[FOM] Falsify Platonism?

[rgheck]

On 04/26/2010 10:00 PM, Timothy Y. Chow wrote:
> Richard Heck raises a more subtle issue, however.
>
>
>> 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.
>
>
In principle, of course, yes: one would really like to know what an 
alternative axiomatization of the theory of the natural numbers might 
look like. (I don't see that "definition" is really at issue now.) But, 
in practice, this strikes me as not a reasonable request. It's like 
asking a physicist to tell us what he'd propose to replace quantum 
mechanics with if there was replicable experimental evidence that 
clearly conflicted with some of its predictions. The right answer would 
be: That is going to depend upon the precise nature of the conflict 
between theory and experiment; moreover, whether one would want, after 
one saw the new theory, to say that, in some sense, it is just a new 
theory about the same things, or whether we've got new things, 
too---there is no reason to suppose that has to be clear in advance.

Similarly in the case of PA. If someone were to discover an 
inconsistency in PA, then it really wouldn't just be one inconsistency. 
As with set theory, there would likely prove to be a whole slew of 
arguments that led to contradiction. We would hope that, after a fair 
bit of work, we could get a sense for where the real issue was. And then 
we would hope to develop a sense of what a principled, rather than 
merely ad hoc, resolution of the problem would be. The right solution 
surely would not be simply to retreat to I\Sigma_n, for some n. That 
really would be ad hoc, and would probably throw lots of legitimate 
instances of induction out with the bad ones. Maybe a proper resolution 
would involve a complete re-thinking of how such theories are properly 
formulated. One just can't say in advance. And, as in the case of 
quantum mechanics, it would be foolish to insist that, no matter what 
the new theory turned out to be, Platonism would survive; I simply 
insist that we do not know, in advance, that, no matter what it turned 
out to be, it could not survive.

Here's another way of putting this point: If PA were shown to be 
inconsistent, that would make every other foundational crisis in the 
history of mathematics look trivial by comparison. It would so utterly 
upset everything we think we know that it seems hopeless even to 
speculate about what PA, or mathematics in general, might look like in 
the aftermath. Of course, like (most?) everyone else here, I'm inclined 
to think we have excellent, perhaps even sufficient, reason to believe 
that PA is consistent, so the speculation may in some sense be idle.

The situation with second-order PA---which is the closest thing we have, 
I take it, to a reasonable formulation of what Tim is calling "the Peano 
axioms"---is very different. In that connection, let me just quote 
George Boolos:

...[I]t is *not* neurotic to think [] we don't know that second-order 
arithmetic...is consistent. Do we really know that some hotshot Russell 
of the 23d Century won't do for us what Russell did for Frege? The usual 
argument by which we can convince ourselves that analysis is 
consistent...is flagrantly circular. Moreover, although we may think 
Gentzen's consistency proof for PA provides sufficient reason to think 
PA consistent, we have nothing like a similar proof for the whole of 
analysis.... We certainly don't have a constructive consistency proof 
for ZF. And it would seem to be a genuine possibility that the discovery 
of an inconsistency in ZF might be refined into that of one in analysis. 
...While we may regret that these theories may well be consistent and 
that it would probably be wise to bet on their consistency, we must not 
despair: we do not *know* that they are and need not give up the hope 
that someone will one day prove in one of them that 0=1. ("Is Hume's 
Principle Analytic?")

There's a hint of playful irony, no doubt, in the last sentence. But, in 
fact, George rightly regarded the foundational crises of the late 
nineteenth and early twentieth century as extremely fruitful and 
believed, I think, that, if we should be so lucky as to set off a 
similar crisis, it could only be similarly fruitful.

Richard Heck


-- 
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University