[FOM] Falsify Platonism?

[Richard Heck]

On 04/23/2010 02:46 AM, W.Taylor at math.canterbury.ac.nz wrote:
> Quoting Richard Heck <rgheck at brown.edu>:
>
>> Why would that falsify Platonism?
>
> Oh sorry, I thought that was obvious enough not to need stating.
>
>> It seems to me that it would simply
>> show that PA is the wrong theory of the numbers.
>
> But it is clear to us what basic properties numbers have.  PA.
> If Platonism is to mean anything worthwhile, it must mean this -
> that knowing "what" numbers are, means knowing their (basic) properties.
>
> So from a Platonistic PoV, PA "can't" be wrong!
>
I'm sorry, but I still don't see this. Platonism is a metaphysical view, 
a view about what the truth of mathematical statements consists in. It 
says that mathematical objects exist objectively, independently of our 
thought (or capacity for thought) about them, and that their properties 
are equally independent of our knowledge (or capacity for knowledge) of 
them. Such a view is in no way committed to any claim about what the 
truth about numbers is.

Granted, we tend to suppose that we "know" what the basic properties of 
the numbers are. But maybe we don't know. Maybe we only think we know. 
That, indeed, is one of the central worries that people have about 
Platonism: It looks at least conceivable that the mathematical facts, as 
they are in mathematical heaven, could come arbitrarily apart from what 
we think we know.

In any event, claims about what we know, about what could or could not 
prove to be false, are epistemological and so their relation to the 
metaphysical claim is not at all straightforward.

Of course, if you want to use "Platonism" as a name for some view other 
than the metaphysical one described, that's up to you. But you're likely 
to confuse a lot of people.

Richard Heck