[FOM] Falsify Platonism?

[Timothy Y. Chow]

Monroe Eskew <meskew at math.uci.edu> wrote:
> Look at the axioms of PA.  I need not list them here.  They are such
> basic and bone-headed statements.  They follow from the concept of
> natural numbers.  If you disagree with this statement please tell me
> which of the axioms could possibly be false while its negation could
> still purport to describe something deserving of the title "natural
> numbers" (whether fictional or real).

I already said that I don't think a contradiction in PA would "falsify 
platonism" (about the integers), but let me say a bit more.  In particular 
let me try to argue the opposite side a little more forcefully, and then 
try to answer it.

As Monroe Eskew says, it's all very well to argue abstractly that a 
contradiction in PA would not cause any kind of crisis for platonism, but 
if we try to think harder about just what such a contradiction would look 
like, I think we can't get off the hook quite as easily as it might seem 
at first glance.

First of all, let's be a little more careful than usual about what we mean 
by "PA."  When I see that abbreviation, I think of first-order arithmetic, 
with the induction schema assumed to hold for all first-order formulas.  
It's worth reminding ourselves, however, that when a mathematician who has 
not specifically studied logic hears the term "Peano arithmetic," he or 
she has a less precise object in mind.  In particular, the induction axiom 
is supposed to hold for "any property" of the natural numbers.  We could 
refer to this version of the Peano axioms as "second-order PA," but I am 
going to resist that term, because the "mathematician in the street" 
doesn't know what "second-order" means.  I'll use "PA" for the usual 
first-order axiom schema and "the Peano axioms" for the vaguer concept.

If you press the mathematician in the street to say exactly what "the 
natural numbers" are, it's quite likely that the response will be 
something along the lines of, "Any structure that satisfies the Peano 
axioms."  In particular, the Peano axioms are thought of as *defining* or 
*specifying* what the natural numbers are.  Unless the mathematician has 
been contaminated by some study of logic or philosophy, the natural 
numbers will not be regarded as being some ethereal object that we somehow 
grasp directly, and some of whose properties we try to capture with the 
Peano axioms.

As evidence for what I just said, let's think about what happens when said 
mathematician is exposed to PA, and nonstandard models of PA, for the 
first time.  Typically, the mathematician is confused when it is 
demonstrated that there are objects that are *not* isomorphic to the 
natural numbers yet still satisfy PA.  I claim that the best explanation
for why people get confused about this point is that

1. their initial impression is that PA is just an attempt to write down 
   the Peano axioms more formally than we usually do; and

2. since they think of the natural numbers as being defined as "anything
   satisfying the Peano axioms," it then doesn't make sense for there to
   be such things as "nonstandard models."

Now what does all this have to do with platonism?  Well, I would argue 
that the typical platonist thinks of the natural numbers in the way that I 
have just said that "mathematician in the street" does.  The natural 
numbers are a definite thing and they are specified by the Peano axioms.  
Now the Peano axioms talk about *arbitrary properties* of the natural 
numbers, and maybe we can't quite formally articulate what that means, but 
*surely* the first-order axioms of PA are a special case.  Surely?

Well, if we're sure about that, then the inconsistency of PA would indeed 
be a bombshell for the platonist.  From a subset of the facts that the 
natural numbers must, by definition, satisfy, we would have found a 
contradiction.  The usual conception of the natural numbers must therefore 
be incoherent.

As a sociological fact, I think that the inconsistency of PA would cause 
consternation among a fair number of mathematicians.  I think it would 
drive some people who don't normally worry about these things to start 
worrying, and probably to be attracted to formalism (in the sense that, 
say, Ed Nelson uses that term).

All right, having said all that, why don't I think that an inconsistency 
in PA would "falsify platonism"?  Well, the way I've laid things out here, 
it's easy to see where the loopholes are.  The one that stands out to me 
is the assumption that all first-order sentences of arithmetic coherently 
express legitimate properties of the natural numbers.  This assumption 
sure seems obvious, but if we had an inconsistency in PA staring us in the 
fact, then I think it would seem less obvious.  Supposing that the 
inconsistency could be avoided by dropping down to a weaker induction 
axiom, that would be a tempting route to take.  Then we could still claim 
that the Peano axioms define the natural numbers, and that induction 
applies to "all properties," but that some of the formulas of first-order 
arithmetic do not, despite appearances, legitimately express actual 
"properties" of the natural numbers.  This route is uncomfortable, 
perhaps, but mathematicians could probably learn to live with it.  After 
all, this is approximately the same approach that mathematicians have 
taken with the set-theoretic antinomies.

Another obvious loophole is to drop the assumption that the natural 
numbers are *defined* by the Peano axioms.  This is easy to say, and 
perhaps especially easy for a philosopher to say, but I think Monroe Eskew 
rightly points out that it is easier said than done.  If the Peano axioms 
*don't* define the natural numbers, then just what *are* the natural 
numbers anyway?  Are we going to turn to set theory to rescue us?  
Shudder.  But what alternative is there?  Philosophers may be able to come 
up with some suggestions, but I think it would be hard to come up with 
something that would satisfy the mathematician in the street.

Tim