# Re: Arithmetical Realism

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Fri, 15 Sep 2006 12:29:52 +0200

Le 12-sept.-06, ¨¤ 19:20, 1Z a ¨¦crit :

>>
>> 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.

All right then. It is an important key point for what will follow.
It will help me to represent the modality "True(p)" by just "p"; that
is useful because correct machine cannot represent their notion of
truth (by Tarski theorem).

> 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...)

OK. So with this phrasing, the consequence of the UDA (including either
the movie-graph argument, or the use of the "comp-physics" already
extracted + OCCAM) can be put in this way:

The appearance of "empirical existence" is explain without ontological
empirical commitment from the mathematical existence of numbers. Indeed
empirical existence, assuming comp, has to be an internal arithmetical
modality.

> The logical case for mathematical Platonism is based on the idea
> that mathematical statements are true, and make existence claims.

Yes.

> 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.

But the whole point is that if you take the "yes doctor" idea seriously
enough, then "empirical existence" appears to be more metaphorical than
mathematical existence.

> That the the word "exists" means different things in different contexts
> is easily established.

Right. Now a TOE is supposed to explain all those notion of existence
and to explain also how they are related.
I take the "simple" math existence as primitive, and explain all other
notion of existence from it. Perhaps you should wait for it, or peruse
in the archive or in my url to see how that works.

> 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.

OK. But after Godel, mathematicians know, (or should know) that the
consistency constrained is not enough.
Simple example: all sufficiently rich and consistent theory T remains
consistent when you add the axiom asserting that T is inconsistent. You
get a consistent but unreasonable and incorrect theory.
Yes: Godel's second incompleteness result is admittedly amazing.

> 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
> but insufficient justification for truth in empircal statements, but
> a sufficient one for mathematical statements).

Alas no. After Godel's second incompleteness theorem (or Lob extension
of it) non-contradiction is insufficient even for the mathematical
reality. Any machine/theory can be consistent and false with respect to
the intended arithmetical reality.
Like Chaitin is aware, even pure arithmetic has some objective
"empirical" features.

>> 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.

It is not given by comp per se. It follows from the UD Argument. Don't
hesitate to ask question about any step where you feel not being
convinced.

> Occam does not support conclusions of impossibility. It could
> be a brute fact that the universe is more complicated than
> strcitly necessary.

You are *trivially* right. This could kill ANY theory. You can say to a
string theorist : what about the particles which we have not yet
discover and which would behave in a way contradicting the theory.

> All the facts about mathematical truth and methodology can be
> established
> without appeal to the actual existence of mathematical objects.

I believe that what you want to say here is this:
[All the facts about mathematical truth and methodology can be
established without appeal to the empirical (or metaphysical, ...)
existence of mathematical object"].
And I agree with this. But you still need mathematical existence. Then
I explain why (UDA) and how (arithmetical UDA, lobian interview) to
extract the other notion of more contextual form of existence.
My problem here is pedagogical (if not sometimes diplomatic): you have
to possess some good understanding of mathematical logic. Even just
concerning the arithmetical propositions p, you have to realize the
differences between

p (p is true, or satisfied by the school-learned mathematical
structure (N, +, *, 0, 1));
Bp (p is provable by the lobian machine M, fixed once and for all)
Bp & p (p is provable by M and p is true in (N, +, *, 0, 1))
Bp & Dp (p is provable by M and p is consistent with M's other
belief/theorems)
Bp & Dp & p (p is provable, consistent and true).

Now if M is a sufficiently simple correct lobian machine compared to
you (like Peano Arithmetic), then *you* can prove that Bp, Bp & p, Bp
& Dp, Bp & Dp & p are all equivalent with respect to the arithmetical
sentences p. But then you can also prove that this equivalence cannot
be proved (Bp) nor known (Bp & p) nor observed (Bp & Dp) nor "felt" (Bp
& Dp & p) by the machine M, or by any correct machine when "B" is its
own provability predicate.

> So the specialness of Time depends on the specialness of nautral
> numbers, depends on the specialness of Robinson Arithemtic ?

You are right and wrong.
Right because RA is a very precise machine/theory which has the
advantage of being both a subset of all rich lobian machine, and at the
same time a turing-complete or universal machine, and this makes it
possible to identify computability with provability in RA, or
Sigma1-provability.
You are wrong because any other machine or language could have been use
instead. Any theory capable or representing the FI and the Wi would
work. I have already try to sell the ontic SK combinator theory which
has other advantage (relating computation theory with computability
theory), but people didn't react to the posts I have send(*) about
them, and I guess the choice of RA will just make things easier if only
by allowing us to treat computability as a special case of provability.
The lobian machine we have to interview are just the extensions of RA
by induction axioms: for any formula F we accept that
[F(0) & An(F(n) -> F(n+1))] -> AnF(n).
This gives PA tremendous introspective ability, making her, not only
able to compute all the Fi and Wi like RA or any universal machine, but
able mainly to reason deeply about those things.

http://www.ltn.lv/~podnieks/gt3.html

An excellent book is the one by Eliot Mendelson:
http://www.amazon.com/Introduction-Mathematical-Fourth-Elliott-
Mendelson/dp/0412808307

Book on the modal logics of provability are the one by Smorynski and
the one by Boolos. Reference:
Smory¨½ski, P. (1985). Self-Reference and Modal Logic. Springer Verlag,
New York.
Boolos, G. (1993). The Logic of Provability. Cambridge University
Press, Cambridge

Bruno

(*) http://www.mail-archive.com/everything-list.domain.name.hidden/msg05961.html

http://iridia.ulb.ac.be/~marchal/

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Received on Fri Sep 15 2006 - 06:31:14 PDT

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