[FOM] Falsify Platonism?
rgheck at brown.edu
Mon Apr 26 08:35:16 EDT 2010
On 04/25/2010 10:14 PM, Timothy Y. Chow wrote:
> This is precisely the kind of proposal that philosophers might find
> satisfying but that is not likely to satisfy the typical mathematician.
> If everything hinges on drawing a sharp distinction between the natural
> numbers and our concept of the natural numbers, then the mathematician is
> going to be perplexed, since this is not the kind of sharp distinction
> that is customarily demanded in mathematical discourse.
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. Surely it is patently
obvious that the stars themselves could not care less about what we
think of them. If this is not obvious, then one need only reflect upon
the fact that our present concept of a star is a fairly recent creation,
whereas the starts themselves are (one would have thought) rather older.
Moreover, if our theory of what stars are proves to be false, then we
will not declare the stars at fault but revise our theory.
Mutatis mutandis then for numbers and our concept of them.
What I am about to say may not be (and probably isn't) quite right, but
it will serve to fix ideas: Concepts are *psychological* in nature.
Stars and numbers are not remotely psychological in nature.
> After all, which mathematical properties are satisfied by the natural numbers but
> not by our concept of the natural numbers, or vice versa? After all, which
> mathematical properties are satisfied by the natural numbers but not by
> our concept of the natural numbers, or vice versa? Are they isomorphic or
> not? If they're isomorphic, or if asking that question is a category
> error, then why should the mathematician care about the distinction?
Not being a mathematical object, our concept of the natural numbers does
not have mathematical properties; so yes, it is a "category error" to
ask these questions. I suppose that the mathematician should care about
the distinction because he or she wants to think clearly and not confuse
facts about us with facts about numbers. E.g., not confuse questions
about numbers with question about what we think about numbers.
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