[FOM] Falsify Platonism?

[Andre.Rodin@ens.fr]

> >> That makes the axioms of PA very unlike the axioms for a group, or a
> >> closed field, or a projective space, or whatever. As I mentioned, even
> >> in those sorts of cases, people can and do struggle to figure out how
> >> best to define the relevant notion. But what is different about the
> >> natural numbers is that, unlike those things, they are given
> >> pre-theoretically, in everyday experience and cognition. That is why
> >> everyone is a "structuralist" about groups, but philosophers argue the
> >> issue about numbers. And it is why one might reasonably think that, if
> >> the axiomatization adopted a hundred or so years ago turns out to be
> >> inconsistent, then we will not conclude that there are no such things as
> >> natural numbers, "because natural numbers just are what the axioms
> >> characterize", but look for another way to characterize the structure in
> >> question, which was known to us long before the axioms of PA were.

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. The geometrical analogy suggests this: alternative theories of
arithmetic will be a result of "practical" work with big numbers rather than a
result of a purely theoretical speculation. Just like a geometrical space,
generally, is only *locally* Euclidean (I'm identifying here a space with a
differentiable manifold) arithmetic may turn to be only locally Classical. Big
(but finite) numbers may have features that remain hidden when one makes
calculations only with small numbers and then says "and so on". One cannot see
the limits of Euclidean geometry in the everyday experience; one needs for it a
more advanced kind of experience like an experience in doing cartography
(Gauss' case).
Such new developments in arithmetic wouldn't falsify a dogmatic metaphysical
Platonism (which is an article of faith and cannot be falsified in principle)
but would show that there is no such things as *the* natural numbers.

best,
Andrei