[FOM] Falsify Platonism?
rgheck at brown.edu
Thu Apr 29 07:36:17 EDT 2010
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
> ...[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
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