[FOM] Falsify Platonism?
Timothy Y. Chow
tchow at alum.mit.edu
Wed Apr 28 14:41:11 EDT 2010
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
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.
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