[FOM] Falsify Platonism?

rgheck rgheck at brown.edu
Sun Apr 25 18:24:13 EDT 2010

On 04/24/2010 11:03 PM, Timothy Y. Chow wrote:
> All right, having said all that, why don't I think that an inconsistency
> in PA would "falsify platonism"?  Well, the way I've laid things out here,
> it's easy to see where the loopholes are.  The one that stands out to me
> is the assumption that all first-order sentences of arithmetic coherently
> express legitimate properties of the natural numbers.  This assumption
> sure seems obvious, but if we had an inconsistency in PA staring us in the
> fact, then I think it would seem less obvious.  Supposing that the
> inconsistency could be avoided by dropping down to a weaker induction
> axiom, that would be a tempting route to take.
If that isn't sufficient, then Q is inconsistent. And if Q is 
inconsistent, then, well, in that case I'd say we have far worse 
problems than the failure of Platonism.

But yes, of course, if PA is inconsistent, then the obvious thing to do 
is to back off to some sub-theory that isn't inconsistent. And then the 
thing to do will be to say that, much to our surprise, some formulas of 
first-order logic don't define true properties of natural numbers.

It is worth remembering that many people, indeed, most people believe 
that there are some predicates of natural language that do not define 
properties of natural numbers and that the failure of induction (in 
fact, of something a bit weaker) for those predicates is what shows that 
they do not define such properties. I have in mind, of course, Wang's 
Paradox and other forms of the Sorities.

> Another obvious loophole is to drop the assumption that the natural
> numbers are *defined* by the Peano axioms.  This is easy to say, and
> perhaps especially easy for a philosopher to say, but I think Monroe Eskew
> rightly points out that it is easier said than done.  If the Peano axioms
> *don't* define the natural numbers, then just what *are* the natural
> numbers anyway?
A good deal of care is needed here about what one means when one says 
that the natural numbers are "defined" by this, that, or the other 
thing. Personally, I'd prefer to talk of how our *concept* of the 
natural numbers is defined; the numbers themselves are not defined any 
more than I am. What a contradiction in PA would show, therefore, is 
that there is something incoherent about our *concept* of the natural 
numbers or, probably better, in our *concept* of the finite, 
since---unless Q is inconsistent again---it is finitude that is really 
at issue here. But it is unclear why such a failure should cast any 
special light on Platonism. As has already been pointed out, if PA is 
inconsistent, so is HA, which makes it all the more likely that the 
issue is our *concept* of the finite.

Richard Heck

More information about the FOM mailing list