[FOM] Falsify Platonism

[Colin McLarty]

"Timothy Y. Chow" <tchow at alum.mit.edu>
Date: Mon, 26 Apr 2010 22:00:12 -0400 (EDT)

wrote

> Let's get down to brass tacks: If we were to find an inconsistency in
> PA, ...  just what would you propose as an alternative definition of
> the natural numbers?  ....
> I just don't see a really viable candidate for an alternative to the Peano
> axioms.  Something based on set theory?  Category theory?

If we simply learned that there was a contraction, without learning
what it was, then I freely say would be at a loss.   But more likely a
discovery of a contradiction would pinpoint certain uses of induction
that lead (astonishingly) to the contradiction.  The alternative
definitions would aim to restrict induction in some way -- presumably
some way unrelated to the restrictions we usually consider today,
unaware as we are of any contradiction lurking in the standard
induction scheme.

best, Colin