Bruno Marchal wrote:
> Le 29-août-06, à 20:45, 1Z a écrit :
>
>
>
> > The version of AR that is supported by comp
> > only makes a commitment about mind-independent *truth*. The idea
> > that the mind-independent truth of mathematical propositions
> > entails the mind-independent *existence* of mathematical objects is
> > a very contentious and substantive claim.
>
>
> You have not yet answered my question: what difference are you making
> between "there exist a prime number in platonia" and "the truth of the
> proposition asserting the *existence* of a prime number is independent
> of me, you, and all contingencies" ?
"P is true" is not different to "P". That is not the difference I
making.
I'm making a difference between what "exists" means in mathematical
sentences and what it means in empiricial sentences (and what it means
in fictional contexts...)
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 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).
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.
(There is also disagreement about some axioms, such as the Axiom of
Choice,
and different mathematicians with different attitudes about the AoC
will
tend to get different answers -- a phenomenon which is easily explained
by the formalist view I am taking here).
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.
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 want sthe 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).
> > Where is it shown the UD exists ?
>
>
> If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if you
> prefer, that the truth of the propositions:
>
> Ex(x = 0),
> Ex(x = s(0)),
> Ex(x = s(s(0))),
> ...
>
> is independent of me), then it can proved that the UD exists. It can be
> proved also that Peano Arithmetic (PA) can both define the UD and prove
> that it exists.
But again this is just "mathematical existence". You need some
reason to assert that mathematical existence is not a mere
metaphor implying no real existence, as anti-Platonist
mathematicians claim. I do not think that is given by computationalism.
> >> Tell me also this, if you don't mind: are you able to doubt about the
> >> existence of "primary matter"? I know it is your main fundamental
> >> postulate. Could you imagine that you could be wrong?
> >
> > It is possible that I am wrong. It is possible that I am right.
> > But you are -- or were -- telling me matter is impossible.
>
>
> Only when I use Occam.
Occam does not support conclusions of impossibility. It could
be a brute fact that the universe is more complicated than
strcitly necessary.
> Without Occam I say only that the notion of
> primary matter is necessarily useless i.e. without explanatory purposes
> (even concerning just the belief in the physical proposition only) .
> This is a non trivial consequence of the comp hyp. (cf UDA).
As is the way with these things, we anti-Platonists appeal
to Occam as well (although not qua impossibilia).
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 actuall seen
and thereby
3) explain the necessity (invariance) of mathematical statements
without
4) performing any other role -- they are not involved in
mathematical proof.
> > But the negative integers exist (or "exist"), so it has
> > an existing predecessor.
>
>
> Yes. But the axiom Q1 "Ax ~(0 = s(x)" is not made wrong just because
> you define the negative integer in Robinson Arithmetic. The "x" are
> still for "natural number". The integer are new objects defined from
> the natural number. All right? To take another example, you can define
> in RA all partial recursive functions, but obviously they does not obey
> to the Q axioms, they are just constructs, definable in RA.
So the specialness of Time depends on the specialness of nautral
numbers, depends on the specialness of Robinson Arithemtic ?
> Bruno
>
>
> http://iridia.ulb.ac.be/~marchal/
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Received on Thu Aug 31 2006 - 16:21:57 PDT