[FOM] Falsify Platonism?

[Monroe Eskew]

On Thu, Apr 29, 2010 at 1:28 AM,  <Andre.Rodin at ens.fr> wrote:
>
> In my view an interesting refutation of Platonism about natural numbers would be
> not the discovery of a contradiction in PA but a creation of alternative
> theories of arithmetic. The pre-theoretical notion of number may be well
> compatible with many different theoretically refined notions of number. I
> expect that such alternative theories of arithmetic will be developed in a near
> future just like alternative theories of geometry were developed in 19th
> century.

I disagree.  PA = Q + Induction.  I'm sure you can find evidence of
mathematicians of various cultures using each of the Q axioms in some
equivalent form dating back to the invention of zero.  Induction was
also used extensively before PA was written down.  People didn't think
of it as induction on first order formulas, since they didn't have
that notion, but some vague notion of property that surely includes
first order properties.  (Can anyone find evidence of a historical
mathematician doubting the validity of induction?)  So PA must be true
of the pre-theoretical notion.

So I think we can take this a step further: Regardless of your
metaphysics, finding a contradiction in PA would pretty much refute
the whole multi-thousand-year collaborative project called
mathematics.