[FOM] Falsify Platonism?

[Timothy Y. Chow]

Daniel Mehkeri wrote:
> For the record, I thought a platonist was more or less someone who, say, 
> has a non-trivial opinion about the continuum hypothesis. I can accept 
> the distinction between "set-theoretic" and "number-theoretic" 
> platonism, as Bill Taylor called it, but even the latter means something 
> stronger than finitism.

I could perhaps be swayed here, because we're starting to split hairs 
about what the word "platonism" means.  However, I still think that 
there's a distinction between being a platonist about the natural numbers 
and being a platonist about first-order logic.  Later on you wrote:

> Set theory wasn't the foundation of anything a century ago. It is
> now, though not necessarily so, and many are not really platonists 
> about them anyway. 
> 
> The natural numbers have always been fundamental and always will be.

The natural numbers, sure, but first-order logic hasn't been foundational 
for nearly that long.  We're not about to give up on the natural numbers, 
but I think there's more room for giving up on the claim that every 
first-order formula of arithmetic expresses a meaningful property of the 
natural numbers.  That's why I still see a distinction.

> Mahlo cardinals are very old, and are often alleged to follow from the 
> iterative concept of set plus the idea that we shouldn't be 
> unnecessarily restrictive about what counts as a set. I think platonists 
> are really quite confident about these, so this would be fairly 
> profound.

Could be.  I'd be curious to poll set theorists on this one.  Maybe it's 
not until measurable cardinals that some acrophobia starts to set in.

Tim