[FOM] Falsify Platonism?

[Timothy Y. Chow]

Despite my drawing a very careful distinction between "PA" and "the Peano 
axioms" and emphasizing that the distinction was very important, Roger 
Bishop Jones and Markus Pantsar have conflated the two in their responses 
to me.  It's absolutely the case that an inconsistency in PA (by which, I 
repeat, I mean the usual axioms for first-order arithmetic, and which is 
*not* the same as what I mean by "the Peano axioms") would cause us to 
abandon PA, but not to abandon the natural numbers.  It's also manifestly 
the case that nobody would regard PA as "defining what the integers are."  
Nonstandard models of PA would quickly cure us of that temptation if we 
ever felt it.  But all of this is beside the point.

Richard Heck raises a more subtle issue, however.

On Mon, 26 Apr 2010, rgheck wrote:
> A corollary of this point is that a definition's status *as* a 
> definition is impermanent.

All this is fine.  The crucial point comes in the following two 
paragraphs.

> The natural numbers are a case in point. Mathematicians have been 
> proving facts about natural numbers for as long as there have been 
> mathematicians, and something like induction has been around since 
> Euclid. But a clear isolation of it as a method of proof doesn't occur 
> until the Renaissance, and nothing like the so-called Peano axioms 
> emerge until the late nineteenth century, in Dedekind. (Frege had 
> something equivalent, but essentially second-order, but definitely later 
> and maybe not independently; Pierce seems to have had something 
> independently, but not really adequate.)
> 
> That makes the axioms of PA very unlike the axioms for a group, or a 
> closed field, or a projective space, or whatever. As I mentioned, even 
> in those sorts of cases, people can and do struggle to figure out how 
> best to define the relevant notion. But what is different about the 
> natural numbers is that, unlike those things, they are given 
> pre-theoretically, in everyday experience and cognition. That is why 
> everyone is a "structuralist" about groups, but philosophers argue the 
> issue about numbers. And it is why one might reasonably think that, if 
> the axiomatization adopted a hundred or so years ago turns out to be 
> inconsistent, then we will not conclude that there are no such things as 
> natural numbers, "because natural numbers just are what the axioms 
> characterize", but look for another way to characterize the structure in 
> question, which was known to us long before the axioms of PA were.

I said earlier that I agree that this is potentially an option one could 
consider.  My point, however, was that this is easier to say than to do.  
Let's get down to brass tacks: If we were to find an inconsistency in PA 
(and again, I emphasize the important distinction between "PA" and "the 
Peano axioms"; throughout this message, I have carefully made sure that I 
have used "PA" when I mean "PA" and "the Peano axioms" when I mean "the 
Peano axioms" and they are *not* interchangeable), just what would you 
propose as an alternative definition of the natural numbers?  I want to 
see one, not just a hand-waving argument that we could in principle look 
around for one.

Yes, people have had a pre-theoretical notion of the natural numbers for 
eons.  But that doesn't mean we can turn back the clock.  Mathematicians 
are not going to retreat to a pre-theoretical notion of the natural 
numbers.  They're going to want some definition of them that they can work 
with.  O.K., you say, what's the problem?  Mathematicians switch from one 
definition to another all the time.  The difference, however, is that 
typically we switch definitions because one or the other definition is 
more convenient, or insightful, or useful for a particular context.  The 
underlying assumption is that all the definitions are fine and that we are 
still talking about the same object when we switch to an alternative 
definition.  In the case of an inconsistency in PA, however, we have just 
raised a red flag that there may be something fundamentally wrong about 
our definition.  It's not just a matter of switching to an equivalent 
definition that is more convenient or insightful.  If the first definition 
was bad, then switching to an equivalent definition is also going to be 
bad; that's surely what "equivalent" means.

I just don't see a really viable candidate for an alternative to the Peano 
axioms.  Something based on set theory?  Category theory?  These are not 
likely to allay any concerns raised by the discovery of an inconsistency 
in PA.  That is why I think the real way out is to stick to the Peano 
axioms but to renegotiate what we accept as a meaningful "property" of the 
natural numbers.  (Once again, I emphasize that I said "PA" once in this 
paragraph and "the Peano axioms" twice and that is what I meant; they are 
not interchangeable.)

> Now, granted, it is hard to imagine that our concept of finitude might 
> have to undergo some radical restructuring. But I do not myself see why 
> the truth of Platonism should pronounce one way or another on the 
> question.

This part of the conversation is, I think, just turning into a semantic 
debate about what the word "platonism" means so I'm not going to argue the 
point any further.

Tim