[FOM] What produces certainty in mathematical proofs?

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Mon Sep 24 01:59:56 EDT 2007

Martin Davis was wondering about certainty in mathematics, and wrote: 

> Harvey Friedman has emphasized the progression of 
> means of proof from the most basic to the highest 
> regions of the transfinite with a decision to 
> draw the line at at particular level a matter of 
> personal idiosyncrasy rather than the result of philosophical
> acumen.

Some time ago, Martin Davis himself wrote, in a different context 
(commenting our knowledge of the truth of the Goedel sentences):

> If the underlying system is PA (first order number theory), 
> I believe that one can claim that the evidence for its consistency 
> is simply overwhelming.  ...
> For higher order systems like type theory or ZFC, I know no reason for 
> believing in their consistency other than the fact that the axioms are 
> satisfied by our intuitive Cantorian picture of sets of sets of sets 
> of ...

Not only I agree, but I am inclined to accept the more general picture 
this suggests about mathematical certainty: that certainty is a matter of 
degree, not a black-and-white issue as many suppose.  

BTW, Heyting (of all!) suggested a similar view. 

Best, Panu

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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