[FOM] Re: Indispensability of the natural numbers

[Timothy Y. Chow]

On Tue, 18 May 2004,  David Isles wrote:
> Roughly speaking, as I understand it, the usual picture of the
> "standard" natural numbers (closed under all primitive recursive
> functions) is to the "true" natural numbers (given by rules R1, R2, and
> R3) as the "nonstandard" model theoretic natural numbers are to the
> "standard" natural numbers.

Off-list, David Isles has clarified to me that "roughly speaking" was an
important rider here, and that there isn't a known way to formalize "There
exist N1 and N2 that satisfy R1, R2, R3, but N1 is closed under primitive
recursive functions while N2 is not."

However, his comments, including the later comments in this email about
the gradual evolution of mathematical concepts, have provided a new (to
me) insight.  This seems like something I should have observed before
because it's simple and obvious, but I had not articulated it to myself
before.  Namely, one way to react to the inconsistency of PA (and similar
matters) is to retain the notion that there is a unique and determinate
entity called "N," but concede that it doesn't have all the *properties*
that we thought it did.  Once stated, this move seems obvious to me, since
it is analogous to how set-theoretic inconsistencies are typically dealt
with; also, we are already used to admitting that there are lots of
properties of N that we're not sure it really has.

Of course, it is still true that the properties in question are rather
basic ones, so it is still difficult for me to imagine that N doesn't
have them.  However, since powerful induction axioms correspond to
exceedingly fast-growing functions that we don't have direct experience
with, I can sort of buy the argument that our pre-theoretic notion of N
does not intrinsically come with the notion of closure under such
functions, and that this notion arose only *after* we had developed a
rather sophisticated theoretical understanding of N.  Later sophisticated
ideas are more likely to be incoherent because of their complexity.

I still think that syntax and arithmetic are (nearly) interchangeable,
so skepticism about the natural numbers still translates into skepticism
about syntax.  However, the above observation indicates that the
skepticism might not need to be about the uniqueness or determinateness
of syntactic rules, but only about what properties they have.

It's easier for me to see now how mathematicians could adapt to an
inconsistency in PA; there would be no need to reject the notion of
the standard integers or to find a simpler replacement (as I was more
or less arguing before).  Only certain assumed *properties* of N would
have to be abandoned.

On the other hand, it still seems to me that a lot of infinitary
mathematics would become suspect.  It would be interesting to investigate
how much of it could be salvaged.

Tim