of two integers next to each other, then we know that it is
true for all the following integers.
And that too is easy, because if we start
with 0, we see that 0 plus 1 gives 1, which is the same as half of 2 times 1.
And, unnecessarily for the purposes of proof: the next step gives 3, which is
half of 3 times two and then we add 3, giving 6, which is half of 4 times 3
etc.
If you like to play with such ideas, you can
find any number of such examples of induction. Mathematical induction is
enormously powerful and versatile; it crops up in all sorts of applications.
If on the other hand you don't like dealing
with numbers, then you might prefer to think of something solid, like a chain.
Think of a chain of links in an unbranched, single chain in a bucket. As long
as you have found the first link, with another link to come, you also can see
that every link is linked to exactly one link more. If there is no next link,
you know that is the end of the chain. And you can change the assertion to deal
with a closed loop of chain instead of a chain with two ends.
And so on. Mathematical induction is a way of
proving things that are true about whole sets of certain categories, one member at a
time, even if you do not do the exercise for each set in a category. For example, in
summing numbers in the way I mentioned, I know, without having to do the
addition, that if I solemnly were to add all the numbers up to say, a million,
I should get 500000500000; a single multiplication and division will do the
trick. Similarly, for the chain: I don't have to work my way along it, nor need
to know how many links there are, to know what the outcome would be if I did.
Where people tend to come unstuck with
mathematical induction, is in one of two places: either they forget to prove
that the statement holds for a suitable starting case, or they forget to show
that it must show for every following case. They tend to pick three or four cases,
and as soon as they fancy they see a pattern, they assume that they have
settled the case.
But it ain't necessarily so. Let's prove that
720720 is divisible by all smaller numbers.
Simple: divisible by 1, yes, 2, easy, 3, OK,
4, right. That proves it, yes? What? Not satisfied? Very well: 5, 6? Obvious,
huh? Oh, you want to be difficult ...?
Well, we carry on a bit and soon it is so
obvious that only a fool could be left in doubt: 720720 is divisible by every
number smaller than itself.
But we have not begun by showing that
divisibility by n implies divisibility by n+1; in other words we have not
applied the mathematical induction proof. All we have is a conjecture. We might
be able to prove it by some other test, but what we have done so far, whatever
it looked like, was not mathematical induction, and proves nothing.
And sure enough, it turns out that 720720 is
not divisible by 17, 19, 323, nor by most other numbers smaller than itself, not
even most numbers smaller than its square root. Our sloppy attempt at
mathematical induction had misled us.
Half‑doing
mathematical induction or formal induction in general, is useless at best, and generally misleading as well.
Empirical inductive reasoning is a different matter.
Unlike mathematical induction empirical
inductive reasoning means trying to find whether a given guess about all
members of particular set is true, by going out to inspect some members of the
set; if none of your examples proves the guess wrong, then you empirically assume
the guess is true. The currently fashionable historical example is the guess
that all swans are white. For thousands of years that is what Europeans
believed, and only with the discovery of Black Swans in Australia, was
that particular example of empirical induction shown to be false.
The black swan debacle led to a lot of fuss
at the time that black swans became known to biologists of the West, because
some European biologists of the day thought the idea of black swans was absurd;
in those days in fact, the very expression "a black swan" was used
for something absurd, much as we might speak of "a mare's nest" or
"hens' teeth"; and in some quarters the first black swan specimens
brought back to Europe were met with accusations of fakery.
Wasn't that silly of the biologists of the
day?
Well, maybe.
But what a lot of people have missed, is the
fact that a lot of the strange animals that explorers of the day brought home
really were fakes: monkey forequarters sewn onto fish tails were sold as
mermaids, and so on. And of course, a duckbilled platypus was an obvious insult
to anyone's intelligence.
So what?
And what is more…
When they found a black object, if they bet
that it was not a swan, they would win nearly all the time. In fact, before Australia was
discovered, they would win all the time.
This is an important principle in science and
sense, and it needs to be taken into perspective in dealing with the real, the
empirical, world.
So, when we apply the same style of thought
to empirical reality, drawing superficial conclusions from a few convenient
examples, it is hardly surprising that we can go badly wrong. We rarely find
anything to assure us that if something happens one way once, then we know for
a fact that the next time the outcome will be exactly the same: shake a stick
at one dog and he will cower; the next dog might tear your throat out.
In empirical induction we generally work on
the assumption that what we seem to see happening a few times with the same
sort of outcome, is always the same thing, and is what always happens. So you
see a plain brown snake and catch it: it bites you. OOOPS! But you come to not
much harm. Ah, so brown snakes are safe to catch. Too bad if that was a mole
snake (which usually are black, but not always) and the next one you catch
happens not to be the same thing at all, but a cobra; cobras often are brown,
but not always.
Suppose you know nothing about firearms and
find a pistol. You pull the trigger and: "bang!" Gosh! Do it again:
bang, bang! Hey! Here is an obvious pattern! And again bang, bang, bang! So by
empirical induction we obviously have a general law here! Pistols go bang!
Sooner rather than later: "click".
Hm .... Maybe we should check on mare's nests
and hens' teeth again.
There is no end to such examples. A
particularly poignant one is the turkey fallacy: every day for months on end
the farmer appears at the door at the same time, carrying food in his hand. The
inductive turkey soon concludes that farmer bearing food is a natural law, and
he strengthens his conclusion every day by successful predictions. Being a statistically
sophisticated turkey, he calculates each day the increasing degree of
confidence he could put into the next prediction. Then the day before
Christmas, the farmer appears as usual, but carrying an axe…
Not the same thing at all ...
The turkey had omitted to begin by proving
that bearing food on day X need not imply bearing food on day X+1.
Failure to understand the underlying
mechanism or situation may be as fatal in empirical induction, as failure to
follow the rules for mathematical induction.
It should be clear that in the examples of
inductive reasoning so far, the players had naïvely taken the underlying
mechanism for granted, much as a savage would accept unquestioningly that
dropped stones fall down — it is a fundamental fact; that is what stones do;
what is there to question?
Is there any
alternative? Well, what was missing was any clear conception of causal
mechanisms; not so much the way things have happened, as what makes them
happen, and how. Refusing to acknowledge that what happens so regularly that to
deny that it is in the nature of things, is commonly perverse, and leads to
painful consequences. I remind you again of Burke's remark in a different connection: "The nature
of things is, I admit, a sturdy adversary ..."
Another aspect of empirical induction, that is not often taken into account, is that even when it is logically invalid to conclude inductively that what one has observed repeatedly must always happen, yet, seeing it happen repeatedly, does suggest that there is a probability that favours its happening. If we at first saw a certain coin land on one face at a time, and inductively concluded that it always would do so, a rare observation might unexpectedly confirm that it is possible for a coin occasionally to land on its rim.
And yet, for most coins, we would find that landing on its face is the way to bet.
Most people accept that it must always land on a face, but that is abuse of induction; naïvely, they could estimate the frequency by tossing the coin thousands of times, but that is seldom fully satisfactory: the frequency of rim landings could change with the lapse of time, say becoming rarer as the rim of the coin wears down and becomes more rounded.
Often in practice we accept the consequences of our reliance on that abuse of logic; we cannot spend all our lives chasing trivialities; but a major, major merit of induction is that it may lead us to make a study that reveals the causal reason why and how and how much the inductively described behaviour occurs in practice.
This leaves us with a need to explain, not
why empirical induction is so fallible, but why it so commonly is successful.
The ways in which things might have happened are hardly limited in general, but
many things, for many reasons, can only happen in a small number of certain
ways, and, of that number, they more often happen in some ways than in others.
So, having watched them happen a few times, we inductively infer that they
behave in certain ways, even though we might lack any cogent proof of why they
do that.
So for instance, we find that a cubical die,
when tossed, soon settles onto one of its six faces, even though its first
contact with the surface is usually by one corner. A coin however, especially
if very thick, though it generally will settle on its face, may rarely settle
on its rim, but never on the edge of its rim. This we can rapidly determine by
empirical induction.
And similarly, when we toss an ordinary coin
a few million times on an ordinary floor in an ordinary room, we wait in vain
to see the falling coin turn into a die, or a soap bubble, or an ostrich, or
float up to the ceiling, or in any way behave other than might inductively be
expected of a coin, especially once we have come to understand the mechanics of the typical behaviour of tossed coins. This happens whether the observer is a modern physicist, or
a naïve member of a rural tribe.
Even when the result is highly unexpected,
this remains true. A pretty example is the so-called tippe top. The ideal tippe
top is more or less mushroom-shaped, and on a level surface it commonly rests
with the cap, the heavy side, beneath. However, once it is set spinning, it
quickly overturns, lifting its heavier end against gravity, apparently
reversing its axis of spin. It then settles into a stable attitude until its
rate of spin decays too far to support it, after which it falls over.
Without the actual experiment, there
would have been the temptation to dismiss such a thing as impossible. After
all, spontaneously reversing its spin clearly violates the principle of
conservation of angular momentum and of energy. And yet, when we try the
experiment in the expectation of more rational behaviour, induction rules
OK! At least until our patience or our
toy wears out or our pistol runs out of cartridges or the turkey farmer brings out his axe ...
When something like that happens, we
can be sure that we need to pay more careful attention to our theory, our
interpretation, or our assumptions. We need to check for trickery, such as
perpetual motion swindles or conjuring. If we can discount anything of that
type, then it is time to reconsider one's preconceptions. Something has to be
wrong somewhere, and if deduction fails to do the trick, then it is time to
look to for abduction and induction as bases for fresh inspiration.
And sure enough, in the case of the
tippe top, more careful observation shows that although the direction of spin
from the point of view of the top inverts, angular momentum is maintained
because, as seen from outside, (most conveniently from above, but suit
yourself) the direction of spin remains unchanged from when it started, either
clockwise or anticlockwise. Also, the rate of spin slows down as the centre of
mass rises, and it does so to match the loss of kinetic energy against the gain
of potential energy.
Very pretty, even elegant, but so far
it gives no reason to abandon applied mathematics in physics.
But the history of science is rife
with discoveries that were contradictory to the received wisdom — think of
Galileo seeing Jupiter’s Moons; Newton's universal gravitation; Schleiden and
Schwann's cell theory; the germ theory of disease that was the fruit of the
labours of many workers; galaxies beyond local space; atomic structure; quantum
theory; special and general relativity; Griffin's discovery of sonar in bats;
there are hundreds of examples. What our successors will make of the
reconciliation of quantum and relativity theory, I would love to know, but I
suspect it will be one of the great examples in future history.
All the same, more and more of our
current scientific discoveries, however sophisticated, arise from a perspective
sufficiently wide to ease the acceptance of new advances. The bits increasingly
hang together.
The limits to the ways in which
things can happen are uncompromising, and every inductive conjecture based on
observation and abduction, constrains the possibilities. Even if the reasoning
from observation is not formally valid, it would be grossly irrational to infer
that the conclusion must be wrong; it accordingly would be unsound to insist
that research ("science") should be, or could be, purely and formally
deductive.
In fact, in the physical world, science and reality never
are purely and formally deductive; our interpretations and conclusions are
plesiomorphic at best. And the underlying reality itself arises from causes far more chaotic than anything we can afford to waste our time on studying.
And such a chaotic nature of reality need not be bad; certainly
it often works well in natural selection. Evolutionary strategies of wide
varieties have turned out to be successful, sometimes for over a billion years.
The best bet is generally that what you see once is the way things generally
happen, and what you see most often is the way things generally happen according to underlying causes.
In the early 20th century
the likes of Karl Pearson and Sewall Wright began to popularise the concept
that "Correlation need not imply causation". Note the "need
not". They were by no means the first to stress the point; it had long
been recognised; but the popular view was to the contrary: that, as Thoreau put
it, “Some circumstantial evidence is very
strong, as when you find a trout in the milk.”
The facile, formally invalid, and
commonly illogical, view, that tends to value, even assent to, circumstantial
evidence, amounts largely to trust in empirical induction. The converse view,
that causation commonly implies correlation, also has been recognised for many
years, even centuries, but Jack and Jill Average commonly fail to sort out the
significance. I discuss some of the considerations under the heading of
"causal webs".
Abduction
The most exciting phrase to hear in science, the one
that heralds new discoveries,
is not 'Eureka!'
but 'That's funny ...'
Isaac Asimov
The further we venture into such matters the
deeper we stray into the field of abduction.
Abduction is a form of reasoning that goes
from an observation or a speculation, to a hypothesis. To illustrate the
concept, I find it entertaining to refer to a story: “Breaking a Spell”, in the
book “Odd Craft” by W. W. Jacobs:
...'e went 'ome one
day and found 'is wife in bed with a broken leg. She was standing on a broken
chair to reach something down from the dresser when it 'appened, and it was
pointed out to Joe Barlcomb that it was a thing anybody might ha' done without
being bewitched; but he said 'e knew better, and that they'd kept that broken
chair for standing on for years and years to save the others, and nothing had
ever 'appened afore.
True, true; but abduction need not be
superstitious or stupid; practically any new scientific hypothesis begins as
abductive speculation. And so does every new line of thought that arises when
encountering a mental obstacle in exploring an emerging field. Examples might
reasonably include the early phases of:
- Newton’s work
on motion and gravity;
- Mendel on genetics;
- Darwin on
natural selection; and.
- Periodic characterisation of
chemical elements.
In fact it is hard to see how early
exploratory work on almost anything could begin without abduction. And
abduction, even in some of the greatest works of genius in our history, is
invariably at least partly wrong. If things were otherwise, research could
hardly be called research.
The various definitions and explanations of
abduction tend towards incoherence and mutual inconsistency. I don't pretend
that mine are compact or compelling but I wish to show at least how, as I see
it, the very concept has several aspects, and that they need separate
consideration. Whether anyone has assigned those aspects to separate categories
with distinct definitions, I do not know, but, for our purposes here, I doubt
it is necessary to do anything of the kind.
Part of the problem is that the different
categories grade into each other. Pure inductive guessing has no compelling
implication or basis to support its basic assumptions; pure deduction proceeds
from premises assumed to be true.
As a concept separate from induction,
abduction is comparatively recent; as far as I can tell, Charles Sanders Peirce
coined the modern usage of the term, and as one of the best of our early modern
philosophers of science, he began well. Popper for no obvious reason (I find it
hard to believe that he never encountered the concept) never seems to have used
the word, much less distinguished it from empirical induction, nor recognised
its importance to research.
As a philosophical strategy, abduction bases
its proposed conclusions on not necessarily perfect assumptions and not
necessarily cogent derivations. It also takes at least three forms:
- proposing causes to explain
empirical information,
- proposing consequences of assumed
causes, and
- proposing mechanisms by which
assumed causes give rise to assumed outcomes.
This is just one point of view: it certainly
looks vague and confusing, largely because it is vague and confusing.
So why even think about it?
Because that point of view is itself a good start to
empirical thinking. Abduction is fundamental to investigation, whether
scientific or not. Before Pierce ever described abduction and its proper use,
human thinkers had been using abductive reasoning for thousands of years.
Abduction is the basis of the starting points of most constructive or
exploratory thought in science, management, common sense, and arguably also in
creative work, whether artistic or technical.
Suppose, not being a turkey, you consider a
visit to an orchard. You pick a fruit from a certain tree: a pear — ah,
very good; pear trees bear pears! You go back for another and tell your friend
how good it is; he asks where you got it from? First tree in that row. He comes
back with a cooking apple, complaining about your misleading him. After
increasingly heated recriminations, you both go back to the tree and sure
enough, one branch bears apples and another bears pears.
WHAT??? How ...?
Harvesting apples and pears from the same
tree flies in the face of your sense of underlying biological mechanisms that
you see as dictating that apple trees bear apples and pear trees bear pears. It
differs from inductive generalisation from repeated observation, in that in the
light of long history and research we also understand something of the
nature of biology, in particular of heredity. We know very well why
apple trees don't bear pears.
Then you discover that the tree was indeed an
apple tree, but one that had had some pear wood grafted onto it.
Time to think again.
As one comes to understand more about the
nature of the phenomena one studies, the nature of one's induction changes
subtly, and mentally we develop into the field of abduction. Our turkey had
been collecting data without comprehension; his conclusion was unthinking
induction. Your assumption about pears and pear trees was at least based partly
on some understanding of the underlying biology. What you knew, and thought you
knew, formed the basis for a coherent hypothesis on which you could base
deductions and predictions.
That is part of the essence of abduction.
The naïve commentator might call it
guesswork.
And the following parable from the WWW is
food for thought:
Researchers put several apes into a cage in
which a banana hung above a ladder. An alert ape promptly went to the ladder to
get the banana, but as soon as it touched a stair, they all were sprayed with
nasty cold water. Every repeated attempt had the same result. Within a few
days, each time a ape went near the stairs, the others would violently prevent
it. The fuss gradually dissipated.
The researchers then replaced one of the apes
with a naive one. The new ape saw the banana, and immediately tried to climb
the steps. All the others attacked him. He soon learnt: forget that banana;
stay away from the stairs, or get beaten up. The researchers then removed a
second experienced ape and replaced it with another new ape. The newcomer
in turn went to the stairs and got beaten up. The previously new ape, who
had never seen the spray in action, but had been beaten, enthusiastically participated
in the correction.
A third ape was replaced and the next novice
learnt the same lesson from fellow‑apes in turn, two of whom had no idea why
they must not permit anyone to get too close to the ladder. This was repeated
till no ape was left who had ever experienced the water spray.
Nevertheless, by then only novices would try to climb the stairs.
. . . One day
a new young ape asks, "But Sir, why not?"
. . .
"Because that's the way we do things around here, my boy."
One could say that the apes had encountered
what Francis Bacon called the “Idols of the tribe”. They had to toe the line
drawn by the community, whether it was comprehensible or not and whether it
made sense or not.
The moral usually drawn from this parable is
a sneer at the mindless way those apes did things round there, but on what
basis might anyone suggest that humans would do better than apes in such a
cage? And apes or humans, clever or stupid, then, unless the researchers had
turned off the waterworks in the mean time, the first novice to buck the system
would have reinforced Pope's lesson the hard way: "a little learning is
a dangerous thing".
If an animal does something they call it instinct.
If we do exactly the same thing for the same reason they call it intelligence.
I guess what they mean is that we all make mistakes,
but that intelligence enables us to do it on purpose.
Will Cuppy
As a matter of simple common sense we learn
to distrust naïve empirical induction, let alone abduction, both of which certainly
are common bases for fallacy; and yet empirical induction is the basis of most
of our dealings with reality. Not only humans, but nearly all our sentient
fellow‑species, work on the basis of learning what usually seems to happen as
an apparent consequence of our actions, and of various types of events around
us. A horse or dog that has experienced an electric fence a few times, may not
understand electricity, but quickly learns to steer clear of that fence.
And it works! Naïve empirical induction works
so reliably that it is the basis, not only of conscious learning and even
apparently mindless reflexes and Pavlovian conditioned reflexes, but also of
physiological adaptation to exercise or food supplies — even of
evolutionary adaptation by mechanistic natural selection. By implication, all
of them, mindless or not, "put their trust" in the consistency of
events.
Induction is not mocked!
I'll discuss this under another heading, when
dealing with the ideas of David Hume.
Meanwhile, the ape parable leads us further
into the topic of abduction. Either as individuals or as a group, they were
unequipped to develop theories about how or why touching those steps led to
dousing or beating. Their induction was pretty nearly pure — it was
one-dimensional. No blame to them; technologically unsophisticated humans would
do no better, and dogs probably would not achieve even the preventive measure
of punishing would‑be transgressors.
First‑world humans in a similar situation
might however, investigate the set‑up for sensors designed to trip the showers,
and for possible means of bypassing or inactivating them. Or they might look
for missiles to knock down the banana. They might speculate on the motives of
the creators and imagine ways to communicate with them, demanding: "Why
are we here? Let us out or improve the comfort level! Stop maltreating us! Find
some better means of communicating with us than squirting us!" They might
brave the shower to get up the ladder in the hope of finding a way out.
Such reactions would require more advanced
levels of abduction aimed at understanding the nature of their situation and of
dealing with it. Less-educated humans might not do as well, or they might
surprise us with unexpectedly more sophisticated reactions or mythology than first-worlders —
I simply do not know.
The terminology is inconsistent, as I already
have emphasised. The very word abduction in this sense is little known
outside circles of professional philosophy.
In science, abduction is the major basis for
initiating the new generation of explanatory hypotheses and for discriminating
between them on the basis of available information or opinion. It is the basis
too, for generating means of learning more about the subjects of speculation in
an underdetermined world.
In technology abduction is the major basis
for diagnosis of problems and developing solutions for them. Such abduction
includes large categories of invention: recognition of the absence of something
desired or presence of something unwelcome, and of options for improving the
situation.
It is not proof in general.
It is not rationally presented as formal
proof at all.
In science:
If you don’t make mistakes, you are doing it wrong
If you don’t correct those mistakes, you are doing it really wrong
If you don’t accept that you make mistakes you are not doing science at all.
Anonymous
Strictly speaking, a lemma is something
proven as a step in the proof of something else. For instance one might prove a
more general proposition before proving that your main, more specific,
proposition is a special case of this concept, and follows accordingly. Suppose
that I needed to prove that there had been a noise in the forest. Suppose
that I can show as a lemma that any tree makes a noise when falling.
Then, as proof that there had been a noise, I can present the observation of a
fallen tree in the forest, where there had been no fallen tree before.
Well, in this essay I do not try to prove
much, but I do at times urge particular opinions. To this end I first offer
conjectures that I propose as persuasive; I might firmly believe them, or
consider them possible or desirable or interesting, but in any case I see them
as illustrative or probable or stimulating. I might for example say:
"Look, I cannot prove that every falling tree makes a noise, but I have
investigated a lot of tree‑felling, and so far every falling tree I have seen
was noisy, so I conjecture that every falling tree makes a sound, and if my
conjecture is correct I may conclude that where I see a fallen tree, there will
have been a sound whether I heard it or not."
An illustrative fable tells of two brothers,
one a pessimist, one an optimist. One Christmas Santa gave each one an
anonymous gift. The pessimist received a case of single-malt Scotch: his
reaction was "Oh no! What a hangover I'll have!" The other got a sack
of horse manure: "Oh goody! Someone's given me a horse!"
When I propose something of either of those
types I might call it a conjectural lemma.
For
one brother, the conjectural lemma was that he could not resist Scotch
and that drinking Scotch leads to hangovers and that he now had enough Scotch
for a monumental hangover; for the other, the lemma was the received wisdom
that where there is horse manure there must be a horse, and he now had the
manure.
Their specific lines of reasoning might be
valid or not, but as long as they suit the point I am urging in the context of
induction and abduction, that is all we need. If anyone encounters such lemmata
and their consequences, and sees fit to categorise them in context, fine. But
my conjectural lemmata are no more than illustrations or proposals, not proofs.
If we knew what it was we were doing, it would not be
called research, would it?
attributed to Albert Einstein
The apparent simplicity of the idea of a
thought experiment, like many apparently simple ideas I discuss here, could
hardly be more treacherous; there are whole books on thought experiments, and
treachery is one point on which the authors agree upon with the least reserve.
Almost any other point is up for debate.
And I agree with those authors because I
derived the same view independently.
So don’t take my definitions and remarks too
rigidly. A thought experiment, loosely speaking, is when one suggests or
assumes, situations or states that actually might or might not be possible, and
one deduces certain conclusions from them, or from particular axioms. The
experienced reader in this topic will recognise several examples that arise in
this essay; in particular, I mention some in the section on magic.
However, such magic is not the only
kind of thought experiment one gets. Even the most obsessive scientist does not
carry out experiments to confirm that every assumption that occurs to him, or
that he relies on, will correctly predict a given material outcome, or will
give a precise result. Instead one often can ask oneself what the result would
be if one changed the assumption to...
Something different.
And the fact that it is something different,
amounts to another thought experiment in its turn.
And more often than not, you can be pretty
sure that you have sufficient reason to accept your rough conclusion as a
working hypothesis. Even if you are not justified in your conclusion, you might
be satisfied enough to skip checking it. Then, whether you have convinced
yourself or not, you might pass on to consider a more ambitious formal or
material experimental programme instead.
But every scientific experiment is
essentially heuristic: you begin with a question and see which answers suggest
themselves. Will hydrolysis of vinyl chloride give you vinyl alcohol? If not,
why not? Will spinning a prolate spheroid around its long axis give a stable
spin? Given a spacecraft manned by a team of astronauts, will shooting it at
the moon from a cannon, be more effective than propelling the craft by rocket?
On Earth, will clear weather permit a mountaineer with a telescope on Everest
to see The Empire State building, as a flat Earth would predict? Will a
pedal-driven propeller above one's seat be a basis for a working helicopter?
Will heavier-than-air flight ever work? Will universal education solve all
social problems in an egalitarian society? Will rotary engines work better than
piston? Will faith move mountains? Will elimination of private ownership lead
to a stable, productive, unselfish, non-competitive, undespotic society? Will
natural selection lead to the emergence of new species? Will injection of
particles into the stratosphere mitigate global warming? Will electric cars
prove better than internal combustion? Will flying too close to the sun with
waxen wings cause the wax to melt…?
And on, and on…
Every new development depends on assumptions
about what we think we know, or what certainly is not yet known, and it is
never certain how many assumptions are implicit, and which of those assumptions
will matter, yielding either frustration or serendipity.
Sometimes just exploring assumptions and
their implications will lead at first to incredible advances in theory, such as
say, non-Euclidean geometry; some will cause gross upheaval of major fields of
science, such as happened with relativity and parts of quantum theory and
information theory. Some will build on partial understanding of reality, and
provide advances and problems variously greater and less than expected, such as
plastics, nuclear physics, and flight engineering.
In all of those examples, thought experiments
played roles throughout.
In all of them, ignorance was a factor:
ignorance of our facts, ignorance of our assumptions, ignorance of their
combinatorial relationships; if ever there were no ignorance, there would be no
need for experiment, whether in thought or in practice.
Thought experiments play their role in all
intellectual advances, whether by deduction, induction, or abduction.
Deductive inference, valid or otherwise, is
arguably the most implicit component of thought experimentation; there is
always an element of: “Suppose this .... then that must follow .... ” The
conception might be accurate, mistaken, or downright misleading, even
meaningless, but the form is at least that of premises followed by conclusion.
The reasoning might be reductio ad absurdum, or direct, but in either form it is formal deduction from
assumptions.
Then there is abduction. You see something,
and you base an idea on it (“Oh, a wooden ball falls more slowly than a stone
ball ... I wonder…”), or an isolated thought occurs to you (“Oh, suppose that
apple tree were taller ...”) or an apparent insight (“Practically every complex
organism passes through different stages of differentiation, growth, feeding,
competition, and reproduction in its life history, so each must undergo
metamorphosis, more or less obvious, not just insects ...”). or “look at the
way arbitrary shapes of rock on beaches or in potholes in streams, are tumbled
into beautifully precise spheroids; the principle must be that salient
irregularities get ground down preferentially ...”)
As for induction, very similar principles
apply, as you can see if you work your way through the examples given in the
section on induction.
In every case you work your way from
assumptions and ignorance to conclusions or hypotheses of various degrees of
usefulness. They might be very crude, though useful, such as flat Earth (works
fine for large-scale local maps) or precise, such as Euclidean geometry
(excellent for carpentry) or Newtonian physics (fine for non-relativistic,
non-quantum systems).
Even now, though in some respects we have
achieved some very high degrees of predictive power, the one thing we can be
most certain of, is that we are nowhere near any finality.
And thought experiments are among the tools
that underline our ignorance.
And do it most cheaply, and sometimes most
quickly.
The existing scientific concepts cover always only a
very limited part of
reality, and the other part that has not yet been understood is infinite.
Whenever we proceed from the known into the unknown we may
hope to understand, but we may have to learn at the same time
a new meaning of the word ‘understanding’.
Werner Heisenberg
You can't prove anything about the physical world by
logic alone.
Anonymous
This is where things start getting messy.
If I had lived in the time of Democritus with
his idea of atoms, I suspect that I would have been one of the sceptics that
rejected his silly assumption. As far as I can make out, he said that there had
to be a particle smallness beyond which it was impossible even in principle to divide an object any into any smaller
particles. The sceptics argued that on the contrary, whenever it was logically
possible to split a big lump of something, it had to be logically possible to
split a small lump. Consider apparently amorphous cheese, or water droplets, or
cleavable salt crystals for example: obvious, isn't it?
Analogous to halving a line in Euclidean
geometry: one can repeat the operation forever, getting new lines or droplets
or crystals half as large each time.
I was grown before I began to change my mind.
I had long since accepted the concepts of molecules and atoms, but still
rejected the reasoning of Democritus, or the lack of it, though what he
actually said was more sophisticated than usually is mentioned in class.
However, the more I saw of how the world
works, the more pervasive the concept of atomism became. Not just in
contemplating the chemical elements, but in biology, physics, logic, and
practically everything. Without going into detail, two modern ideas concerning
atomism are of immediate interest. Neither of them deals with what we commonly
call atoms (which strictly speaking are not "atoms", anyway —
and certainly not in the sense intended by Democritus and some of his
associates).
The first of the two modern ideas deals with ultimately
indivisible particles. I suggest as a matter of nonessential opinion and
no more, that such particles might (or might not) correspond to at least
some of our currently perceived elementary leptons, quarks, bosons, and the
like.
I might be partly or completely wrong here,
but, as I see it, the validity of the concept of ultimately indivisible
particles does not greatly matter in our context: all I am concerned
with is the idea that particles in our world are not turtles all
the way down, that not every particle is a
structure of sub‑particles and then of sub‑sub‑sub particles or fragments.
That denial is not a doctrinal
assertion — it is no more than an opinion, with its implied assumptions
about our world of perception and whatever passes for the reality underlying
it. I quietly take it that there is a stage beyond which the behaviour of what
we see as points or physical point‑like particles, goes no further. There is
where Good Ol' Dick, my chosen bottom turtle, finds his level. Underlying
realities reduce to whatever constants reign at that ultimate level and Nature
itself goes no further.
I offer that view without proof: it is not
formally axiomatic, but it is assumptive. I comfortably reject any opposing
assumptions until their proponents can present them cogently, or at least with
strong experimental support. After they achieve that, I shall be appropriately
astonished and happy to acquiesce.
For example, last I heard, no one had yet
managed to cleave an electron neutrino or an electron, not even in the likes of
a double slit experiment. In contrast some discussion was under way, about
whether quarks were compounded of sub‑particles. Whether they are or not, I
shall assume that there are such things as particles that are as much
indivisible and point‑like as might be possible in our world. That will do well
enough for our purposes.
That is what I assume in full awareness of
concepts such as the attributes of wave‑like and particle‑like
behaviour; although those concepts are no longer respectable in their naïve
form: atomicity is the concept I am referring to here, and atomicity is
fundamentally unaffected by those considerations.
So, such particles amount to
"atoms" — for our purposes: "indivisibles".
At this point I introduce a neologism: I
started writing about splittable items. as being “non‑atomic”, but the
clumsiness of talking about what amounted to “non‑non‑splittable” items became
irksome, so I have changed all those references to “tomic”. I cannot
find that usage anywhere in physics textbooks, but the term is convenient, so I
present it here. My apologies to anyone who hates the word, but feel free to
choose your own terms when you are the author!
More relevantly however, there are at least
two separate and distinct senses in which I use the term “atomic”, and
they do not have much to do with each other:
. . . firstly: not being tomic in any sense, and
. . . secondly: being the fundamental particle of a chemical element.
So for example, an alpha particle is the
nucleus of a helium atom, which does not mean that it cannot be split by
suitable application of force.
This is where the second of the two modern
concepts concerning tomicity arises. I will discuss it in more detail later,
but let's introduce it here: some things that one can split physically, none
the less cannot be split without changing their nature. Our familiar atomic
nuclei certainly are atomic in the sense of being atoms of elements, but
not in the sense of not being physically tomic: not only do their
nucleic structures consist of hadrons that can be separated or can clump or
interact in particular circumstances, but the atoms' nuclei can shed, shift,
collect, or share outer electrons in chemical reactions or electric fields, and
their nuclei can spall or split or grow when hit by suitably energetic
particles of the appropriate nature, so they are decidedly messy structures.
But if you split a nucleus you do not get the
same sort of thing that you started out with. Suppose you evenly split
an atomic nucleus of sulphur (possible in principle, though hardly practicable)
you do not get two smaller helpings of sulphur — you probably do get at
least two atoms all right, but probably two atoms of oxygen, not smaller atoms
of sulphur. One atom of sulphur is the smallest helping of sulphur you can get,
much as half a pair of gloves is not a smaller pair of gloves.
The whole topic is very messy in fact,
because, while some of the candidates for atomic status, some leptons in
particular, might be indivisible for all I know, most of the particles
of interest are compound, whether divisible or not.
Let's not go into that — not till later
anyway.
"The fool hath said in his heart that there is no
null set.
But if that were so, then the set of all such sets would be empty,
and hence, it would be the null set. Q.E.D."
Bas van Fraassen
Now I descend into hand waving, fables, and
speculation. Imagine some empty space of indefinite extent: in effect a universe
that has nothing within its event horizon, except space. Whether that
space has a horizon, and whether the horizon is a yoctometre or a yottametre
away, or whether it has any form of horizon at all, we do not yet specify. It
is not even clear to me that distance, or even direction or time, could be
meaningful concepts in such a universe, so I do not discuss them here.
It also is not at all clear that “space” is
"nothing", but let that pass for now.
That imaginary feat demands more (or perhaps
less) imagination than you might like to deal with; I am not even sure whether
the whole idea in itself is meaningful or not —for example, would such a
space accommodate vacuum fluctuations or not? And if it did, what sort of
particles could one expect could fluctuate into and out of such a vacuum? If it
could, then could that universe accommodate more than one fluctuation at a
time? And if it could, could multiple fluctuations interfere and create complex
structures that could no longer fluctuate? If you happen not to be familiar
with vacuum fluctuations, don’t let that bother you — this is an abstract
exercise, so bear with me.
In such a universe outside observers are
among the things that are excluded, and so are photons or other particles that
could carry information, so we could never really see anything. After all, if
we were there to watch, that universe would no longer be empty space, right? I
rely on a magic God's‑Eye‑View (call it G‑E‑V), an ability to see things
without needing to reach in and disturb the things we look at, and to do so
without needing to transmit information. To the best of my belief, this is
absolutely impossible even in principle and I remind you that it is purely a
thought experiment.
Now, to begin with, imagine that into this
imaginary universe we release a population of identical notional particles,
each of which behaves in ways consistent with Newtonian momentum: if a particle
is moving in a given direction, it continues in that same direction at a
constant velocity. That is very much as particles behave in free space in our
universe, but in our thought experiment the particles are not subject to
the effects of gravity or any other accelerating influence from each other. The
only way that the behaviour of the particles in our thought experiment is
unfamiliar to us is that they have absolutely no effect on each other's
presence. They don't bump, or exchange photons, and their paths do not curve in
each other's electromagnetic or gravitational fields. If they have spin or
anything like it, they ignore each other's spin. They certainly are not
Fermions, and it is not clear to me that they could be Bosons.
In short, each particle behaves as though all
the others do not exist at all — not even slightly. In fact, in such a
notional universe none of those particles really does exist as
far as the others could be concerned: they do not exchange information; no
events result from their mutual interaction or existence. Only an observer with
a suitable G‑E‑V could recognise that they all have coordinates inside the same
space.
All the same, with our G‑E‑V, we can observe one
important and impressive thing about those particles in that space: they do
have intrinsically consistent behaviour.
It follows that their behaviour is constrained:
there are things that they do, and things that they do not do.
In any universe, anything that happens,
anything that is done, is an event, a change: a change in time, with
some situation before and some different situation after. I could state as an
article of faith, that the change in situation will imply a change in entropy
and information, and I suspect that its very occurrence has to do with the
definition or creation of the passing of time, of time's arrow. But I am so
vague about the very meaning of the terms and arguments, that I shall not urge
them. Instead I pass them on as not much better than suggestive hand waving.
But I do so without much apology. Key
questions and key suggestions often are more important than key answers, which
is an idea that I am not the first to suggest. Since I cannot know in advance
which suggestions and questions are key to anything of value, I ask first and
argue after, if at all.
Note that it does not follow from the fact
that these particles do not affect each other, that they must lack all other
attributes that could affect any different notional particles at all. For
example, in principle we then could introduce some different kinds of particles
that do affect the behaviour or trajectory of one or more of the inhabitants of
that space, and in doing so are affected in turn. Such new particles might
combine with the original particles in forming structures that, unlike the
original unstructured particles, could indeed exist from each other's point of
view, meaning that their compound structures do affect each other, whereas the
simple particles did not affect each other.
Notionally one could imagine particle types
that cause all the other particles into mutual recognition and interaction.
Somewhat like fastening little magnets or scraps of camphor to floating corks
that otherwise would ignore each other.
This last point, of particles that enable
other particles to interact in particular ways, is not as fanciful as it
sounds — if the idea interests you, you might like to read about say,
Higgs' bosons and gravitational attraction, possibly in Wikipedia.
It would not be difficult to program cellular
automata that model some such universes, but note this important reservation:
one could program similar behaviour as resulting in different ways from
different rules. Given such a possibility, any such universe might resemble an
indefinite number of other model universes with similar behaviour, without
really being the same. Each such a universe with its own set of visible rules
might look exactly similar to the others, but if one does not know the
underlying rules, one could not be sure whether they are invisibly different.
Accordingly we would indefinitely remain uncertain whether our prediction of
the next move would be correct, or whether two automata suddenly would diverge
from behaving identically.
Just bear that in mind for now; I discuss the
grue/bleen paradox later on.
Anyway, in such a universe one could not
distinguish, constrain, or characterise the alternatively possible underlying
programs ("realities") by inspecting the behaviour of the individual
independent entities unless one either could experiment with them, interfering
with their behaviour, say by introducing new classes of particles that interact
differently with the different components, or by varying the entities'
interactions, or could observe a wide range of interactions to seek effects
that constrain the range of possible explanations or mechanisms.
When we cannot control the interactions, we
are reduced to observation; that is the situation in cosmology and astrophysics
for example: we cannot go out there to say, smash stars together, so we just
have to watch more and more stars in the hope of seeing something that narrows
the range of reasonable explanations.
Down here near Earth's surface, we are in a
better position to interfere, and such interference for the sake of gaining
information, we call by names such as “measurement” and “experiment”.
By way of analogy, the behaviour of floating
corks with suitably attached magnets might be difficult to distinguish from
corks with suitably attached electric charges, and lights that flash so that
they seem to jump about in the dark might be very difficult to distinguish from
lights that do actually jump from place to place, unless one indeed undertook
research by physical intervention (experiment) or by mathematical analysis.
Be that as it may, all such ideas are based
on the assumption that some form of underlying reality supplies the
constraints. However, all visible patterns of behaviour are underdetermined,
meaning that there always are multiple distinct possible underlying realities
or preceding events that notionally could in principle account for what we see.
Effective research would involve varying the conditions so as to change the
behaviour in ways that would exclude some of the possibilities, or reduce some
probabilities, thereby reducing the underdetermination to some extent.
Another complication is the question of what
counts as an underlying reality or origin of the system we see about us. There
are whole classes of such concepts and suggestions. For one thing, the rules
could be temporal, that is, time‑based: whatever exists, there must be
something that existed before it, so either there never was a beginning
(turtles all the way down), or it was Good Ol' Dick (turtle number 666666, with
nothing below), or there was a Big Bang, before which there never was anything,
not even time. Then there certainly could be no more turtles, so there never
could have been a "before", any more than there could have been
anything north of the North Pole.
And the same could be true of a West‑ or East
pole. (Try working out why!)
Or an underlying reality could be
topological, a looping history in which the universe reaches back to create (re‑create?)
its own beginning. Like Ouroboros?
Or a Phoenix?
Why ask me? I don't
know…
You might find it amusing, perhaps even stimulating, to contemplate a concept of which I was not the author: Why did the chicken cross the road?
To get to the other side.
Very well, then why did the chicken cross the Moebius strip?
To get to the other. . . err. . .to stay on the same. . . Well, never mind!
Again, underlying realities could be rule based: the very concept of
nothing existing might in fact prove not to be self‑consistent in practice, so
that there always had to be some universe, no matter where, why,
how, or what.
We might find that it is terribly difficult
for nothing to happen, or nowhere to be — ever.
And also, the very idea of a universe with no
rules, whether self‑generated or not, might in itself be self‑inconsistent. And
the rules in question might simply follow from whatever it turned out to be
that constituted that universe and those rules.
For example, consider our first little toy
universe with its non‑interacting particles: with our G‑E‑V we could see all
the particles travelling forever at the local equivalent of constant velocities
in their geodetic paths: the logical equivalent of straight lines.
That sounds intuitively minimal, but in fact
it takes for granted more assumptions than we might think of at first; for
instance coordinates: for particles to move, they must change
coordinates — changing coordinates is part of what moving means. And
if they do not exist relative to each other or the magic outside
observer, they don’t really have coordinates in their own universes. And it
assumes continuity of the identity of at least some classes of entities
("that particle moving over there, is the same one that we saw over here
before its coordinates changed: it did not in itself change in nature or identity
when it moved"). Shades of Zeno’s paradoxes!
This raises thoughts of Heraclitus with his:
"No man ever steps in the same river twice". It assumes some aspects
of conservation of inertia, some aspects of the flow of time, and certain
constraints on the patterns and states of the behaviour of the particles (such
as that the change of coordinates is continuous: a moving particle's next
coordinates will be right next to the current one, no matter how fast it
moves).
Heraclitus also said: "panta rhei"
(everything flows). Personally I suspect that to be an even more evocative line
of thought.
But anyway, suppose that instead of the
minimal attributes that we had assumed for our particles, we add a few more
assumed attributes, such as individual masses, charges, rigidity, fields,
radiation (electromagnetic, gravitational, and so on). Then all sorts of new
things happen in the behaviour of the particles. They change their velocity
(that is to say, they undergo acceleration, and may deviate from straight
paths), they exhibit effects of forces and momentum. They attract or avoid or
repel each other, collide and recoil or fuse, and generally begin to exhibit or
participate in all sorts of events; they exhibit causal behaviour,
doing things that, rightly or wrongly, but not necessarily unreasonably, we
come to think of as the physical consequence of their respective natures,
states, and coordinates.
These things happen because they begin to
find things in their universe that for them had not existed before. Such
constraints and events and manifestations give rise to concepts such as of causality
and the consequences of the natures and individual circumstances of the
entities.
In short, we find ourselves able in principle
to predict following events (to extrapolate inductively, if you like) in
the light of earlier behaviour: we observe, and can infer: cause and effect, or
at least event and outcome, with precision limited by the amount of information
and computation at our disposal or in existence in that universe. We find
degrees of consistency of behaviour within circumstances, from which we can
deduce, or at least conjecture on, some of the constraints on their behaviour,
even if we cannot guess how far down the stack of turtles we need go to
establish a full comprehension and explanation.
For all we know, we might not need any
stacked turtles, just a very few self‑defining consequences of nothing.
The important thing in science is not so much to
obtain new facts
as to discover new ways of thinking about them.
Sir William Bragg
Now, think again of our magical empty
universe that we have postulated, or possibly created: suppose that somewhere
in that universe we magically release a solitary electron (as a
convenient example of a presumably fundamentally atomic particle). With only
that one particle in our toy universe, it still is not clear that direction or
distance or time as such have any meaning at all. Without distance the very
concept of a line, a one‑dimensional line, makes very little sense.
Without any points of reference, we cannot say much about that electron, except
perhaps that it is notionally immortal as far as we can tell. We cannot in
principle say where the electron is nor even say meaningfully when
it is, or what its momentum is, because in its space and time its only location
is where it is; there is no other point of reference from which we can
say: "That way!" or “Then!”, let alone give any coordinates that
would amount to Euclidean points.
Whether "space" has any meaning in
a universe without that first electron, I cannot say. I cannot even say whether
it makes sense to speak of space at all where there is exactly one
particle, never mind no particle. And even that is on the assumption that we,
outside that universe, have some sort of G‑E‑V of our universe, a view that
enables us to see our particle or particles wherever they are, without any
observer effect or information creation to mess things up.
In other words magic.
In such a universe, I do not see how we even
can say whether that lonely particle is moving or not, except that it is hard
to imagine how it could be moving from where it is. I suspect that we cannot
even say when it is. I am unsure how to give it any sense of time or
vibration or anything in a universe in which no events can occur, but I leave
such puzzles and definitions to the physicists, or perhaps to philosophers of
physics. I am not even sure whether in such a universe concepts such as force
or energy would be meaningful. Probably we would need extra concepts to specify
them.
As you can tell, thought experiments are not
necessarily as simple as one might expect.
If instead I had chosen a proton, things
might not have been so simple (or possibly they would have been simpler),
because a proton is complex: it contains three quarks and their associated
gluons at least. Accordingly, a proton is not truly point‑like. But I had not
chosen a proton, but instead, a notionally atomic lepton, so let that wait. And
I am ignoring the spin of the electron, because particle spin always did
leave me confused, even in our empirical universe.
It still does.
Given also that the wave‑like behaviour of an
electron is hard to conceive in an empty universe, I ignore it similarly.
Now let us give that notional solitary
electron a friend: another electron also not at any particular location or with
any particular velocity, but so that each is well within the other's
observational horizon. That means that they can interact, and will do so
according to particular rules. Causal rules if you like. Perhaps those two
electrons are parsecs apart, and perhaps microns apart, but they are not in the
same place and in the same state at the same time. To simplify that concept,
let’s assume that they have the same spin, which, according to the Pauli
exclusion principle, will forbid them from being exactly in the same place.
In any case they will affect each other's
trajectory and velocity, and will react appropriately to each other's spin and
mass. Before there were two they did not even have anything like trajectory or
velocity, or even history (whether spin and charge would have any meaning in
isolation, I cannot even guess). But given two electrons, they must interact by
gravity and electromagnetism and spin at least. Whether they do so by the
intervention of Higgs bosons or other obscure particles, and whether those
particles existed before there were two of our electrons, or whether just
having a universe implies that they can pop in and out of existence by vacuum
fluctuations, I cannot guess either, and will not pursue.
What does matter is that having more
than one particle suddenly lends new dimensions to the universe.
New dimensions? Were there fewer dimensions
before there were two particles?
In a universe of exactly zero particles, or
exactly one particle, I am not sure how to make sense of the concepts of
dimensions of up, down, sideways, forward and back, of rotation, pulsation, and
passage of time, but with two particles there certainly is conceptual room for
things that perhaps made no sense before there were two particles, particles
that might love or loathe each other, might shove each other apart or draw each
other together — effects that could result in accumulation, collision,
repulsion, or annihilation. Distance now begins (only begins!) to gain meaning,
in the sense of being represented by the line one notionally could draw between
the particles.
And similarly time begins to gain meaning,
whether it had had meaning before, or not. Momentum and energy might begin to
make sense once there are two particles, even if the only momentum were nothing
more than momentum towards or away from each other.
However, such terms still lack some of the
meanings that they have for us in our richer universe. Even if you have not
realised it yet, just these consequences of adding a second particle have
changed that universe in ways more complex than I for one can assess. Whether
one could argue that the universe began with the potentiality for such things,
I cannot say, and it is not clear to me that anyone could say it with
authority — certainly not without first stating some fundamental
assumptions, assumptions that as far as I know, are not yet compelling —
anywhere.
Magic is
treacherous stuff.
I disclaim any sophisticated mastery of art
or appreciation of art, but some of my favourites among artwork are those of
Maurits Cornelis Escher. Many professional artists sneer, and jolly good luck
to such in their naïveté, but I am unable to think of anyone else’s work that
rivals his for sheer substance in various aspects that I personally value. And
of these, the one I find most striking is the ability to suggest and capture
dimensions and universes.
Consider these two works:
The first shows a tessellation of angels and
devils, an art form of which Escher was a master. The striking thing about such
tessellations is that, although the two populations occupy the same
surface — the same space, as it were, then when one looks at any
individual, whether angel or devil, the other population as it were, fades into
the background; it is hard to see both clearly at the same time.
One can imagine this as being like two
universes interpenetrating each other, each being unaware of the other.
* * * * *
The Rimpeling (rippling) picture is more
subtly creative, each addition contributing to new aspects. It begins with a
disk of white on grey; this could represent almost anything. Then a tangle of
black branches partly obscures the disk, suggesting the moon behind trees. But
the trees are upside down: that suggests a reflection — well and good,
except that there is nothing in the picture to represent a reflecting surface.
Also, if it is a reflection of trees, those trees themselves are not in the
picture; any trees the picture suggests must be beyond the top of the picture.
But the reflections of some of the branches are distorted in ways that suggest
that we see their reflections in water, just after the surface was disturbed by
waves created by two drops of water. But neither the drops, nor the waves, appear
in the picture, any more than the water surface does.
All the effects we see in the picture are
vivid and precise, but they emerge as indirect products of items that one tends
to overlook. To me they seem analogous to the way that introducing entities
into an empty, or at least sparse, universe, can create structures and
processes not at first foreseen.
The best book on programming for the layman
is "Alice
in Wonderland";
but that's because it's the best book on anything for the layman.
Alan J. Perliss
Why should all this fuss, this groping after
fundamentals, be worth the bother?
Because it puts us in a better position to
think in terms of developing classes of algebras of physics. As I
defined algebras before, whether mathematically or formally, an algebra is a set of objects or object
types, plus a set of operations on those objects. One could regard games
such as Chess, or Conway's
Life, or Go, as algebras; the objects are the pieces, the boards and so on, and
arguably the players; the operations are the rules that define the valid moves.
If we extend the idea of algebras to physical
universes of objects and operations, or interactions that affect their
behaviour, then that can enable us to do some things that make it possible to
deal with some stubborn philosophical challenges from past centuries.
For an example of an algebra of physics,
consider the universe of particles that I imagined as acting only according to
their momenta. That would represent an algebra, though an impoverished one. A
more substantial example would be the behaviour of matter according to
Newtonian physics, such as we might find in an introductory textbook.
Note: the fact that we use a lot of numeric
algebra in Newtonian physics is not the reason why I speak of an
"algebra of physics": my reason is that Newton's physics involves
items of matter and energy (sets of objects) and rules for how such items
interact in particular types of events (operations on the objects).
And the question of whether Newtonian physics
represents our universe perfectly or not, is irrelevant to whether it may be
seen as an algebra, or whether it is useful to express it as an algebra.
To think in terms of forms such as algebras
helps in dealing with concepts of: information, measure, logic, implication,
hypothesis, truth, probability, numeric description of physical realities, or
consequences of potential states and events. It also gives us a basis for
reasoning inductively and abductively in science, instead of shackling our
conceptions by restricting ourselves to deduction (and commonly invalidly at
that).
All of those concepts are manifested or
modelled in the states and disposition and nature of the physical objects,
states and events: this is why I assert that mathematics, logic, philosophy,
and information, are manifestations of physics and attributes of entities in
physics — they are not the basis of physics. They are instead implied by
the nature of events in physics. Even if we abstract them (meaning that we copy
aspects of them, or something sufficiently close to those aspects) into
isomorphisms or plesiomorphisms of the primitive entities, those morphisms need
physical representations if they are to exist at all, ever, anywhere, anyhow,
whether accurately or otherwise.
Is that important, do I hear you ask?
Think about it: for one thing, such a set of
objects and operations puts us in a position to speak meaningfully in terms of
what to expect from past events, and what to extrapolate into future events. It
also permits us to estimate the relative probability of past events being the
cause of present states. In other words it enables us to attribute causes and
effects and implications and developments, instead of waffling ineffectually
about statistically correlated observations.
It permits us to model situations, either
symbolically or materially. It does not guarantee that our conceptions are
exact, though it might suggest how far from exact they might be, and commonly
it permits us to deduce to what degree our interpretations are incomplete,
imprecise, or no better than convenient fictions.
And, as I shall show later, they introduce
the concept of emergence, emergent behaviour, emergent phenomena, and
similar terms.
Importantly though, a physical algebra need
not be deterministic, even when it implies causal behaviour: even the most
precise physical interactions between physical entities involve certain
physically non-zero uncertainties, whether because of quantum mechanical
considerations, or because of classical limits to the availability of the
information necessary to determine the outcome. So, for example, in
symmetry-breaking events, such as the toppling of a sharp needle that had been
balanced on a smooth, hard surface, a typical physical algebra tells us that
the needle will topple, but cannot tell us in which direction.
A sufficient reason why it cannot tell us is
because neither the physical algebra, nor any conceivably pre-existing
information, that is to say: any parameters, can offer any implication; until
the symmetry is broken, there is no future for that event — the reason
that the future does not yet exist, is that the information defining it does
not yet exist. Whether it exists after the event, is another matter, but
underdetermination reigns there as well.
As I see it, fundamentally non-determinate
events, such as symmetry breaking and quantum-random events — that is to say:
truly random events, make nonsense of any proposal, whether physical or
philosophical, that space-time already determines any subjective future.
As for predictability in QM itself, a quantum
particle in a superposition, contrary to common belief, is not really in two
(or more) states at once. Rather, a superposition means that there is one
state, with more than one possible outcome of a measurement, but that
successive measurements of the same state have a very low probability of
contradicting each other. That is one form of the consequence of the
non-existence of the information prior to its creation by the passing of the event
that eliminates the alternatives.
That all sounds very simple of course, but it
may become a little confusing from the points of view of observers who exist
many light years — or light-aeons — apart, but remain in contact by
continual exchange of messages.
Thus it seems Einstein was doubly wrong when he said,
God does not play dice.
Not only does God definitely play dice, but He sometimes confuses us by
throwing them where they can't be seen.
Stephen Hawking
Cause, causality, and implication constitute
a very vexed field. About all that one can be sure of is that whatever claims
anyone makes about them, will cause some other people to contradict the claims.
In fact the contradictions commonly will be so various and categorical that it
is hard to characterise the field of discourse coherently. There is a growing
field of study known by various terms such as "causal inference", but
of course academic philosophy is slow in assimilating it.
So I won’t yet try to deal with the field
definitively — not very hard anyway. And I certainly lay claim to very
little originality in my discussions. The closest I come to originality is by
not agreeing fully with anyone.
Including any of my own earlier conclusions.
The
concept of cause
"Who
art thou that weepest?"
"Man."
"Nay, thou art egotism. I am the scheme of the universe.
Study me and learn that nothing matters."
"Then how does it matter that I
weep?"
Ambrose Bierce
Cause is such a common concept that it seems
too obvious for definition. As a concept however, it is treacherous. Many
philosophers argue, though not generally compellingly, that it is not
compellingly definable at all.
Really serious arguments on the topic began
more than 2000 years ago and some of those are not settled yet; in fact, very
little about cause and causality is completely uncontroversial even today. I do
not try to outline the field, let alone cover it; I just deal with aspects that
occur to me here.
First let's discuss David Hume's argument of
more than two centuries ago. He creatively questioned the views of major
writers who preceded him. His own ideas developed during his career, which
makes it hard to attribute any fixed views to him, and a generation gap of more
than two centuries makes it tricky to interpret some of his terminology. For
example, his use of the word "induction" seems to have been vaguer
than I, for one, prefer.
Be that as it may, Hume argued that our only
support for the concept of cause, in particular the concept of specific cause
as we would see it today, is inductive: inductive along the lines of those apes
in the cage. This was not all he said on the subject, and what he did say was a
good deal more sophisticated than that, but it is the relevant aspect here. He
rightly pointed out that, as we saw when banging away with our pistol, the fact
that some particular thing appeared to happen in one particular way in the
past, is not in itself any proof that it will happen in the same way in the
future.
However, in his chapter: "Rules by which
to judge of causes and effects" Hume at the same time asserted essentially
the contrary, among other things: "The same cause always produces the same
effect, and the same effect never arises but from the same cause." His
wording in such topics was barely coherent in our current terminology however,
and so was his reasoning.
Part of the problem is that a lot of physical
and logical concepts that we now take for granted, were not well conceived in
his day, so I am not sure how literally he meant that. After all, in his day
such errors were at least partly reasonable, coming as they did, before
subsequent advances in physics and mathematics; maybe in philosophy too. At
that time they tended to take infinity for granted as a single concept,
their concept of information was poorly defined, and entropy was
not yet a word, let alone a posy of concepts.
In our day, when we assume an algebra of
physics, or guess at such a thing, we can hardly be more than partly right at
best, and perhaps dead wrong. To me, Hume's writings in reducing the concept of
cause to something like mindless induction, seem to be opposed to the very idea
of that sort of algebra. But even if his views had no merit it need not follow
that rival theories of his day, or of our day for that matter, are closer to
anything like reality; there always are more ways of being wrong than of being
right.
And yet, it also seems to me that he was
groping in the direction of what would have amounted to something very like an
algebra of physics.
Not that I think he would have been happy
with the concept of there being more than one type of algebra; he wrote his
"Enquiries Concerning the Human Understanding" and related works
roughly between 1748 and 1777; In those days the authoritative view of Algebra
was largely that of the first edition of Encyclopaedia Britannica, whose
article opened with the following description (pretty well matching the
concepts we were taught when I was in high school):
Algebra is a general method of computation
by certain signs and symbols, which have been contrived for this purpose, and
found convenient. It is called an Universal ARITHMETIC, and proceeds by
operations and rules similar to those in common arithmetic, founded upon the
same principles But as a number of symbols are admitted into this science,
being necessary for giving it that extent and generality which is its greatest
excellence, the import of those symbols must be clearly stated.
In geometry, lines are represented by a
line, triangles by a triangle, and other figures by a figure of the same kind:
But, in algebra, quantities are represented by the same letters of the
alphabet; and various signs have been imagined for representing their
affections, relations, and dependencies ...
Not actually wrong as it stood, of course, but a far more limited concept than
we apply today.
One way or another, Hume, a brilliant thinker
in his day, rejected the view that was dominant in the mid‑eighteenth century:
as he saw it, the concept of what I have called empirical induction was formally
invalid.
And in his formal terms it certainly was
invalid, but as I have pointed out, to apply formal principles in empirical
science is hazardous at best — it tempts one into unsupported assumptions.
Note that Hume himself cautioned his readers:
"And as the science of man is the only solid foundation for the other
sciences, so the only solid foundation we can give to this science itself must
be laid on experience and observation."
Let's think about how he blundered in his own
key conclusions. If your formal assumptions, conclusions, and predictions are
at odds with your observations, we cannot necessarily tell immediately which
are wrong, but we can be sure that something needs adjustment.
Hume was thinking in terms of causality, the
principle of cause and effect. He did not clearly distinguish between causality
and determinism in the way that we do in our time, and it would have made
little difference to his views if he had. I will speak of causality only as the
idea that: if any of a certain class of things happens or a particular
condition obtains, then a particular class of event, a particular type of
outcome, is likely, as implied by the nature of the algebras of
physics.
And furthermore, the outcome generally will
differ from what would have been the case in the absence of the notionally
causal event. It need not follow that every such outcome will be identical, and
in practical circumstances it is not possible to define identical initial
circumstances, so the question is largely academic, but we certainly can apply
the principle effectively enough to live in a world that seems fairly
comprehensible.
Make no mistake: it does not follow that
every such correlation is causal, nor that correlation has to be 100% if
it is indeed causal as we understand causality: when I breathe pepper it might
not make me sneeze every time, but I still regard the, say 90%, correlation as
causal, even though I either may or may not sneeze whether I have recently
breathed pepper or not; and I can support my opinion by studies of the
stimulant effects on the nerve endings in my nose, of certain essential oils
and other influences.
In other words, I can adduce successful
predictions in support of the idea that the pepper commonly makes me sneeze.
This might sound like ignorance of Popper's
principle of falsification, but, because I reject the principle, as I shall
justify in due course, that is not relevant here.
The first and most basic concepts of what I
called algebras of physics are the modes of behaviour of sets of recognised
object types, plus sets of recognised operations on those objects. The
operations in this connection might amount to actions internal to an entity:
the things it does by its own nature, or, on the other hand, we might be
looking at interactions between entities: the things they do to each
other.
Assume for example that we observe empirically
that under given circumstances, particular particles interact in apparently
more or less consistent ways, that is to say, with characteristic changes of
state or interrelationships: throw potassium into water, and it fizzes, bangs,
and flashes; hit a glass with a hammer and it breaks; permit an electron and a
positron to collide, and they annihilate each other. No such example happens exactly
the same way twice, even if we could repeat the action exactly in the same way,
which in fact we cannot. But still, they do happen after their kind,
time after time.
In principle we therefore can base an algebra
on the assumption that such particles and their interactions represent some set
of objects and operations that may constitute our relevant algebra.
Now, this is partly open to criticism on the
grounds of terminology, or if you like, semantics. I shall not go into the
matter in detail (there are whole books on the subject, books on semiotics and
on the philosophy of science) but I discuss the main points superficially in
the following paragraphs.
Notional examples of physical constants or
theorems in the algebra of physics, variously defensible, might include the
likes of "electrostatic like charges repel", "unlike charges
attract", "gravitational masses always attract", “their strength
of attraction is inversely proportional to the square of their separation”,
“F=ma”, “rigid bodies cannot occupy the same space simultaneously”, and so on.
From this point of view, cause in an algebra of physics is the relationship
between the objects operated upon or participating in an operation given their
states at the start of the operations. The effect comprises the
output states, the results, after the operations. In some cases, in refutation
of Hume's pronouncement, the output states are not precisely
predictable, even in principle, and then it might be better to speak of
something like cause and outcome, rather than effect.
But it is not clear how much more useful that
might be; not clear to me anyway.
But the fact that such an algebra might not
be precisely predictive in all cases does not invalidate the concept of an
algebra. There are indefinitely many examples, even in formal algebras, where
outcomes of operations are not unique.
This resembles, or is analogous to,
operations in formal mathematics, in which say, integer division of any natural
number by any natural divisor results in a quotient and a remainder which might
or might not be zero, depending on which numbers were chosen, or which tangent
to a line might be chosen by a given operation at a non-differentiable point.
Analogously, suspension of any suitable
object, such as an apple, in a suitable gravitational field, such as we
experience in an orchard, results in ("causes") the proverbial
effect: the fall of the apple once its suspension fails. Failure of such
suspension is a complex process or event, and is not precisely deterministic.
As I shall discuss, this view obviously is simplistic
in various ways, quite apart from the naïveté of such convenient ideas as
"falling down", rather than "accelerating along a segment of an
elliptical trajectory" or something similar. Nor does it in everyday terms
address concepts such as precision, unambiguity, and noise ("neglecting
air friction" and the like). However, it is just an illustration, so I beg
patience, if not pardon — there are more immediately relevant
considerations.
There are all sorts of logical objections
that conceivably could be raised against the idea of establishing, or even
defining, cause, and indeed many have been raised from time to time, more or
less independently, with various degrees of success, by many people, in many
contexts. Examples of such objections include: what looks like cause might be
coincidence, or delusion, or misinterpretation, or a transient or ill‑defined
effect.
Such examples and counter‑examples teem in
the literature. One impressive specimen is the grue‑bleen paradox, introduced
by Nelson Goodman in the 1950s: suppose someone produces four objects, say
glass balls. Two look blue and two look green and they are not otherwise
distinguishable. The cause of their seeming to be of one colour or another, is
the way they interact with ordinary white light and the way that light that has
passed through or reflected from the balls interacts with our eyes and nervous
systems.
However, we happen to be in error in our
assumption about the colours of some of these balls in particular: in reality,
though one ball is blue, and one green, the other pair are one bleen, and one
grue.
A bleen object is one that looks blue till
the first day of the year 2100, and thereafter looks green; a grue object is
one that looks green till the first day of the year 2100, and thereafter looks
blue. Each of those glass balls had been inspected for centuries daily and
never given any cause for anyone to see them as other than their apparent
labelled colour. How can we tell which will change its apparent colour, except
by waiting till the dawn of the 22nd century?
Equally, after the start of 2100, how can we
tell which had changed colour? If we showed the balls to a bright young
physicist born on the morning of the first day of the century, and when he was
old enough, assured him that the apparently green one really was bleen, the
resulting discussion might well be fraught with frustration.
One thing that we certainly could agree,
whether we regard anything like that as plausible or not, is that if it could
occur at all, it would upset our expectations of both causality and our algebra
of physics — if it did not destroy them outright.
Another idea that would make nonsense of a
standard view of consistent causality is the 19th century
"Omphalos" hypothesis of Philip Henry Gosse: he pointed out, for
reasons that have nothing to do with our topic here, that we have no way of
knowing whether the world we see now was not created as a running concern,
politicians and all, just a minute ago, complete with our mountains and
geological strata, complete with light on its route through space, complete
with our fossils, and the year rings in our trees and the memories in our
brains, and histories in our books. If that were correct, then our impressions,
both of our past and of causality in our world so far, however convincing when
we examine them, would be illusory.
Again, suppose that a snooker player
discovered that at the start of 2100, some of his balls started behaving
peculiarly: some would change in density and elasticity. Some could change in
inertia without any change in density; is that possible? Would it require any
change in our algebra of physics? Think about that one...
Or yet again, our world could be totally
chaotic, and our perception of even partial consistence, future or past, could
accordingly be a delusion. If so, it is hard to see how even a delusion of
consistence makes sense, or how anything matters at all, even temporarily. If,
on the other hand, we do care, then that is why anything matters.
As Bierce pointed out in the quotation in the epigraph: if it does not matter
at all, then why should it matter whether it matters to us or anything else?
There are plenty of other examples of similar
reasoning, some more charming than others, but, as well as we can, let's pass
causally on for now: let us see howsofar Hume's criticism of the concept of
cause is justified. We may at once grant his assertion that we depend on
empirical inductive reasoning whenever we assume the reality of cause in the
usual naïve sense: when we see:
- that certain things keep happening
in more or less the same way, and
- we get a pretty good idea of why
and how that happens in terms of a physical algebra, then :
we have some pretty good grounds to suspect
that we have identified an example of what we reasonably might call causation.
As a matter of common sense however, we
always need to remember how exposed we are to the risk of errors of various
types. Remember the black swans: we seldom can tell whether we have examined
enough examples, or examined the right population of examples, or are able to
be sure that our sample is random or "fair", or in general non‑misleading.
What we really need, to justify any such
categorical conclusion, is some sort of categorical implication of the nature of
the set we are considering. And as a rule, in fact arguably invariably, such an
implication is just what we cannot get; it might not even exist. Instead, we
are reduced to concepts of induction, abduction, approximation, and
probability.
Fortunately we have been able to achieve
amazing results with such rickety tools lately.
Still, we cannot always tell whether we have
examined our phenomena deeply enough; Newton's brilliant reasoning, abductive,
inductive, and deductive, not only gave remarkable precision, but also gave us
far greater power in dealing with our realities, than anything we had had
before (and far and away, more than most things since). The tendency then was
to assume that we had discovered what I would call a complete algebra of
physics. When Einstein's work and the follow-up in relativity and quantum
mechanics were developed, we found that, for some applications, we had been
working on insufficiently precise and insufficiently sophisticated assumptions.
In a way, this is very like our erstwhile
assumptions that Euclidean geometry was all there could be to geometry, whereas
non-Euclidean geometries of various sorts hovered undiscovered all around us
for thousands of years.
We do need to recognise the merit of Hume's
view that formal reason alone cannot prove the reality of what we call
efficient causality. However, he may have gone too far in appealing to custom
and mental habit, observing that all human knowledge derives solely from
experience — from several points of view a questionable idea at best.
The first thing to bear in mind is that
Hume's criticism amounts to pointing out the failure of the causal hypotheses
to meet the standards of formal proof.
In this he was quite correct from several
points of view.
However, he never proved the contrary: that
efficient causality was necessarily a delusion; let us grant that we cannot
prove formally that there is such a thing as cause in the everyday
sense — but that gives us no special reason to doubt the material reality
of cause in general. The general power of abductive and inductive evidence
strongly support cause.
Although, as
far as I could tell, Hume never mentioned Occam's razor by name, he implied the
same principle: that we should not multiply essences unnecessarily. For
example, he said in part: "And tho' we must endeavour to render all our
principles as universal as possible, by tracing up our experiments to the
utmost, and explaining all effects from the simplest and fewest causes, 'tis
still certain we cannot go beyond experience ..."
And yet that very principle could be applied
in the opposite direction, cutting the hand that wields it, so to say. Anyway,
no such a razor ever is proof in itself: it is a rule of thumb, not a formal
justification, and a loose rule at that. And it leaves room to wonder whether
all multiplications of essences are equally unacceptable: if, in the dark I
hear what sounds like hoofbeats, am I better off assuming as essences: six
black horses, three zebras, or one unicorn?
We cannot always tell which rival hypotheses
multiply essences most economically. And if rival hypotheses notionally invoke
equally many essences, they may be invoking different essences. And some
essences, some concepts, are more believable, more powerful, or more
productive, than others: which hypothesis is more valuable — the
ancient assumption of four elements, or our current assumption of something
like ninety, not counting synthetic elements?
Or, suppose we assume solipsistically that
random essential concepts in our minds, or in my mind at least, create what
seems to be a consistent, coherent causal world. Such creation could be argued
to require a significantly larger multiplication of essences than to assume
that what we seem to see necessarily reflects, within our capacity, something
much like the actuality we fancy we see.
Or close enough for jazz anyway.
Yet, as I see it, even this is not Hume's
fundamental false step. That lies deeper than denying our ability, by
application of formal logic, to support our claims concerning cause. What was
worse, was the assumption that in applied logic, we can claim formal proof
at all. Even our formal proofs of formal theorems commonly are impure: we
implement them by physical, mechanical, procedures, and by mechanisms such as
brains and calculators, and by models, visual symbols and constructions. Even
if we grant for argument's sake that they were valid yesterday, it is
arbitrarily inductive to assume that they will be valid again today.
We also must recognise that some of those
anti‑inductive fingers point both ways: to be sure, formal proof cannot
formally prove general assertions in applied logic, but formal proof cannot
prove formal proof either. Formal proof depends directly or indirectly on
derivation of conclusions from arbitrary axioms; it is possible for workers on
such proofs to make mistakes; possible for them to propose inappropriate
axioms, possible for the axioms themselves to conceal inconsistencies that
render whole classes of proof invalid, or at least insufficient. For that matter,
Hume's own assertions could hide errors.
Again, our inability to prove that a
particular apparently (or even obscurely) causal correlation really is causal,
is no better formal proof that it really is non‑causal than inductive reasoning
can prove formally that it is in fact causal.
In short, even in mathematics, let alone in
science, formal proof is at best a very, very tricky field, as Charles Dodgson
pointed out in his brilliant sketch: "What the tortoise said to
Achilles".
Now what was it that established our
conceptions of "rational" causality in the first place? It was a
combination of empirical observation, with consistent patterns of events that made
algebraic sense in terms of empirical observations and hypotheses. In our
historic and classical past, we routinely invoked associations of events with
"supernatural" causes such as those fabricated in superstitious
belief in witchcraft or divinities.
It took millennia of time, and hosts of human
sacrifices, tragedies, and false starts, for us to develop anything that begins
to look like an empirically rational toolkit. Only then could we critically
analyse and investigate causal hypotheses in anything like a commonsense
manner, let alone rationally in the light of the results of scientific work,
false starts, and progress. It was uphill work, because at every level we had
to fight vested interests in quackery and delusion.
But, in the light of Hume's criticisms, in
what way were such empirically rational tools any better than previous
delusions?
Firstly they were no worse than Hume's
arguments themselves: as I discussed earlier, Hume was applying his accusations
to material systems, not formal systems, so that his axioms themselves really
were no better founded than any other material assumptions. Reasoning about
material systems implies empirical investigation and conclusion, and empirical
work has little to do with formal proof. For one thing it is widely accepted,
if not actually common cause, that empirical systems are underdetermined, so
that ultimately one is limited to selection of hypotheses on a basis of
principles that differ from formal proof, and assumptions that differ from
formal axioms.
Curiously, given his rejection of naïve
versions of "cause" Hume seems not to have thought in terms compatible
with our current views on underdetermination; to the contrary, he said: "
...the same cause always produces the same effect, and the same effect never
arises but from the same cause: this principle we derive from experience
..."
As I already have quoted Feynman speaking
some two centuries after: “A philosopher once said, ‘It is necessary for the
very existence of science that the same conditions always produce the same
results’. Well, they don’t!”
To some extent we deal with
underdetermination by empirically scientific disciplines, generally along the
lines of hypothesis, prediction and verification/falsification. Such approaches
intrinsically cannot banish underdetermination completely, so they cannot prove
that our new ideas are correct, but they can eliminate large classes of invalid
guesses.
In this way abduction, hypothesis,
prediction, and associated expedients, have made the science of the last few
centuries unprecedentedly successful in giving us power over our world.
In particular, empirical science does not
even aim at formal proof, only at the selection and generation of
hypotheses concerning empirical observation and the generation and comparison
of rival hypotheses. Proofs of Hume's type are not relevant in physical
science; physical science is a field in which working hypotheses change
continually, although, as far as we can manage to achieve it, progressively.
As a matter of historical context, the recent
record of the working hypotheses of science, though spattered with instances of
errors, blunders, bad faith, and downright stupidity, has been one of success
unprecedented in our entire human past, and in fact of continued success on an
unprecedented scale during our most recent two to four centuries, depending on
who is counting — and there is no end in sight; mainly just
continuing acceleration.
Causal
chains and webs
You
can’t proceed from the informal to the formal by formal means.
Alan J. Perliss
First, before trying to discuss “cause” in
the philosophical sense, let’s clear up some confusion that arises from
simplistic assumptions. People speak of some one thing “causing” some other
thing, and thereby they immediately introduce difficulties. Commonly however
they fail to recognise the difficulties that they themselves have created by
their unspoken assumptions.
No matter what people assume, whole chains
and webs of events must necessarily happen first or in parallel for practically
anything to happen at all.
I already have discussed some aspects of
correlation and causation, under the heading: "Induction", but the
topic has so many aspects to that it inevitably remains treacherous. For one
thing, causation itself is a slippery concept, and full of intellectual traps.
Practically any of those items I discuss is the
“cause” in the sense that, if we had prevented that item, something different
would happen instead. We could say that what we did that "caused"
that different event, became the “cause” of what happened instead. Sometimes it
might be possible to rescue a change by adding other "causes", but
generally at the cost of greater changes elsewhere. This reflects the view that
we cannot destroy information.
We can illustrate such principles beautifully
with cellular automata such as variations on the game of “Life”; detailed description
would take too long for our immediate purposes here, but I cannot offhand think
of any sufficiently large, finite, non‑trivial pattern in the game of Life, for
which it is impossible to change its behaviour by addition or removal of a
suitably chosen single cell, though it sometimes would be possible to change
some cells without affecting the outcome. It certainly is possible to find an “Eden” pattern, one that
cannot be “caused” in the sense of arising from any preceding pattern,
so that it must be specified in its entirity from outside; but that is a
different matter. If such concepts of cellular automata are unfamiliar to you,
you might find it helpful to begin by reading "Conway's Game of Life" in Wikipedia.
So ignore cellular automata for now, and
imagine a more familiar, proverbial example instead: a farrier's assistant
loses his grip, causing his hammer to fall; the falling hammer distracts the
farrier, causing the improper installation of a nail into a horseshoe; loss of
the nail causes loss of the horseshoe, causes loss of the horse, causes loss of
the rider, causes loss of the message, causes loss of the battle.
But what caused the loss of the nail? A
generation before the falling of the hammer, the father of the assistant of the
farrier met the assistant's mother‑to‑be by accident because her scarf blew
away. If that had not happened the assistant never would have been born, and
instead a different assistant farrier would have stood by, who probably would not
have dropped the hammer at that exact moment and the shoe would not have
been lost. So that gust of wind that blew the scarf perhaps twenty years
earlier, was what really caused the loss of the battle, right?
And yet, any of an indefinite number of still
other causes could have meant that there was a different rider, who lost the
message without losing the horse, or lost the horseshoe without losing the
horse. And furthermore, the non‑birth of that particular farrier's assistant
would have an indefinite number of other consequences; for example, there might
not have been a farrier or a battle in those exact places and at those times at
all.
Or suppose that the hammer did fall and the
horse was indeed lost, but the same accident sent another rider on an inferior
horse after the fallen rider. He stops to see what had happened to his
predecessor, and offers his horse in exchange for the lamed, though otherwise
better, horse and saddle, because they will be worth more after care and
healing. So the message does get through in time to have the crucial effect,
and causes the general to win the battle. Now what won the battle?
As you can see, assigning a unique and
independent cause to any event rarely is practical, if ever, and the
same is true for the assignment of a unique and independent effect to any
cause. An indefinite number of other "causes" of such types could
have lost or won the battle. Another similarly trivial accident could have
caused a stray shot to kill the general, or the opposing general, just before
the message arrived. In my personal opinion, it is flatly impossible for any
noticeable event in real life to have only one cause, or for any one event to
cause only one effect. For a simple causal event one would have to resort to a
notional universe containing a very small number of particles. That is not us,
and certainly not on a planet like Earth.
One could of course do some finicky
redefinition of what a cause is, dismissing indirect causes: the dropped hammer
only caused the noise, and the noise, not the dropped hammer, was what
distracted the farrier; so a musket fired behind the fence could have caused a
similar defect in the horseshoe — and so on.
We usually expect big effects to have big
causes, but that is not always the case.
Examined in such terms the very concept of cause
tends to fall apart. Remember Aesop’s fable of the mountain that laboured, only
to bring forth a mouse? Analogously, we seldom think in terms of something
large specifically causing something small; as a rule we are correct; something
large might produce many small things — a major earthquake might
destroy many tonnes of eggs in many places — or it might produce
something large: a quake might collapse a mountain to destroy a
city — or a mouse.
But a big quake causing nothing more
significant than a broken egg? I cannot
imagine that.
Ambrose Bierce, with characteristically
perverse penetration, wrote the following amended fable on that theme:
A Mountain was in labour, and the
people of seven cities had assembled to watch its movements and hear its groans.
While they waited in breathless expectancy out came a Mouse.
"Oh, what a baby!" they cried in derision.
"I may be a baby," said the Mouse, gravely, as he passed outward
through the forest of shins, "but I know tolerably well how to diagnose a
volcano."
What caused what? At a pinch, we might
imagine a combination of minor events causing a big event, but as a rule a big
event either needs a big event to cause it, or a potentially big event to be
triggered to cause it. For example, a shout can start an avalanche, but only if
the snow had built up in advance. And pushing the snow back up would not
unshout the shout.
Call that the trigger effect — and do
not expect unpulling the trigger to undo the effect.
Some legal systems recognise associated
difficulties in ascribing the operative cause among a combination of causes:
two pedestrians collide and one of them staggers into the street; a passing car
knocks him down and passes over him before the driver can stop. The car
immediately behind doesn’t see him in time either and also goes over. The
pedestrian is found to be dead. Was the fatal injury caused by the first car or
the second? How does one allocate the blame? First car or second car, or the
victim himself, or the other pedestrian in the collision? Or all? Or none?
Various legal systems would differ in their interpretations and actions. But
for each one, the resolution of the problem would be an arbitrary compromise at
best.
For an example of an uncompromising
resolution, consider certain aboriginal groups in Africa at least, in which, I
understand, every death was asserted to have been caused by witchcraft,
and the culprit had to be identified by the local witchdoctor, with varying
penalties.
In his book: “Demon Haunted World”, Carl
Sagan pointed out that we should beware of too smug a view of such naïve
savagery: witch hunts in our mediaeval world were hardly less uncivilised or
unintelligent; many a victim of witch-hunts was burnt or hanged on no better
evidence than that someone’s cow had died. After all, why else should it have
happened?
Not all "civilised" legal systems
are clean of such taints; compromise has its points — think of say,
the abuses that belatedly led to "good Samaritan" laws in some
countries.
For a more complex scenario, try this: Alph,
Billy, Charlie, and Dennis crashed unhurt in an aircraft routed over the
desert, landing among remote dunes.
Alph so bullied the others that each
privately decided not to hold out for rescue while Alph lived.
Billy surreptitiously put a slow leak into Alph's
water canteen. Charlie did not know of the leak, so he dissolved some cyanide
in the water. Dennis did not know of the others’ attempts, so he dissolved some
gelatine in the water while the others slept. This solidified the water once it
had cooled, so that it could not leak, but also could not be poured out, giving
the impression of a vessel containing no liquid, though if it were dug out with
a suitable rod and the jelly were swallowed, it could have kept Alph alive just
as effectively as liquid water would have, if it had not been for the cyanide.
Alph set out to walk towards the river,
telling the others to wait by the wreck, or he would skin them when he
returned, but as soon as he had gone they hurriedly left in another direction,
taking the rest of the water from the wreck and abandoning him to his fate.
Each of them, for his own reasons, expected Alph never to return, but each
guiltily refrained from telling their mates what they had done. Alph was within
walking distance of the river if only he had had enough drinking water in the
canteen to get him so far, but not knowing of the cyanide or the gelatine, he
died of thirst before he could reach the river, or reach the water that would
have been in the wreck if the others had not removed it.
One way or another the story came out, and
Billy, Charlie, and Dennis were charged with Alph's murder. Each pleaded not
guilty, though some reserved pleas of attempted murder, or arguing that nobody
had done anything that killed Alph, so that in fact there was no murder at all.
Billy argued that his leak had done Alph no
harm because it was not the leak that had deprived him of water, and in fact he
had had enough water with him all the time, and that his leak would in fact
have saved him from the cyanide that would have killed him if it had not been
for the gelatine.
Alph had not died of cyanide, or even known
of its presence in the water, and Charley had done nothing but administer
cyanide, so he could not possibly have been guilty of harming Alph in
any way.
Dennis argued that that the jellied water
could have kept Alph alive till he reached the river, so that actually his
jelly had conserved the water from leaking out, prolonging Alph's life from a
quicker death from the cyanide that Dennis had not known about anyway — if
only Alph had elected to be saved by it. Alph had died of his own ignorance,
not murder by cyanide or absence of water.
And yet, though Alph could not have survived any
of those attempts on his life individually, he had not died of them
collectively, but of unrelated thirst.
Still — only if the bungling
killers all had refrained, could he have survived. But no individual
attempt had succeeded more than any other and there had been neither collusion
nor conspiracy. And what about the plea that some of the actions had prolonged
Alph’s life, however briefly? Ultimately, which of the four were guilty of
what, if of anything? Alph of suicide, or any of the others?
Arguably the moral thing to do would be to
arrest the desert as the murderer. Or the mother of the mechanic whose
negligence had caused the plane's failure: if she had not had the affair with
the mechanic's biological father ...
To assign guilt of murder, you need to
identify deliberate actions that had caused the death directly. For a would‑be
murderer simply to be happy to celebrate the death, does not comprise murder,
nor even attempted murder.
Now, there are several aspects to making
sense of dilemmas of those various types. One such aspect is the difficulty of
defining what a direct cause might be, given the problem of the defining the
string of causes within the web of causes. Another is working out what it is
about one event or circumstance that from the viewpoint of a particular
observer makes such an event amount to a cause in our everyday sense, and a
third is how we can define a cause in our sense of scientific empirical
investigation.
The first of those problems of definition is
partly psychological and practical, given that the relationships are clear and
understood. Our view of any situation is limited because we can only see or
predict a small number of factors as relevant, which is why we cannot say how
many things could reasonably affect any outcome.
So we see a particular action or event as a
simple cause of a particular event: “I pulled the alarm cord to prevent the
train smash.” or “His leaky fuel‑tank caused the forest fire that killed 97
people.” Or: “She stuck pins into the wax model, and that was the cause of his
dying of prostate cancer.” And so on.
Such examples, that might not even be
accurate, let alone reliable, deal with poorly‑defined events that affected
multiple outcomes of poorly defined circumstances, most of which weren’t even
suspected by the witnesses.
It is not that the events were not real, nor necessarily
that the concepts of the causal connections were not real, but that they were
poorly understood, and their causal connections were hardly ever more than part
of an indefinitely complex web of causal interrelationships and
confounding factors. And generally the outcomes were grossly unrecognised or
minor effects in a major complex.
For example, if one of those pins in the wax
model had pierced a finger, that might have been the unrecognised cause of the
death of the spell weaver by septicaemia; the cancer might have had nothing to
do with the pin, or even with the murder victim's death at all; unless there
was an autopsy possibly no one would have suspected any possibility of cancer.
Or instead he might have died of a stroke when he was told that someone was
sticking pins into a model of him.
Things become far more troublesome in
empirical science, in which we teach the young researcher to use controls.
The idea of the control is to reduce the number of differences between the
cause of the effect under investigation, and any alternative. In particular we
wish to reduce that difference to a single variable, or a coherent combination
of variables. If we cannot reduce it in such a way, then we need more controls.
The intention is to be sure that we are correct in asserting that the variables
are what cause the predicted effect.
That is of course a hopeless
oversimplification of experimental theory, but I am not trying to cover the
entire field, so please be tolerant.
Unfortunately, even as it stands, that is not
sufficient. There are so many variables that there are many ways of producing
wrong predictions or even nonsensical interpretations. One of my personal
favourites is the schoolbook demonstration of the lighted candle standing in
the basin of water. Invert a glass vessel over the candle and lower it to stand
over the candle in the water. When the flame goes out, the surrounding water
gets sucked up into the cylinder, proving that the oxygen absorbed by the
candle flame had reduced the volume of the atmosphere in the cylinder.
That experiment, though charming to school
children, is radically misleading. The oxygen consumed in combustion in the
cylinder has hardly anything to do with the gas volume and pressure, because it
is largely replaced by the same volume of carbon dioxide, but that uncontrolled
experiment has remained a favourite for generations. The experiment ignores the
heating of the air in the glass by the flame, followed by its cooling when the
flame is extinguished; and it ignores the reduction of the volume of carbon
dioxide by its solution in the water. That experiment could be put to better
use as an example of the various misleading assumptions that one may encounter
in science, and of why notionally strict controls are not generally possible
and of why experimentation theory can become horrendously complex, demanding
other means of analysis, such as sophisticated statistics.
For the least arguable idea of cause, one
might confine the discussion to a single event or situation, such that it
reflects a given physical principle, say Newton's
assertion that F=ma, that is to say: Force equals Mass times Acceleration. So,
we could refer to that principle in considering say, the causal event of a club
hitting a golf ball. We could apply a few concepts such as the ball's
trajectory and derive the force that the club head exerted on the ball. The
question of the force that the golfer's arms applied to the club we then would
regard as a separate and different causal event, and only of indirect relevance
to the stroke of the ball.
But all the causal events leading up to the
ball's trajectory would amount to a causal chain.
And all the events that affected the causal
chain indirectly, so‑called confounding events, such as whether the
golfer was distracted by his caddy's munching on an apple, would be parts of a causal
web of events.
But we would not generally undertake to
demonstrate the whole causal chain or web, only the last, most local, very few
events or states.
Philosophically this approach greatly
simplifies the concept of cause: to the extent that we could estimate or
calculate the various items in the causal web, we can derive the outcome of
interest meaningfully.
However, there are important principles
involved in this view. We find that:
- we can accept the concept of webs
of observed or predicted events being causal; we need not assert it
as being inarguable, let alone deterministic.
- differences in parameters such as
forces or coordinates cause differences in outcomes.
- such differences commonly have
quantitative aspects: big differences tend to cause bigger changes
in outcomes sooner than little differences do.
- the natures of such differences in
the parameters amount to whatever information affects the outcomes.
- in practice the information
available to any observer, sentient or otherwise, usually is a small
fraction of the physical forces that materially lead to the outcome of any
event in a causal web.
- in theory a great deal of
variation is flatly unavoidable because quantum events making up reality
involve intrinsic unpredictability; this implies non‑determinism because
in the nature of things some of the information necessary to determine
outcomes intrinsically never could exist at all, whether it would have
been available to any observer or not.
- even without quantum
unpredictability, as I show elsewhere, limits to the information existing
in classical physical systems, let alone to information accessible
to measurement, imply limits to the precision of physical determination.
This applies both in Newtonian and Einsteinian physics.
- such effects do not forbid
causality, but they inescapably affect such things as determinism, the
creation of entropy, and the arrow of time.
Now, back to correlation and causation.
We must first bear in mind the foregoing
limitations on the concept of causation in any sense. Having done that, if we
find that a given preceding event occurs frequently, even invariably before the
consequent event, then we need to consider it strongly as being causal. That
alone is not proof; it also could be common cause. Say I buy some eggs every
Thursday at the same shop as Ken, whom I happen not to know, and he buys some
an hour later, and this happens repeatedly. One observer might conclude that I
cause Ken's purchase. However, the primary cause is simply that we both buy on
Thursdays because we both know independently, that that is when fresh eggs are
delivered to the shop.
We might not even know of each other's
existence, but notionally a G‑E‑V could see the correlation and understand its
cause, a cause that would not have much to do with one causing the other. It
does not follow of course, that in a case such as this one, there was no mutual
cause, or at least causal influence: for example, by behaving as regular
customers, we might be encouraging the shop to stock fresh eggs on Thursdays.
At the same time, there could be all sorts of
contributory minor causes that Ken and I were hardly aware of: our respective
reasons for wanting eggs at all, fresh in particular. But our proximate reason
would be the freshness of the Thursday eggs: one common cause for two
independent effects.
Not all causes have multiple effects in that
form; notionally some could simply be caused in a simple one-to-one sequence.
This is strictly true only in principle, because it is arguably impossible for
any material cause to have just one effect, but, to get the idea, imagine a
series of toppling dominoes, each knocking the next one. Here we deal with what
we might call transitive causation:
- If domino A falls it knocks down
domino B
- If domino B falls it knocks down
domino C
- Therefore, transitively, knocking down A causes knocking down of C, and
indeed knocking down of any following dominoes.
- In the latter case, transitively
knocking down B, and any other intervening dominoes, is a collateral
effect, whether desired or not.
- Though they are not generally
taken into account in considering the transitive knocking down, there
always will be other collateral effects, such as the noise made by the
falling dominoes. And it is possible for one domino to knock down two
dominoes, causing the line of fallen dominoes to split into two chains of
falling dominoes.
Yet again, some effects intrinsically require
multiple simultaneous contributory causes. Think of the string of a musical
instrument: to be of use it needs to be under equal tension from two ends at once. Pulling on just
one will not do.
Those three types of component causes:
linear, propagating, and combining, are the major contributors to the
structures of what I call causal webs. I do not develop the theme much here,
but the concepts seem to be of increasing importance in fields such as decision
theory.
We know very little, and yet it is
astonishing that we know so much,
and still more astonishing that so little knowledge
can give us so much power.
Bertrand Russell
This might sound like a prudent time to get
out of here, but really, it is just the beginning. One idea to deal with
straight away is the very concept of entity. Another one is emergence.
Here again I exceed my own imaginative
capacity. Whether “entity” had any meaning in the universe without that first
particle, or even whether an empty universe necessarily contains itself in
itself as an entity, I cannot say. I cannot even say whether it makes sense at
all to speak of entities in a universe where there is exactly one entity,
assuming that such a thing might be possible in principle. And even that is on
the assumption that we have some sort of magic G‑E‑V that enables us to
observe without defining any location or having any other observer effect that
messes things up.
Pure magic again, of course. Until
someone can demonstrate some such thing in principle at least, I do not
believe in any real form of G‑E‑V. In practice as opposed to principle,
I flatly disbelieve it.
There are confusing rules about speaking of
different electrons in a universe where electrons are all identical as far as
we can tell, but there are two kinds of plain vanilla electrons: negative
electrons and positive (that is to say: positrons), and their charges and the
fields of those charges are consistent enough to label them for our purposes,
and their spin is a story in itself. Already we can recognise each of the two
electrons as an entity, and we also can recognise the two electrons as a pair;
from that very fact, that pair is an entity too. Aspects of their positions and
trajectories already can be discussed as entities.
Entities essentially imply information.
Exactly how that information can be conserved in a universe with only an
indefinite space plus two electrons, I am unsure (much less with a single
electron), so I will not follow that line of thought, but I suspect that just
by starting with a universe with at least two particles, necessarily with
different coordinates, we are breeding space at a huge rate, something to do
with c, the speed of light. And I bet that in the process we are
breeding energy and particles as well.
It seems to me that I am describing something
like the Big Bang, which notionally had emerged from some sort of point‑like
event.
But I am not sure of that either. And it is
another subject, not vital to my theme, as far as I can tell, so let’s leave it
there.
The main thing at this stage is that by
adding a second particle we have created more than just two of one kind of
particle in a universe. Things can happen in that new universe, things that
could not happen in what previously was a less complex universe.
As soon as we have at least two entities, we
get what I call emergent effects. For one thing, we get
emergence; and besides that effect of emergence, a new kind of entity now
exists: a pair of electrons. There will be much more to say later about
emergence. You need not bother to interrupt to tell me that what I say contradicts
many prominent philosophers who lay down the law about emergence; they
contradict each other too, and there is plenty more that they are
catastrophically wrong about as well, and I don't have the time to nursemaid
them in everything.
Mind you, I myself am not sure about everything;
for example: it strikes me that one might argue that even before adding
the second electron, adding the first particle already produced emergent
effects that were not possible before. But I do not immediately insist upon
that. I still am unsure about what space itself might be, or what all its
implications are.
Anyway, the effects that those two electrons
have on each other may be slight (but how much is one to expect from a universe
of just two particles anyway?) and it might be hard to put a finger on some of
those effects, but if we add a third particle to our universe, then yet
more, completely new, effects and entities emerge.
For one thing, the presence of three
point-like particles that are not necessarily on the same straight line,
introduces the concept of a plane as well as a line. And they bring the n‑body
problem into existence, and an n‑body system is as far beyond the 2‑body system
as having a second particle is beyond a single‑body universe.
Arguably it is vastly further beyond.
It is not clear to me that a three‑particle
system can define space and matter as we experience them, more realistically
than a two‑dimensional universe. So it might be good to imagine a fourth
particle as well. And suddenly we have a new emergent concept: three‑dimensional
space as well as line and plane. I am of course omitting concepts such as
Riemannian spaces and the like, because I cannot see that they are as yet of
much relevance in the present limited context.
"The geometry, for instance, they
taught you at school is founded on a misconception."
"Is not that rather a large thing to expect us to begin upon?" said
Filby,
an argumentative person with red hair.
"...You know of course that a mathematical line, a line of thickness NIL,
has no real existence... Neither has a mathematical plane. These things are
mere abstractions...
Nor, having only length, breadth, and thickness, can a cube have a real
existence."
"There I object," said Filby. "Of course a solid body may exist.
All real things —"
"So most people think. But wait a moment. Can an INSTANTANEOUS cube
exist?"
"Don't follow you," said Filby.
"Can a cube that does not last for any time at all, have a real
existence?"
Filby became pensive.
H.G.Wells. The Time Machine
Time replies on "Killing Time"
There's scarce a point whereon mankind agree
So well as in their boast of killing me;
I boast of nothing, but when I've a mind
I think I can be even with mankind.
Voltaire
In discussing any of these concepts of
emergence, I have not been counting time as a dimension; this is not
because I deny time the status of a dimension, but because I do not feel up to
that topic here. Inclusion of time in the system would entail the concept of
world lines (or time lines if you like). That is an interesting field in
itself; in his brilliant novelette, “The Time Machine”, H.G. Wells anticipated
world lines by introducing the concept of “an instantaneous cube”, and I
commend it to your attention as one example of how far ahead of his time some
of his ideas really were.
But let’s steer clear of such concepts for
now, and take the time dimension for granted. If you omit the time dimension,
then things get messy! We need time to bind all sorts of concepts together. I
suspect that time too, is emergent from the existence of particles, or perhaps
fields, but I cannot support that strongly, even as a concept, let alone a
conjecture.
Some schools deny time except as a delusion,
because most of the basic physical descriptions of events work without taking
time into account, and most of the rest work equally well if you reverse the
direction of the flow of time. However, there are concepts and classes of event
that do inescapably work in one direction only, so, whatever time is and
however it works or behaves, I do accept its existence. More on that later.
But this line of thought and its implications
do not stop there. As I write, our model of three or four bodies by now
comprises a few-body system, where "few" could have many meanings,
but certainly would mean more than two bodies.
If not all our particles are the same
particle, nor of the same particle type, but there are say, of the order of a handful
or a score of different types, and they can interact by interrelationships in
different ways, then we commonly find complex structures emerging. Just as we
can build cities from buildings (architecturally, anyway — real cities
need humans and other components as well) and we can build buildings from
bricks, and bricks from mineral particles, so quarks and gluons and the like,
can compose hadrons, and hadrons and leptons can compose atoms, and atoms can
compose molecules.
And molecules can compose the mineral
particles that compose bricks.
And if there are enough particles for gravity
to overcome their separate momenta, the particles can combine into planets.
Or stars.
Or solar systems, nebulae, or galaxies.
And all those things are entities, as
I defined entities earlier.
In most cases entities imply perceivers in
various roles, but the perceiver need not be a conscious entity, any
more than the tree falling in the forest needs a Berkeley or a god to perceive
the fall or its consequences, nor than a supernova needs an astronomer a
million light years away to observe it a million years after its explosion.
This denial of time is peculiar, not least
because it is implicit in one of the most fundamental of Newton's equations: F=ma; force equals mass times
acceleration, or equivalently, a=F/m. This tells us how to calculate
acceleration, but acceleration itself is defined in terms of time and change of
velocity.
Time then, can be expressed in suitable units
as:
where S
stands for time (seconds, if you like) and a
and d for acceleration and distance.
The details don't matter: the point is that such basic concepts in physics define time directly whether in Newtonian or
Einsteinian terms.
So we don't have to go
far to find justification for time as a physical concept. The fact that in many
concepts we can eliminate the time term is irrelevant; what matters is not that
in some relationships time disappears, but that in some basic relationships time
appears directly.
Another point is that
the very fact that c, the speed of light in vacuum, is taken as arguably the most
fundamental constant in physics, means that time, as determined in terms of the
distance that light travels in determining the measurement, can be defined
unambiguously.
Killing time is a crime.
But words are things, and a small drop of
ink,
Falling like dew, upon a thought, produces
That which makes thousands, perhaps millions, think;
'T is strange, the shortest letter which man uses
Instead of speech, may form a lasting link
Of ages; to what straits old Time reduces
Frail man, when paper — even a rag like this,
Survives himself, his tomb, and all that 's his.
Lord Byron. Don Juan
Though radio signals from a transmitter
consist of photons, the transmitter is not made of photons: it is of metals and
other components. And the stone from a catapult is not a catapult. Such
entities, such systems, differ in nature from what they emit, and differ from
the behaviour of the humans that receive the stone many metres away, or the
signal from possibly thousands of kilometres away.
The photons themselves are not the message
either: the same photons could convey an arbitrarily different message, much as
the ink on paper that expresses the private affection of a love letter, could
equally convey a declaration of a war that kills millions. The effect of any
such message might take the form of behaviour, say laughter, lunching, or
launching a missile, and each of those effects is emergent in that none
is a component of a source or emitter or radio set or a catapult or ink, nor is
it even comprised of such things.
The cliché that the medium is not the
message, is largely unexceptionable, though not necessarily universally true in
all respects, but the medium itself is not necessarily the materials composing
it, nor limited to a conveying only one single message, nor is the message
limited to one medium. Different forms of emergence might produce different
emergents, or produce functionally identical, emergents in indefinitely
different ways and contexts.
This has important implications for the
nature of our universe, in that in a huge range of contexts, it is the basis
for the concept of underdetermination.
So a lot of that dismissal of the concept of
the nature of emergence arose from differences in the assumptions on which the
definitions were based. And a lot of the assumptions were arbitrary,
inconsistent, incoherent, unjustified, or flatly wrong. One argument against
the concept of emergence is that it adds nothing to ordinary physical cause and
effect. That however is breathtakingly unperceptive. "Ordinary physical
cause and effect" transcend ordinary comprehension and imagination in
so many ways that it leaves one, if not actually speechless, at least
inadequately coherent. One might as well argue that art, architecture, science
and technology are indistinguishable form the components that go to make up
painting daubs and masterpieces, lumps and sculptures, rockpiles and hovels and
palaces, illiterate scribbles and sublime literature, dogma from living
science, and shards from atomic force microscopes.
More not only is different, but is different
in so many different ways as to beggar my imagination at least ...
And such differences are the inescapable and
essential basis of emergence from "ordinary physical cause and effect"
by the universal interrelationships between entities.
We realise thus that:
. . . . big whirls have little whirls that feed on their velocity,. . . .
. . . . and little whirls have lesser whirls and so on to viscosity
in the molecular sense.
. . . . . . . . Lewis Fry Richardson
So far we have been discussing the types of
emergence that result from simple introduction of additional entities. And we
encounter emergent effects or emergent entities
wherever we put entities together; furthermore, we commonly get
different emergent effects and entities if we put them together differently.
Introducing more entities than there were before, or conversely, introducing
entities where we separate entities from each other, whether physically or
conceptually, intrinsically introduces emergence, or if you like, emergent
entities.
For a familiar example consider the story of
the old man who walks into the barber shop: "Haircut and styling
please!"
"Certainly sir; please sit down and let
me take your hat."
On removal of the hat, it emerges that the
old man has just three hairs.
In a triangle on top of his head.
Each hair standing erect and separate.
"Errm, how do you want them cut,
sir?"
"Flat top! All the same length!"
Snip!
"How's that Sir?"
"Fine. Now comb them; path on the
left"
The barber gingerly applies the comb between
two hairs, and Oh! One hair pops off the scalp. "Sorry, sir! It was very
loose"
"Never mind; better part it in the
middle."
Another attempt. Another hair lost.
"And how do you want me to comb it now,
sir?"
"Oh, never mind combing it; just leave
it all tousled."
That was an example of emergence in reverse. At
each loss, scope for other classes of possible relationships either emerges
or vanishes. With less than three hairs, parting on the left is hardly
meaningful — indistinguishable from parting in the middle; with less than
two hairs, parting is hardly meaningful at all. With just a single hair or
less, tousled hair is hardly meaningful either. So we get the emergence of a
class of hairstyles that cannot tousle.
Or vice versa.
Emergent effects, as I use and define
the term are precisely those that we could not get without one or
more of the following:
- putting entities together or
separating them, or considering them together or separately
- in general changing the
interrelationships of entities in ways that produce effects other than the
effects to be expected from the entities in isolation, or
- changing the numbers of
entities in any relationship.
What is more, we commonly get differently
emergent effects by combining or splitting or rearranging identical sets of
entities in different ways. In short the concept of emergence also deals
with aspects of:
- the mutual relationships of
entities and their combined effects in those relationships, and
- the information that determines
the relationships between the component entities; for example new entities
may emerge without addition or subtraction of a single elementary
component, just from rearrangement of component entities that are
themselves all emergent entities. One illustration might be the J. M.
Barrie play ("The Admirable Crichton") in which different
community structures emerge, change, and re-emerge successively, while
retaining the same community members.
Nor, in general, conversely, could we make
such changes without emergent effects.
It is important to recognise that emergent
effects are themselves entities. A river is more than just a lot of water
molecules plus a channel in bedrock; freeze that water to make a glacier, and
it changes many things, including the form of that channel in the bedrock. A
glacier channel is not a river channel, and once the water and ice are gone,
the two dry channels are easy to tell apart.
And a galley is more than just a lot of
planks fastened together.
These are nearly fundamental concepts; some
might be primitives for all I know.
If you think that I am over‑stating the case,
reducing the concept to meaninglessness through over-application, then think
carefully about this: the very concept of thermodynamics depends on emergent
effects — without large numbers of particles, generally gigantic
numbers, such things as entropy and temperature have little meaning at best;
the concepts become degenerate, like the arithmetic sign of sin(1/X) as
X approaches 0.
Or like parting the two hairs on the left,
perhaps?
Another bee in the bonnet of many of the
early writers on emergence, was that to be emergent, an effect must be
unexpected, in fact unpredictable in principle. That idea however, is
incoherent to the point of meaninglessness; for one thing it introduces the
concept of predictability, and does so without considering context in its
definition. Either it takes the observer into account in defining the
phenomenon, whereas the observer commonly is extrinsic to the
phenomenon, or it assumes that the unpredictability is part of the phenomenon,
which would imply that the phenomenon is non-causal, which it clearly need not
be. For example, asserting the impossibility of predicting oceans from a study
of the water molecule, is a favourite onset of supporters of that assertion,
but we find that whenever enough water molecules get together in a basin large enough,
we get an ocean.
Plainly the molecules know well enough in
advance how to make an ocean, even if the philosophers find the prediction
beyond them; such emergent outcomes are causal.
And the same is true for most other emergent
effects, such as crowds, clouds, crystals, and nuclear explosions; the details
differ, but the generic patterns remain true to form, even though the details
vary more or less randomly according to type.
If the participating elements can work it
out, it plainly is not impossible in principle to work it out, even if it is
not humanly feasible.
When quantum randomness plays a part, then
that affects matters of determinism, but patterns of causality remain in force.
From such points of view we find ourselves
steered into concepts of semiotics. Whether we find it easier to regard
semiotics as a branch of physicalism or the other way round, physicalism as a
branch of semiotics, might be regarded as a matter of perspective, but either
way, the connection is unavoidably intimate.
Here I am not concerned with the versions of
semiotics that have become popular topics, often superficial or vacuous, in
art, literature, sociology, and similar soft fields. I deal with the basic
concepts of physicalism, causality, information, syntax, semantics, and other
intimately interrelated topics.
For example, a single A4 sheet of blank paper
plus some milligrams of ink from a pen, offer scope for wide ranges of emergent
effects. The ink might be distributed at random, or in no readable language or
recognisable picture, with no effect other than a waste of ink and paper. But
that same ink could be distributed onto the paper in such a pattern as to
express an indefinitely vital message. Or it could produce distinct, different,
even opposite, effects if the same ink is rearranged to express different
messages or pictures. Depending on the ink's distribution, alternative messages
could present more information, or less, but no message or picture would
be possible without the paper and ink, nor, as a rule, without the pen and the
writer and the readers.
As for what the messages might be, that
is itself open to definition and interpretation. Consider these examples:
These three patterns all have the same number
of visible characters and spaces and "ink"; do they all convey the
same meaning? Do they all embody the same amount of information?
The next three also have the same amount of "ink" in their characters
on about the same amount of "paper" and their characters are in the
same arrangement as in the former example; do they all convey the same meaning?
Do they all embody the same amount of information?
The main point remains constant: information
embodied in mutual relationships between entities includes aspects that could
never have existed without those mutual relationships.
That is the essence of emergence as I see
it and as I use the concept in this essay.
By this definition it sounds stupidly simple,
but, as I shall show, and as the previous arrangements of characters hint, in
practice the concept is potentially almost indefinitely complex, and in
principle very, very important, especially in identification of entities.
Hardly anything in material existence happens without emergence of many sorts.
Emergence may be seen as a class or aspect of consequence or implication in
causality, yet emergence is not generally deterministic, nor yet
fully determined in its consequence or implication.
In the course of writing this document I
necessarily did some peripheral reading, and found in particular a great deal
of argument about what emergence is or is not. The poverty, confusion and
question‑begging of much of such discussion was so great that the very term
“emergence” became widely derided. Some authors still dismiss the concept as
meaningless or at best useless.
Such authors grossly misunderstand and
misrepresent concepts of emergence from information and entropy. What emerges
cannot come about without the subvenient, and sometimes supervenient,
relationships; therefore the class of emergent effects is inescapably real and
important.
As you may see, a large part of what I have
come up with amounts to an independent formulation of a version of physicalism:
its metaphysical basis is that everything is physical or depends on
("supervenes on") the physical; however, I regard even that very
dependence as physical, so to mention it separately might be seen as redundant.
Notice that this version of physicalism neither
asserts nor denies that emergence from a physical system implies that
what emerges is of the same nature as the generating system. I reject, as
unjustified and incoherent, the demand that the product of the emergence must
be "novel". I reject too, as an incoherent and unnecessary denial of
a negative, the view that for anything to be classed as newly emergent, it must
be unpredictable by any analysis focused exclusively upon on the component
parts. Some aspects of some examples might or might not be thus predictable other
than in retrospect, but such unpredictability neither need apply to all
emergence, nor need it be fundamental to the general concept of emergence.
For one thing, that idea fails to distinguish
between predictability to the viewer, and predictability to the system itself.
Perhaps a human without foreknowledge could not be expected to deduce and
predict the various emergent aspects of water molecules that lead to the
emergence of oceans, but the molecules seem to have no such constraints; they
simply do what water molecules do, which amounts to calculating the nature of
oceans on the fly. The same applies to any computationally challenging examples
of emergence. The algebra of physics is
not much given to compromise.
Another claim, made by some very prominent
parties, is that there is no such thing as emergence at all, because everything
can be represented as interactions between quantum mechanical particles. This
however is about as sensible as saying that a pylon is no different from a
cable because both may be made of steel, so that if you know enough about rigid
steel pylons, you accordingly know all about flexible steel cables.
Those same steels are made from largely
similar molecules of similar atoms made from hadrons and leptons that comprise
all our elements; to claim that therefore there is no difference between them
might have merit in a neutron star's quark soup, but not on the surface of our
planet, and the fact that at the quantum level all of them are products of the
same kinds of causal events, while true, does not affect the concept that, not
only is it true that "More is Different", but that "different
arrangements are different".
Or on similar assumptions, you might argue
that because poetry and other art and literature cannot exist without the
appropriate media in appropriate relationships, and all such media are comprised
of the same ultimate particles and quanta in their appropriate relationships in
turn, therefore we know all about art if we understand quantum mechanics. By similar
reasoning, a bold quantum theorist could undertake to play unbeatable go or
chess, or pentominoes, or tennis, on the basis of quantum mechanical realities.
That simply is not how it works.
Many principles are involved here, but one of
the most salient is that it would be a fallacy of composition to assume that
because the ultimate components of an entity are elementary, the entity too, is
elementary and consists in nothing but simple components. Not only can one see
complex structures built of simple, or notionally simple, items, such as
bricks, Lego blocks, or atoms, instead of leptons and quarks, but one also can
see complex sub‑assemblies such as prefabricated walls, windows, or engines,
built into yet more complex entities such as factories in which one performs
operations different from the operations that are appropriate to assembling the
factories.
And as for breaking eggs to make omelettes,
or, if one is a hen, assembling eggs to make chicks .... Try mentally tracing
the nature of the operations back to the original atoms, and the atoms back to
the QM elementary particles. One repeatedly needs to change mode at different
levels throughout the hierarchy.
Many examples are possible at many levels. If
one builds a house of bricks, the bricks in the structure are not emergent from
the house; the “house‑ness” of the house is emergent. In fact, we could build
functionally identical houses from widely different bricks. As long as the
bricks do not undergo changes in the building process (say by being fused together)
they remain bricks, but the house still remains the house, irrespective of the
nature of the bricks or other components as long as they are adequate. And
there are all sorts of considerations if one replaces, adds, moves or removes
say, just one brick: the house‑ness is hardly affected, but the house itself is
no longer all the same in every possible sense.
Two years research can often save you ten minutes in
a library
Anonymous
Similar considerations arise from the parable
of the Ship of Theseus. For clarity I retail it here in my own version: Theseus
commissioned a ship to take him and his companions on the expedition to deal
with the Minotaur. On taking delivery, he took the ship for a trial run. His verdict
was that the ship was fine, except for one of the trunnels that needed
replacing. The shipwright privately thought it was nonsense, but it is ill
arguing with formidable, short-tempered princes, so he extracted the trunnel
and put in an identical one. At this point no reasonable person would deny that
the ship had been changed just a little, or, contrariwise, argue that
therefore the ship was not the same ship as ever.
After another trial, Theseus wanted a spar
replaced. The shipwright prudently replaced it rather than annoy the prince,
and the process was repeated on one component after another till not a single
component remained that had been part of the original ship.
Now is it
still the same ship?
If not, when did it stop being the same ship?
When a plank was replaced that marked the point upon which more than half the
original ship material was no longer the same, perhaps? Are we then to say that
that plank made the difference between one ship and another? That plank looked
just like most of the others, including in particular the plank that it had
replaced, so that sounds like special pleading. Once again, any reasonable
person would say that there never had been a point at which the ship
stopped being the same ship; all the replacements were just like those that
might have been replaced in routine maintenance; we don't find ourselves with a
new ship every time a rope abrades a few microns of wood off a block.
Well then it would follow that the ship is unchanged.
It looked no different and at no point stopped being the same ship.
Irrespective of its components, its
shipness had at no point changed.
That ship was an emergent entity, and
remained the same emergent entity until there might come a definitive change,
say, if someone erases the name (say “Argo”? Tradition is not clear on the
point) from the stern, and writes a new name, say “Ariadne”. Whether that makes
it a new ship or not is a matter of preference in context; some ships are
repeatedly renamed for one reason or another. In fact, the name might not have
been written on the ship at all; in the pre-classical past not all ship names
were written physically on the vessel at all, in which case any change in name
would have been only in the minds and mouths of the people; it would not
directly have changed the ship at all.
Meanwhile, as Theseus finally sailed off, the
disgruntled shipwright reflected that there really had been nothing wrong with
any of the parts, so, as he had stored them, he reassembled them exactly as
they had been, economically getting the exact ship he had started out with,
with the same materials in the same places as before.
Which is not strictly true of the vessel in
which Theseus had left.
But before the reassembly, no one would have
called the pile of parts on shore a ship at all: its material components were
unchanged, but its shipness had
vanished; its emergent nature would have been a pile of scrap, not a ship.
And at no time had the replacement ship
stopped being the same ship. At no point had it forfeited the relationships
that had constituted its shipness. In fact, not one of the changes would have
been different from changes that could have been applied during routine
maintenance.
Suppose the shipwright had surreptitiously
marked each component of the original ship with an inconspicuous
identification, and put an equally inconspicuous distinguishing mark on the
replacement part as he installed it; then for anyone properly informed, but a
newcomer to the scene, it would be possible to identify the reassembled ship as
the first ship and the second one as a different ship. But any observer who had
recorded the whole history of the process, would have said that the ship that
had sailed off was the same ship as in the first place, and the reassembled
parts now emerged as a new ship.
Suppose again, that after a while the alpha
ship, the one that had originally been assembled, and the parts of which,
Theseus had dismissively left behind, was no longer wanted as a ship, but its
components were disassembled and reassembled as a house. And later, the house
might be dismantled and reassembled as a bridge over a narrow chasm.
Still the same physical materials, but in
neither case would Theseus, nor even the shipwright, confuse the new
construction with anything that had preceded it, neither in its form as a new
ship, or as a rival for recognition as any ship at all.
By this time we are in a position where we
are tempted to assert that if we adhere to standard concepts, the question of
the identity of objects is one of semantics rather than of physical fact, and
the semantics are context-sensitive. And that is a valid example of one
semantic view, but it is not the exclusive essence of the matter; in fact it is
not the operative concept in this ship's history; in each case the difference
would have been the effects of emergence, and the continuity of the
interrelationships between the components participating in the emergent
effect.
Many writers in this field (I do not assert
that those ones are philosophers) either dismiss the Ship of Theseus problem,
or make heavy going of it, practically mystical. And yet, pretty problem though
it is, they generally fail to recognise its significance.
The problem is the failure to recognise that
many of the entities that we recognise in our world are emergent, and, in their
emergence, are distinct from their parts. A novel is not a jumble of letters. A
book containing the novel is not the novel either, and the letters are not the
book. If I erase a suitably chosen “E” from the text of a copy of “The War of
the Worlds” it will not cease to be that novel, and if I then replace the “E”
it does not suddenly turn into a new novel.
Entities that are comprised of arrangements
of other entities, and that we recognise as particular entities, exist in their
own right (in case you have forgotten my definition of existence, better return
to re-evaluate it). That means that while the cloud in the fixed position over
the mountain remains, it is the same cloud even if the water inside it is no
longer the same as it has been just a few minutes ago, and now forms another
cloud downwind. The ship still has the same shipness, the same identity, if a
plank has been replaced, and even if a new mast has been added.
Such emergent essences as the shipness,
houseness, cloudness, or other attributes of compound entities have their own
world lines, and the world line of a compound entity changes in its three-dimensional
cross section or silhouette as its history proceeds. The Sequoia tree that now
towers over the surrounding forest, having grown from a seed that germinated possibly thousands of years ago,
never discontinued its treeness, its Sequoia-ness, in all that time, but if we could look at
photographs taken of that tree at five-hundred-year intervals, we easily might
think them to have represented different trees. It started as a seed with a
mass of a few milligrammes, and now masses roughly a billion times as much.
You taught me language; and my profit on't
Is, I know how to curse. The red plague rid you
For learning me your language!
William Shakespeare: The Tempest
In this essay I use the term “emergence” in a
small number of limited and largely mechanistic senses that each time suffice
for the then current context. For my purposes, largely contemptuously, I ignore
other senses of the word. I also ignore arguments on the point, as being
irrelevant, whether sound or unsound. In fact, I had intended to ignore the
controversies altogether, but some are so widespread and so unsound that I
found I had at least to concede them the discourtesy of explaining why I
dismissed them and accordingly why they are irrelevant here.
Among the commonest delusions about
emergence, is that, to be emergent, effects must be intrinsically
unpredictable by deduction from the nature of the component entities of any
system that produces any emergent effect. This is mental confusion with the
basic concept, which is that: an emergent effect cannot emerge from any
system simpler than the minimum necessary to generate it, whether the
emergence is regarded as predictable or not.
Furthermore, predictability is not a
well-defined term in this context. Roughly
speaking, it is a relationship between the subject (the predictor) and the
object (the emergent effect). Just because one predictor cannot see a world
in a grain of sand, does not mean that no one can; or that no one could inspect
a hydrogen atom and an oxygen atom, and predict an ocean.
Consider our paper and ink for example: omit
either, and no writing emerges, and without writing, no message emerges. Extra
paper plus extra ink permit greater ranges of messages to emerge. When the
shipwright was putting the alpha vessel together, it would not have occurred to
him either to predict or wonder whether he was in fact working on the
components of a house or a bridge.
Extra confusion could arise from the fact
that some emergent effects might be real and distinguishable all right, but not
be observable or explicable to all recipients without special abilities or
equipment.
We already have considered the remote
islander who is unequipped to imagine the function of a radio transmitter, of
which he can see nothing more functional than some dim indicator lights; and he
has no idea even that the lights are there for information, not illumination.
How much better-off are we in our turn? We
observe subjective consciousness as an emergent effect of the human brain. No
one has yet produced a cogent and coherent explanation of the nature or
function (if any) of that subjective consciousness, but we routinely demonstrate
in multiple ways, that any physical effect on the physical function of the
brain, such as from drugs, sounds, darkness, education, or violence, affects
the subjective consciousness. In that respect we are not much superior to our
islander who lacks comprehension of the radio transmitter.
One popular example of the impossibility of
predicting the nature of an emergent effect, is denial that anything one could
observe about an isolated water molecule could enable anyone to predict that
having huge numbers of such molecules could give rise to liquid, to turbulence,
to solid ice, oceans, flow, waterfalls and all the emergent effects we observe
in the amazing behaviour of one of the simplest molecules in nature.
Assertions of such impossibility or
possibility, I see as not only arbitrary and. intrinsically nonsensical, but
also irrelevant to the question of emergence as I view it. The denials are not
logically compelling, either as a matter of principle or as a matter of fact or
even of non-trivial interest. Such things cannot be established by bare
assertion.
They also are not useful distinctions between
emergence and non‑emergence. No matter how complex the emergence, whether
intrinsically predictable or not, emergence never has been demonstrated to
occur in any way other than by physical and causal relationships and
interactions. Accordingly the predictability of emergence is limited only by:
- the knowledge of the nature of the
interactions,
- the information available, and
- available computing power.
Given those, the interactions are necessarily
as predictable or observable as anything else in physics, and the available
information limits nothing but the precision of the outcome. The fact, for
example, that water molecules behave consistently in all circumstances, implies
that they behave according to their operations in the algebra of physics, and
that our inability to predict all the consequences of their behaviour, reflects
own limitations of information or of computing competence; the molecules, and
the systems in which they participate, are not similarly limited, so the
predictability of the emergence is not intrinsically absolute.
My rejection of the assertion of the
impossibility in principle, of the prediction or explanation of emergence,
leaves me with the need to clarify where that leaves us with prediction of
emergence, and to explain emergent events where they occur. Commonly such
prediction and explanation are qualitative rather than precise, but that
limitation applies to other physical predictions and explanations as well.
For example, in principle a rainstorm as an
emergent event can be predicted from sufficient knowledge of the physics of the
atmosphere, including the physics of water, and that knowledge explains the
event as well. But in practice rainstorms are not nearly predictable in fine
detail, such as where each drop will fall, though we nowadays can predict
weather with accuracy that would have been incredible in my own youth. However,
whether we, as humans, are able to predict or explain rainstorms, does not
define the principle of their emergence.
Analogously, consider a simpler system: we
can predict that, given a chessboard and 32 dominoes, each of a size to cover
two adjacent squares, it is possible to cover the board with them. However, we
cannot predict exactly which pattern any particular person will choose to cover
the board with; the number of possible patterns is very large. We know that if
we remove certain combinations of squares from the board, then we can cover the
remaining squares with fewer dominoes and with every domino on the board fully
covering exactly two squares, but that removing certain other combinations of
squares will render impossible, any pattern that exactly covers every square
and no more. However, of those combinations that do permit perfect covering by
dominoes, by far the most will permit many patterns of dominoes.
Most also will require a certain amount of
experimentation or insight to exclude though all will depend on the
relationships between squares, with the outcomes being emergent from those
relationships. To prove it by my criteria, changing the relationships would
change the problem, but in all cases, with sufficient thought one can tell
which arrangements would meet the challenge.
Another form of emergence can arise from
situations in which it really is not in principle possible to predict certain
aspects of the outcome, because determinism is not involved. We can predict
confidently that a sharp, symmetrical chromium pin balanced vertically on a the
tip of a similar pin in a vacuum, will tumble sooner rather than later, and
that it will in fact tumble rather than falling smoothly and vertically, but we
can neither in practice nor principle
predict the direction.
We would have enough difficulty no more than
balancing the pin in the first place.
This pin problem is an example of symmetry
breaking, a very general class of problems in physics and philosophy.
Commonly their precise outcome is unpredictable either in practice, or in
principle; this is because the information necessary for the prediction simply
does not exist, not because existing information just happens to be
unavailable.
Furthermore, even when we can manage some
partial prediction and explanation, it does not follow that we can do so completely.
We cannot generally predict the exact extent and time of a rainstorm, nor how
many drops shall fall in each yard. But that inability, much like our inability
to predict the details of the falling of our chromium pin, does not deny our
ability to predict certain aspects of emergent effects.
Another question is that of the concept of
"downward causation", the question of whether an entity whose
existence is the consequence of the interaction of more elementary entities,
can affect the nature or behaviour of its own component entities. This question
has paralysed and fascinated generations of philosophers, causing them to waste
much ink and bandwidth. They tend in particular to be obsessed with “mind”
as an emergent effect (which I do not dispute in several senses at least,
though I do not undertake to define mind, and though I do not accept that any
concept of mind that I have as yet seen, is either definitive or fully
explanatory).
The relevant point here however is that mind,
whatever it might be, in some sense affects the behaviour of body. In
other words mind and body physically affect the nature and actions of their own
components, and those include examples of downward causation. An example of an
observable difference in behaviour, as a result of downward causation, is that
harbouring of a conscious mind (or not, as the case might be) could affect the
choice of reply by the body that might host the mind, to a question such as “are
you aware of any subjective impression that you have a conscious mind?”
The question of the comprehension of the
question by the entity questioned, and of the honesty of the reply amount to a
different matter; what is relevant is that if an entity has such
a conscious impression, and can process the question in terms appropriate to a
reply, it can answer "yes", and if not, it can answer "no".
It then would be reacting according to its own emergent subjectivity or lack of
any such thing. That would be another example of downward causation.
Let us ask ourselves: is downward causation,
or feedback, surprising? The principles of feedback, both positive and
negative, are very familiar. Let’s not waste time on the emergence and downward
causation of mind as such — mind and its causation being too poorly
defined, and possibly too difficult, for us to deal with at present; we can
however model downward causation with simpler emergent systems.
And, surprise! In practice we find plenty of
examples of both downward and upward causation of different levels, intensities
and kinds. In the epigraph I quoted Lewis Fry Richardson’s remark from a
century before my time of writing, concerning turbulence in the atmosphere.
Turbulence of all kinds is a typical example of emergence: you cannot get
turbulence from isolated particles. But when you have a fluid of huge
numbers of particles in turbulent motion, that turbulence consists in numbers
of vortices, where each vortex is an entity, an element, that emerges from the
relative motion of large numbers of molecules. Each vortex affects the behaviour,
not only of elements at higher levels, but of elements at lower levels in the
turbulence as well. Their interactions cause effects of momentum, charge
transfer, density, refractive indices, temperature, density, state, and more.
In short, feedback in every possible
direction. That is hardly surprising, in the light of fact that it is arguable
that every interaction in nature affects both entities in the interaction.
Now, some argue that if we have upward as
well as downward causation, then that will imply circular causation, but that
is only true if one is slovenly in the definition of entities and instances of
entities, not to mention circularity. I say more about entities elsewhere,
atomic entities such as electrons and humans, and highly tomic entities, such
as crowds and clouds, but let that stand for now.
Meanwhile, when critics attribute circular
driving of events, or circular causation, to emergent systems that entail both
upward and downward causation, they overlook the question of the sequential
nature of causation. Although some people deny the concept of cause‑and‑effect
entirely, no one I know of denies that whatever we might reasonably describe as
cause, does precede effect, and that each effect in turn is a new cause that
will have new effects. But when the new effect is of the same nature as the
original effect, does that imply circular causation?
Hardly — the effect might indeed be
of the same kind as the cause, but it
is not the same instance as the
cause. So such a system is not circular, just a sequence of events in a cyclically
driven system: a different concept — stop driving the system, and it will
eventually stop or diverge from its cyclic pattern of behaviour. For examples,
consider eddies in a stably turbulent stream, or the behaviour of a piston in
an engine: stop the stream or stop the fuel supply, and the sequence of events,
of emergent effects, stops as well.
There is nothing circular about such a cyclic
nature.
So,
naturalists observe, a flea
Hath smaller fleas on him prey;
And these have smaller fleas to bite 'em,
And so proceed ad infinitum.
Thus every poet, in his kind,
Is bit by him that comes behind.
Jonathan Swift
The relationship between emergence and
epiphenomena is another popular topic. As with most concepts in this field,
definitions of epiphenomena vary, but roughly speaking, an epiphenomenon is
something caused by a system, but that does not affect the system that caused
it. Again, some definitions of emergence exclude epiphenomena from emergent
effect status, but this too, I reject categorically, as being irrelevant as
well as questionable. A phenomenon either emerges (that is to say, as I see it:
is emergent from) its system, or it does not. Whether it has any functional
or causal connection to the system from which it emerges, is neither
here nor there. It is in any case difficult to demonstrate that something is
absolutely epiphenomenal in the sense of being functionally or causally
independent or irrelevant.
For example consider an early example of an
epiphenomenon, namely T.H.Huxley’s steam whistle on a locomotive. It is a
favourite example attributed to him, though he never used the term
"epiphenomenon"; in his day it was not much used in that sense as far
as I know, usually being used as a medical term for irrelevant signs or
symptoms. But to be sure, Huxley used the whistle in the sense of an
epiphenomenal item: he asserted that the whistle does not affect the function
of the engine, though anyone unfamiliar with the concepts might think it more
essential to the train than the far less striking and dramatic internal bolts
and pistons.
Well, the whistle certainly is not necessary
for the propulsion of the locomotive, and in fact you would notice no effect on
the propulsion of the train if you removed the whistle, except perhaps an
increase of delays due to collisions, but if the steam jet out of the whistle
were directed either forward or back it could in principle have some propulsive
effect, either positive or negative, and in practice the whistle’s consumption
of steam has a greater effect on the propulsion of the locomotive than the
sound of its whistling has.
And neither the whistle nor the steam
propulsion would have occurred at all if there were no subvenient system such as
the steam locomotive. The subvenient system is the system on which the
supervenient system depends. Without the locomotive and its steam supply, the
whistle cannot whistle. The example is trivial I accept, but to claim that to
be emergent, an effect must neither be epiphenomenal nor include downward
causality, is pure mental confusion, non‑cogent and unsupported.
Nor may we be too confidently dismissive of
such notionally epiphenomenal effects. Some of the early small steamships
actually could exhaust their steam supplies by sounding their horns too freely,
and remain helpless till their steam pressure built up again.
The
only thing to prevent what's past is to put a stop to it before it happens
Boyle Roche
More: in contradiction to another class of
claim, there certainly can be multiple levels of emergence. Consider water
molecules once more: isolated molecules behave fairly simply according to the
local environment of radiation, gravity and the like. Add enough molecules, and
there can be collision, attraction, and repulsion effects, and more. When
enough molecules are close enough to each other, then, as the concentration of
molecules increases, gaseous behaviour emerges, including convection,
transmission of sound, and turbulence. With sufficiently increased
concentration of molecules the emergent effects include condensed matter in
various forms of liquid or solid: droplets or crystals of one type or another.
We also could elaborate the discussion into consideration of solution or chemical
reaction.
Droplets have their own emergent behaviour in
several forms that differ from the emergent behaviour of individual molecules.
The emergent behaviour types of the droplets also differ from each other as
they grow in size and as their environment changes. At first surface tension
and cohesion dominate gravity, momentum, volatility and various resonances. As
the number of molecules increases, the other effects begin to dominate the
surface tension and we pass through other effects that, among other things show
both upward and downward causation, though still with little effect on the
nature of the molecules, and still less on the nature of the atoms and their
sub‑atomic particles. First we get larger drops, then masses of water that will
flow through sieves and tubes, and form puddles, lakes, and oceans with waves
and surf.
Eventually the sheer gravitational effect
leads to downward causation through pressure. Given enough pressure, the very
molecular structure begins to break down; at its centre, the behaviour of an
Earth mass of water would be very different from its behaviour near the
surface. In a Jupiter mass the behaviour would differ even more, and a solar
mass of water would turn into a star, in which the deeper molecules will generally
break down into states other than water molecules, and in fact other than
hydrogen and oxygen atoms.
Again, although at any level of emergence,
one can trace the limiting or defining effect of the ultimately atomic
components, more complex systems require more complex components, even when
those components have the same ultimate components. We could use the same
cellulose to make a rope or a piece of paper, but if you asked for paper and I
gave you rope, you would not be grateful.
Yet again, the same million tonnes of water
in a few thousand cubic kilometres of air has drastically different effects on
the local weather when it is present as vapour, from when it forms a raincloud.
The same biochemical and physiological materials, when present as a gorilla, or
a man, or a woman, or a different person, cannot in each social, political,
physical, or legal situation, fulfil interchangeable roles. There are entity
classes in which a woman or a gorilla might play equally appropriate component
roles, but in most everyday entities, where the required component is
"woman", "gorilla" will not do.
In short, at every level of the combination
of the elements of the system, or the combination of emergent entities in a
system, new classes of emergence are possible and occur commonly, whether they
result in downward causation or not; new levels of emergence quite commonly run
counter to previous levels by negative feedback, destroying the very emergent
effects that created them. For example, a flame might consume the supply of
fuel that led to its own ignition, or its products could smother the reactions
that support the combustion.
Those items include examples of emergence by
both upward and downward causation.
And yet there are certain classes of effect
that do not appreciably emerge in most conditions that we can easily imagine.
To be sure, in compression effects, electrons are squeezed closer together and
may be accelerated, released, or confined, but their individual charge and mass
are not affected, probably not even in a neutron star. The fact that there is
some downward causation does not mean that it is all downward causation. Some
of the emergent effects affect in one direction and some in the opposite
direction; some no doubt in both. But in either case, denial of the possibility
of predicting any particular effect is something that cannot in general be
justified in principle by arbitrary assertion.
Now, given an adequate description of
any entity or class of entities, it is possible in principle to predict certain
aspects of its emergent behaviour. For example, we can predict that regular
polygons when fitted together according to certain rules can form only certain
classes of regular polyhedra. We also can predict of certain classes of
irregular or partly irregular polyhedra that they cannot form in two or three
dimensions. Such exercises are at first simple, but related exercises can
become very tricky: one example is the tessellation of planes with irregular
polygons such as pentagons or Penrose darts and kites. At the time I write
this, the tessellation of pentagons has not yet been fully described
mathematically.
Generalised polyominoes too, produce
combinatorial problems of rapidly growing complexity, and so does the adoption
of shapes of organic molecules, especially of very complex organic molecules.
However, except in their computational
complexity none of those presents any obstacles in principle. Though their
effects are undebatably challenging to our currently standard computational
tools, we can construct or imagine tools or toys that model such emergence.
Accordingly, the problem is by no means incomputable or unpredictable in
principle. Bouncing marbles can compute sphere packing to some degree of
precision, and soap bubbles can produce shapes that are killingly difficult to
model mathematically or algorithmically.
The fact that some of them are too complex
for humans to compute, or that it would be too expensive, is a separate issue.
Consider NP‑complete problems such as the travelling salesman problem: we
generally can solve them by some algorithm or other, but as the problem grows
larger, the resources required, especially the time needed, explode, and to
provide and prove a perfect solution to a travelling salesman problem of just a
few hundred towns would take more than the estimated age of the universe. In
other words it is to a good first approximation impossible.
But it is simply mathematically possible.
We can improve our efficiency by improving
our programming techniques such as annealing or genetic algorithms; we can
combine astronomic numbers of processors to work in teams to solve the problem
in a fraction of the time, but given a surprisingly small problem, we still
would need billions of years to get the theoretically optimal answer.
But none of that changes the fact that all finite NP problems are computable in principle
even if not in practice.
However, there remain other aspects to
emergence — to at least some classes of emergence.
How hard is it to solve problems if you
only care about
getting the right answer most of the time?
For everyday problems, answers that work most of the time
are often good enough.
We plan our commutes for typical traffic patterns, for instance,
not for worst-case scenarios.
Anonymous
To clarify the practical significance of at
least one aspect it is necessary to distinguish between generic and specific
events, effects, or entities.
Perhaps surprisingly, the difference between
generic and specific effects is arbitrary: in a nondeterministic,
underdetermined world such as ours, every apparently specific effect really is
a whole class of indistinguishable outcomes. To a high degree of precision we
know the shape of a pile of sand formed by grains trickling through a tiny hole,
and the shape of the splash when a falling drop of milk strikes the surface of
still water, and the trajectory of a steel ball bouncing on a concave steel
surface, but to regard their apparent precision as perfect would be delusional:
on a sufficiently microscopic scale each such event is effectively unique.
None the less, the effects are close enough
to deterministic to satisfy most everyday needs. When the billiard ball goes
into the pocket, we do not much concern ourselves with details of impacts or
vibrations, with how it bounced on the cushion on the way in, or which way up
or touching which particular fibres it came to rest.
Generic effects might include the
gravitational, geometric and electromagnetic attributes of water molecules
leading to their attracting each other and forming liquids with surface
tension, and droplets and oceans, and crystals and so on.
But which droplet, ocean, or crystal
in particular? In quite modest systems we have more options than any realistic
computational power could predict or model, not even if we ignore that many
such outcomes may be ephemeral, lasting only fractions of a second before settling
into stable emergence.
Certain classes of emergent effect might
not in principle be predictable from the source entities one at a time; in
fact, some philosophers shear a lot of hogs in addressing that assertion and
those particular philosophers furthermore assert that an effect is emergent only
if it could not have been predicted from the nature of its components. To
be emergent, it must be "novel" in nature, they say.
I of course, deny that any such a stricture
is logically necessary or even necessarily coherent. As I see it, if the
impossibility of prediction of every form of emergence necessarily did
follow, then to establish that impossibility would be no easier than many other
negative assertions.
I accept that a particular outcome
might not be predictable in advance, especially in chaotic systems: one might
not have predicted the emergence of life on the surface of a particular partly
watery planet or, granting that one did foretell its emergence, fail to predict
which life forms would emerge, or how long the emergence would take; but that
follows from the scale of the number of forms that are reasonably possible, not
the principle of emergence. One also could not precisely predict the form of an
accumulating pile of sand, but still be pretty confident of its plesiomorphic
form.
The impossibility of predicting the emergence
of something like life on a given young planet does not follow cogently from
the general nature of planetary accretion. By way of analogy, the possibility
of computer viruses was predicted before they emerged, and before most people
outside the field could comprehend the concept, and in fact some programmers
even denied the possibility for some time after the first viruses were
“released into the wild”.
But the very fact that computer viruses were
predicted, and that those actual predictions turned out to be a major factor
leading to the very creation of the viruses, shows that they could in principle
be predictable, even though the predictors could not have predicted the
particular code or objectives of the various viruses that would emerge.
And that is enough for my purposes. Their
predictability in principle, given their possibility, is not generally an
attribute of the emergent events themselves, but of the predictors.
Furthermore, the term “prediction”, is ambiguous: it could refer to. a specific
event, such as an unstable atomic nucleus decaying in a particular way at a
particular time, or a generic event, such as sand forming a mound as it lands
in the lower bulb of an hourglass, or a liquid. forming a flat surface or a
meniscus, depending on the nature of the liquid and other factors.
The very concept of predicting emergence of
types of behaviour is fuzzy, but what most matters here is how well we can
predict the nature of events, not the details. We might be able to predict the
resting position of a marble rolling into a hollow, but not all predictions are
equally easy. If from the nature of isolated entities of particular units, such
as molecules, one might predict the nature of their interaction to produce
effects such as liquid, gaseous and solid behaviour, then, whether chaotic or
contingent, details matter less; in predicting splashing, we do not undertake
to predict the trajectory of each droplet.
Given that one knows enough about the
mechanisms involved, we might well predict that some items are not possible to
predict. For example we might indeed know enough to predict some events or
“effects” (generic events, classes of events, or “trends”) emergent from the
reaction between atoms and molecules. We might indeed know enough to predict
how putting them together could produce surface tension and gases, liquids, or
solids. A pile of marbles does certain types of things that a single marble or
a scattering of separate marbles will not do.
The same is true of a pile of bricks, but
different piles of bricks can have more complex emergent effects than piles of
marbles can. Looking at a pile of bricks could put one into a better position
to imagine a cathedral, or propose its building, than looking at a pile of
marbles could. And we can predict that the friction of gently tumbling a pile
of sound bricks for long enough will first erode the bricks into prolate
ellipsoids or spheroids plus dust, depending on the shapes and constitution of
individual bricks, whereas continued tumbling could produce accurate spheres.
For practical purposes, no matter how
predictable or unpredictable from first principles they might be, all such
effects can be viewed as variously emergent, either specifically or
generically. And what emerges can differ in indefinitely various and marvellous
ways.
Let us consider some classes of those
emergent effects.
A scattering of scrabble letters on tiles
does not have the same effect as the same letters arranged into a legible
statement. A classical example is Hooke’s anagram of his announcement of his
discovery of his law of elasticity: “ceiiinosssttuv” for “ut tensio, sic vis”. We might equally well reassemble the letters as “sosiitctuseivn”
or “cute visionists”, or in Laputan: “Isi etsti sco vun, each
with its own significance in a suitable context.
Whether to regard such outcomes of
arrangements of the same components as emergent or not is largely a semantic
question. I insist on calling them emergent because they would not be possible
with fewer or different components. Furthermore, if the set of components is
large enough, it is combinatorially infeasible to predict every possible
outcome.
For illustration consider the following block
of letters:
AAAAAAAAAAAAAAAAAAAAAAAAAAABBBBCCDDDDDDDD
DDDEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEFFFFFG
GGGGHHHHHHHHHHHHHHHHHHHHHHHHHHHIIIIIIIIIIIIIIIIII
IIIKKKKKLLLLLLLLLLLMMMMMMMMMMNNNNNNNNNNNO
OOOOOOOOOOOOOOOOOOOPRRRRRRRRRRRRRRRSSSSSS
SSSSSSTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTUUUUVVVVWWWWWWLet’s ignore punctuation, formatting, and
similar complications as needless distractions. They could be included without
affecting the generality of the question, but can those letters be rearranged
to make a sequence of words? Could they be English Words? Could the words be
arranged to make sense? Could the arrangement be anything memorable? Would it
be an emergent effect if it turned out that it could?
Well, they certainly could; as it happens,
those are the letters in an Authorised Version Old Testament passage from
Habakkuk, for convenience omitting spaces and other punctuation:
WHAT
PROFITETH THE GRAVEN IMAGE THAT THE MAKER THEREOF HATH
GRAVEN IT; THE MOLTEN IMAGE, AND A TEACHER OF LIES, THAT THE
MAKER OF HIS WORK TRUSTETH THEREIN, TO MAKE DUMB IDOLS? WOE
UNTO HIM THAT SAITH TO THE WOOD, AWAKE; TO THE DUMB STONE,
ARISE, IT SHALL TEACH! BEHOLD, IT IS LAID OVER WITH GOLD AND
SILVER, AND THERE IS NO BREATH AT ALL IN THE MIDST OF IT.
Good luck to anyone who would seek alternative arrangements that
make sense or convey message or passion. To argue that that passage of
invective was not an emergent effect flies in the face of anyone who claims
that it could.
Conversely, it is easy to see that the same
letters could not be rearranged to produce say:
Power
tends to corrupt and absolute power corrupts absolutely. Great men are almost
always bad men, even when they exercise influence and not authority: still more
when you superadd the tendency or the certainty of corruption by authority.
There is no worse heresy than that the office sanctifies the holder of it.
However, the same ink and paper certainly could
be arranged to say either one passage or the other, or any arbitrary
message of an appropriate size.
Some philosophical schools sniff at the
concept of emergent effects, rejecting the idea as too vague, but I
unapologetically assert that the foregoing principles adequately justify the
term and the concept. The very vagueness they refer to reflects the thinking of
the critics as strongly as that of the emergentists.
The first thing that it is helpful to
understand, is what it is that emerges in or from an emergent event:
what emerges is an entity: an entity that had not been there before, or had had
fewer components, and probably lower complexity.
There is nothing that living
things do that cannot be understood from the point of view
that they are made of atoms acting
according to the laws of physics.
Richard Feynman
Furthermore, the nature of what emerges may
change rapidly with increasing scale.
The concept of scope emerges. To begin
with, think again of "few". What does “few” mean: ten particles? Or
ten to the ten? Ten to the ten may sound like a lot: ten thousand million, but
in a cubic metre of space, ten to the ten particles such as hydrogen molecules
is very small. A fairly good vacuum in fact. Keep pumping particles in,
however, and all sorts of new entities emerge progressively.
Interestingly, the first aspects of gas‑like
behaviour, as opposed to the behaviour of independent particles, begin to
appear when there are something like a dozen particles in close proximity. By
then the behaviour is not critically affected by the addition of an extra
particle; we already are dealing with a "substance" rather than just
a set of identifiable particles. This is different from when we changed to
successively a two, three, or 4‑body ensemble and each new addition changed the
rules. True, every addition does affect the potential effects, but with each
addition the relative magnitude and effect of the change grows less
dramatic.
Let us first consider everyday gases. In a
vessel of modest size we might have say, ten to the 25th gas
molecules.
And gases in modest‑sized containers imply
pressures, and gases under pressure can support convection, currents, laminar
flow — winds, if you like. And sufficiently intense currents imply
turbulence. You cannot get winds or turbulence where there are not relatively
many particles close together. Nor, similarly, can you get condensation that
creates phase changes to liquids or to solids. Nor crystallisation.
Conversely, where there are enough particles,
you can get gravitational attraction, and planets, and suns, and solar systems,
and galaxies ...
All those are emergent effects that are very
different from what we would expect from the same numbers of particles in a
space in which the individual particles have no special effects on each other,
or are on average far enough apart to behave effectively independently.
Furthermore, the nature of entities emergent
from given sources may differ in various ways from that of the entities from
which they were derived, and from each other. The parent entities might remain
unchanged or not: for example, in emergently forming a tetrahedron, ten stacked
balls remain obvious as balls. Dry sand poured from a leaky sack might form a
pile with a normal bell‑curve profile, but it visibly still is sand. A house of
bricks (ignoring various practicalities) still may consist of typically visible
bricks, undeniable though their houseness may be. A surface of Scrabble tiles
placed randomly form a plane but, placed deliberately, they can also form an
arbitrary message. And that message can be copied in ways that have nothing in
particular to do with scrabble or tiles.
However, in their intrinsic nature some
emergent effects have little in common with the parent entities. Carbon,
hydrogen, nitrogen, and oxygen can be combined in various ways to form various
compounds, most of them unremarkable, but in a particular form, suitably
supplied with free energy, they can combine to form TNT. TNT is a substance
from which one can form bricks that do not look special, but, a structure
formed from TNT, suitably ignited or detonated can produce an event such as a
fire or an explosion. Neither a fire nor an explosion is TNT; in fact, without
proper information, one might not be able to tell whether the transient
emergent event that comprises the explosion or fire, did emerge from TNT or any
one of many other compounds or mixtures.
In contrast, from those same carbon,
hydrogen, nitrogen, and oxygen atoms, one could produce any of the amino acids
that the human genome encodes for, except a few that also contain sulfur. None
of them share much resemblance with TNT, none is as toxic or as explosive for
example, and in one form or another we need them or certain of their precursors
or compounds in our food.
Different yet again, but still consisting of.
carbon, hydrogen, nitrogen, and oxygen atoms, are the various types of nylon
polymers. None of them is a human food, nor explosive.
It is all in the attributes that emerge from
combination of even identical elements.
There are even more abstract examples.
Consider a massive accumulation of electric charge in the air, say a mass of
electrons. Beyond certain limits the system breaks down and the charges escape
through the air, causing a flash of light and peal of thunder. Neither the
light nor sound resembles electrons in any particular way.
Note that, among the types of emergence, we
get entities with limits of various sorts: boundaries in various dimensions. As
a rule the individual atomic particles that we originally considered would be
notionally or potentially immortal, but consider the things that happen when
enough particles get together to form a gas cloud.
In a gas cloud one commonly might get
vortices if the conditions are right. But a vortex is not generally immortal
and occurs only when enough (potentially immortal) molecules with enough of the
right individual momentum jostle each other suitably. And a vortex lasts only
as long as the right energy is applied in the right way to maintain it. This
means that among the limits to the dimensions of some entities, one common
limit is temporal: meaning in the time dimension, even if the material entities
that compose the dynamic entity are immortal.
In short, entities in combination beget, not
only new entities, but new classes of classes of entities. First one gets
relationships between entities and later one gets relationships between
relationships. Think of all the so‑called "abstract" entities one
gets in maths and logic. Operations, values, variables, functions, relations,
theorems — the list goes on.
Note that the concept of boundaries
does not imply that a boundary is necessarily indefinitely sharp nor simple,
nor that every example of a boundary is of the same type as every other
boundary. For example, a boundary might be in location: go beyond that boundary
and you are in another yard, another state, another country, in water instead
of on shore. Go beyond a boundary in time, and you are old instead of
(relatively) young. Go beyond a boundary in temperature and time, and you have
burnt your toast or melted your ice cream or frozen your feet. Go beyond the
limits of your neighbours’ toleration, and you are in a state of conflict.
One might employ fuzzy theory to decide where
or how strongly to apply the idea of a boundary. All it comes down to is that
in a given set of dimensions, as you pass further from one coordinate to
another, you are more definitely on one side of a given boundary than on the
other.
Now consider again the concept of splitting.
It is of course related in some ways to the concept of boundaries, because
every split involves some part ending up on one side of a boundary, and the
rest on the other. But I am not in this part of the discussion emphasising that
correspondence in most examples.
I suggested earlier that there are presumably
fundamentally unsplittable entities (electrons, neutrinos etc). If we take a
lump of lithium for example, and halve it, and cut one of those halves in two
again, then after perhaps some 70 or 80 halvings, depending on the size of your
lump, you are left with a lump of two lithium atoms. No conceptual problem.
Halve that, and you get two separate atoms.
So far still so good, but what now?
Well, you can cut one of those atoms again,
but this time you don’t get neat halves: on one side one, two or three
electrons will go shooting off, and on the other side you would get a lithium
ion with a net charge that depends on how many electrons you have removed.
But you have to ask what you mean by calling
that splitting. Certainly you have done some separation, just as
you had with the previous splittings, but each of the previous splittings gave
you a lump of lithium on either side, each lump consisting of lithium atoms.
Now, with this last splitting, you no longer have a lithium atom on either
side. A lithium atom has six or seven (depending on the isotope) hadrons in its
nucleus, plus three electrons in orbitals around that nucleus. Split off any of
the electrons, and from some points of view you no longer have a lithium atom,
though any freely passing electrons could quickly repair it by latching onto
the nucleus.
What you have is an ion. And ions
behave differently from atoms.
Some people would regard that as a quibble in
most contexts, but rather than waste time on further quibbles, suppose you next
split the nucleus. The nature of the nucleus (specifically, its number of
protons) is what makes it lithium instead of any other kind of atom, much as
having four corners distinguishes a square from a triangle or a pentagon.
Splitting a nucleus, such as that of lithium,
is physically possible, For example, you can strike it with a sufficiently
energetic proton or neutron. You wind up with two smaller atomic nuclei, say
one deuterium and one helium. Yes, you have split that kind of. atomic
nucleus, and what you have got is two atomic nuclei.
But these two new nuclei emphatically are not
lithium nuclei, let alone lithium atoms.
In this sense the lithium atoms were
indivisible. Like a soap bubble. Try splitting a soap bubble, and unless you
are very skillful, you generally wind up with no bubble at all, just some
droplets or flakes.
Try another example. Suppose you take a
nation like Britain.
You split it into two populations of some 32 million or so. Then split one of
those into 16 million.
You see where this is going.
After some 28 splits you are left with one
man or woman. Splittable?
Perhaps, in a sense, but previously, every
time you split a population you got two populations, each with about half as
many British citizens as before you did the splitting. This time if you
continue to split into less than one citizen, all you get is no citizens plus
some pieces of carcase.
In that sense a human is atomic.
Splitting a human does not generally give you two humans. Even when a woman
gives birth you do not wind up with two of what you started out with.
We could define. an atomic entity as: one
that cannot be split in any sense (like a neutrino perhaps) or
that changes its character when split at all, or when. split in other than
particular ways, depending on the sense.
So what then?
I believe in calling a spade
a spade, but that is no reason
for calling everything a spade
Anonymous
So here we run into some really messy fields
of concepts.
As I use the word, an entity is
something one can refer to meaningfully.
As I see it, a tomic entity
could be a mass of. substance or of multiple parts of such a nature that if one
splits it, each part is another entity of the same type in the same context,
such as the lump of. lithium I mentioned before. Same with a lump of glass.
Break it and you get smaller lumps of glass. But if that glass had been a
windowpane or a goblet? The glassy substance would be tomic: you could split it
into smaller pieces of glass. But the goblet would not split into two goblets.
And by the time you get to a single molecule,
you no longer have a glass; the glassy state depends on your having multiple
molecules adhering to each other in suitable ways; it in no way demands any
particular change to any particular molecule.
Again, consider a cloud above a mountain
peak. It is effable: one can refer to it verbally: “Look at that cloud.”
“Which one?”
“The one over the rounded peak on the left, not
the little one over the sharp peak.”
Such a cloud is something one can refer to
without obvious difficulty, and the listener can identify it easily — for
a while anyway.
Similarly: “Look at that Manhattan crowd coming down Seventh on the
way to Second”
“What about the crowd coming up Second avenue on
the way to Seventh?”
We can tell that both speakers agree that
there are two crowds. The two crowds seem likely not to know about each other
and are moving at right angles to each other. Each crowd might be shedding some
members on the way, and gaining other members, probably without affecting the
situation very drastically. In this they resemble the clouds passing over
mountains. But what happens when the two crowds meet at Second and Seventh?
There are several possibilities:
- The crowds might merge and
continue as one.
- They might interpenetrate and
continue on their independent paths. While they were together, moving
largely independently, would they be one crowd or two, and which observers
would decide one way or the other?
- Assuming that they do
interpenetrate and then separate to pass on their ways, it is possible
that not all of the members of each crowd stay with their original
companions; each crowd might lose some members and gain some. Now
how many crowds do we recognise? And are any of those crowds the same as
they were before they met?
- The crowds, or their members,
might notice that they were carrying rival banners, then fight and in the
fighting disperse into many smaller crowds on largely independent paths.
Some member or small group of one crowd might find himself or themselves
isolated in the fight, and run away. Do we call each such isolated
individual a crowd too?
- They might on the other hand find
that they were carrying the same banners, and all join into one larger
crowd.
To save anyone the trouble of raising the
topics of fuzzy logic and sorites, yes, I do understand them; let
that go! Theirs are not the current points at issue.
Now, a local, isolated cloud over a peak
commonly is created by a wind blowing up one side and cooling as it rises, till
its water vapour content condenses into the component droplets of the cloud.
That is what makes the cloud visible: a single droplet is not much of a cloud
and in most ways does not behave like a typical cloud at all.
Coming out on the other side of the peak, the
leading edge of the cloud generally drops down, and, as a result, it heats up,
so that the droplets evaporate again, and the altitude at which they vanish is
the downwind limit of the cloud. As a rule, the life of any droplet in such a
small cloud with the wind blowing through is just a few seconds, and within a
minute or so, not a single one of the original droplets still is in the cloud.
Is it still the same cloud?
That question is related to classic questions
such as George Washington’s axe and the Ship of Theseus, but to pursue the
theme here would be too much of a digression.
Such a cloud might grow, till two
neighbouring clouds’ boundaries collided and merged, much as our Manhattan crowds merged.
Both those clouds and crowds definitely did
not always exist. They might disperse, or evaporate or otherwise become
unavailable to our observation.
And yet their behaviour patterns that
resemble each other so temptingly, involve some radically different principles
as well as some essentially nearly identical principles.
One thing that such clouds and crowds have in
common is that they have boundaries in time and boundaries in space: wait long
enough and they pass; run far enough and fast enough, and you leave them
behind.
But still, their boundaries are imprecise and
the identities of their components commonly are imprecise. The droplets in the
cloud might look sharply defined, but they are growing or shrinking,
oscillating or attracting solutes all the time. No man’s regard ever has
contemplated the same droplet twice, as Heraclitus might have said, but never
did say. Droplets or people near the boundaries of clouds or crowds might be
seen as members by some observers, but other observers might exclude them. A
droplet or a member leaving a cloud or crowd could be seen as a splitting
event, or as dispersal.
As for their origin and dispersal, in the
clouds and crowds, those resemble not only each other, but the origin, merging,
and extermination of species or nations. Such populations begin to collect or
grow from particular times and places, they expand, and move through time and
space;. seconds or minutes or years, or millions of years later, they tail off
or die gradually or catastrophically.
When we try to pick out anything by itself, we find
it hitched to everything else in the
universe.
John Muir
So what all those types of entities have in common is:
- they have mutual relationships
between themselves and their generating circumstances, if any, and
- they have mutual relationships
between their various parts, if any, and
- they have particular relationships
to the rest of the universe, and
- they have particular relationships
to their observers, which might be counted. as any part of the universe,
not just conscious observers. The tree that falls in the forest has
implications for the creepers and mushrooms, no matter whether anyone
failed to hear the crash, or thought the crash was thunder. The atom that
decayed in the box tripped the hammer that broke the cyanide vial whether
the cat was there or not. And the independent viewer looking through the
glass back of the box would see what happened even if the front of the box
prevented other viewers from seeing anything.
And relationships are fundamentally dependent
on and comprise information. One even could define relationships as
being information themselves, but I’ll not get into arguing about that.
Everything should be made as simple as possible, but
not simpler.
Albert Einstein, as paraphrased by Roger Sessions
In some circles "reductionism",
with its twelve letters, is regarded as three times as bad as a four‑letter
word: at least the traditional four‑letter words have lost much of their power
and value lately, by being adopted into common use. Reductionism in contrast,
still is poorly understood by most people, and accordingly remains a term of
abuse among naïve critics. For one thing the word gets applied in various
senses to a variety of not‑very‑closely related concepts, and most people who
use the term fall abjectly into the trap of their own confusion. Some of the
concepts are mutually independent, some mutually inconsistent, and some are
blatantly fallacious.
None the less, without reductionism of sorts,
the very idea of science could hardly be meaningful; in fact any claim to
understanding anything concerning the “real world” would be futile. Take
anything macroscopic and consider it in its relationship to the universe, and there
is no practical limit to what there is to know about it; you can't know
everything about anything. This implies that trying to investigate
or even speak of it without simplification is impossible.
"Simplification"???
In science???
Yes, certainly. In this sense simplification
means leaving out whatever you can leave out without thereby talking
nonsense or otherwise producing nonsense.
In science we have the concept of parsimony:
A principle that has been variously
attributed, is that the supreme goal of the construction of a theory is to make
its elements as simple and as few as possible, but no simpler, nor fewer.
Now, various schools of thought that have
names such as "holism", or incline to use words such as
"holistic", correctly point out in their various ways, that
leaving out the wrong parts invariably lets in error and nonsense. Many such
schools would like to regard the omission or neglect of any part of
anything as a defeat or betrayal. Such holism is tempting, because,
demonstrably, not only is the whole more than the sum of its parts, but in one
sense or another, every whole is part of a greater whole, all the way up to the
entire observable universe.
Observations of that sort commonly lead to a
type of defeatism, in which the conclusion is that we cannot ever understand
anything at all, because to understand anything you need to understand
everything.
In this conclusion defeatists conclude too
much and understand too little.
For one thing, as I shall demonstrate, not
even the whole observable or inferable universe, what we might call:
"nature", either "understands" or even "knows" or
has information about, everything in its own self, and yet things all keep
rolling along, so it does not follow that perfect knowledge or understanding or
consistency, conscious or otherwise, is necessary to realism or reality.
Fortunately our universe seems to be so
constituted that functional, "rational" simplification is in fact
possible and indeed is a practical option, certainly for most purposes. One
consequence of this option is that we do not have to consider everything about
anything before it becomes possible for us to make certain classes of
meaningful statements about any particular thing. Our assertions might be
uncertain, probabilistic, wrong, or even meaningless, but may be useful all the
same. As Babbage put it, though possibly too optimistically: "Errors using
inadequate data are much less than those using no data at all."
This is one part of the vital intellectual
tool that one may call “reduction” or "reductionism". Commonly, meaningful
reductionistic statements intrinsically have much in common with generalisations.
Generalisation is another evil word, according to the unthinking horde, and yet
it is a component essential to all human understanding.
In essence what generalisation means is that
by leaving out certain classes of facts about some things, one can draw
conclusions that apply to all of those things together, instead of having to
make separate statements about each one every time. I might say: “humans are
animals”, or “our lunch vegetables are in the pot”, it does not rob the
statements of meaning if I omit to mention that some particular human is called
Pat and another has a sore toe and that those things are not true of all
humans. Similarly, I need not say that some vegetables in the pot are green and
some red, or that there exist vegetables that are not in that pot, or that the
pot contains meat as well and is cooking a stew that is not yet done. Those are
not points that need be discussed in reply to a question such as “What happened
to the vegetables?”
That kind of thing, in which we enable
ourselves to draw conclusions and convey information without trying to cover
matters that are not relevant to the current concern, is generalisation.
Failure to generalise competently is likely to lead to conveying unwanted
information, and in information theory, unwanted information interferes with
the processing of required information, or “signal”. We call unwanted
information “noise”, and the more noise, the harder to deal with the signal. An
essential concept in information theory is the “signal to noise ratio”.
Unfortunately it also is possible, indeed
easy, to fall into the trap of forgetting, or failing to realise, what one has
left out. If I say “Humans are nothing but animals” and. “Pigs are nothing but
animals” then my wording implies (erroneously) that humans are pigs and pigs
are humans because there is nothing to distinguish them. In fact, pigs have
many attributes, or combinations of attributes, that no animal other than pigs
could share; any animals that could not share those, accordingly could not be
pigs, and humans too, have many attributes that no other animal shares, not
even pigs, or either we could not be humans or those “others” must be humans
too.
That is why this use of the expression
“nothing but” constitutes a fallacy so important that it has its own name: the reductionistic
fallacy. That term is most beloved of lazy thinkers, most of whom have no
idea of what reductionism is, whether valid or otherwise. They thereby make
themselves victims of their own reductionistic fallacy.
Such fallacies are dismayingly common among
people with no better excuse than smugness, but there is another, more
insidious, form. When most people speak of reductionism they think of reducing
the universe of discourse to the simple items while ignoring the combinatorial
complexities of the entire system — the whole being more than the sum of
its parts. That certainly sounds bad.
However, in retreating from reductionism in
that sense, they back into basic reductionistic unsoundness of the opposite
kind. Concentration on the big picture seems easy and obvious when the big
picture is the easy one to see (it often is not!) It tempts those who have not
done their reductionistic homework, into smugly ignoring the details of the
system.
How small‑minded of me! Fancy my niggling about the details when the
glory of the full system stands out, inviting, commanding, one's
attention!
All the same, by leaving out those details
one is being far more erroneously reductionistic than by concentrating more
obsessively on the details than on the big picture. At least by concentrating
on the details one is in a better position to understand the things one is
working on. Conversely, if one reductionistically omits the basics, one
guarantees missing important foundations to the system; foundations that one is
at best ill-equipped to understand.
One becomes like the innocents who abuse the
filling station attendant when there is no more fuel to be had: why does the
malicious oaf not fill up my vehicle?
Why doesn’t he stop nattering on about technical details about his tanks
and stuff? All I am asking him to do is to stick the spigot into the hole, grip
the handle and fill ‘er up! Why should I
care about his tanks and things?
There we have true largeness of spirit: no
reductionistic concentration on details such as how it comes about that
commonly one does get fuel out of the pump!
Comparing mythologies, the rival errors of
rival errers, tends to be unproductive. And yet, while I hold no brief for
either, if I must take sides in this battle of the reductionists, I am inclined
to prefer those reductionists who at least know enough about little enough to
know what they are talking about.
I already have mentioned the concepts of “top‑down”
and “bottom‑up”, either of which is important when used correctly, and
disastrous when abused. And both can be seen in terms of reduction versus
holism.
Reduction and generalisation are important at
all levels, but they are most striking when they lead to changes of type, at
changes of scale. Commonly it is at such stages that they also are most
inclined to tempt us into fallacies, especially into various forms of
oversimplification. Sometimes the originator of such a fallacy was not
personally misled by it, only intending it as convenient imagery, but the first
publication of the concept led to public confusion and illusion. Take for
example Rutherford’s announcement that atomic
matter is “mostly empty space”; public reactions ranged from mystical pseudo‑philosophical
pronouncements, to commonsense dismissal of scientists as not being sane.
Only a minority were equipped to make sense
of it in context. In fairness, the concept is a lot more difficult than it
seems in everyday English; the very definition of “empty space” is commonly
taken for granted, and yet it might be fundamentally meaningless. Certainly
what seems to a neutrino like empty space, might well look like a block of
granite or a planet, either to an electron or to your spaceship.
Such fine distinctions work out very
differently for say, schoolroom Newtonian physics, or General relativity, or
for Quantum Theory. In each case there is a great deal of possibly unspoken
reductionism that works out differently in each body of theory. Each in turn is
internally robust, but their generalisations tend to unravel as one approaches
problem areas.
This is not a criticism from the point of
view of application of the bodies of theory in their own fields, but it does
mean that the lay public should be very careful in drawing conclusions when
things get tricky. It is similarly difficult for the physicists when they
approach the boundaries of their theories and experiments. In such regions of
theory and research, emergent effects play skittles with theories. What the
research workers can be confident of, is that we still have a great need for
more breakthroughs.
My adversary's argument
is not alone malevolent
but ignorant to boot.
He hasn't even got the sense
to state his so-called evidence
in terms I can refute.
Piet Hein, “The Untenable Argument”
I have encountered intelligent, educated
people who deny existence, though I never have encountered one who was able to
explain that reasoning satisfactorily, let alone cogently. The best I have
heard is that “existing” is not an attribute of an entity and that “to
exist” is not an activity.
That sounds well, even trenchant, but I am
not satisfied that the argument is valid or, if valid, that it is useful, let
alone correct — even grammatically. Whether a concept is dealt with
in a given language or not, and if it is expressible, then whether it is to be
expressed as a particular expression, noun, verb, or interjection, need not
reflect on its logical content or validity.
It seems to me that such objection to verbs such as "exist" arise largely from failure to recognise the respective natures of stative and dynamic verbs. Most of us, most of the time, are comfortable with dynamic verbs because they plainly are "doing" words. So they do not complain about "dig" as a verb because they can see "digging" as a present continuous dynamic activity, and they use the participle form for a "digging dog" because they can see things happening.
However, languages and perceptions deal with stative verbs as well, verbs such as in "This stone weighs too much for me to lift". "That canal joins the oceans." and "I exist to confound the deniers." are examples of verbs of a type that is widely recognised among extant languages, and recognised for good reasons.
And many verbs may be stative in some contexts, and dynamic in others. "I live here." "You call that living?"
And to ignore or forbid such attributes of
languages seems to me evocative of aspects of E-prime; which amounts to English
shorn of all forms of the verb "to be". I regard the idea as an
arbitrary whim, a waste of time and effort, and unrewarding. Interested readers
may explore the concept, starting with the article in Wikipedia. For my part I
have more entertaining concepts in which to invest my time.
What is more rewarding in this discussion is
how our language treats existence words,
so till further notice I happily exploit the semantic convenience of the
existing convention — or the established convention, if you
object to my begging the principle. And that principle is what I shall exploit
anyway. English verbs are simple compared to those of some other languages but,
to put it crudely, they can refer to events and states as well as actions.
And in at least the languages that I have
been able to examine, each had an infinitive verb meaning "to exist".
Such languages enable the speaker to refer to events, states, and actions as if
they were entities, much as in English.
And in my opinion all such things
comfortably accommodate the concept of the concept of the existence of
existence, including "exist" as a verb. For my part, those who
disagree and deny that they and their existence exist, may do so in good
health, because I can comfortably ignore whatever does not exist.
Elsewhere.
Still, to express exactly what might amount
to existence, or define existence, remains a tricky conceptual problem; there
are about half a dozen major lines of thought on the matter, plus umpteen minor
variants, each with cohorts of partisans squabbling about the details. This is
not the place to debate those details, and as I see it, the discussions that I
have read are mutually inconsistent to the point of incoherence. They are full
of question begging about such items as: which properties or attributes are
intrinsic or extrinsic, what constitutes a property, or what constitutes an
individual or an object, or what their various opponents said, or should have
said in terms anyone could refute.
As a rule such discussions do not even
consider the inconsistency of asserting whether one can step into the same
river twice, while at the same time they assume that the person stepping into
that notionally second river, is the same as the person who stepped into the
first, un‑stepped‑into river.
As I already have pointed out, entities have
boundaries in time as well as space, and not necessarily sharply defined
boundaries. An entity can be an evolving state, such as a cloud, a river, a
species, or a human, rather than a statically defined object, such as...
Well, such as what? To present any inarguable
example of a statically defined empirical object is not easy; I am not so much
as sure whether it is at all meaningful. Not even rivers, diamonds, rocks, or
stars or George Washington's axe, are forever. And a lot of the debate seems to
ignore such concepts as the limits to identity.
So forgive me for unregenerately ignoring
such topics, not because there is nothing substantial to them in themselves,
but because there is nothing substantial to the views on which so many writers
have expended spittle and ink.
For my part, seeing that this is my
essay, I accept the existence of existence and naïvely assign to its
reality a criterion of sorts. Anyone dissatisfied with that is welcome to my
sympathy, but I do not expect his satisfaction to approach the top of my
attention stack in the near future.
Some aspects of the concept I already have
hinted at. It seems to me that whatever exists, even for a while, must be an
entity, commonly a complex of entities at that, and even things that don’t
exist although their concepts exist as entities, such as the concept of ten‑tonne
green swans, or the number two to the power of its own reciprocal, or odd
perfect numbers smaller than 1000000. Even if they never existed before, such concepts
do exist now that I have conceived and recorded them; existing not as
themselves (there definitely is no such swan, nor any such number) but as the information
it takes to represent their specification.
You might wonder why I bother with anything
so trivial, but it leads to questions such as whether numbers exist at
all, and if so, in what sense they could exist? I am inclined to argue that in
some senses they do not, but that is a complex matter in its own right, so let
it pass for now.
More relevant to us in this discussion, is
the question of what it means for a material
entity such as a cloud, a crowd, a species, an ocean, a nation, a molecule, an
electron, a tune, or a diamond, to exist in the naïve sense.
I propose that it does exist, at least if its
presence or influence affects the course of events in some ways differently
from how its absence would affect the course of events. Let us consider an
example.
Charitably assuming that I exist,
suppose that I walk into my bedroom in the dark. It is my intention to retrieve
the book I left on my bedside table. I step on something that rolls underfoot
and I crash down painfully on my coccyx. I get up and step forward and bump my
nose equally painfully. At that point my subjective sensations fully persuade
me of my own existence. A real “cogito ergo sum” moment. Or perhaps “sentio
ergo sum”.
Nothing feels as much like feeling to persuade one that one does exist!
I also deduce that something round on the floor
had existed where it had rolled underfoot; never mind cogito ergo sum: Si impediat, existit! If it impedes, it exists.
A dropped pencil perhaps? Falling had
disoriented me so that on rising I then had bumped my nose against an also
existing wall, cupboard, or the like.
Further experiments could locate the light
switch and gain me more information, but the principle already should be clear:
if something exists in the sense of being “real”, it has consequences that
differ in some ways from the consequences of some alternative thing (including
the possibility of the absence of any thing) being real in its stead.
Consequences? What might that mean? In
everyday terms it generally would be something observable in principle, some
event or combination of events, some object or objects. In submicroscopic
terms, where quantum considerations come into play, that trivially might affect
the probability of one event rather than another, instead of its definite
occurrence. Many quantum effects depend on probabilities rather than specific
consequences. Examples include the passing or reflection of a photon by a semi‑reflective
mirror or polariser.
“Existence” defined in such terms might be
seen as being causally significant (not deterministically significant,
because I reject determinism. But causal significance would mean that the
existence of any entity would change some probability somewhere).
This might seem very academic, but it is the
basis of many thoroughly material effects. (In fact, at the quantum level,
conceptually all effects.) Consider a practical example: in a nuclear reactor
energy is produced by the splitting of fissionable atomic nuclei (usually of particular
uranium or plutonium isotopes). In splitting they give off some neutrons, and
there is a probability that some of those neutrons will hit another nucleus at
the right speed either to split it to release more energy in a chain reaction,
or stick to that nucleus and change its mass and neutron number.
By adjusting the probability of these events,
we can adjust the heat that the reactor gives off, to match the power that the
users desire. Among the atoms in the fuel of say, an HTR (High Temperature
Reactor) there will be some uranium‑238, which in itself is too inert to have
much effect on the reaction; the probability of a neutron hitting a U‑238 atom
hard enough to be absorbed is too low to count for much, so the neutrons go
bouncing around till they decay or can cause another fuel nucleus to split.
But suppose the reactor begins to overheat;
then the hot U‑238 atoms themselves bounce around much harder, which causes some
of the collisions with the neutrons to become much harder, more energetic. This
increases the probability that the neutron gets absorbed, so that it cannot
cause any immediate splitting. In turn this slows down the chain reaction
automatically — a highly material and important consequence of the adjustment
of probabilities.
All by managing the probability of particular
quantum reactions; cause and effect of the existence of particular entities can
be subtle and, in my view, often beautiful.
If a given entity has no existence, then
whatever does exist in its stead must have consequences that differ in some way
from its absence. In that dark room, if the wall or something equally solid had
not existed there I would not have bumped my nose, even if my nose existed. If
there had been no similarly round object on the floor, I probably would not
have fallen in the first place. If something flat had been there, say a
booklet, a thing that, unlike a pencil, would not roll, I still would have been
affected, but in other ways. Even if that same existing pencil had been there,
in a slightly different place or orientation, things might have happened
differently. The consequences of such items can be interrogated for their
information and its implications for what exists, in what form and place.
To perform such interrogation is to make use
of consequences in measurement. I feel around in the dark for the wall
say, and locating it gives me the measurement I need if I am to know where not
to put my nose.
Similarly, if the U‑238 was too cool to
absorb the neutrons with sufficient probability to interfere with the
reactions, then the chain reaction would go faster and hotter, but if
sufficient heat of reaction existed in the reactor, the probability of
absorption would reduce the production of unwanted heat: negative feedback.
Subtler examples with more obscure or extreme
outcomes are easy to multiply. For example, consider a meteoroid in
interstellar space. Suppose it has a diameter of some 20 kilometres, a mass of
some ten trillion tonnes, and a trajectory that, all other things being equal,
would strike the planet Earth some 66 million years ago. Suppose its impact
caused the K/T event that ended the reign of the dinosaurs, eventually leading
to the presidency of Trump, that in turn established the incorporation of America into
the Chinese empire.
But how did that meteoroid’s journey begin?
Perhaps in a planetary collision some 6000000000 years ago, many light years
away, perhaps in another galaxy. How small an influence could in our
imagination have prevented its striking Earth?
Imagine a single uranium atom in free space,
remnant of a neutron star collision. U238 has an enormously long fission half‑life
but this particular atom happens to fission spontaneously at a suitable time,
producing, as it happens, a highly accelerated barium atom. These things do
happen in uranium, and like all similar events in quantum interactions, such
fission is not deterministic as far as we can tell, nor predictable, not
even in principle.
A few years later that speeding barium atom
strikes a microscopic chip of silica that in turn deflects a milligram grain of
iron, that a few thousand years later drifts in turn into the path of our
meteoroid, minutely affecting its trajectory and its spin. Had that uranium
fission been a microsecond earlier or later, the meteoroid would have struck
Earth on schedule, though of course all sorts of things would have happened to
it on its four‑billion‑year journey, all of them in the first couple of billion
years necessary to direct it on its way to Earth, so in fact, as a result,
Trump gets elected anyway and China celebrates.
But suppose that little grain of iron
affected the meteoroid’s momentum just enough to change its trajectory by some
fraction of one trillionth part. Suppose that event microscopically changes the
meteoroid's passage in slingshotting past some massive body a few million years
later, and some five billion years after, and after some large number of
intervening events, our meteoroid skips off the atmosphere of Earth and passes
fairly harmlessly on through space. The dinosaurs survive and the mammals
remain subordinate along with other animals such as frogs.
Sixty million years after that, a fat, bald,
orange lizard with a comb‑over yellow crest gets voted into the presidency of a
leading world power, and starts a nuclear war that blasts all higher forms of
life off the planet, and democracy triumphs again.
If such a thing could happen, and we magically
could know about it, we would say that that uranium atom, and its choice of instant
and direction in which to fission, had existed in terms of our definition. That
did not happen, so we exist instead of that lizard.
That was a sizeable consequence for a single
splitting atom.
Or even for two atoms.
Because, suppose that if the first uranium
atom had in fact split a second later and had no measurable effect on the
meteoroid’s path, then the meteoroid might have passed close to another quantum
event that it otherwise would have missed, say from a vagrant Thorium atom, and
that this one caused it to proceed on schedule.
We should then be unable to allocate a unique
minimal cause to either the collision, or non-collision with the planet; this
being an example of underdetermination.
You will recognise this as just another
example of the webs and chains of cause that we dealt with earlier. Precisely
how we would become aware of such an event or train of events is a separate
question, and non‑essential, because we do not demand that we in particular
must be aware of every event or probability or existence in nature, whether in
our history or pre-history. One thing we can be sure of however, is that untold
examples, equally trivial in themselves, and equally drastic effects, have
happened throughout our past, and continue to happen now.
Anyway, you might like to wonder about how
significant such thoughts might be to the question of what ‘exist’ might mean,
and what the questions of implication in physical algebra might suggest for the
concept of existence or nonexistence.
Investigations need not tell or even
determine all the details of the nature of the entity’s existence, but often
they can suggest the existence of something somewhere. And in principle the
more closely that something can be interrogated, the more information about it
can be determined, progressively excluding more and more alternatives.
The
boundaries that Napoleon drew have been effaced;
the kingdoms that he set up have disappeared.
But all the armies and statecraft of Europe
cannot unsay what you have said.
Ambrose
Bierce
It is a commonplace that information cannot be destroyed. That could
be interpreted in various ways, but one way is in the assumption that every
consequence or outcome of something existing or happening will change something
in the universe. That change in turn, if it may be said to exist in its own
right, will change something else, and so on, commonly with changes of increasing
complexity as time goes by.
In general, no operation of the physical
algebra of our universe will ever decrease entropy except trivially,
temporarily, or locally, but not globally.
We have seen that the choice of the instant
that a single radioactive atom happens to decay, can affect the history of life
on a planet, or even the continued survival of the planet itself; could we
imagine an event so small as unable to affect contemporary human
history?
I say it is difficult. Very difficult indeed.
Some people say that no single person can affect history, and that great events
require great movements to direct or change them. Tolstoy was one of those with
such views. His view I deny and dismiss categorically, except for trivially
local and short‑term events. If Szilard had stepped under a bus at the
moment of conceiving fission chain reactions, Hitler or Stalin might have got
the Bomb instead of the US.
If so, the history of the second half of the 20th century and all
our currently foreseeable future history certainly would have been drastically
different.
I argue that a world without Bach or Newton or Hitler or
Einstein would have been different too, both qualitatively and quantitatively.
After a few generations, if we had some clairvoyant means of comparing the
possible alternatives, we could hardly recognise the human situation on the
planet. Now what would it have taken to prevent their birth?
Very, very little.
Suppose that on the evening that Hitler was
conceived, his father lit his pipe as usual, but that a match failed to strike
smoothly and it took an extra strike to light his pipe. If he noticed the
incident at all, Herr Hitler very likely would have forgotten it in less than a
minute, and anyway, it certainly would be too trivial to affect human history,
right?
Maybe, but it could have made him shift in
his chair, perhaps unnoticeably.
An hour or two later in bed, not the same
sperm that would lead to Adolf, but another sperm ten microns to one side, was
the one among tens of millions of rivals, that got through to fertilise the
ovum. Its DNA certainly would have differed from the DNA in the sperm that won
the race in our world line. The Adolf Hitler known and loved by all would never
have been born; the sibling actually born certainly would have been different;
might even have turned out to be a daughter. That child in turn would have done
all sorts of things more significant than one extra strike of one match. The
doings of those different children in their turn would have caused the birth of
hundreds or thousands of children other than the children that actually were
born during Hitler's lifetime. Some better no doubt, some worse. Within a
century or so after the double match strike, not a single human on the planet
might be one who, in the event, actually was born, and if there were one, that
one necessarily would have done things different from what instead got done.
Whether the resulting world would have been any happier or more miserable than
ours, we never can know.
But such potential tiny differences always
could have had big, big consequences.
Franz Joseph Haydn having been unborn in
similar circumstances would never have written the music that later was
appropriated for the German national anthem. For Tolstoy’s siblings to have
written “War and Peace” would have been unlikely, and depriving the world of
“War and Peace” would have changed the world more dramatically, if not more
drastically, than the failure to strike a match at the first stroke.
Never mind monstrous macroscopic objects such
as matches; any of these people could have been born different as the result of
a single phosphene caused by a single quantum event in any one parent’s eye,
say from a cosmic ray produced from an event tens of millions of light years
away, at the height of the age of the dinosaurs.
And as things stand, we live on a planet that
has been deprived of millions of people greater than any we have known, greater
than Newton, Maxwell, Bach, Alexander, Caesar — anyone you care to
mention, all because of events far smaller than that double match strike, and
at the same time we have been afflicted with the appalling stupidity and
cruelty that might have arisen instead of greatness.
Of course, we can be sure that we have missed
more fools and deadly tyrants than heroes, but we have no shortage of those in
our current world anyway, so that is not much consolation.
Be all that as it may, you will recognise
these as examples of the webs or chains of cause and outcome that we dealt with
earlier in this essay.
And anyway again, we live in a chaotic
universe; it fundamentally is not possible in general to predict the detailed long‑term
effect of the smallest event, as long as the relevant light cones overlap
sufficiently.
Speaking empirically, we live in a world of
physical implication that produces information by the development or
succession of events. Whether “production” of information can imply creation or
destruction of information, is another matter. One common view is that it
cannot, but let’s not discuss that here and now.
The immediate point remains: whatever
physically exists (including information) intrinsically has consequences, and
what does not exist has none, or at the least, its non‑existence, that is to
say the presence of what does exist in its stead, has consequences; different
consequences. It is a vague point, poorly defined, but a good place to start,
in trying to define existence. If we accept it, it makes nonsense of
Schroedinger’s cat dilemma.
And anyway, I like cats.
There are important implications to this line
of thought. Not only does humanity have no G‑E‑V, but no G‑E‑V seems possible
even in principle: certainly no G‑E‑V is possible to us and none to nature
either, not as we understand any related concepts. We only can observe anything
through the physical consequences of physical events, in other words, the
implications of those events.
As far as we can tell, the cause and course
of any event are controlled only by what information affects its circumstances,
and because information is more limited than the theoretically possible ranges
of outcomes of events, the courses and outcomes of events are not fully
determined, but are partly random: they are “underdetermined” as the
term is. In fact they are never at any stage fully “determined”.
Even after the event the outcome is not fully played out before it has
made its contribution to all the events that it possibly could affect directly
in turn. And those events in turn have no clear limit to their own ultimate
consequences.
If it is indeed true that information cannot
be destroyed, then that is true only in a special sense. A quantitative sense
if you like, suggesting that the universe never grows simpler. And I am
suspicious even of that. But qualitative information certainly can be
lost indefinitely; even if, in wiping out a poem composed on a slate, one does
not decrease the physical information in that system, the information now
embodied in the medium certainly no longer conveys the same message.
It might be easier to envisage the principle
if you think instead of a board on which the poem had been composed in Scrabble
letters. If we scramble the letters the new arrangement of letters, whether
intelligible or not, would still contain the same amount of information. However,
that original message would be gone forever. To retain the original message
would require that a separate copy had been taken before the scrambling.
We see therefore that it is possible to
change the universe by changing information in such a way that, though our
universe has not become any simpler in consequence, and does not contain any
less information, it never again will yield up particular information
that once had existed.
Imagine for example a super‑Einstein sitting
on a wide, deserted sandy beach at low tide. He suddenly stumbles on a
completely new line of thought accidentally initiated by a trick of light.
Hastily he writes out the derivation of a TOE, a Theory of Everything.
Not having paper with him, he scratches the TOE into the smooth surface of the
beach. It only takes him several square metres of beach sand.
Hastily he turns and runs for his rural
hotel, intending to photograph the text before it is lost. Dashing across the
road all unmindful, he gets squashed flat by a bus. No one knows of his work,
which eventually gets wiped out by the rising tide.
Now, none of that, however sad, suggests
either loss or much creation of physical information in the sense I
already have discussed, but what is certain is that however little
difference it made to the behaviour of the tide or the flattened skull of the
dead genius, that TOE is gone and
remains unsuspected for say, the next few thousand years. And when it is
rediscovered, that does NOT happen by an ebbing tide recreating the information
of the work that our squashed genius once had written in the sand.
In fact, if by some miracle some tide did
leave some legible script somewhere in the sand, it would stand hardly any
chance of being meaningful or in the right language, and if it did, it would be
likelier to spell the lyrics of the song “Love Letters in the Sand”, than the
several square metres of the TOE.
And if the tide did reproduce the TOE,
the chances of a passer‑by noticing it and recognising its message, let alone
recording it, would be negligible.
Of course, I am being a little unfair, given
that the nature of ebb‑tide waves is not suited to producing legible script at
all, let alone anything useful. But the same principle would have applied if
the formulae had been produced by someone scrambling pebbles on a board, or by
chickens scratching in the sand for seeds.
Actually, though what I just wrote remains
valid, the meaningful information expressed in the script is not in principle
lost to the universe immediately. An aircraft passing overhead could have
photographed it a few microseconds after the photons left the beach, and in
principle, light that passed the aircraft could still carry the same message
upwards into space for as far as optics could in principle resolve the message.
Those travelling photons are a special case of information storage. The light
in flight might be regarded as a sort of delay-line storage mechanism.
But the cameraman would have to hurry. No
reasonable photographic equipment could carry the message recognisably even a
few light seconds into the sky. The signal would get lost in quantum noise,
probably in less than a light minute (the sun being some eight light minutes
away from Earth).
It is easy to multiply such examples. Think
of a message or a work of art written in soluble ink on a slab of sugar. In
simple physical terms you lose no information if you dissolve that sugar in hot
water, but though you were to wait till the sun cooled, that message or art
work never would reappear from that solution whether the sugar crystallised
out, collecting the pigment as it did so, or not.
And if it did reappear, and its significance
were to be recognised, no one would be able to tell that it had existed once
before, perhaps trillions of years before.
Furthermore, if such a message did
crystallise readably at any point in the future, there would be no way to tell
whether it was the same as some message that had been written there before or
not. The chances would be hugely against its even vaguely resembling any
particular message written before or at all.
The information written in such volatile
media patently existed as one or more entities as long as it was not erased in
such a manner: in suitable circumstances it might build or destroy whole
civilisations. Once erased it could do nothing in particular except in the
negative sense of causing something to happen, other than its presence once
could have caused. Ironically, it might not even be the original or intended
message, or it might not have had any intention at all. For example, another
genius catching a brief glimpse of part of such a message might misread it and
create a totally unconnected, equally influential, message, but with an effect
substantially different or even opposite. An artist might be inspired by the
vanishing glimpse of the picture, and create something equally great, but quite
different.
Nothing of any such kind would be a good bet,
but it still would be possible in principle.
The original entity might not have been any
intended message at all: an artist might break open a lump of malachite and be
confronted for just one second with a pattern of shapes and shades that shakes
him so passionately that he drops it and it shatters. He tries to recreate it
mentally, so as to copy it, but fails drastically. Instead he does his best to
recapture his vision, but even if what he then produces is his finest work
ever, he never knows how true it is as a copy. Certainly, whatever he produces,
insofar as it resembles the original, was not information originally formulated
by any subjective mind.
But that information in each case undeniably existed
in the sense of causing things to happen that would not otherwise have
happened, and by the same test, stopped existing once disrupted to the point
that no such entity was once again distinguishable.
There again the test for existence in any
particular sense is what the entity can cause: what its effect would be on the
web of causes and events. The ink that got dissolved certainly still exists, if
only as its molecules or even atoms, just as the sand that got scratched into
the representation of the TOE theory still existed after the tide came in. But
the semantic message content? Suppose the ink settled down again and
miraculously produced a legible message again. Physically this is
hypothetically possible, though not to be expected in several lifetimes of our
observable universe; however the chances of its having content with the same
effect as the original are incomprehensibly improbable.
Nor is the principle limited to writing or
graphic art. Entities created from other entities could take the form of Rupert
Brooke’s “keen unpassioned beauty of a great machine”. Having seen a poem or
seen a construction such as a machine, one might be able to recreate it. Many
inventions of biological brains are repeatedly presented at patent offices
around the world, some resembling others, and some unexpectedly different. To
be sure, plagiarism is nothing unusual, but the same is true of convergent
originality. Someone has referred to the effect as: “the congruence of great
minds”.
Much the same is common in nature, in which
biological evolution repeatedly produces mechanisms and structures that
resemble each other positively eerily. The effect is so marked that it has its
own technical term: convergent evolution.
Convergent evolution is marvellous,
repeatedly, beautifully marvellous, but it is not miraculous: given
circumstances offer similar or analogous evolutionary opportunities to similar
or analogous structures in different organisms, so there is nothing mysterious,
however breathtaking, about the outcome. The reason is what we might call causal
structure: the structure of the causal circumstances “causes” the form of
the outcome.
One might as well be astounded at the
repetitively marvellous formation of repetitively marvellous glass shards when
we break glass, or the complex routes that streams trace in eroding their beds
down a mountainside.
Think again of a species. Kill every
reproductive specimen, and it is extinct.
But suppose a genetic engineer had captured
its full genome and sequenced it. Suppose he had stored the information in a
computer's storage medium. If so, is the species still extinct? If he is
sufficiently advanced or the species sufficiently simple, he could recreate the
living species from his recorded data. Well, suppose that he records the data
in a hologram in a block of solid silica that could last for millions of years.
Unfortunately he ships the block overseas, and it falls into the deep sea and
is buried in a kilometre of ooze.
Now is the
species extinct? Potentially, but we never can know whether some future
palaeontologist, whether from Earth’s future or from an alien planet, would
stumble on the block and re‑create that species.
Euclid taught me that without assumptions there
is no proof.
Therefore, in any argument, examine the assumptions.
Eric Temple Bell
This is a topic of such virulent disagreement
that I have no hope of imposing any definitive views on anyone. The best I can
do is ask any serious reader to ignore his own views and accept mine for the
context of this essay. I do try to present them cogently, but entire
philosophical systems have been founded on conflicting views of the matter.
Let us begin with Plato’s cave and the idea
of the existence of ideals. I have little patience with them, and refer
interested readers to Wikipedia articles on the topics.
Nor am I happy with Russell’s definition of
numbers in the form of “three being the set of all sets that have three
elements”. As it happens I far prefer the likes of Conway’s Surreal Numbers, but never mind
that. I have no quarrel with the construction of number concepts, or for that
matter any other formal concepts by way of formal axioms and theorems derived
from them, but when it comes to defining the concept of the “existence”
of such concepts, we need to examine the concept of existence for latent
ambiguity and the risks of inappropriate contexts.
So, when we say “there exists an integer X
such that 2<X<8” we are happy to accept that integer as being entailed by
the axioms of integer arithmetic plus any necessary intermediate theorems. We
also might be happy with “there exists no integer X such that 2>X>8”.
Slightly trickier would be the question of
whether this has any meaning in our physical world (the purely formal
mathematician might not care, but that does not imply that their indifference
disqualifies the question from anyone else’s consideration).
And in fact, there are some material
implications. Every axiom, theorem, proposition, sentence, symbol, operation,
or derivation in such a system, only can be demonstrated to exist in the form
of information. With no information it is not meaningful to speak of any
entity at all, because without information no entity could be distinguished
from any other entity, in particular by its effect on the causal web of events.
In a statement describing integer X:
2<X<4 for example, the information was sufficient to determine X=3
uniquely. If it had stated that 2<X<7 then at least two bits more of
information would be required to distinguish X within the set 3, 4, 5, and 6.
Information is physical and refers to
physical things, and the significance of any particular information
resides in the distinction between alternative physical states, realities, or
anything similar.
And no formal concept, axiom, or theorem can
be instanced, communicated, derived, stored, or operated on without some
physical manifestation, whether of mass or energy, even if only in the form of
photons in space or sound waves in matter, ink on paper, toner on a drum, or
states in a brain.
So I assert that in that sense to begin with,
formal disciplines are subject to physical constraints. No formal relationship
or object can exist in the absence of information. In fact, even formal
errors such as “X=(X+1)” in standard arithmetic require information for their
manifestation or existence.
I do not state this as a formal axiom, please
note, but I invite counter‑examples. It leaves us with thorny questions of the
distinctions between truth, error, and existence. A truth value,
whether formal or applied, may be TRUE or FALSE in a binary universe of values,
but in larger ranges of values it also could be MEANINGLESS or UNDETERMINED.
Consider for examples:
X=X,
1=0,
=+=, and
X=Y.
The facile reply, that it is meaningless to
suggest that something can be true without existing, cannot be trusted: since
it patently follows that if a true statement or fact can exist, then in a
similar sense its negation, or its various possible misrepresentations, can
(must?) exist as well. In fact one could make a persuasive case for the assertion
that vastly more expressible statements, or structures of signs, are untrue or
meaningless, than are true. In particular, it also is true that the number of
possible statements, true or false, unique, distinct, or synonymous, is finite.
If that were not true, then one could argue that true and false statements were
equally numerous.
And not only in politics, courts and
businesses.
A common reaction is that this is nonsense
because every theorem in every formal axiomatic structure is entailed by other
theorems or axioms, and therefore is true (otherwise it is not a theorem), but
there are at least two difficulties. In context the less interesting difficulty
is the principle that some true statements are Gödel‑undecidable, as
established by Gödel’s first incompleteness theorem; it might be interesting to
speculate on what the smallest intelligible Gödel‑undecidable statement in
simple numerical algebra might be, but let that pass for now.
A more interesting difficulty might be to ask
how in the sense I am discussing here, any Gödel‑accessible theorem
might be proved, preferably by the shortest possible route of formal
derivation. Assume that derivation occurs by achieving each new step by
algebraic operations on the results of one or more earlier steps, in other
words, on axioms or theorems.
If you know of a better way, please present
proofs or examples of its validity.
Now, the result of each valid step, each
operation, in a proof is necessarily a statement in some relevant notation
within a relevant medium and convention. It might be an operation such as
addition, negation, inspection, comparison, or enumeration, or it could be an
assertion such as that a given identified value differs from some other value.
Any of those implies relationships, and accordingly that the statement
has information content and requires that information to be supplied;
one cannot provide random noise as proof or derivation (except perhaps as the
logical equivalent of “which is absurd”, but I am uncertain even of that
possibility). Furthermore, the application of each operation is a special case
of information processing, which intrinsically involves energy and entropy.
The implication is that implication is a
physical process or relationship. Without physics there can be no statement,
no derivation, no storage of information. And arguably, no existence in the
sense that existence implies, depends on, imposes, consequences.
Still, there is at least one more hiding
place for the existence of formal reality: physical relationships. As
long as physical things can happen in threes or minus thirty threes, or
(roughly) in pi’s or tau’s or e’s, such numbers might be said to “exist” in
various senses. That is certainly reasonable, because each such relationship
involves masses or energy levels or their states and coordinates, and those in
turn demand energy and changes in entropy to change.
As I see it, such things are about as real as
anything can get, and I am not inclined to debate the niceties.
Well then, whatever we can write down, or
derive from axioms, or observe by counting or measurement, or can be taken to
be embodied in some physical entity or situation, can be taken as existing, as
being real in some sense, if you like. One could for example demonstrate the
reality of the number 1729 either as a row of 1728 bolts plus a pencil, or a
cube of 10*10*10 blocks of wood plus nine rhombi of 9*9 pennies.
But what about numbers one cannot
represent in some such manner?
Let us consider an academic example. The
first part of the decimal representation of pi is 3.1415926535897932.
Let us express that by saying that the numbers 14159 and. 35897
occur “in the decimal expansion of pi”. Now, those two happen to be
primes, each expressed as a prime number of significant decimal digits and we can
say that they are the first two five‑digit primes that occur in pi.
This is not of intrinsic mathematical
interest, please note, just an illustration to clarify our terminology. There
is nothing special about this; we expect both numbers to appear an indefinite
number of times: for example, the number 14159 occurs again after the digit in
position 6954. The prime 35897 also recurs, some 209390 decimal digits after
its first occurrence.
Well then. To the best of our available
evidence, the sequence of the decimal digits of pi are of maximal entropy,
which implies that they are truly random in every sense other than their
occurrence in pi, so that every possible sequence of digits of any given length
will occur somewhere. I do not assert this, but will assume it as a matter of
convenience. It is just an example, right?
Now imagine taking say, the first googol‑digit
prime in pi (if you prefer to look for the first googolplex‑digit prime, suit
yourself; it will not affect the argument). Add to that prime the following
single digit of pi. For example, if we did this with 14159, we should add 2,
that being the next digit in pi, giving 14161. Or if we did it with 35897 we
should add 9, giving 35906.
But we are considering not 5‑digit, but
googol‑digit primes. Call this first googol‑digit sum your starter. Then
continue looking for the next googol examples of such googol‑digit primes. Each
new prime you find, you raise to the power of the current value of the starter,
then add the next digit of pi, and that becomes your new starter.
Call the final number you get p.
Now, of all the insane, pointless numbers I
could have chosen, p just about has to take the cake.
That is exactly why I have chosen it.
It does however have several points of
interest — not mathematical interest perhaps, and as far as I can tell,
nothing even of arithmetic interest, but of conceptual interest. Let us
consider some of them.
Pointlessness has nothing to do with the
“existence” or meaning of a number. Nothing about the calculation I have described
is in principle impossible; it works as well if, in the description, you
replace the word “googol” with the word “one”, though it does become
impracticable if you replace “googol” with “two”.
And yet, it is not possible to calculate the
full‑sized p at all; in fact, it is not even possible to find and write any
googol‑digit prime in pi whatsoever, and never will be, for the adequate reason
that, as far as we can tell, there are fewer than a googol atoms in the
observable universe, so we could never find enough ink to represent it, even if
we used just one atom for each digit in the number. Nor could such a number
ever exist in any relationship in our universe at all, let alone meaningfully.
As for calculating p itself ...
In other words, with relatively few trivial
exceptions, everything we do with or about p is magic! The
only clear exceptions are mathematical facts that are independent of the exact
value of the integer, such as p0=1, p*0=0, p‑1<p
and so on. And those are not facts that distinguish p from most other
integers, so they have little to do with its value or meaning or nature or
existence.
Firstly, there are all sorts of things we do
not know about p, and never will know about it; in our observable
universe “nature” itself does not “know” them: nothing in nature depends on the
precise value of p, nor can determine nor represent them.
Let us think of some of those things.
Everything we know about p is in the discipline of formal
mathematics, because the number itself is not physically real.
Firstly, we do not know, and never can know,
the value of p, nor of any particular one of its digits,
not even the first nor the last digit unless it is represented in binary
notation. We do not even know how many digits p has, or whether p
is odd or even. It is extremely unlikely to be prime, but any firm statement
about its being prime is necessarily a conjecture based on the expected
frequency of primes in the neighbourhood of p, for which we do not have any
precise figures anyway. We know at most one of its divisors, namely 1. The
other divisor we can assert is p itself, but then we don’t know the
value of p either, so that is hardly interesting.
We can guess a lot about p, such as
that its digits are about as random as those of pi, but such things are no more
than conjectures. But such conjectures may be such as to give one pause.
Formally we
can deduce some things validly, from the formal nature of number theory. We
know that the process of generating p is deterministic, so that every
integer either is p or is a divisor of p, or it is not, and every
integer except p is not p. Given our algorithm for generating p,
p is unique. We can be sure that p is a unique finite
integer, which implies that p > p‑1, that if we change or
remove or insert or append even any single digit of p, then we get a
different number, not p.
We also know that with negligible
exceptions, p is the smallest of all finite integers.
Next, consider the ratio between the next
larger prime and p: call that ratio "r"; r is a
simple rational number, with all the attributes of rational numbers: for
example we know that r has a repeating decimal expansion. And yet, it is
not physically possible to distinguish r from an irrational number, or
even a non-algebraic number; there simply is not room in our observable
universe to accommodate the first repetition.
This is an example of working with purely finite
numbers, and of the limitations of finite systems working with finite numbers.
In many ways I find them more interesting than infinite numbers.
As I see it, there is no way that p
really exists; it is quite possible that no one has ever discussed this exact
number before, and possibly no one ever will again. No relationship within our
physical universe is described by the exact value of p, and if it could
be, we could not write it down to express the fact. Even in formal
mathematics we cannot distinguish between p and an indefinite set of
other impracticably large finite numbers. We can describe the algorithm
that (though only by magic) formally would generate p, and we can
speak of numbers relative to the notional value of p, such as p+1,
but really, I suspect that we know less about trivially small little p
than we know about aleph null, the cardinal number of the integers.
For example, we know quite a bit about what
we get if we divide aleph null by a finite integer or multiply it by any
integer, or by itself. Or if we raise 2 (or p) to the power aleph null.
But aleph null itself, and in fact any other
infinity, cannot exist in our observable universe except as the implication of
a particular class of formal axiomatic structures.
In other words, magic.
And in my opinion to claim that even such a
small number as p exists, on the basis of formal concepts, is a
meaningless claim. Not to mention much larger integers. For example, instead of
raising each new prime to the power of the starter, we could raise it to the
power of the starter, starter times. Interestingly however, the formal
existence of the class of impracticably large numbers is as real as the formal
existence of say, the finite integers in general.
That was the good news. On a similar basis,
most of the integers smaller than p (those negligible exceptions) do not
exist either. With even fewer negligible exceptions, they all are too large to
fit into our observable universe.
Formally it is likely that one could in
principle calculate numbers that are smaller than a googol, but that happen
never to represent any real event or relationship either: not because they could
not be calculated or written, but because there just never happened to be
any relationship that they describe, that happened to occur in the universe.
The p‑analogue one would get if you replaced "googol" in the
generation algorithm with, say 7 or 13, might be examples.
To claim that such numbers exist because we
can conceive them formally is not cogent. You might as well claim that unicorns
or non‑Goldbach even numbers or odd perfect numbers exist because we can draw
pictures of them or include them in our stories, or claim that snowflakes of
frozen water exist on the sun because we can imagine them.
If we call that existence, then it is
time to re‑examine our semantics.
So, as an conjectural lemma, I propose that
the fact that a concept can arise out of a formal system, even if it is a
convenient fiction in practice, need not imply that it is literally true in
practice. One might speak of “Platonic meta‑existence”, where the idea of
their existence, as opposed to their actual existence, demonstrably exists
at least in our minds, and consequently has real, physical effects in our
empirical world. But that does not imply their independent existence as
amounting to more than the existence of unicorns.
Consider: in principle a zoologist
could contemplate the nature of horses, and of horns, and describe, and
possibly depict, a viable, true‑breeding species that has most of the physical
attributes of fictional unicorns. Even if the head and neck and horn were less
graceful than most of the pictures, no one would look at the result and call it
anything but a unicorn, nor confuse it with an Asian rhinoceros, but we could
hardly claim on that basis, that such a species exists.
Consider again: imagine somewhere in space,
say between our galaxy and Andromeda, somewhere near their common centre of
mass, a diamond of high quality, in free fall, with roughly the mass of the
planet Earth; it could not be solid, because that would create internal
pressures that would destroy the diamond crystal lattice near the centre, but
it could be say, a hollow sphere, a bubble with a diameter many times that of
Earth, and with a wall thickness of say, one kilometre, and with negligible
rotational velocity or similar sources of stress. Except for the difficulty of
imagining any plausible mode of its creation, nothing about such a diamond
demands any violation of any laws of physics; we could calculate all sorts of
things about it in great detail, and there is no shortage of the carbon
necessary for its construction — if we were to create such an object,
the material budget would hardly dent the carbon content of a single dwarf
high-carbon star.
And yet, we are confident that no such
physical object exists. We can justify our opinion in various ways, and on a
basis of common sense reject any claim that it exists outside Plato's cave,
which I reject as unsupported mysticism. It is unfalsifiable at best and
arguably meaningless.
I propose that whatever else a number or any
other abstract mathematical object is, it is information, and not in any imaginary
abstract warehouse, but only where it materialises in some conceptual or
material relationship.
Let us consider some purely formal examples.
We already have considered p, which is a relatively small finite number,
but what about infinite numbers? Infinite numbers could include the lengths of
non‑terminating fractions, say 1/7, which is a repetition of 0.142857. Well,
that certainly cannot be written out fully, because it does not have p
digits, which could never fit into our universe; for most formal purposes it
has aleph null digits, and such a string of digits never could fit anywhere.
However, the information content of the
decimal expansion of 1/7 actually is remarkably low, and we can deduce a
formally exact value from simple arithmetic and we can tell the value of any
digit in the notional expansion, and we can convert the base decimal radix to
another. For example, in base 49 the value of 1/7 is 0.7 precisely.
Which we easily can write as an exact,
obvious value.
Well, consider some different numbers in
decimal notation, such as 0.1010010001 ...
Here we have another number that formally has
a unique definite value, and accordingly if you change any of its digits you
change its value, making it either less or greater. We furthermore can describe
it exactly and succinctly, so its information content is very small. It also is
known that the number is transcendental, and, unusually among transcendental
numbers, we can easily predict the value of any of its digits. However, if the
exact value of that number is known, or whether it has any special attributes
of interest, such as e or pi would have, I do not know of it.
And we certainly cannot improve matters by
changing the base or anything of that kind.
Which seems curious for a number that we can
define so easily, and find the value of each digit so easily.
And yet, we not only cannot write it as an
exact value, but could not fit its exact value into the observable universe.
Not as far as I know, not yet anyway.
Still, compare that value with say e
(2.7182818 ...), or π (3.14159265 ...). Formally these two
numbers have precise values with plenty of well‑understood contexts, but
written in any base not related to their own value, they not only will not fit
precisely into any number of digits, but we cannot simply tell the value of a
particular digit without calculating it.
In spite of their being definable in formal
mathematics, we can fairly say that physically they do not exist.
“But,” you might argue, “I can draw a circle
with a compass, and add its diameter with a straight edge, and thereby
immediately produce two lines with a precise mutual ratio of π.”
Oh no you can’t!
You would be doing very well to get four
digits of precision, and no chance whatsoever of achieving forty digits, let
alone conserving it if you could. And neither four nor forty digits would be
significant in comparison to the actual decimal expansion, which certainly
could not fit into the observable universe, neither as a number in positional
notation, nor as any other representation of the same amount of information.
I claim that in terms of existence, such
numbers no more exist than our spherical diamond bubble exists.
All of which amounts to much ado about not
much?
Possibly, but watch this space.
The
mathematical theory of structure is the answer of modem physics
to a question which has profoundly vexed philosophers.
But, if I never know directly events in the external world,
but only their alleged effects on my brain, and if I never know my brain
except in terms of its alleged effects on my brain,
I can only reiterate in bewilderment my original questions:
"What sort of thing is it that I know?” and ‘‘Where is it?
What sort of thing is it that I know? The answer is structure.
To be quite precise, it is structure of the kind defined and
investigated in the mathematical theory of groups.
It is right that the importance and difficulty of the question should be
emphasised.
But I think that many prominent philosophers, under the impression
that they have set the physicists an insoluble conundrum,
make it an excuse to turn their backs on the external world of physics
and welter in a barren realism which is a negation
of all that physical science has accomplished in unravelling
the complexity of sensory experience.
The mathematical physicist, however, welcomes the question
as one falling especially within his province,
in which his specialised knowledge may be of service
to the general advancement of philosophy.
Arthur
Eddington
So much, I said, for the non‑existence of
numbers or mathematical values of random and indefinite representation. What
about numbers of a more tractable form, such as integers or simple fractions?
It takes very little information to represent say 2, or ½ to indefinite
precision, yes? Say 2.0 and 0.50. Right?
Well no, not really. Not if that precision
really counts.
Consider: π begins
3.14159265358979323846264338327950 .... Usually such precision would be insane
in any material system, and even in most formal contexts, but formally
we know that if we changed that first 0 by adding 1 to it, we would no longer
have π, but a different number. Not just partly different,
but formally speaking, a totally different number, as truly different as
if one had changed every known digit in the number. And the same would apply
equally to adding 1 to the millionth 0 in the decimal expansion of π.
And in that respect of formal definition
of numbers, how does 2.0 or 0.50 differ from π?
They do not differ at all.
If we change their 32nd digit from
a 0 to a 1, then both of them change as inevitably as π did. To
define them absolutely, both of them should in theory be represented as being
expanded into a repeated 0. There also is the option of their being represented
as 1.99 .... and 0.499 ..., but that still makes no difference. A 1 added at
the same position in any representation changes their values by the same
amount.
This is different from the case of physical
objects; our diamond bubble would remain practically indistinguishable from
perfection even with imprecision of the order of tonnes rather than
femtogrammes.
And yet, we go ahead and use the formal
numbers as if they were absolutely correct, and formal maths works in theory
and applied maths works in the face of our use of numbers that do not exist;
what does that tell us?
It tells us all sorts of things, including
the fact that the amount of material we need in our aircraft wings, and how
much fuel we need in the tanks of our vehicles if we wish to have a successful
journey, are based on rough data and rough calculations, not indefinite
precision.
Just suppose that someone said that we needed
the wing to be able to take a loading of 159.265358979 tonnes, such precision
would imply a load accurate to roughly one billionth of a tonne. It would mean
that another milligram one way or another could make all the difference between
success or failure, or a waste of material that would be unacceptable even
though it happened to be too small for anyone to feel it in his hand.
Which is of course nonsensical.
In practice we do not, and cannot, and would
not want to, make anything exactly big enough and strong enough and no more. We
rarely make it even as little as 50% stronger than we expect the demands to
require, and engineers often make things several hundred percent stronger than
is formally necessary. Oliver Wendell Holmes was only joking when he wrote of his
“wonderful one‑hoss shay, that was built in such a logical way, it ran a
hundred years to a day” and then collapsed into a pile of dust when its period
of duty had expired.
Similarly, when we measure something, it is
seldom that we measure to an effective precision of more than four decimal
places, and I believe that the record is about 14 decimal places. One part in
about one hundred million million.
Pathetic.
But suppose we could get it down to beyond
Planck’s constant, say to fifty places;. wouldn’t that be exact?
Not compared to a million places, and
certainly not to a formally infinite number of places.
Our universe decidedly does not work that
way.
Everything is made of particles in some
sense, and to measure beyond the finest‑grained limits that atomic (“non‑splittable”)
particles permit, just will not happen.
Now, by the time that anyone with any idea of
quantum mechanics (QM) has read as far as this, he no doubt will be seething at
such nonsense, because QM simply does not work that way either.
Yes, yes, relax. I know. But I have a
deeper object in view, so I largely ignore QM in this essay. For the most part
the “atoms” I consider are fictional: rigidish, roundish particles.
But there is one item of quantum theory that is
too good to ignore. We cannot specify any real value to infinite precision, can
we?
Oh so? Then what about 0?
Sorry, no luck. 0=0.000 .... Change any
digit and you have a different number.
Right? Then how can we have empty space, with
0 mass and reality at any point?
A very logical question, and very correct
too: and as it happens, we cannot.
And we do not.
Every Planck volume in space, as nearly
as we can get to characterising it, keeps oscillating about the presence and
absence of a particle or charge or mass‑energy effect, and the oscillation has
all sorts of physical implications — real physical consequences, no
unicorns required.
For example, from the most perfect vacuum
that we could get, or that we could find in space, we get vacuum noise, "vacuum
fluctuation" or "quantum fluctuation", that can be
physically measured electronically, and that has various established effects.
There is nothing magic or imaginary about that.
For another example, we get black hole
radiation (read Stephen Hawking’s “A Brief History of time”. if you want more
detail).
Another effect of the consequences of vacuum
fluctuations is that if you strip say, a uranium atom, of all its electrons,
which would give you a U+92 ion, the local space‑time concentration
of positive charge around that uranium nucleus is so intense that it soon
strips electrons from adjacent vacuum fluctuations, and emits the matching
positrons that necessarily maintained the overall neutral charge. The positrons
soon combine with ambient electrons and vanish in the form of gamma rays.
Meanwhile, as the uranium orbitals fill up, the ionic charge reduces, so that
say, a U+6 ion is reasonably stable, practically unable to snatch
vacuum electrons.
Some people even suggest that the Big Bang
that resulted in our observable universe, started out as a vacuum fluctuation,
but I cannot comment on that and it is a bit off topic, so I’ll pass it by.
Whether you regard such things as frightening
or beautiful or both, is up to you.
And whether it helps you in wondering about
why there is something instead of nothing, is also up to you.
And so is wondering about what "nothing"
might be.
Euclid’s and Others’ Excesses
The point is there ain't no point.
Cormac McCarthy
This will be a frustrating section for
everybody: anyone knowing basic mathematics will be grinding his teeth at
obviosities, and anyone else will be wondering what it is all about. For anyone
who would like to know more, read Wikipedia’s article: Interval
(mathematics). It is a good one and will tell you far more than I will be
dealing with.
Now let’s get back to Euclid.
Euclid was one of the pioneers of formal mathematics, though I doubt he
realised it at the time. There is a tendency to sneer at him nowadays, but I
count him as a heroic genius of classical days. He posited all sorts of things
that he called axioms, propositions and what not, but as I see it, in modern
terms they all can be classed most reasonably as axioms and theorems. Exactly
which axioms one chooses for given purposes depends on the formal disciplines
one is working in (and applied disciplines too, where appropriate).
Now, much of the Euclidean style of thought
is very basic to much of our current everyday science and technology. It is
simple, convenient, and for practical purposes, generally close enough for
jazz. Much as a map for navigation of any small region of a planet can
cheerfully be based on the implicit assumption that the Earth is flat, so we
can assume that a given position on the map is a point and a given path is a line.
However, for some fields of study, not only are such views inadequate, but even
where they are adequate, they are not even metaphors for reality,
but just rough pictures, placeholders if you like, or analogies.
For example if I am illustrating the theorem
of Pythagoras and draw freehand lines with chalk on a board, or in sand on a
beach, or in pencil on paper, then the "points" I mark are not
points, and the lines, apart from not being straight, are not lines at all. A
point in physical reality is not at all a point in any sense that Euclid used. It is not
even an infinitesimal, as we encounter in calculus. Nothing in everyday physics
can possibly be anything of the kind
Consider: a formally true, genuine point
according to Euclid, a Euclidean point, has a measure of zero in all
dimensions except time (such as the point is, it lasts for some time, so, by
that very fact, any true point has a world line, or, depending on the frames of
reference of notional observers, has indefinitely many world lines or world
volumes).
As far as I know, Euclid never thought in such terms as time
being a dimension, nor of the world line of a point, but, also as far as I
know, he never contradicted them either. Still, the Euclidean point
unambiguously and explicitly has zero length, zero breadth, and zero
height. Zero measure in every way but time. Not just small, not tiny, not
microscopic, not negative, fuzzy, or hollow either, but absolutely, literally,
and precisely zero. No more, no less. No digit in its expansion
to any base can be anything but 0, either in practice or theory.
The definition of a Euclidean point is not
without ambiguity. In most of mathematics, including, as far as I know,
Euclidean geometry, there is no definition of a point or a line. Some works do
use a rather hand‑waving definition that I have relied on for as long as I can
remember: a point is a position. To put that differently, possibly more
precisely, though still informally: as an entity a Euclidean point cannot
comprise anything more than coordinates in whatever happens to be
the relevant number of dimensions. And only one infinitely precise coordinate
in each dimension.
If this conflicts with any particular usage,
then that usage is an arbitrary deviation and we are no longer thinking in
terms of what Euclid’s
ideas were.
And after all, Euclid got there before we did.
I am not sure where that leaves us with
fractals, but you can't have everything. If you could, then, as Steven
Wright said about having "everything": "Where would you put
it?"
Similarly, a line is defined as having
precisely zero measure except in length (and of course time), and a plane
has zero thickness.
Now, all that sounds pretty simple, but some
people tend to miss some of the implications.
First, think of a line, and assume for
convenience that it is straight.
Euclid assumes that it is possible to select an arbitrary point on that
line, say by putting the point of your compass on it. The way that you can
recognise that point is by its coordinate on that line.
So select your point, and label the
coordinate of that point 0. (After all, why not 0? It's a nice number!)
Then select a different point on the
same line (meaning that it has a coordinate different from 0, on that same
line). Call that newly selected point 1.
We call the line segment or interval
from point 0 to point 1, a closed interval, meaning that
it includes the two end points (0 and 1 in this case) and also includes all
and only the points in between 0 and 1. In particular it includes no
points outside those end points.
Furthermore, every coordinate along the line
matches exactly one single, unique point. No point lacks a coordinate,
or needs more than one coordinate to define it, for each of the number of
dimensions of the space occupied by the points in question. For example, a line
being 1‑dimensional, only one coordinate is required per point. In a plane (2‑dimensional)
we would need two coordinates to define or label any point.
Now imagine that we delete just points 0 and
1, and no other points, from our chosen line segment. This leaves
an open interval: it is called “open” because it has no
first and no last point to cap either end.
This applies in any field of study in which a
point in a continuum is a concept. In some disciplines that do not deal with
continua, such as versions of lattice theory or of finite sets of discrete
elements, in which isolated points are defined, it is not relevant, but I am
dealing with continua only, for now anyway.
But, one feels, that is nonsense anyway,
because having removed those two points, we have neither added nor removed any
other points that had been in the line before we had defined our 0‑to‑1
interval, and what is more, there had been nowhere along the line without its
unique point. So the next point along must now be the last or first point on
the interval in the place of the previously removed point.
Yes?
That sounds very sensible, but it does not
work out. After all, what was the length of the points you removed?
Zero. Points have zero length. Not nearly
zero: not more nor less than 0: exactly zero. Remember? Otherwise they are not
points.
Well then, if we had contemplated a line
segment say, 3 units long, then the coordinates of the end points would have
differed by 3 units, including the lengths of the two end points; the length of
the segment would be that of the line (3 units) plus that of the two end
points, giving 0+3+0, which adds up to 3 units exactly, not even a proton
diameter out. But if two points are next to each other, with 0 length of
segment between, then the length that they span is 0+0+0, which still adds up
to 0; by simple arithmetic their coordinates differ by 0: the length of one
point.
So the coordinates differ by zero. (In formal
mathematics they call zero the additive identity: adding the additive
identity to anything, including adding zero to zero, gives what you started
with; it does not change anything. That is exactly why they call it the
additive identity.) But it follows that if the coordinate of point A differs by
zero from the coordinate of point C then they have the same coordinate. But a
point has nothing except its coordinates, so any number of references to
exactly the same coordinates are references to exactly the same point, and no
other point, though possibly under different descriptions, say for example:
"two" and "the even prime" and "the cube root of
8" would be the same coordinate.
In other words, those coordinates define, not
nearly the same point, not close, not next to each other, but precisely
the same point: in talking of points, “next to” makes no sense unless it means
the same point, which in most senses makes very little sense indeed.
So it also makes no sense to speak of the
first or last point of an open interval.
There is nothing new or obscure about this;
it certainly never was my invention.
Well then, let us speak of two points with different
coordinates. "Different coordinates" means "not in the same
place, but a greater‑than‑zero distance apart, even along a straight
line". Somewhere in those two points’ coordinates, at least one digit in
one of the coordinates will differ from the digit in the matching position of
the other coordinate. Call the differing points A and C. No matter how close
they might be, it always is possible to name another point halfway between: add
their coordinates together, and divide by two. That gives a new coordinate: say
coordinate B. If A is at coordinate 0 and C at coordinate 1, then B would be at
0.5 and if you repeat the operation, the new point would be at 0.25, then if we
repeat the process, at 0.125 then 0.0625, 0.03125, and so on.
Very important: notice that at each halving
the coordinate grows longer, requires more digits, or changes in digits. This
is an example of the need for more information in finding
something smaller and smaller.
Imagine you were looking for someone in say,
the United States:
well, it is easy to find the country: it takes possibly eight bits of
information to pick that country out of a list. But that person: where to find
him at home in the whole of the US?
Not very helpful! Narrow it down by telling us the state. It takes us perhaps
six more bits of information to pick a state from a list: say California;
California is smaller than the United States
and the fourteen bits are more helpful than those first eight. Well, California still is a
bit big. How about the town? Say another eight bits? How about the street?
Possibly ten more bits .... and so on.
And suppose that instead of looking for that
person, you were looking for a particular freckle somewhere on his skin. That
might require another 20 bits or so.
But in these material examples, we always
have a final non‑zero target, a final turtle to aim for in the stack. Looking
for a final coordinate along a formally Euclidean line however, we keep getting
smaller and smaller line intervals, but because the length of each of the
points along that line is zero, this means that a short line contains
exactly as many points as a long line. No more, no fewer.
This is something that is easy to demonstrate
in even the most naïve Euclidean geometry.
It follows that formally there is no final
turtle in specifying something smaller and smaller. One keeps having to add
at least one more digit of information to specify it, to address it. The
information doesn’t have to be binary; we get the same effect if we chose each
new point one tenth of a unit away from the previous point instead of one half
of a unit. Information is information, no matter what the medium or notation might
be. We need not even be using information expressed in digits at all, but we
still need the same amount of information to locate anything sufficiently small
in a sufficiently large space.
We might use a ruler to identify the segment
when we begin searching for a particular line segment, but looking for smaller
line segments, soon that would no longer suffice and you would need say, a
Vernier calliper, then a micrometer, then a microscope.
Startlingly soon, no possible instrument
could be powerful enough to supply all the information you would need to find
really small segments.
And apart from addressing segments,
ultimately to address a specific point would take infinite
information.
In no observable universe is there room for
infinite information.
You might argue that the example is
artificial. In looking for little things we don’t always need to write out
long, long strings of numbers.
No, that certainly is true, but you certainly
do need the necessary information one way or another. Information is
information, no matter how you convert it.
But we select points on lines every day
without all that fuss! Think of a construction on a chalkboard!
You do, do you? Fat chance!
First of all, a chalk line isn’t a line at
all; it is a layer of powder with fuzzy width, thickness, length, and even
mass. A chalk line is about as real as a mountain range, and about as hard to
map. In fact, if you look carefully, you even can see a fringe of separate
particles around the so‑called chalk line, like boulders around a mountain.
Some “line”!
Well then, let’s use a diamond point or a
laser to score a microscopically invisible line on a polished diamond surface.
We would have to use an atomic force microscope to see it.
Sorry, not categorically better. Even if your
invisible scratch shifted only a single row of carbon atoms, that row still
would be tens of picometres wide and deep, or high, depending on how you worked
it. Formally a picometre‑long line contains as many points as an exametre‑long
line. Never mind carbon, even a single hydrogen atom is some 100 picometres in
diameter, and you are no closer to selecting a specific point even across
your "line", never mind along its length. The point you thought you
selected is no point at all; it is a great big smudge or heap or hollow.
And notice that we are not even considering
quantum effects here: we are behaving as though atoms were neat, clearly
defined spheres. We are ignoring practical problems such as the floppiness of
material on microscopic scales, or their constant stretching or shrinking or
vibration in response to changes in pressure, Brownian movement, or
temperature.
In other words we are assuming magic,
and even the magic isn’t helping us here.
All right, so who cares? What possible
practical difference could that make in our lives? So the stripe I draw isn’t a
line and the dot I choose isn’t a point. So the real point would require
infinite information, but I only get a few bits worth of information, so what?
Aren’t a few bits all I need?
Not necessarily. The point is that, no matter
how valid his assumptions might have been in formal terms, in physical terms
Euclid was wrong about being able to choose a point on a line (no point
makes sense, and neither does any line make sense). And if we could choose a
single point magically, we never could get back to that same point again
without more magic. Watch to see how that destroys the very concept of
determinism.
As we shall see, this arises from confusion
of formal with applied, empirical principles — everyday reality.
Where we use mathematics to deal with
physical realities, we perpetrate a fiction. We pretend for the sake of
convenience that the entities we work with are mathematical objects, whereas
they are not; the point that we make with our pencil on the paper is a pile of
graphite or similar pigment. To be useful to us it must behave sufficiently
similarly to the behaviour of the mathematical point. We speak as though we
were dealing with an isomorphism, which it is not; it is a plesiomorph,
something sufficiently nearly of the same logical form, to suit particular
needs; for example, I calculate where the centre of a circle is on my drawing
board, or construct it with straight edge and compass. Then I mark it with a
fine pencil. Is that mark the centre? Mathematically no. The centre is a point,
not a pile of graphite. Is it an approximation? No. An approximation is a
figure that approaches to a desired result with desired precision — so many
decimal digits. The spot of graphite (if you worked carefully enough) may cover
the mathematical centre of your notional circle, but it is a picture, a
fiction, much as the drawn circle also is a fiction. The pencilled dot has
area, has volume. But as a fiction, a plesiomorphism, it commonly is good
enough for drafting purposes.
In the set of natural numbers or ordinal
numbers one can arrange things to be physically meaningful, because one only
needs enough information to distinguish one value from another, even if the
values barely exceed Planck dimensions, but as soon as one deals with a
continuum, such as a line, area, or volume, there is nothing that can
physically identify a point as distinct from every different point, because the
required information is infinite.
Synergy
is the only word in our language that means behavior of whole systems
unpredicted by the separately observed behaviors of any of the system’s
separate parts
or any subassembly of the system’s parts.
There is nothing in the chemistry of a toenail that predicts the existence of a
human being.
Richard Buckminster Fuller
Let us magically construct a
laboratory. It is isolated from all vibrations or sources of noise. Except for
precisely vertical weight, all gravitational effects, including tidal effects,
including the tidal forces of experimenters’ heartbeats, are neutralised by
precisely distributed masses or whatever might be most appropriate in
principle.
In a perfect vacuum in one chamber we have a
perfectly hemispherical, massive, rigid body, immune to abrasion, vibration,
dust, and impact. Suspended exactly above its centre in terms of the local
gravitational field we have a perfectly spherical and symmetrical ball of
similar material. Around the chamber the masses we have distributed would
neutralise any gravitational attractions other than directly up and down.
Remember, this is magic; we could not
do anything of the kind in practice!
Furthermore, we ignore our inability to make
a perfectly spherical ball in physical fact, because all real materials of
visible sizes, are made from atoms, and that fact forces them to be submicroscopically
lumpy.
Now, we release the suspended ball, and it
hits the convex surface below and bounces vertically above the unique Euclidean
point where the tangent plane above the lower ball and below the bouncing ball
is exactly horizontal. According to Newton’s
F=ma it cannot do anything else, because anything else would require some
sideways force to make it strike anywhere else. And it will keep bouncing up
and down on the spot indefinitely till it runs out of restitution, after which
it will remain balanced stationary on top of the lower ball.
Right?
In fact we could get swanky with our magic,
and could equally well balance a vertical stack of rigid, frictionless balls
above each other without their rolling off — if they all were of the
same mass, the effect would be the same, with only the top ball bouncing. If
you find such ideas amusing, you might like to imagine what would happen if
their masses differed, or if you bounced more than one ball at a time in the
same stack. Newton's
cradles would pale in comparison to our virtuosity.
Of course, anyone with a knowledge of Quantum
Theory will be muttering about Planck’s constant and Brownian motion and so on,
but remember that we and our magic can afford to ignore those.
All the same, magic or no magic,
some facts remain. Mathematically speaking, when one ideally rigid ideal ball ideally
free of horizontal forces, rests or bounces on another ideally spherical
surface, there is precisely one Euclidean point at which it can balance
without rolling off or bouncing away. No matter how little it wanders from that
point, or in which direction, it will on every subsequent bounce wander further
and further and faster and faster in the same direction away from that point.
In real life, you will be doing well if you can get a steel ball to bounce on a
lower ball even twice, let alone three times. To get it to bounce repeatedly
and come to rest on the lower ball, is literally (not figuratively)
impossible without interference, even with just two balls, one above and one
below.
As for a vertical stack of multiple balls.
What is more, the direction in which
the bounces finally would diverge, would be genuinely random if the experiment
as described is competent and honest, with even a little magic.
Consider: we said we had managed it by magic.
For the first bounce at least. This means that when it comes back it must hit
that same point exactly. Exactly, not approximately, meaning precisely,
mathematically, zero deviation.
But if we only assumed magic for the first
bounce, then by now we have run out of magic, and we would need more
magic to hit that Euclidean point again, to hit it even once more,
because we also know that to hit any one point with another point, we need infinite
information.
For which our observable universe has no
capacity, remember? Not without magic anyway.
So on the first return the impact will be off
centre by anywhere from zero to some small fraction of an atomic radius.
Exactly how far and in which direction we cannot say, because we would need
more information to make any slight guess with anything better than zero
justification.
The very fact that the information is lacking
(it is in fact non‑existent in any meaningful sense) means that when
the path of the ball deviates, the deviation must be random, because non‑randomness
in any respect would imply matching information. Given any relevant
information, it becomes possible in principle to predict the outcome to some
matching, but limited, degree of precision.
That is why precise selection of any unique
point is not possible even in a notionally non‑quantum, non‑atomic world: not
possible to humans, and not possible to whatever amounts to the nature of our
world.
Of course this whole exercise is a drastic
over‑simplification, because in real life there are too many sources of noise,
some notionally (meaning “not really”) deterministic, most not even that. That
is part of the reason that we need magic for our exercise, though it is
not the fundamental reason,
The “observer effect” in Quantum Mechanics (which
from the very first I heard of it as a youngster, I always have been convinced
has nothing whatsoever to do with anything like consciousness in an observer)
is only one example of noise. The atomic nature of matter, with the Brownian
motion of particles, together with the granular nature that it implies, is a
related example. Not even to mention vacuum fluctuations.
But for reasons that I hope to establish, I
ignore such things even though in practice they dwarf the effects that I
discuss.
The conceptual effects of the principle are
dramatic, even though they are not easily observable in practice. More than two
centuries ago Laplace showed that a literal
application of Newtonian mechanics implied that every motion in nature,
including all the motions in the entire universe, would in principle be
perfectly reversible, so that if we magically reversed the momentum of
every particle in the universe, time would in effect begin to run in reverse
from that instant. This is of course something of a parody, because it does not
allow for various forms of hysteresis, phase changes and symmetry breaking, but
there are other, more fundamental considerations as well, some of which I shall
discuss.
Remember in particular the impossibility of
specifying a formal mathematical point:. it would require infinite information
just to determine a future straight‑line trajectory of any particle, never mind
its magically reversed trajectory. Secondly its magically
reversed trajectory would not be the reverse of its originally forward
trajectory, because that too would equally necessarily require infinite
information.
All this repeatedly implies that the formal,
classically Euclidean and Newtonian predictions of trajectories are no more
genuinely representative of real trajectories than the lines we draw on a paper
are representative of Euclidean lines with zero thickness and selectable points
of zero length. They are not even metaphors, arguably not even abstractions;
they amount to impressionistic pictures, or at best maps.
And proverbially, the map is not the
territory.
One of the consequences of all this so far,
is that references are not perfectible, and therefore meanings are not
perfectible either; they depend on references: ‑ we cannot identify anything
perfectly because perception is mechanism; we cannot conclude explanations; and
meanings can only be as secure as teaching or learning permits.
Also, the quest for fundamentals is grounded
in the process of teaching and learning. We learn QM from observations of the
real world and to the extent that we can, we explain the real world, largely in
terms of QM. Similarly, we try to derive our conceptions of the fundamentals of
the real world from our observations and then explain the real world in terms
of those conceptions of the fundamentals. Such a process cannot formally
demonstrate indisputable fundamentals ‑they can only approach fundamentality to
the extent that they are the best and most effective that we have.
From our observations of the world and our
interpretations and preconceptions, we arrive at impressions and notions of
determinism and probability; we can think of them as patterns, and we hold to
these notions and patterns for as long as they remain our best working
hypotheses. The idea that the world actually is deterministic or probabilistic
is an intellectual convenience. It is as close as we have gotten so far, to
perceiving the world as it actually is, always assuming that it actually is
anything that we actually can perceive. This is as good as it gets, as far as
we have been able to tell so far.
The
Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.
Omar Khayyam
By now I think I have said more than enough
about Brownian motion, QM, vacuum fluctuations, and finite information, to
dispose of the idea of true determinism. altogether.
However, the way the world works entails a
good deal more than determinism. Certainly our world has chaotic aspects, to
the extent of justifying “chaos theory” as a special branch of science; all the
same, the way it works brings forth a great deal of order as well. There is
enough of such order to support impressively useful practical predictions. In
particular, in spite of the howls of certain holists, even though we would need
to know everything about the universe to know everything about anything at all,
we do get on fairly well, just knowing very little indeed about anything at all.
We can plan, we can build, we can aim, we can
adjust and correct, and we can achieve results that generally satisfy us as
being what we were trying to achieve, whether it was to build the skyscraper,
or pot the billiard ball or hit the bullseye or toss the die. Not everything
requires infinite precision; ten or twenty bits will suffice for most everyday
purposes. For golfing and fishing stories even ten bits will usually be
excessive.
Now how does this come about? People speak
airily about "approximation" as if the word explained everything.
Actually, apart from not explaining anything, the word commonly is not even
used correctly at all. But what is it about our world that permits vaguely
causal results to have even vaguely satisfactory outcomes and successful
predictions?
Consider a billiard ball hanging peacefully
in its pocket. That was the state we had aimed for, was it not? Aiming
carefully with the cue at another ball, we had caused it to follow a planned
path so that it struck this ball that in turn followed a planned path that
deposited it into that pocket and nowhere else.
This is not easy to argue consistently, but
our difficulty arises from our simplistic view of the problem. As a rule we
assume that there are two possible states: ball‑in‑correct‑pocket, and ball‑not‑in‑correct‑pocket.
The fact is that "a billiard ball in the pocket" is not a well‑defined
unique situation, any more than a dead (as opposed to a live) cat in a box is a
well‑defined unique situation, or even class of situation. Physically there are
indeterminately many distinct ways for the ball to be in the pocket (such as
which way up it lies, and which fibres of the pocket it is in contact with) and
even more distinct states for it to pass through in getting into the pocket. Any
of these would suit the ball‑in‑pocket case, but they form a set of
acceptable outcomes, not a definable unique case. They differ from a
spin‑up or spin‑down electron in a given orbital of a given, isolated atom in a
given location. We could regard the spin as a binary case — a case of
precisely two possibilities. Even that is simplistic, but it is close enough,
compared to the macroscopic cases that involve tens to the tens of particles in
particular states.
All the same, there are indefinitely more ways
for the ball to miss the pocket than to land in the pocket. Otherwise there
would be little point to such games at all.
And the amount of information required to pot
the ball, is related to how many times more ways it can go elsewhere than into
the pocket, rather than how many ways it could rest in the pocket after a
successful shot.
It might be easier to visualise instead a
game of darts in which the winner is the first to put a dart into the inner
bullseye. Again there is no simple limit to how many states there are that
comprise a winning throw, but there are many times more ways to perform a
losing throw than a winning throw. And the more sound information one can apply
to directing the throw, the smaller the probability of a losing throw.
Now, the control of the transition from one
state to another depends on the available information and energy. Energy we can
ignore in this toy exercise, but information is of the essence. The required
choice is any one of the winning states. If there were no information available
to direct the shot, then the probable outcome would be a losing state, and in
most games there would be many more losing than winning states.
However, an effective player generally can
apply some information to direct his shot, and the result would be to bias his
result towards a winning shot. Let us suppose that he needs one bit of
information to hit the board at all; then the chances are equal for any point
on the dartboard, with say, roughly one throw in 4000 hitting the bullseye. But
then suppose that a few bits more would improve the probability of getting near
the centre, giving a normal curve centred on the bullseye. Immediately the
bullseye becomes the area with the highest frequency of hits of any similar
size of area on the board, even though an actual bullseye hit still
would be fairly rare.
The more bits of information that affect the
precision of the shot, the steeper the normal curve and the sharper its peak.
My description and assumptions are too vague for detailed prediction, but
depending on the physical details, somewhere near ten or twelve bits should be
enough to put most of the darts in the bullseye.
Now, for most purposes that sounds very
persuasive, and we count each dart as landing on a particular point, preferably
in a the bullseye, ideally a circle of about four millimetres in diameter, but
the reality is nothing of the type. The "point" is actually about two
millimetres across; the player is trying to get a rough circle where the point
lands, to overlap a larger rough circle of the bull’s-eye by a sufficient
margin, and for the dart to strike at an effective angle with effective force.
When there are a large number of possible ways for the dart to strike, then the
smaller the proportion of acceptable numbers of ways for the dart to land, the
more information one needs for an encouraging chance of a winning throw.
In some mathematical descriptions we speak of
the set of values that might be acceptable or unacceptable (or of any other
relevant parameter) as the "spaces" that overlap or not, as the case
might be.
An old joke is that of a desirable girl in
the middle of a large floor. An engineer and a mathematician are offered the
opportunity to approach her, but each minute they must just halve the distance
between themselves and the girl. The mathematician indignantly refuses, because
any fool can tell that 0.5 to the power n never reaches 0. The engineer accepts
eagerly because after a few minutes he will be close enough "for all
practical purposes".
Ilya Prigogine was one of the most cogent
exponents of the concept of an arrow of time. I have repeatedly been taken
aback to find how seldom his name crops up in such discussions. Granted, he did
not have much to say about clocks in particular, but he did show the nature of
irreversibility in physics, which intrinsically implies an arrow of time.
Add the tocks and you get the clocks.
Now, one version of the tocks is the ubiquity
of events (a tendentious remark, given that some definitions of
"event" are a lot more constrained, but I use the term here. to mean
anything distinguishable that can result in a change in entropy). In other
words, wherever anything can happen, time "flows",
"passes".
But even in "empty space" vacuum
fluctuations happen. They might (or might not) involve entropy, but I speculate
that their happening could be enough to make time pass even in an otherwise
empty universe. Time does not need a clock to pass, any more than a river needs
a flow meter to flow.
And even if vacuum fluctuations won't cut the
mustard, in a universe as full as ours, there could be enough happening in the
universal increase of entropy to keep time on the go. Anywhere that the
physical effects of an event could be experienced as information, time would
pass, whether quantified or quantised or not.
Like the quantum physicists, I am as
exercised as ever about how the time of quantum mechanics can be reconciled
with the notion of time as the fourth dimension in Einstein’s general theory of
relativity, but I'll watch this space.
Remark
At the following link there is an article on
a topic related to this one. It has a long string of comments at the end,
including several of my own:
https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/
