[FOM] Falsify Platonism?
Timothy Y. Chow
tchow at alum.mit.edu
Wed Apr 28 14:59:30 EDT 2010
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
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