[FOM] Falsify Platonism?

[Daniel Méhkeri]

Timothy Chow writes:

> However, I think it's still fair to use the term "platonism" if one 
> decides that the problem lies with "logic" in the sense of the 
> meaningfulness of quantifiers, or at least with too many 
> alternations of quantifiers. "Platonism" to me means the notion that 
> the natural numbers have an objective existence and that meaningful
> statements about them have a definite truth value.

If the class of meaningful sentences can be weakened to exclude 
quantifiers, then even finitists are "platonists". Yes, you are right 
that it would follow that platonism isn't falsified by an 
inconsistency in PA. The inconsistency would need to be at the level 
of primitive recursive arithmetic at least. 

Has the word ever been used this way, though?

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. 

> The inconsistency of ZFC + Mahlo would be startling, but what kind of 
> effect do you think it would have?  Mathematicians who aren't set 
> theorists typically don't even know what Mahlo cardinals are; it's 
> hard to believe that they would be perturbed.  

Sure, but CH is independent of ZFC + Mahlo isn't it? They probably 
say CH is not meaningful (or perhaps false, because they doubt the
continuum can really be well-ordered). They wouldn't be perturbed 
because they aren't really (set-theoretic) platonists.

> And set theorists have enough experience with this kind of thing to 
> be able to take it in stride (for example, I have in mind Kunen's 
> inconsistency theorem).  They would promptly dissect the argument 
> to extract its essence and would make all the necessary adjustments.
> It's precisely because set theory is a mature subject that the 
> practitioners know how to exercise "damage control" if necessary.

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.

As well, Mahlo cardinals make sense in constructive set theory, but 
in that context they are equiconsistent with a fragment of second-
order arithmetic. Assuming the latter were not also shown 
inconsistent, we would have a situation: axioms which were invented 
by platonists, which were alleged to follow from the concept of a 
set, and which stood for about a century, would be inconsistent with
classical logic, but consistent without. The "adjustment" could be 
a logical one.

Reinhardt cardinals were not around very long, nor were there or are
there many (any?) people who are sure about the largest of cardinals.
Reinhardt cardinals have no known constructive analog and are not 
known to be inconsistent with ZF.

> I think it is a reasonably fair analogy to compare an inconsistency 
> in PA today with, say, Russell's paradox back when Russell found it.  
> As far as most mathematicians are concerned, their view of the 
> integers is about as naive as the general view of set theory was about 
> a hundred years ago.

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.

Regards,
Daniel


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