[FOM] What produces certainty in mathematical proofs?
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
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.
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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