[FOM] Falsify Platonism?

[Julian Rohrhuber]

On 23.04.2010, at 18:17, Richard Heck wrote:

> 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
> 



Yes, I find this quite a good point. Without reference to metaphysics (whatever this turns out to be eventually), I would suggest to ask a simple question: What is an axiom? If to accept an axiom means to accept a convention and nothing more, one is not a Platonist. If to accept an axiom is to admit an independent existence, one is a Platonist. The problem however is that either view is "metaphysical", and I doubt that one can escape the choice unless simply ignoring it. Think only of the disciplinary question whether mathematics belongs to natural sciences or to the humanities. I'd be glad to hear alternatives though.

Of course, historically, certain ontologies were connected with certain methodological views, e.g. Brouwer's programme being based both on constructibility and Anti-Platonism. Perhaps one should account for a certain feedback: ontologies influence what axioms count as valid.

The problem about the question "Falsify Platonism?" is that the notion of proof itself is informal; its formalisation are axioms and deduction rules, which in turn are defined/admitted.

Julian Rohrhuber