[FOM] Falsify Platonism?

[Timothy Y. Chow]

On Mon, 26 Apr 2010, rgheck wrote:
> I was not trying to draw any complicated distinction that should be 
> controversial. Surely we can all agree that there is a rather large 
> difference between stars and our concept of stars.

The distinction you're drawing is not at all complicated or controversial 
*to a philosopher*.  It's something you expect your better undergraduate 
philosophy students to grasp without much difficulty.  But I maintain that 
this is *not* the sort of distinction that a mathematician is accustomed 
to having to draw in order to think clearly about mathematics.  As a 
mathematician, I know that I have to distinguish carefully between 
different but isomorphic copies of a field F embedded in another field K.  
I have to distinguish between a manifold and its embedding in R^n.  I have 
to distinguish between a random variable and the value it takes in a 
particular trial.  These are subtle distinctions but I know it's important 
to keep them straight.

But why should it be important, in the practice of mathematics, to 
maintain a distinction between N and the concept of N?  Why are the 
philosophers telling me that I can't define N to be "anything that 
satisfies the Peano axioms" and that only the "concept of N" can be so 
defined?  This represents an intrusion of philosophy into mathematics that 
does not currently exist.  I still say that if you're right about the 
distinction between N and the concept of N being of critical importance, 
then some version of platonism is falsified by PA, even if it's true that 
a slightly more sophisticated version of platonism isn't.

Tim