both instances, one is powerfully driven to ~sk, "But which one of these two
rival theories is correct? Which is thl' truth?" In a certain sense, there is no
answer to such a question. (And yet, in another sense-to be discussed
later-there is an answer.) The reason that there is no answer to the
question is that the two rival theories, although they employ the same
terms, do not talk about the same concepts. Therefore, they are only
superficially rivals, just like Euclidean and non-Euclidean geometries. In
geometry, the words "point", "line". and so on are undefined terms, and
their meanings are determined by the axiomatic system within which they
are used.
Likewise for number theory. When we decided to formalize TNT, we
preselected the terms we would use as interpretation words-for instance,
words such as "number", "plus", "times", and so on. By taking the step of
formalization, we were committing ourselves to accepting whatever passive
meanings these terms might take on. But-just like Saccheri-we didn't
anticipate any surprises. We thought we knew what the true, the real, the
only theory of natural numbers was. We didn't know that there would be
some questions about numbers which TNT would leave open, and which
could therefore be answered ad libitum by extensions of TNT heading off
in different directions. Thus, there is no basis on which to say that number
theory "really" is this way or that, just as one would be loath to say that the
square root of - 1 "really" exists, or "really" does not.Bifurcations in Geometry, and PhysicistsThere is one argument which can be, and perhaps ought to be, raised
against the preceding. Suppose experiments in the real, physical world can
be explained more economically in terms of one particular version of
geometry than in terms of any other. Then it might make sense to say that
that geometry is "true". From the point of view of a physicist who wants to
use the "correct" geometry. then it makes some sense to distinguish be-
tween the "true" geometry, and other geometries. But this cannot be taken
too simplistically. Physicists are always dealing with approximations and
idealizations of situations. For instance, my own Ph.D. work, mentioned in
Chapter V, was based on an extreme idealization of the problem of a crystal
in a magnetic field. The mathematics which emerged was of a high degree
of beauty and symmetry. Despite--or rather, because of-the artificiality of
the model, some fundamental features emerged conspicuously in the
graph. These features then suggest some guesses about the kinds of things
that might happen in more realistic ~ituations. But without the simplifying
assumptions which produced my graph, there could never be such insights.
One can see this kind of thing over and over again in physics, where a
physicist uses a "nonreal" situation to learn about deeply hidden features of
reality. Therefore, one should be extremely cautious in saying that the
brand of geometry which physicists might wish to use would represent "the
(^456) On Formally Undecidable Propositions