[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 

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 Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki

E-mail: panu.raatikainen at helsinki.fi

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