knowledge of their properties could help with problems in surveying.
The properties of the functions discussed in Arrow’s theorem are clearly
motivated by the problem Arrow initially started to investigate—how to
translate the preferences of individuals (as expressed by voting) into the
results of an election.
The utility of mathematics is due in large measure to the wide variety of
situations that are amenable to mathematical analysis. The following tale
has been repeated time and time again—some mathematician does
something that seems of technical interest only, it sits unexamined for
years (except possibly by other mathematicians), and then somebody
finds a totally unexpected practical use for it.
An instance of this situation that affects practically everyone in the civi-
lized world almost every day would have greatly surprised G. H. Hardy,
an eminent British mathematician who lived during the first half of the
twentieth century. Hardy wrote a fascinating book (A Mathematician’s
Apology), in which he described his passion for the aesthetics of mathe-
matics. Hardy felt that he had spent his life in the search for beauty in the
patterns of numbers, and that he should be regarded in the same fashion
as a painter or a poet, who spends his or her life in an attempt to create
beauty. As Hardy put it, “a mathematician, like a painter or a poet, is a
maker of patterns. If his patterns are more permanent than theirs, it is
because they are made with ideas.”^3
Hardy made great contributions to the theory of numbers, but viewed
his work and that of his colleagues as mathematical aesthetics—possess-
ing beauty for those capable of appreciating it, but having no practical
value. “I have never done anything ‘useful’. No discovery of mine has
made, or is likely to make, directly or indirectly, for good or ill, the least
difference to the amenity of the world,”^4 he declared, and undoubtedly felt
the same way about his coworkers in number theory. Hardy did not fore-
see that within fifty years of his death, the world would rely heavily on a
phenomenon that he spent a good portion of his career investigating.
Prime numbers are whole numbers that have no whole number divisors
other than 1 and the number itself; 3 and 5 are primes, but 4 is not be-
cause it is divisible by 2. As one looks at larger and larger numbers, the
primes become relatively more infrequent; there are 25 primes between
1 and 100, but only 16 between 1,000 and 1,100, and only 9 between 7,
and 7,100. Because prime numbers become increasingly rare, it becomes
extremely difficult to factor very large numbers that are the product of
two primes, in the sense that it takes a lot of time to find the two primes
that are the factors (a recent experiment took over nine months with a
large network of computers). We rely on this fact every day, when we type
xiii Introduction