the atomic and subatomic level. The problem is that relativity only mani-
fests itself in the realm of large objects, whereas quantum mechanical
effects are significant only in the world of the really, really, really small.
Many physicists agree that the single most important theoretical problem
confronting physics is the construction of a theory (generally referred to
as quantum gravity) that subsumes both these theories. Current contend-
ers include string theory and loop quantum gravity,^5 and part of the dif-
ficulty in picking a winner is devising or discovering phenomenological
results that will help distinguish between the two. After all, if the two
theories predict different results when five black holes simultaneously
merge, it might be a long wait for such an event to occur.
The melding of theories in mathematics is seamless by comparison.
Probably the first to achieve success in this area was Descartes, who
wrote an appendix to his Discourse on Method which laid the foundation
for analytic geometry. In terms of utility, the few pages Descartes wrote on
analytic geometry far outstrip the volumes he wrote on philosophy, as ana-
lytic geometry enables one to apply the precise computational tools of alge-
bra to geometric problems. Ever since then, mathematicians have been
happily co-opting results from one area and applying them to another. To-
pology^6 and algebra are, on the surface, disparate fields of study. How-
ever, there are important results in topology obtained by using algebraic
tools such as homology groups and homotopy groups (the precise defini-
tion of a group will be given in chapter 5) to study and classify surfaces,
and there are equally valuable results obtained by taking advantage of the
topological characteristics of certain important algebraic structures to
deduce algebraic properties of these structures.
Part of the charm of mathematics, at least to mathematicians, is how
results in one area can often be fruitfully used in another apparently un-
related area. My own area of research in recent years was fixed-point the-
ory. A good example of a fixed point is the eye of a hurricane; while all
hell breaks loose around the hurricane, the eye experiences not even a
gentle zephyr. Many fixed-point problems are nominally placed in the
domain of real analysis, but at the same time that I and a colleague sub-
mitted a solution to a particular problem that placed heavy reliance on
combinatorics, the branch of mathematics that deals with the number
and type of arrangements of objects, a mathematician in Greece submit-
ted a paper solving the same problem, also using combinatorics, but an
entirely different branch of combinatorics than the one that I and my col-
league employed. It hasn’t happened yet, but I wouldn’t be surprised to
see a conference on combinatorial fixed-point theory at some stage in the
future.
Reality Checks 35