[FOM] Is Godel's Theorem surprising?
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Tue Dec 12 02:41:50 EST 2006
Antonino Drago <drago at unina.it>:
> According to Poincaré's criticism to formalists methods, we cannot prove
> the
> consistency of the totality of theorems of arithmetics unless we apply
> the
> induction principle, which however essentially belongs to this totality
> (according to van Heijnoorth From Frege to Goedel, Hilbert never answered
> to
> this criticism).
Hilbert did respond that the alleged consistency proof for the infinistic
mathematics (with unrestricted induction), which is to be carried out in
finitistic (meta-)mathematics, only uses restricted induction, and
therefore is not circular.
- For example, one should prove Cons(PA) in PRA. We now know that this
cannot be done, but one can't conclude this by pure philosophical
reflection.
I am afraid I did not understand your idea about a possible philosophical
component in Goedel's theorem... It is certainly a strictly mathematical
result, right?
> Actually what was surprising for me in studying the history of Goedel
> theorem is why intuitionists never commented in an attentive way this
> theorem (I was capable to discover some remarks only in A. Heyting:
> Intuitionnisme, Gauthier-Villars) although they could proclaimed it as
> their indisputable victory.
Yes, it is striking. I myself think - if we forget Brouwer and his truly
radical views on logic, language etc - that for many variants of
intuitionism, Goedel's results cut both ways. On the one hand, they
arguably undermine formalism (but who said that formalism is the only
alternative for intuitionism?)
On the other hand, if we consider intuitionistic logic (which, for
Brouwer, was only of secondary interest anyway) and its standard proof
interpretation, we must conclude, because of Goedel's results, that the
notion of proof presupposed there is highly abstract, for it cannot
coincide with provability in any possible formalized system - however
strong. This much is sometimes admitted, in passing, by contemporary
intuitionists, but the situation would certainly deserve more
philosophical reflection.
Goedel had some critical remarks about this, arguing that the notion of
proof involved is insufficiently clear. And I am inclined to think that
the situation poses more problems for those contemporary intuitionist who
give a central role for intuitionistic logic than is generally recognized.
(I mean philosophical problems; I am not suggesting that it provides a
strict refutation of such intuitionism.)
All the Best
Panu
Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm
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