[FOM] Constructive reactions to Goedel Incompleteness

[praatika@mappi.helsinki.fi]

giovanni sambin <sambin at math.unipd.it>:
 
> Why have constructivists been so little interested in Goedel's 
> theorems? (For example, the proof mentioned above is absolutely 
> elementary, but I could not find anything similar in the literature.) 
> It seems to me that the explanation of this is shown by Brouwer's 
> beliefs, as reported in Mark van Atten's posting. For Brouwer, 
> incompleteness was neither as surprising nor as dramatic, as it was 
> for Hilbert.

Now Brouwer thought - not only that proofs are mental constructions which 
are prior to and independent of language and logic - but also that many 
proofs are infinite, whereas language is essentially finite. It was for 
this reason he thought that language cannot adequately express proofs, and 
that any formalization must be incomplete. 

Without taking any stand on this, I wish to emphasize just how original 
and radical Brouwer's view is. It is my impression that many (most?) 
contemporary intuitionists and constructivists want to distance themselves 
from these views of Brouwer. And for them, it seems to me, the 
completeness should be (without Goedel's proof) an open question. 
 
> Upon reflection, he was right. Actually, I personally see in the 
> incompleteness phenomenon a good reason to become a constructivist. 
> Incompleteness (and unprovability of consistency) is just part of a 
> dynamic view, which is fully compatible with constructivism, while it 
> is very disappointing when truth is seen as fixed, as is common to 
> the classic view.

I don't think that incompleteness phenomenon in any way speaks for 
intuitionism and against realism. Both think that there is more to 
mathematics than language and formalizations. 

> If something is accepted as true only when a proof is given, then it 
> is obvious that no fixed system of rules will give all truths: one 
> can immediately obtain a new true proposition (e.g. a statement of 
> underivability) in which those rules are taken as objects. Note en 
> passant that, since mathematics is not reducible to logic, one still 
> can consider intuitionistic logic as central and see that Panu 
> Raatikainen's fear is unfounded.

I must say that I failed to understand the argument here. 

But let me try to spell out my worry in more detail (it would require much 
more, but here is a bit). A number of qualifications are needed. 

To begin with, let us focus on intuitionists who do not accept Brouwer's 
radical views mentioned above; such intuitionists (I assume there are some 
(many?)) do not think that proofs may be in principle inexpressible in 
language, and unformalizable. Such a brand of intuitionism gives (unlike 
Brouwer) a central role for intuitionistic logic.   

Now Brouwer also thought that a statement is true only if it has been 
proved. Intuitionistic logic, on the other hand, is based on the idea that 
truth amounts to provability in principle (i.e., the mere possibility (in 
some (what?) sense) of proof). That is, the kind of intuitionism I am 
interested in here is essentially based on the notion of absolute 
provablity. 

This notion of provability, which is the foundation of intuitionistic 
logic, cannot coincide with provability in any formal system, however 
comprehensive. This is a logical fact. 

On the other hand, it seems to be vital for intuitionism that one should 
be able to recognize a proof when one sees one – that is, to require that 
the proof relation (and axiomhood) must be decidable. 

But these views do not harmonize well. Unless the human mind can see the 
truth of infinitely many independent mathematical facts - an alternative I 
find highly implausible, and in various ways contrary to the general 
spirit of intuitionism -  the notion of provability assumed by 
intuitionistic logic has very little to do with the notion of provability 
in any intuitive sense, and in particular with the provability by us 
humans (even in a somewhat idealized sense). 

This is my worry. Obviously, it does not concern the truly radical 
Brouwerian view. 

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