Bifurcations in Number Theory, and MetamathematiciansSo bankers, cloud-counters, and most of the rest of us need not worry
about the advent of supernatural numbers: they won't affect our everyday
perception of the world in the slightest. The only people who might
actually be a little worried are people whose endeavors depend in some
crucial way on the nature of infinite entities. There aren't too many such
people around-but mathematical logicians are members of this category.
How can the existence of a bifurcation in number theory affect them? Well,
number theory plays two roles in logic: (1) when axiomatized, it is an object
of study; and (2) when used informally. it is an indispensable tool with which
formal systems can be investigated. This is the use-mention distinction once
again, in fact: in role (1), number theory is mentioned. in role (2) it is used.
Now mathematicians have judged that number theory is applicable to
the study of formal systems even if not to cloud-counting, just as bankers
have judged that the arithmetic of real numbers is applicable to their
transactions. This is an extramathematical judgement, and shows that the
thought processes involved in doing mathematics, just like those in other
areas, involve "tangled hierarchies" in which thoughts on one level can
affect thoughts on any other level. Levels are not cleanly separated, as the
formalist version of what mathematics is would have one believe.
The formalist philosophy claims that mathematicians only deal with
abstract symbols, and that they couldn't care less whether those symbols
have any applications to or connections with reality. But that is quite a
distorted picture. Nowhere is this clearer than in metamathematics. If the
theory of numbers is itself used as an aid in gaining factual knowledge about
formal systems, then mathematicians are tacitly showing that they believe
these ethereal things called "natural numbers" are actually part of reality-
not just figments of the imagination. This is why I parenthetically re-
marked earlier that, in a certain sense. there is an answer to the question of
which version of number theory is "true". Here is the nub of the matter:
mathematical logicians must choose which version of number theory to put
their faith in. In particular, they cannot remain neutral on the question of
the existence or nonexistence of supernatural numbers, for the two differ-
ent theories may give different answers to questions in metamathematics.
For instance, take this question: "Is -G finitely derivable in TNT?" No
one actually knows the answer. Nevertheless, most mathematical logicians
would answer no without hesitation. The intuition which motivates that
answer is based on the fact that if --G were a theorem, TNT would be
w-inconsistent, and this would force supernaturals down your throat if you
wanted to interpret TNT meaningfully-a most unpalatable thought for
most people. After all, we didn't intend or expect supernaturals to be part
of TNT when we invented it. That is, we-or most of us-believe that it is
possible to make a formalization of number theory which does not force
you into believing that supernatural numbers are every bit as real as
naturals. It is that intuition about reality which determines which "fork" of
number theory mathematicians will put their faith in, when the chips are
458 On Formally Undecidable Propositions