[FOM] Falsify Platonism?

[Timothy Y. Chow]

On Tue, 27 Apr 2010, rgheck wrote:
> In principle, of course, yes: one would really like to know what an 
> alternative axiomatization of the theory of the natural numbers might 
> look like. (I don't see that "definition" is really at issue now.) But, 
> in practice, this strikes me as not a reasonable request. It's like 
> asking a physicist to tell us what he'd propose to replace quantum 
> mechanics with if there was replicable experimental evidence that 
> clearly conflicted with some of its predictions. The right answer would 
> be: That is going to depend upon the precise nature of the conflict 
> between theory and experiment; moreover, whether one would want, after 
> one saw the new theory, to say that, in some sense, it is just a new 
> theory about the same things, or whether we've got new things, 
> too---there is no reason to suppose that has to be clear in advance.

I grant that it's not reasonable to demand to know full details of a fix 
for something that hasn't been broken yet.  Nevertheless, there are *some* 
things we can say a priori.  We're not going to say, "Hmmm...we need a new 
definition of the natural numbers?  Well, I've always been fond of pigs. 
Why don't we define the natural numbers to be any mammal that goes oink?" 
Without knowing the details of an inconsistency in PA, I can still be 
confident that such a pig-proposal wouldn't fly.

Whatever definition we settle on, it's going to have to be recognizable as 
picking out the same thing we currently call the natural numbers.  I just 
don't see any way we could come up with anything that doesn't involve 
induction on properties in some sense.  We can debate which "properties" 
are really properties, for sure.  And without knowing which properties go 
bad, we can't delimit the correct boundaries for what constitutes an 
acceptable property.  But to say that we're going to find some other way 
to define the natural numbers that does not involve induction at all 
strikes me as bizarre.  If you can't even vaguely sketch what that might 
look like, then I have to believe that no such thing exists.  It's 
supposed to be something that we will gladly accept as defining the same 
thing we now call the natural numbers, yet we can't even hint at what it 
might look like?  If this alleged definition is so radically inconceivable 
then I can't believe that we'll all agree that it's still defining the 
same object.

Tim