Le 12-sept.-06, à 19:20, 1Z a écrit :
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.
(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).
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.
Perhaps this links could help:
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��ski, 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@eskimo.com/msg05961.html
http://iridia.ulb.ac.be/~marchal/
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