[FOM] Falsify Platonism?

[rgheck]

On 04/28/2010 02:59 PM, Timothy Y. Chow wrote:
>
> 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?  [snip]
>
> 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.
>
>    
I agree with all of this: The new definition will be essentially like 
our current definition, except that it will restrict the class of 
formulae that define "properties" of numbers; I have not said otherwise.

As far as that goes, then, I'm in full agreement with Colin McLarty, who 
wrote:

> ...[A] discovery of a contradiction would pinpoint certain uses of induction
> that lead (astonishingly) to the contradiction.  The alternative
> definitions would aim to restrict induction in some way -- presumably
> some way unrelated to the restrictions we usually consider today,
> unaware as we are of any contradiction lurking in the standard
> induction scheme.
>
>    
The most ipmortant part of this, to my mind, is the part beginning 
"presumably". Restricting induction to I\Sigma_n, for appropriate n, 
will work (unless it is Q that is inconsistent), but it is clearly ad 
hoc and leads naturally to the question: Why on earth should \Sigma_22 
formulae define properties when \Sigma_23 formulae do not? So the 
restriction would presumably be very different, in ways we cannot now 
imagine.

Richard Heck