Bruno Marchal wrote:
> Le 30-oct.-06, à 00:40, David Nyman wrote (to Peter Jones (1Z)):
>
> > Name your
> > turtle. Can't we just get on with investigating what either theory
> > explains or predicts, and stop arguing over words - isn't this why no
> > agreement is ever reached on this?
>
>
>
> Peter, I think that David is right. We are in a loop. On the FOR list
> we would have been moderated out a long time ago :). Tell us your
> theory please.
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
Well, I think numbers don't exist AT ALL....
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Metaphysics of Mathematics
* Introduction: Why Mathematics Works
* Six approaches to the Philosophy of Mathematics
*
o Approach 1: Empiricism: The "maths is physics" theory.
o Approach 2: Platonism: Objectivity and objects.
o Approach 3: Psychologism.
o Approach 4: Formalism: Mathematics as a game.
o Approach 5: Constructivism: How Real Are The Real Numbers?
o Approach 6: Quasi Empiricism
* More on Platonism
*
o Penrose on Platonism
o The Source of Mathematical Certainty
o Platonism and Causation
* More on Formalism
*
o Formalism and Godel
Why Mathematics Works
1. Mathematics is the theoretical exploration of every possible kind
of abstract structure
2. The world has some kind of structure; it is not chaos or
featureless or void
3. Therefore, it is likely that at least some of the structures
discovered by mathematicians are applicable to the world
(It would be strange if everything mathematicans came up with was
empirically applicable, and might tend one towards Platonism or
Rationalism, but this is not the case. It might be possible for
mathematicians working completely abstractly and theoretically to fail
to include any empirically useful structures in the finite list of
structures so far achieved, but maths is not divorced from empiricism
and practicallity to that extent -- what I say in (1) is something of
an idealisation).
Six Approaches to the Philosophy of Mathematics
* Approach 1: Empiricism: The "maths is physics" theory.
* Approach 2: Platonism: Objectivity and objects.
* Approach 3: Psychologism.
* Approach 4: Formalism: Mathematics as a game.
* Approach 5: Constructivism: How Real Are The Real Numbers?
* Approach 6: Quasi Empiricism
Approach 1: Empiricism: The "maths is physics" theory.
Mathematical empiricism is undermined by developments since the 19th
century, of forms of mathematics with no obvious physical application,
such an non-Euclidean geometry. I should probably say no obvious
application at the time since non-Euclidean geometery -- or curved
space -- was utilised in Einstein's General Theory of Relativity
subsequent to its development by mathematicians. So M.E. has two
problems: the existence of mathematical structures with no (currently)
obvious physical application, and the fact that the physical
applicability of different areas of mathematics varies with time,
depending on discoveries in physics. Todays mathematical game-playing
may be tomorrow's hard reality. It is also undermined by mathematical
method, the fact that maths is a chalk-and-blackboard (or just
thinking) activity, not a laboratory activity.
Approach 2: Platonism: Objectivity and objects.
Both Platonism and Empiricism share the assumption that mathematical
symbols refer to objects. (And some people feel they have to believe in
Empiricism simply because Platonism is so unacceptable). Platonism gets
its force from noting the robustness and fixity of mathematical truths,
which are often described as "eternal". The reasoning seems to be that
if the truth of a statement is fixed, it must be fixed by something
external to itself. In other words, mathematical truths msut be
discovered, because if they were made they could have be made
differently, and so would not be fixed and eternal. But there is no
reason to think that these two metaphors --"discovering" and "making"--
are the only options. Perhaps the modus operandi of mathematics is
unique; perhaps it combines the fixed objectivity of discovering a
physical fact about the external world whilst being nonetheless an
internal, non-empirical activity. The Platonic thesis seems more
obvious than it should because of an ambiguity in the word "objective".
Objective truths amy be truths about real-world objects. Objective
truths may also be truths that do not depend on the whims or
preferences of the speaker (unlike statements about the best movie of
flavour of ice-cream). Statements that are objective in the first sense
tend to be objective in the second sense, but that does not mean that
all statements that are objective in the second sense need be objective
in the first sense. They may fail to depend on individual whims and
preferences without depending on anything external to the mind.
Approach3: Formalism: Mathematics as a game.
Both Platonism and Empriricism share the assumption that mathematical
symbols refer to objects. An alternative to both is the theory that
they do not refer at all: this theory is called formalism. For the
formalist, mathematical truths are fixed by the rules of mathematics,
not by external objects. But what fixes the rules of mathematics ?
Formalism suggests that mathematics is a meaningless game, and the
rules can be defined any way we like. Yet mathematicians in practice
are careful about the selection of axioms, not arbitrary. So do the
rules and axioms of mathematics mean anything or not ?
The reader may or may not have noticed that I have been talking about
mathematical symbols "referring" to things rather than "meaning"
things. This eliptically refers to a distinction between two different
kinds or shades of meaning made by Frege. "Reference" is the
external-world object a symbol is "about". "Sense" is the kind of
meaning a symbol has even if does not have a reference. Thus statements
about unicorns or the bald King of France have Sense but not Reference.
Thus it is possible for mathematical statments to have a sense, and
therefore a meaning, beyond the formal rules and defintions, but
stopping short of external objects (referents), whether physical or
Platonic. This position retains the negative claim of Formalism, that
mathematical symbols don't refer to objects, and thus avoids the
pitfalls of both Platonism and Empiricism. Howeverm it allows that
mathematical symbols can have meanings of an in-the-head kind and thus
explains the non-arbitrary nature of the choice of axioms; they are not
arbitrary because they must correspond to the mathematician's intuition
-- her "sense" -- of what a real number or a set is.
So far we have been assuming that the same answer must apply uniformly
to all mathematical statmentents and symbols: they all refer or none
do. There is a fourth option: divide and conquer -- some refer and
others don't.
Approach 4: Psychologism: Nubers are Concepts
Psychologism is the view that mathematical propositions are "about"
ideas. (As opposed to physical objects, or Platonic objects) The
counterargument is that there are inifinities in maths, but we do not
have infinite numbers of ideas.
All of these theories assume mathematical statements must refer to
something, and all can be undercut by the hypothesis that mathematical
statements can get by purely on "sense" and do not need "reference" at
all (the sense/reference distinction is Frege's). So the
Counter-counter argument is: we can have an idea of infinity without
having an infinite number if ideas.
"A fourth and final version of the third strategy, developed by
Balaguer (1995, 1998) (and see Linsky and Zalta (1995) for a related
view), is based upon the adoption of a particular version of platonism
that can be called plenitudinous platonism, or as Balaguer calls it,
full-blooded platonism (FBP). FBP can be intuitively but sloppily
expressed with the slogan, 'All possible mathematical objects
exist'; more precisely, the view is that all the mathematical objects
that possibly could exist actually do exist, or that there exist as
many mathematical objects as there possibly could. Balaguer argues that
if platonists endorse FBP, then they can solve the epistemological
problem with their view without positing any sort of
information-transferring contact between human beings and abstract
objects. For since FBP says that all possible mathematical objects
exist, it follows that if FBP is true, then every purely mathematical
theory that could possibly be true (i.e., that's internally consistent)
accurately describes some collection of actually existing mathematical
objects. Thus, it follows from FBP that in order to attain knowledge of
abstract mathematical objects, all we have to do is come up with an
internally consistent purely mathematical theory (and know that it is
consistent). But it seems clear that (i) we humans are capable of
formulating internally consistent mathematical theories (and of knowing
that they're internally consistent), and (ii) being able to do this
does not require us to have any sort of information-transferring
contact with the abstract objects that the theories in question are
about. Thus, if this is right, then the epistemological problem with
platonism has been solved."
http://plato.stanford.edu/entries/platonism/#2
Not really, because there is now no role for Platonic objects. They are
not involved in the actual process of verifying the truth of
mathematical claims. A daemon could wish Platonia and all its contents
away, and nothing would change.
Approach 5: Constructivism: How Real Are The Real Numbers?
Mathematical Platonism is the view that mathematical objects exist in
some sense (not necessarily the sense that physical objects exist),
irrespective of our ability to display or prove them.
Constructivism is an opposing view, according to which only objects
which can be explicitly shown or proven exist.
Some constructivists believe in constructivism as an end in itself.
Others use it as a means to an end,namely the elimination of
infinities.
Real numbers (numbers which cannot be expressed in a finite number of
digits) are a particular problem for both kinds of constructivist. For
the first, because most real numbers cannot be shown. Some are well
known, such as pi and e (the base of natural logarithms). These can be
generated by an algorithm; a sort of mathematical machine, where if you
keep turning the handle, it keeps churning out digits. Of course, real
numbers are not a finite string of digits, so the process never
finishes. Nonetheless, the fact that the algorithm itself is finite
gives us a handle on the real number, an ability to grasp it. However,
the real number line is dense, so between any of these well known real
numbers are infinities of intransigent real numbers, whose digits form
no pattern, and which therefore cannot be set out, either as a string
of digits, or as an algorithm.
Or, at least, so the Platonists would have it. What alternatives are
available for the constructuvist ?
* 1. All numbers are equally (ontologically) real.
* 2. The accessible real numbers (the transcendental numbers) are
(ontologically) real, and the other (mathematically) "real" numbers are
not.
* 3. No (mathematically) real number is (ontologically) real.
1. is the position constructivists are trying to get away from. 3 is
hardly tenable, since some irrational numbers, such as pi, *can* be
constructed. And (2) contradicts the idea of density -- it suggests the
real number line has gaps. Moreover, something like the requirment of
denstity can be asserted without making any strong ontological
commitments towards the reality of the real number line.
Approach 6: Quasi Empiricism
A number of recent developments in mathematics, such as the increased
use of computers to assist proof, and doubts about the correct choice
of basic axioms, have given rise to a view called quasi-empiricism.
This challenges the traditional idea of mathematical truth as eternal
and discoverable apriori. According to quasi-empiricists the use of a
computer to perform a proof is a form of experiment. But it remains the
case that any mathematical problem that can in principle be solved by
shutting you eye and thinking. Computers are used because mathematians
do not have infinite mental resources; they are an aid. Contrast this
with traditonal sciences like chemistry or biology, where real-world
objects have to be studied, and would still have to be studied by
super-scientitists with an IQ of a million. In genuinely emprical
sciences, experimentation and observation are used to gain information.
In mathematics the information of the solution to a problem is always
latent in the starting-point, the basic axioms and the formulation of
the problem. The process of thinking through a problem simply makes
this latent information explicit. (I say simply, but of ocurse it is
often very non-trivial). The use of a computer externalises this
process. The computer may be outside the mathematician's head but all
the information that comes out of it is information that went into it.
Mathematics is in that sense still apriori.
Having said that, the quasi-empricist still has some points about the
modern style of mathematics. Axioms look less like eternal truths and
more like hypotheses which are used for a while but may eventualy be
discarded if they prove problematical, like the role of scientific
hypotheses in Popper's philosophy.
Thus mathematics has some of the look and feel of empirical science
without being empirical in the most essential sense -- that of needing
an input of information from outside the head."Quasi" indeed!
Peter D Jones 5/10/05-26/1/06
More on Platonism
Penrose on Platonism
"our individual minds are notoriously imprecise, unreliable and
inconsistent in their judgements" TBD
So when we do maths we are not using our minds, but something else? Or
maybe we are just using our minds in disciplined way -- after all, the
discipline of maths has to be painfully learnt. TBD
"Does this not point to something outside ourselves with a reality that
lies beyond what each individual can achieve?" TBD
What is achieved by individuals is achieved by individuals. How can
some transcendent realm speak through the mouth of an individual? TBD
"Nevertheless, one might take the alternative view that the
mathematical world has no independent existence, and consists of
certain ideas which have been distilled from our various minds, and
which have found to be totally trustworthy, and are 'agreed by all',
for example, or 'agreed by those in their right mind's, or 'agreed by
those who have a PhD in mathematics' (not much use in Plato's day), and
who have a right to venture an authoratiative opinion".
Which mathematical world? The world of mathematics so far proven and
understood, or the world that is waiting to be explored? However few
mathematicians there are, there is no contradiction that they carry the
world of known mathematics around in their heads. The fewer there are,
the less there is to carry round. Perhaps Penrose thinks they must
carry the whole Platonic world around to act as a "standard", but
mathematical proof doesn't work by direct inspection of Platonia, it
works slowly and painfully by axioms and deductions.
"There seems to be a danger of circularity here; for to judge whether
someone is in his or her right mind requires some external standard"
External to them, yes...but when we make such judgements, we ue finite
and earthly resources. The idea that Platonia provides a
once-and-for-all absolute standard is not of much use unless it can be
explained how the standard is applied. The actual standard against
which a proposed arguemnt is tested involves the community of
mathematicians checking it with their flawed and finite minds. Platonia
might provide a higher standard, but it is not one anyone has ever
succeeded in employing.
"What I mean by this existence is really just the objectivity of
mathematical truth".
And what does thatmean? Existence is existence, truth is truth. People
who disbelieve in Platonism can still believe in the objectivity of
mathematics.
Platonism and Causation
Maths has no causal effect on the mind in the sense of an event in time
causing a subsequent one. Arguably, there are other forms of causation.
Arguably the laws of the universe "cause" events in the sense that if
they were different, events would be different. Can the same argument
be applied to mathematical truth? The problem is that mathematical
truth is logically necessary -- it could not have been different!
More on Formalism
Formalism and Godel
'The incompleteness results affect the philosophy of mathematics,
particularly viewpoints like formalism, which uses formal logic to
define its principles. One can paraphrase the first theorem as saying
that "we can never find an all-encompassing axiomatic system which is
able to prove all mathematical truths, but no falsehoods."'
'On the other hand, from a strict formalist perspective this paraphrase
would be considered meaningless because it presupposes that
mathematical "truth" and "falsehood" are well-defined in an absolute
sense, rather than relative to each formal system.'
"Minds and Machines"
* "Is the Brain A Machine?"
* The Chinese Room and Consciousness
* The Chinese Room and Semantics
* Artificial Intelligence and Computers
* Computationalism
* Is the computationalist claim trivial -- are all systems
computers?
* Algorithms, recordings and counterfactuals
* Turing and Other Machines
* Time and Causality in Physics and Computation
* >Maudlin's Argument and Counterfactuals
"Is the Brain A Machine?"
John Searle thinks so . Is he right ? To give a typically philosophical
answer, that depends on what you mean by 'machine'. If 'machine' means
an artificial construct, then the answer is obviously 'no'. However.
Searle also thinks the the body is a machine, by which he seems to mean
that it has been understand in scientific terms, we can explain biology
by in terms of to chemistry and chemistry in terms of physics. Is the
brain a machine by this definition ? The problem is that the job of he
brain is to implement a conscious mind, just as the job of the stomach
is to digest. The problem is that although our 'mechanical'
understanding of the stomach does allow us to understand digestion we
do not, according to Searle himself, understand how the brain produces
consciousness. He does think that the problem of consciousness is
scientifically explicable, so yet another definition of 'machine' is
needed, namely 'scientifically explained or scientifically explicable'
-- with the brain being explicable rather than explained. The problem
with this stretch-to-fit approach to the meaning of the word 'machine'
is that every time the definition of brain is broadened, the claim is
weakened, made less impactful.
PDJ 03/02/03
The Chinese Room
According to the proponents of Artificial Intelligence, a system is
intelligent if it can convince a human interlocutor that it is. This is
the famous Turing Test. It focuses on external behaviour and is mute
about how that behaviour is produced. A rival idea is that of the
Chinese Room, due to John Searle. Searle places himself in the room,
manually executing a computer algorithm that implements
intelligent-seeming behaviour, in this case getting questions written
in Chinese and mechanically producing answers, without himself
understanding Chinese. He thereby focuses attention on how the
supposedly intelligent behaviour is produced. Although Searle's
original idea was aimed at semantics, my variation is going to focus on
consciousness. Likewise, although Searle's original specification has
him implementing complex rules, I am going to take it that the Chinese
Room is implemented as a conceptually simple system, in line with the
theorem of CS which has it that any computer can be emulated by a
Turing Machine.
If you think a Chinese Room implemented with a simplistic, "dumb"
algorithm can still be conscious, you are probably a behaviourist; you
only care about that external stimuli get translated into the
appropriate responses, not how this happens, let alone what it feels
like to the system in question.
If you think this dumb Chinese Room is not conscious, but a smart one
would be, you need to explain why. Any smart AI can be implemented as a
dumb TM, so the more complex inner workings which supposedly implement
consciousness. could be added or subtracted without making any
detectable difference. This seems to add up to epiphenomenalism, the
view that consciousness exists but is a bystander that doesn't cause
anything
If you think that complexity makes a difference, so dumb AI's are
definitely not conscious, you are probably a (strong) emergentist: the
complexity is of such a nature that it cannot be reduced to a dumb
algorithm such as a Turing machine, and is therefore non-computable.
(A couple of points: I am talking about strong, Broad-style emergentism
here, according to which the emerging property cannot be usefully
reduced to something simpler. There is also a weaker sense of
'emergence' in which what emerges is a mere pattern of behaviour, like
the way the waves 'emerge' from the sea. This is not quite black-box
behaviourism, since some attention is being paid to what goes on inside
the head, inasmuch as the right high-level pattern needs to be
produced, but it is closer to behaviourism than to strong, irreducible,
Broad-style emergentism.
Also: I casually remarked that mental behaviour 'may not be
computable'. This will shock some AI proponents, for whom the
Chuch-Turing thesis proves that everything is computable. More
precisely, everything that is mathematically computable is computable
by a relatively dumb computer, a Turing that something can be simulated
doesn't mean the simulation has all the relevant propeties of the
original: flight simulators don't take off. Thirdly the mathematical
sense of 'computable' doesn't fit well with the idea of
computer-simulating fundamental physics. A real number is said to be
mathematically computable if the algorithm that churns it out keeps on
churning out extra digits of accuracy..indefinitely. Since such a
algorithm will never finish churning out a single real number physical
value, it is difficult to see how it could simulate an entire universe.
Yes, I am assuming the universe is fundamentally made of real numbers.
If it is, for instance finite, fundamental physics might be more
readily computable, but the computability of physics still depends very
much on physics and not just on computer science). Peter D Jones 8/6/05
The Chinese Room and Semantics
The CR concludes that an abstract set of rules is insufficient for
semantics.
The objection goes: "But there must be some kind of information
processing structure that implements meaning in our heads. Surely that
could be turned into rules for the operator of the Chinese Room".
The Circularity Objection: an abstract structure must be circular and
therefore must fail to have any real semantics. (It is plausible that
any given term can be given an abstract definition that doesn't depend
on direct experience. It is much less plausible that every *term* can
be defined that way. Such a system would be circular in the same way
as: "present: gift" "gift: present" but on a larger scale. If this
argument, the Circularity Objection is correct, the practice of giving
abstract definitions, like "equine quadruped" only works because
somewhere in the chain of definitions are words that have been defined
directly; direct reference has been merely deferred, not avoided
altogether.)
The objection continues: "But the information processing structure in
our heads has a concrete connection to the real world: so do AI's
(although the CR's are minimal) (This is the Portability Assumption)".
But they are not the *same* concrete connections. The portability of
abstract rules is guaranteed by the fact that they are abstract. But
concrete causal connections are non-abstract, and are prima-facie
unlikely to be portable -- how can you explain colour to an alien whose
senses do not include anything like vision.
If the Circularity Objection is correct, an AI (particularly a robotic
one) could be expected to have *some* semantics, but there is no reason
it should have *human* semantics. As wittgenstein said: "if a lion
could talk, we could not understand it".
Peter D Jones 13/11/05
Artificial Intelligence and Computers
An AI is not necessarily a computer. Not everything is a computer or
computer-emulable. It just needs to be artificial and intelligent! The
extra ingredient a conscious system has need not be anything other than
the physics (chemistry, biology) of its hardware -- there is no forced
choice between ghosts and machines.
A physical system can never be exactly emulated with different hardware
-- the difference has to show up somewhere. It can be hidden by only
dealing with a subset of a systems abilities relevant to the job in
hand; a brass key can open a door as well as an iron key, but brass
cannot be substituted for iron where magnetism is relevant. Physical
differences can also be evaded by taking an abstract view of their
functioning; two digital circuits might be considered equivalent at the
"ones and zeros" level of description even though they physically work
at different voltages.
Thus computer-emulability is not a property of physical systems as
such. Even if all physical laws are computable, that does not mean that
any physical systems can be fully simulated. The reason is that the
level of simulation matters. A simulated plane does not actually fly; a
simulated game of chess really is chess. There seems to be a
distinction between things like chess, which can survive being
simulated at a higher level of abstraction, and planes, which can't.
Moreover, it seems that chess-like things are in minority, and that
they can be turned into an abstract programme and adequately simulated
because they are already abstract.
Consciousness. might depend on specific properties of hardware, of
matter. This does not imply parochialism, the attitude that denies
consciousness to poor Mr Data just because he is made out of silicon,
not protoplasm. We know our own brains are conscious; most of us intuit
that rocks and dumb Chinese Rooms are not; all other cases are
debatable.
Of course all current research in AI is based on computation in one way
or another. If the Searlian idea that consciousness is rooted in
physics, strongly emergent, and non-computable is correct, then current
AI can only achieve consciousness accidentally. A Searlian research
project would understand how brains generate consciousness in the first
place -- the aptly-named Hard Problem -- before moving onto possible
artificial reproductions, which would have to have the right kind of
physics and internal causal activity -- although not necessarily the
same kind as humans.
Computationalism
Computationalism is the claim that the human mind is essentially a
computer. It can be picturesquely expressed in the "yes, doctor"
hypothesis -- the idea that, faced with a terminal disease, you would
consent to having your consciousness downloaded to a computer.
There are two ambiguities in "computationalism" -- consciousness vs.
cognition, process vs programme -- leading to a total of four possible
meanings.
Most people would not say "yes doctor" to a process that recorded their
brain on a tape a left it in a filing cabinet. Yet, that is all you can
get out of the timeless world of Plato's heaven (programme vs process).
That intuition is, I think, rather stronge than the intuition that
Maudlin's argument relies on: that consciousness supervenes only on
brain activity, not on counterfactuals.
But the other ambiguity in computationalism offers another way out. If
only cognition supervenes on computational (and hence counterfactual)
activity, then consciousness could supervene on non-counterfactual
activity -- i.e they could both supervene on physical processes, but in
different ways.
Platonic computationalism -- are computers numbers?
Any computer programme (in a particular computer) is a long sequence of
1's and 0's, and therefore, a long number. According to Platonism,
numbers exist immaterially in "Plato's Heaven". If programmes are
numbers, does that mean Plato's heaven is populated with computer
programmes?
The problem, as we shall see is the "in a a particular computer"
clause.
As Bruno Marchal states the claim in a more formal language:
"Of course I can [identify programmes with numbers ]. This is a key
point, and it is not obvious. But I can, and the main reason is Church
Thesis (CT). Fix any universal machine, then, by CT, all partial
computable function can be arranged in a recursively enumerable list
F1, F2, F3, F4, F5, etc. "
Of course you can count or enumerate machines or algorithms, i.e.
attach unique numerical labels to them. The problem is in your "Fix any
universal machine". Given a string of 1's and 0s wihouta universal
machine, and you have no idea of which algorithm (non-universal
machine) it is. Two things are only identical if they have all*their
properties in common (Leibniz's law). But none of the propeties of the
"machine" are detectable in the number itself.
(You can also count the even numbers off against the odd numbers , but
that hardly means that even numbers are identical to odd numbers!)
"In computer science, a fixed universal machine plays the role of a
coordinate system in geometry. That's all. With Church Thesis, we don't
even have to name the particular universal machine, it could be a
universal cellular automaton (like the game of life), or Python,
Robinson Aritmetic, Matiyasevich Diophantine universal polynomial,
Java, ... rational complex unitary matrices, universal recursive group
or ring, billiard ball, whatever."
Ye-e-es. But if all this is taking place in Platonia, the only thing it
can be is a number. But that number can't be associated with a
computaiton by another machine, or you get infinite regress.
Is the computationalist claim trivial -- are all systems computers?
TBD
Computational counterfactuals, and the Computational-Platonic Argument
for Immaterial Minds
For one, there is the argument that: A computer programme is just a
long number, a string of 1's and 0's.
(All) numbers exist Platonically (according to Platonism)
Therefore, all programmes exist Platonically.
A mind is special kind of programme (According to computaionalism)
All programmes exist Platonically (previous argument)
Therefore, all possible minds exist Platonically
Therefore, a physical universe is unnecessary -- our minds exist
already in the Platonic realm
The argument has a number of problems even allowing the assumptions of
Platonism, and computationalism.
A programme is not the same thing as a process.
Computationalism refers to real, physical processes running on material
computers. Proponents of the argument need to show that the causality
and dynamism are inessential (that there is no relevant difference
between process and programme) before you can have consciousness
implemented Platonically.
To exist Platonically is to exist eternally and necessarily. There is
no time or change in Plato's heave. Therefore, to "gain entry", a
computational mind will have to be translated from a running process
into something static and acausal.
One route is to replace the process with a programme. let's call this
the Programme approach.. After all, the programme does specify all the
possible counterfactual behaviour, and it is basically a string of 1's
and 0's, and therefore a suitable occupant of Plato's heaven. But a
specification of counterfactual behaviour is not actual counterfactual
behaviour. The information is the same, but they are not the same
thing.
No-one would believe that a brain-scan, however detailed, is conscious,
so not computationalist, however ardent, is required to believe that a
progamme on a disk, gathering dust on a shelf, is sentient, however
good a piece of AI code it may be!
Another route is "record" the actual behaviour, under some
circumstances of a process, into a stream of data (ultimately, a string
of numbers, and therefore something already in Plato's heaven). Let's
call this the Movie approach. This route loses the conditional
structure, the counterfactuals that are vital to computer programmes
and therefore to computationalism.
Computer programmes contain conditional (if-then) statements. A given
run of the programme will in general not explore every branch. yet the
unexplored branches are part of the programme. A branch of an if-then
statement that is not executed on a particular run of a programme will
constitute a counterfactual, a situation that could have happened but
didn't. Without counterfactuals you cannot tell which programme
(algorithm) a process is implementing because two algorithms could have
the same execution path but different unexecuted branches.
Since a "recording" is not computation as such, the computationalist
need not attribute mentality to it -- it need not have a mind of its
own, any more than the characters in a movie.
(Another way of looking at this is via the Turing Test; a mere
recording would never pass a TT since it has no
condiitonal/counterfactual behaviour and therfore cannot answer
unexpected questions).
A third approach is make a movie of all possible computational
histories, and not just one. Let's call thsi the Many-Movie approach.
In this case a computation would have to be associated with all related
branches in order to bring all the counterfactuals (or rather
conditionals) into a single computation.
(IOW treating branches individually would fall back into the problems
of the Movie approach)
If a computation is associated with all branches, consciousness will
also be according to computationalism. That will bring on a White
Rabbit problem with a vengeance.
However, it is not that computation cannot be associated with
counterfactuals in single-universe theories -- in the form of
unrealised possibilities, dispositions and so on. If consciousness
supervenes on computation , then it supervenes on such counterfactuals
too; this amounts to the response to Maudlin's argument in wihch the
physicalist abandons the claim that consciousness supervenes on
activity.
Of ocurse, unactualised possibilities in a single universe are never
going to lead to any White Rabbits!
Turing and Other Machines
Turing machines are the classical model of computation, but it is
doubtful whether they are the best model for human (or other organic)
intelligence. Turing machines take a fixed input, take as much time as
necessary to calculate a result, and produce a perfect result (in some
cases, they will carry on refining a result forever). Biological
survival is all about coming up with good-enough answers to a tight
timescale. Mistaking a shadow for a sabre-tooth tiger is a msitake, but
it is more accpetable than standing stock still calculating the perfect
interpretation of your visual information, only to ge eaten. This
doesn't put natural cognition beyone the bounds of computation, but it
does mean that the Turing Machine is not the ideal model. Biological
systems are more like real time systems, which have to "keep up" with
external events, at the expense of doing some things imprefectly.
Time and Causality in Physics and Computation
The sum total of all the positions of particles of matter specififies a
(classical) physical state, but not how the state evolves. Thus it
seems that the universe cannot be built out of 0-width (in temporal
terms) slices alone. Physics needs to appeal to something else.
There is one dualistic and two monistic solutions to this.
The dualistic solution is that the universe consists (separately) of
states+the laws of universe. It is like a computer, where the data
(state) evolves according to the programme (laws).
One of the monistic solutions is to put more information into states.
Physics has an age old "cheat" of "instantaneous velocities". This
gives more information about how the state will evolve. But the state
is no longer 0-width, it is infinitessimal.
Another example of states-without-laws is Julian Barbour's Platonia.
Full Newtonian mechanics cannot be recovered from his "Machian"
approach, but he thinks that what is lost (universes with overall
rotation and movement) is no loss.
The other dualistic solution is the opposite of the second:
laws-without-states. For instance, Stephen Hawking's No Boundary
Conditions proposal
Maudlin's Argument and Counterfactuals
We have already mentioned a parallel with computation. There is also
relevance to Tim Maudlin's claim that computationalism is incompatible
with physicalism. His argument hinges on serparating the activity of a
comptuaitonal system from its causal dispositions. Consciousness, says
Maudlin supervened on activity alone. Parts of an AI mechansim that are
not triggered into activity can be disabled without changing
consciousness. However, such disabling changes the comutation being
performed, because programmes containt if-then statements only one
branch of which can be executed at a time. The other branch is a
"counterfactual", as situationthat could have happened but didn't.
Nonetheless, these counterfactuals are part of the algorithm. If
changing the algorithm doesn't change the conscious state (because it
only supervenes on the active parts of the process, not the unrealised
counterfactuals), consciousness does not supervene on computation.
However, If causal dispositions are inextricably part of a physical
state, you can't separate activity from counterfactuals. Maudlin's
argument would then have to rely on disabling counterfactuals of a
specifically computational sort.
We earlier stated that the dualistic solution is like the separation
between programme and data in a (conventional) computer programme.
However, AI-type programmes are typified by the fact that there is not
a barrier between code and programme -- AI software is self-modifying,
so it is its own data. Just as it is not physically necessary that
there is a clear distinction between states and laws (and thus a
separability of phsycial counterfactuals), so it isn't necessarily the
case that there is a clear distinciton between programme and data, and
thus a separability of computational counterfactuals. PDJ 19/8/06
How to Build a Universe
1. Mysticism I: Is anything knowable?
2. Mysticism II: The Unreality of Real Things
3. Mechanism: The Simulation Argument
4. Mechanism, Organism and Knowability
5. Mathematics, the Universe and Everything
1. Why Mathematics Works
2. Gods and Monsters
3. For and against Mathematical Platonism
4. For and Against Matter
5. The Case Against Mathematial Monism
6. Logic, possibility, and Necessity
Mysticism: Is Anything Knowable?
(Based on a Dialogue with Bill Snyder)
"Thinking about existence is quite other than existence itself. There
is no such thing as "concrete" thinking; thinking always deals in
abstractions and, hence, cannot contain existence itself".
Well, when you think about rocks, you don't have rocks in the head. But
does that put you at a disadvantage?
"Thinking of rocks and being a rock are very different activities, or
more broadly thinking about existence and existence itself are very
different activities".
That is true, but it does not show there is anything missing from our
thinking. If there is nothing it is like to be a rock. there is nothing
to stop us grasping all there is to know about rock-ness from our
non-rock persepective.
Of course, we feel that we ourselves have inner thoughts and
(especially) feelings that cannot be understood by others, and we
assume others do, even bats, as in the famous paper by Thomas Nagel.
(More here). But the claim that is being examined language and thought
are necessarily limited when it comes to grasping the world -- not that
they are possibly limited depending on the nature of the world.
It is logically possible, if coutnerintuitive, that there something it
is like to be a rock, or anything else , and if so, there is something
we will never know about anything. This is the conclusion being sought,
but it is far from a necessary truth, only a possibility.
But if there is something it is like to be a rock,. that is a
metaphysical fact: the point being that we cannot discover the limits
of thought from the nature of thought alone.
"As for being able to describe something (event or thing or whatever),
being able to refer to it is irrelevant. Referring to the keyboard on
which I am typing (I just did it) is not to describe it. Describing it
would be to state its colors, shape, and things like that. But to do
that we must use words which have no precise meaning".
Words only need to be precise enough to pick one thing out form
another. It is probably overkill to describe every last atom.(But
again, that is as much about what the world is like as the limitations
of langauge).
"To put my point broadly, describing perceptual objects is never
completely possible, but only approximative because we use abstract
concepts in the making of our description (like words which refer to
shape and color). When one comes to the objects of wholly abstract
thought such as the "objects" referred to in advanced scientific
theories, then one is living in a dream world if one thinks that one
can describe them or that the theory in question describes them".
That's a rather different issue. The objects of fundamental science
tend to be quite simple. The problem is not one of information
overload, is the lack of familir reference-points. (Reference *does*
matter).
"The language used to refer to the objects MAY invite or suggest a
certain descriptive content, but it is more MISleading than helpful in
doing so. Calling an electron a particle may suggest that it is a very
tiny hard little grain of sand like thing. But as we have come to know
that is a very misleading way of thinking about it".
Yes, the real description is mathematical.
"It can just as well be thought of as a field or a wave; when thought
of as a field, it is a field which at times generates particle like
behavior and at other times wavelike behavior and at other times
neither".
The verbal descriptions are unified mathematically.
"If QM is correct, its maths describes electrons completely. There is
no hinterlan of additonal detail -- electrons have no 'hair'".
"Language necessarily uses abstractions even in the description of
perceptual objects. When one is dealing with scientific language
(certainly in the advanced physical sciences) one is dealing with only
abstractions"
In what sense of "abstraction"?
* The sense that fails to capture what-it-is to-be-like a thing?
* The sense of omitted detail?
* Or the sense of abstruse mathematics?
The salient point is whether the thing described has any kind of
hinterland which is left undescribed by the language employed. That is
almost always the case with ordinary-langauge descriptions of everyday
objects containing billions and billions of atoms -- but we only know
that to be the case because of the success of scientific descriptions!
It could be argued by analogy that just as the atom-by-atom picture is
left undescribed by the ordinarys term "rock" or "tree", so there might
be a further hinterland behind the scientific explanation. But that is
only a "maybe", not necessity, and as usual, metaphysics, the nature of
the world, is involved.
"when one attempts to connect those abstractions with features drawn
from the perceptual world and thereby to "describe" the objects, one
only bamboozles any possibility of undertanding what is being referred
to".
There a various kinds of potential problem depending on the language
being used, and on what it is being used to describe. Lack of
understanding or communication is certainly possible, but is not a
foregone conclusion.
"Therefore, as long as one operates wholly within the conceptual
structure one has no chance in hell of getting hold of existence
itself.
I think this is case of being bewitched by language -- specifically
being misled by a very unforced choice of metaphor. One is not
literally encased by ones own beliefs. The whole point of a concept is
to refer to things other than itself.
What does "getting hold" mean? As we have seen, knowing things from a
third-personse, second-hand, arm-length perspective does not by itself
imply that one is at a cognitve disadvantage. It depends on the thing
being understood.
"There are people who appear to me to operate wholly within one
conceptual structure or another and live caged within their own
thoughts, confusing what they think the world to be with existence
itself".
They may, as individuals be representing things incorrectly (insofars
as they can be represented correctly). Or they may suffer from a more
genreral problem of being out of contact with reality. Of course, if
the general problem obtains in a strong way -- if philosophical
scepticism is true -- there is probably not much difference between
individuals. Everybody is equally wrong and right. (This is the classic
"error" argument against solipsism).
"But existence itself is necessarily beyond any thinking or conceiving"
There is no necessity to it at all. How knowable a thing is depends on
what it is and how you intend to know it.
Peter D Jones 15/10/2006
Mysticism II: the Unreality of Real Things, and the 3-Level View of
Reality
Some forms of mysticism claim that everything is a mystery to us, that
we do not really know anything. Others claim that the problem lies in
things themselves, that they lack reality rather than knowability.
There is presumably some sense in which horses exist and unicorns do
not. Thus the claim about the unreality of "conventional objects" seems
to imply that they fall short of some higher standard. Is it a purely
theoretical standard, or does it imply that there is some higher,
transcedent reality , beyond conventional or empirical objects?
A commmon-sense thinker might well claim that in the absence of any
demonstrable higher reality, conventional existence is existence
simpliciter, and the objects we conventionally take to be real are
real.
A famous proponent of this view was the Buddhist philosopher Nagarjuna.
His philosophy was critical in nature, enabling him to establish the
"nothing exists" claim using their ready-made criteria for existence of
the philosophers he was criticising.
Mechanism, Organism and Knowability
Sceptical arguments often confuse being unable to know anything with
being unable to know everything. But is that really a confusion? It's
(logically) possible that things could be so constituted that the
nature of every thing is intintimately dependent on the nature of every
other thing. This is illustrated by an old parable called the "net of
indra".
FAR AWAY IN THE HEAVENLY ABODE OF THE GREAT GOD INDRA, THERE IS A
WONDERFUL NET WHICH HAS BEEN HUNG BY SOME CUNNING ARTIFICER IN SUCH A
MANNER THAT IT STRETCHES OUT INDEFINITELY IN ALL DIRECTIONS. IN
ACCORDANCE WITH THE EXTRAVAGANT TASTES OF DEITIES, THE ARTIFICER HAS
HUNG A SINGLE GLITTERING JEWEL AT THE NET'S EVERY NODE, AND SINCE THE
NET ITSELF IS INFINITE IN DIMENSION, THE JEWELS ARE INFINITE IN NUMBER.
THERE HANG THE JEWELS, GLITTERING LIKE STARS OF THE FIRST MAGNITUDE, A
WONDERFUL SIGHT TO BEHOLD. IF WE NOW ARBITRARILY SELECT ONE OF THESE
JEWELS FOR INSPECTION AND LOOK CLOSELY AT IT, WE WILL DISCOVER THAT IN
ITS POLISHED SURFACE THERE ARE REFLECTED ALL THE OTHER JEWELS IN THE
NET, INFINITE IN NUMBER. NOT ONLY THAT, BUT EACH OF THE JEWELS
REFLECTED IN THIS ONE JEWEL IS ALSO REFLECTING ALL THE OTHER JEWELS, SO
THAT THE PROCESS OF REFLECTION IS INFINITE
This metaphysic does validate what otherwise be a non-sequitur; we
cannot know anything because we cannot know everything. Any finite
belief-system must be fall short of ultimate truth because it is
finite. But without the holistic metaphysics of the Net Of Indra, the
inference is simply a mistake. This is another mystical belief that
requires a metaphysical posit, not simply a consideration of the
limitations of knowledge.
Science uses a methodology or reductionism, of divide and conquer, of
breaking things down into their simplest components. A universe that
plays ball with this is a mechanical, rather than organic universe. For
our purposes, the most pertinent aspect of mechanism is the fact that
it involves linear chains of causation. A causes B , and B causes C. C
is not directly related to A and does not carry traces of A within it.
This is an advantage, because it reduces the number of things one has
to understand in order to understand C. (In the Organic universe, that
is of course everyhting). It is also the source of pretty well every
variety of scepticism, since A could have been replaced by A' without
changing C. If C is some effect worked on my sense-organs by an
external cause, then C could be the same in a scenario where it was
caused by A' rather than A. I could be a brain in a vat, I could be the
dupe of a Descartes Evil Spirit, I could be in a computer simulation.
All these scenarios are variations on the theme of my percepts, my
sense-data, being causes mechanically by something other that I think
is causing them. In the organic universe, they are impossible because I
already contain the whole world within myself, so substitions are
impossible without changing everything else.
Both Mechanism and Organism, then, lead to paradoxes. Mechanism makes
the world simple enough to understand (structurally, from a 3rd-person
perspective) at the expense of making its "in itself" nature uncertain.
Organism means that every being contains the nature of the whol eof
reality within itself -- but at the expense of making the Whole
incomprehensible to linear, 3rd-person, structural knowing.
Is Reality real ? the Simulation Argument
The Simulation Argument seeks to show that it is not just possible that
we are living inside a simulation, but likely.
1 You cannot simulate a world of X complexity inside a world of X
complexity.(quart-into-a-pint-pot-problem).
2 Therefore, if we are in a simulation the 'real' world outside the
simulation is much more complex and quite possibly completely different
to the simulated world.
3 In which case, we cannot make sound inferences from the world we are
appear to be in to alleged real world in which the simulation is
running
4 Therefore we cannot appeal to an argumentative apparatus of advanced
races, simulations etc, since all those concepts are derived from the
world as we see it -- which, by hypothesis is a mere simulation.
5 Therefore, the simulation argument pulls the metaphysical rug from
under its epistemological feet.
The counterargument does not show that we are not living in a
simulation, but if we are , we have no way of knowing whether it is
likely or not. Even if it seems likely that we will go on to create
(sub) simulations, that does not mean we are living in a simulation
that is likely for the same reasons, since our simulation might be rare
and peculiar. In particular, it might have the peculiarity that
sub-simulations are easy to create in it. For all we know our
simulators had extreme difficulty in creating our universe. In this
case, the fact that it is easy to create sub simulations within our
(supposed) simulation, does not mean it is easy to creae simulations
per se.
Mathematics, the Universe and Everything
1. Why Mathematics Works
2. Gods and Monsters
3. For and Against Mathematical Platonism
4. The Case Against Matter
5. The Case Against Mathematial Monism
Why Mathematics Works
1. Mathematics is the theoretical exploration of every possible kind
of abstract structure
2. The world has some kind of structure; it is not chaos or
featureless or void
3. Therefore, it is likely that at least some of the structures
discovered by mathematicians are applicable to the world
(It would be strange if everything mathematicans came up with was
empirically applicable, and might tend one towards Platonism or
Rationalism, but this is not the case. It might be possible for
mathematicians working completely abstractly and theoretically to fail
to include any empirically useful structures in the finite list of
structures so far achieved, but maths is not divorced from empiricism
and practicallity to that extent -- what I say in (1) is something of
an idealisation).
Peter D Jones 8/12/04
Gods and Monsters
In the meditations of Marcus Aurelius, a distinction is drawn between
two basic attitudes to philosophy, the Gods and the Monsters. Monsters
are down-to-earth and commonsensical, Gods are high-minded and
abstract. >
gods Monsters
Epistemology Rationalism Empiricism
Metaphysics Idealism Materialism
Logic Necessity Contingency
TBD
The Mathematical Monistic Many-World Model Introduced
1. Single (physical) universe theories
2. Physical Multiverse theories.
3. Mathematical Multiverse Theories
Platonism is the idea that mathematical objects have an independent
existence of their own, in addition to the physical world. The
Mathematical Monistic Many-World Model idea keeps the Platonic realm of
numbers, but drops the physical world, claiming we are somehow "inside"
the world of numbers. One alleged benefit of this view is that it
removes the contingency, the "just so" quality of a universe consiting
of specific physical laws. Another is the claim that the success of
physics shows everything is mathematical, or at least has a
mathematical description. Another is the claim that matter adds nothing
explanatorily to maths.
For and Against Mathematical Platonism
The case for Mathematical Platonism needs to be made in the first
place; if numbers do not exist at all, the universe, as an existing
thing, cannot be a mathematical structure. (solipsists read: if numbers
are not real, I cannot be mathematical structure).
The claim that nothing exists but numbers is not just the claim that
the world is the kind of mathematical structure described by physics,
because that is only one of a multitude of mathematically possible
structures. The nothing-but-numbers claim entails the claim that a
multitude of mathematically consistent worlds exist which do not
correspond to anything that is observed. To put it another way, physics
is based on picking the mathematical structure that corresponds to
observation, so that only a subsection of physics is mathematically
useful. Thus the success of physics as a mathematical science does not
prove the nothing-but-numbers claim.
Theories of Everything (TOEs) seek to avoid explain away the apparent
(contingency) of the world. They are exercises in armchair rationalism,
and as such contingent facts that can only be derived from observation
cannot be permitted. Having said that, the armchair rationalist is
allowed to appeal to some very broad facts, such as his own existence,
and the *appearance* of contingency -- the fact that he sees some
things in the world around him, and others just are not there. Without
his own existence, the simplest non-contingent theory of reality is
that nothing exists! Apparent contingency, the finiteness, of the
experienced world is a trickier issue.
The Case Against Mathematical Platonism
The epistemic case for mathematical Platonism is be argued on the basis
of the objective nature of mathematical truth. Superficially, it seems
persuasive that objectivity requires objects. However, the basic case
for the objectivity of mathematics is the tendency of mathematicians to
agree about the answers to mathematical problems; this can be explained
by noting that mathematical logic is based on axioms and rules of
inference, and different mathematicians following the same rules will
tend to get the same answers , like different computers running the
same problem.
The semantic case for mathematical Platonism is based on the idea that
the terms in a mathematical sentence must mean something, and therefore
must refer to objects. It can be argued on general linguistic grounds
that not all meaning is reference to some kind of object outside the
head. Some meaning is sense, some is reference. That establishes the
possibility that mathematical terms do not have references. What
establishes it is as likely and not merely possible is the obeservation
that nothing like empirical investigation is needed to establish the
truth of mathematical statements. Mathematical truth is arrived at by a
purely conceptual process, which is what would be expected if
mathematical meaning were restricted to the Sense, the "in the head"
component of meaning.
The logical case for mathematical Platonism is based on the idea that
mathematical statements are true, and make existence claims. That they
are true is not disputed by the anti-Platonist, who must therefore
claim that mathematical existence claims are somehow weaker than other
existence claims -- perhaps merely metaphorical. That the the word
"exists" means different things in different contexts is easily
established.
For one thing, this is already conceded by Platonists! Platonists think
Platonic existence is eternal, immaterial non-spatial, and so on,
unlike the Earthly existence of material bodies. For another, we are
already used to contextualising the meaning of "exists". We agree with
both: "helicopters exist"; and "helicopters don't exist in Middle
Earth". (People who base their entire anti-Platonic philosophy are on
this idea are called fictionalists. However, mathematics is not a
fiction because it is not a free creation. Mathematicians are
constrained by consistency and non-contradiction in a way that authors
are not. Dr Watson's fictional existence is intact despite the fact
that he is sometimes called John and sometimes James in Conan Doyle's
stories).
A possible counter argument by the Platonist is that the downgrading of
mathematical existence to a mere metaphor is arbitrary. The
anti-Platonist must show that a consistent standard is being applied.
This it is possible to do; the standard is to take the meaning of
existence in the context of a particular proposition to relate to the
means of justification of the proposition. Since ordinary statements
are confirmed empirically, "exists" means "can be perceived" in that
context. Since sufficient grounds for asserting the existence of
mathematical objects are that it is does not contradict anything else
in mathematics, mathematical existence just amounts to concpetual
non-contradictoriness.
(Incidentally, this approach answers a question about mathematical and
empirical truth. The anti-Platonists wants the two kinds of truth to be
different, but also needs them to be related so as to avoid the charge
that one class of statement is not true at all. This can be achieved
because empirical statements rest on non-contradiction in order to
achive correspondence. If an empricial observation fails co correspond
to a statemet, there is a contradiction between them. Thus
non-contradiciton is a necessary but insufficient justification for
truth in empircal statements, but a sufficient one for mathematical
statements).
The forgoing establishes that Platonism is dispensible. What
establishes that anti-Platonism is actually preferable? One point in
its favour is that it is simpler.
All the facts about mathematical truth and methodology can be
established without appeal to the actual existence of mathematical
objects. In fact, the lack of such objects actually explains the
objectivity and necessity of maths. Mathematical statements are
necessarily true because there are no possible circumstances that make
them false; there are no possible circumstances that would make them
false because they do not refer to anything external. This is much
simpler than the Platonist alternative that mathematical statements:
1) have referents
which are
2) unchanging and eternal, unlike anything anyone has actually seen
and thereby
3) explain the necessity (invariance) of mathematical statements
without
4) performing any other role -- they are not involved inbr />
mathematical proof.
For and Against Matter
Matter is a bare substrate with no properties of its own. The question
may well be asked at this point: what roles does it perform ? Why not
dispense with matter and just have bundles of properties -- what does
matter add to a merely abstract set of properties? The answer is that
not all bundles of posible properties are instantiated, that they
exist.
What does it mean to say something exists ? "..exists" is a meaningful
predicate of concepts rather than things. The thing must exist in some
sense to be talked about. But if it existed full, a statement like
"Nessie doesn't exist" would be a contradiction ...it would amount to
"the existing thing Nessie doesnt exist". However, if we take that the
"some sense" in which the subject of an "...exists" predicate exists is
only initially as a concept, we can then say whether or not the concept
has something to refer to. Thus "Bigfoot exists" would mean "the
concept 'Bigfoot' has a referent".
What matter adds to a bundle of properties is existence. A non-existent
bundle of properties is a mere concept, a mere possibility. Thus the
concept of matter is very much tied to the idea of contingency or
"somethingism" -- the idea that only certain possible things exist.
The other issue matter is able to explain as a result of having no
properties of its own is the issue of change and time. For change to be
distinguishable from mere succession, it must be change in something.
It could be a contingent natural law that certain properties never
change. However, with a propertiless substrate, it becomes a logical
necessity that the substrate endures through change; since all changes
are changes in properties, a propertiless substrate cannot itself
change and must endure through change. In more detail here
The Case Against Mathematial Monism
Mathematical monism is both too broad and too narrow.
Too broad: If I am just a mathematical structure, I should have a much
wider range of experience than I do. There is a mathemtical structure
corresponding to myself with all my experiences up to time T. There is
a vast array of mathematical structures corresponding to other versions
of me with having a huge range of experiences -- ordinary ones, like
continuing to type, extraordinary ones like seeing my computer sudenly
turn into bowl of petunias. All these versions of me share the memories
of the "me" who is writing this, so they all identify themselves as me.
Remember, that for mathematical monism it is only necessary that a
possible experience has a mathematical description. This is known as
the White Rabbit problem. If we think in terms of multiverse theories,
we would say that there is one "me" in this universe and other "me's"
in other universes,a nd they are kept out of contact with each other.
The question is whether a purely mathematical scheme has enough
resources to impose isolation or otherwise remove the White Rabbit
problem.
Too narrow: there are a number of prima-facie phenomena which a purely
mathematical approach struggles to deal with.
* space
* time
* consciousness
* causality
* necessity/contingency
Why space ? It is tempting to think that if a number of, or some other
mathematical entity, occurs in a set with other numbers, that is, as it
were, a "space" which is disconnected from other sets, so that a set
forms a natural model of an *isolated* universe withing a multiverse, a
universe which does not suffer from the White Rabbit problem. However,
maths per se does not work that way. The number "2" that appears in the
set of even numbers is exactly the same number "2" that appears in the
list of numbers less than 10. It does not acquire any further
characteristics from its context.
The time issue should be obvious. Mathematics is tradionally held to
deal with timeless, eternal truths. This is reflected in the metpahor
of mathematical truth being discovered not found (which, in line with
my criticism of Platonism, should not be taken to seriously). It could
be objected that physics can model time mathematically; it can be
objected right back that it does so by spatialising time, by turning it
into just another dimension, in which nothing really changes, and
nothing passes. Some even go so far as to insist that this model is
what time "really" is, which is surely a case of mistaking the map for
the territory.
Consciousness is a problem for all forms of materialism and physicalism
to some extent, but it is possible to discern where the problem is
particularly acute. There is no great problem with the idea that matter
considered as a bare substrate can have mental properities. Any
inability to have mental properties would itself be a property and
therefore be inconsistent with the bareness of a bare substrate. The
"subjectivity" of conscious states, often treated as "inherent" boils
down to a problem of communicating one's qualia -- how one feels, how
things seem. Thus it is not truly inherent but depends on the means of
communication being used. Feelings and seemings can be more readily
communicated in artistic, poetic language, and least readily in
scientific, technical language. Since the harder, more technical a
science is, the more mathematical it is, the communication problem is
at its most acute in a purely mathematical langauge. Thus the problem
with physicalism is not its posit of matter (as a bare substrate) but
its other posit, that all properties are physical. Since physics is
mathematical, that amounts to the claim that all properties are
mathematical (or at least mathematically describable). In making the
transition from a physicalist world-view to a mathematical one, the
concept of a material substrate is abandoned (although it was never a
problem for consciousness) and mathematical properties become the only
possible basis for qualia. Qualia have to be reducible to, or
identifiable with, mathematical properties, if they exist at all. This
means that the problem for consciousness becomes extreme, since there
is no longer the possibility of qualia existing in their own right, as
properties of a material substrate, without supervening on
mathematically describable properties.
The interesting thing is that these two problems can be used to solve
each other to some extent. if we allow extra-mathemtical properties
into our universe, we can use them to solve the White Rabbit problem.
There are two ways of doing this: We can claim either:-
* White Rabbit universes don't exist at all
* White Rabbit universes are causally separated from us (or remote
in space)
The first is basically a reversion to a single-universe theory (1).
Mathematical monists sometimes complain that they can't see what role
matter plays. One way of seeing its role is as a solution to the WR
problem. For the non-Platonist, most mathematical entitites have a
"merely abstract" existence. Only a subset truly, conceretely, exist.
There is an extra factor that the priveleged few have. What is it ?
Materiality. For the physicalist, matter is the token of existence.
Maerial things, exist, immaterial ones don't.
The second moves on from a Mathematical Multiverse to a physical one
(3). The interesting thing about the second variety of
non-just-mathematical monism is that as well as addressing the White
Rabbit problem, it removes some further contingency. If the matter,
physical laws, and so on, are logically possible, then the general
approach of arguing for a universe/multiverse on the grounds of
removing contingency must embrace them -- otherwise it would be a
contingent fact that the universe/multiverse consists of nothing but
mathematics.
Peter D Jones 05/05/2006
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Received on Tue Oct 31 2006 - 13:37:37 PST