true geometry'.', for in fact, physicists will always use a variety of different
geometries, choosing in any given situation the one that seems simplest and
most convenient.
Furthermore-and perhaps this is even more to the point-physicists
do not study just the 3-D space we live in. There are whole families of
"abstract spaces" within which physical calculations take place, spaces which
have totally different geometrical properties from the physical space within
which we live. Who is to say, then, that "the true geometry" is defined by
the space in which Uranus and Neptune orbit around the sun? There is
"Hilbert space", where quantum-mechanical wave functions undulate;
there is "momentum space", where Fourier components dwell; there is
"reciprocal space", where wave-vectors cavort; there is "phase space",
where many-particle configurations swish; and so on. There is absolutely
no reason that the geometries of all these spaces should be the same; in fact,
they couldn't possibly be the same! So it is essential and vital for physicists
that different and "rival" geometries should exist.Bifurcations in Number Theory, and BankersSo much for geometry. What about number theory? Is it also essential and
vital that different number theories should coexist with each other? If you
asked a bank officer, my guess is that you would get an expression of
horror and disbelief. How could 2 and 2 add up to anything but 4? And
moreover, if 2 and 2 did not make 4, wouldn't world economies collapse
immediately under the unbearable uncertainty opened up by that fact? Not
really. First of all, nonstandard number theory doesn't threaten the age-old
idea that 2 plus 2 equals 4. It differs from ordinary number theory only in
the way it deals with the concept of the infinite. After all, every theorem of
TNT remains a theorem in any extension of TNT! So bankers need not despair
of the chaos that will arrive when nonstandard number theory takes over.
And anyway, entertaining fears about old facts being changed betrays
a misunderstanding of the relationship between mathematics and the real
world. Mathematics only tells you answers to questions in the real world
after you have taken the one vital step of choosing which kind of mathemat-
ics to apply. Even if there were a rival number theory which used the
symbols '2', '3', and '+', and in which a theorem said "2 + 2 = 3", there
would be little reason for bankers to choose to use that theory! For that
theory does not fit the way money works. You fit your mathematics to the
world, and not the other way around. For instance, we don't apply number
theory to cloud systems, because the very concept of whole numbers hardly
fits. There can be one cloud and another cloud, and they will come together
and instead of there being two clouds, there will still only be one. This
doesn't prove that 1 plus 1 equals 1; it just proves that our number-
theoretical concept of "one" is not applicable in its full power to cloud-
counting.On Formally Undecidable Propositions 457