to anyone but a specialist, and totally useless for any practical purpose.
My appreciation for abstract expressionism, as well as my understanding
of it, is limited—but I might look at it anew, considering that a Jackson
Pollock recently sold for $140 million. Maybe this analogy is not so bad,
because highly abstract areas of mathematics have turned out to have
significant—and unexpected—practical value, and $140 million is a lot
of practical value.
The successes of physics are extraordinary—but its failures are extraor-
dinary, too.
One of the early theories of heat was the phlogiston theory. The phlogis-
ton theory states that all f lammable substances contain phlogiston, a
colorless, odorless, weightless substance that is liberated in burning. I
strongly doubt that anyone ever produced a truly axiomatic theory of
phlogiston, but if anyone did, the moment that Antoine Lavoisier showed
that combustion required oxygen, the phlogiston theory was dead as the
proverbial doornail. No further treatises would be written on phlogiston
theory because it had failed the acid test—it did not accord with observa-
ble reality. This is the inevitable fate that awaits the beautiful physical
theory that collides with an ugly and contradictory fact. The best that can
be hoped for such a theory is that a new one supersedes it, and the old
theory is still valid in certain situations. Some venerable theories are so
useful that, even when supplanted, they still have significant value. Such
is the case of Newton’s law of gravitation, which still does an admirable
job of predicting the vast majority of everyday occurrences, such as high
and low tides on Earth. Even though it has been superseded by Einstein’s
theory of general relativity, it is fortunately not necessary to use the tools
of general relativity to predict high and low tides, as those tools are con-
siderably more difficult to use.
Mathematics rarely worries about reality checks. There are exceptions,
such as the tale related to me by George Seligman, one of my algebra in-
structors in college, whose classes I greatly enjoyed. The real numbers—
the continuum discussed in the previous chapter—form a certain type of
algebraic system of dimension 1.^4 The less-familiar complex numbers
(built up from the imaginary number i�1) form a similar structure of
dimension 2, the quaternions one of dimension 4, and the Cayley num-
bers a structure of dimension 8. Seligman said that he had spent a couple
of years deriving results concerning the structure of dimension 16 and
was ready to publish them when someone showed that no such structure
existed, and that the four known structures described above were all
there were. Interestingly, at that time two manuscripts had been submit-
ted for publication to the prestigious Annals of Mathematics. One of the
32 How Math Explains the World