[FOM] Falsify Platonism?

[Timothy Y. Chow]

Daniel Mehkeri wrote:
> All of this seems true, except for the implicit claim that what is left 
> is "Platonism" in any sense. It's not even constructivism. To deny 
> meaning to quantifiers over all of N is to adopt some form of finitism, 
> if not ultrafinitism. No?
[...]
> because wouldn't it be felt that "logic" as we know it had been shown to 
> be nonsense?

Yes, I find your argument here pretty compelling.  It seems that 
*something* has to give.  Further, if one takes the route of ultrafinitism 
then I agree that "platonism" is not a very good word any more.  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.

> This analogy was made by others, but I'm not sure. Set theory was very
> young and not established at the time. Also that was long ago and those
> mathematicians are all dead. Wouldn't an antinomy in even ZFC+Mahlo
> have a completely different effect now?

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.  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.

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.

Tim