in a password or take money from an ATM, because this difficulty in fac-
toring large numbers that are the product of two primes is the corner-
stone of many of today’s computerized security systems.
Like number theory, each of the Big Three has had a profound, although
somewhat delayed, impact. It took a while, but the uncertainty principle,
and the science of quantum mechanics of which it is a part, has brought
us most of the microelectronic revolution—computers, lasers, magnetic
resonance imagers, the whole nine yards. The importance of Gödel’s
theorem was not initially appreciated by many in the mathematical com-
munity, but that result has since spawned not only branches of mathe-
matics but also branches of philosophy, extending both the variety of the
things we know, the things we don’t, and the criteria by which we evalu-
ate whether we know or can know. Arrow did not receive a Nobel Prize
until twenty years after his theorem was first published, but this result
has significantly expanded both the range of topics and the methods of
studying those topics in the social sciences, as well as having practical
applications to such problems as the determination of costs in network
routing problems (how to transmit a message from Baltimore to Beijing
as cheaply as possible).
Finally, a surprising common element uniting these three results is that
they are—well, surprising (although mathematicians prefer the word
counterintuitive, which sounds much more impressive than surprising).
Each of these three results was an intellectual bombshell, exploding pre-
conceptions held by many of the leading experts in their respective fields.
Heisenberg’s uncertainty principle would have astounded Laplace and
the many other physicists who shared Laplace’s deterministic vision of
the universe. At the same mathematics conference that David Hilbert, the
leading mathematician of the day, was describing to a rapt audience his
vision of how mathematical truth might some day be automatically ascer-
tained, in a back room far from the limelight Gödel was showing that
there were some truths whose validity could never be proven. Social sci-
entists had searched for the ideal method of voting even before the suc-
cess of the American and French Revolutions, yet before he even finished
graduate school, Arrow was able to show that this was an impossible
goal.
The Difficult We Do Today, but the Impossible Takes Forever
There is a fairly simple problem that can be used to illustrate that some-
thing is impossible. Suppose that you have an ordinary eight-by-eight
chessboard and a supply of tiles. Each tile is a rectangle whose length isIntroduction xiv