papers outlined the structure of the object of dimension 16; the other
showed that no such object existed. For Seligman, it was two years of
work down the drain, but despite that setback he had a long and produc-
tive career.
For the most part, though, mathematics is extremely resilient with re-
gard to the question of how many angels can dance on the head of a pin.
While the question is open, mathematicians can write papers in which
they derive the consequences of having a particular number of dancing
angels, or of placing upper or lower limits on the number of angels. If the
question is eventually answered, even the erroneous results can be viewed
as steps leading toward the solution. Even if it is shown that this is a ques-
tion that cannot be answered, a perfectly reasonable approach is to add an
axiom regarding the existence or nonexistence of dancing angels and to
investigate the two systems that result—after all, this was the approach
that was followed when it was shown that the continuum hypothesis was
independent of the axioms of Zermelo-Fraenkel set theory. The physicist,
ever mindful that his or her results must accord with reality, is indeed
like the portrait photographer; whereas the mathematician, like the ab-
stract expressionist, can throw any array of blobs of paint on a canvas and
proudly proclaim that it is art—as did the English mathematician G. H.
Hardy, whom we encountered in the introduction.
The Difference Between Mathematical and Physical Theories
The word theory means different things in physics and mathematics. The
dictionary does a good job of explicating this difference—a theory in sci-
ence is described as a coherent group of general propositions used as ex-
planation for a class of phenomena, whereas a theory in mathematics is
a body of principles, theorems, or the like belonging to one subject. My li-
brary contains books on the theory of electromagnetism and the theory of
groups. Even though the theory of groups is not my area of expertise, I
have little difficulty navigating my way through it. On the other hand, I
got a D in electromagnetism in college (to be fair, that was the first semes-
ter that I ever had a girlfriend, and so my attention to the course in electro-
magnetism was unquestionably diverted), and one of my goals on
retirement is to read the book through to its conclusion. In my spare mo-
ments, I have picked it up and started reading—it’s still really tough sled-
ding. It’s not the mathematics that’s the problem—it’s the juxtaposition of
mathematics and an understanding of, or a feel for, physical phenomena.
A mathematical theory generally starts with a description of the objects
under investigation. Euclidean geometry is a good example. It starts with
the following axioms, or postulates.
Reality Checks 33