opens doors for us to learn fascinating things and, even better, to use
what we learn to make our lives richer beyond imagining.
However, the twentieth century also witnessed three eye-opening re-
sults that demonstrated how there are limits—limits to what we can
know and do in the physical universe, limits to what truths we can dis-
cover using mathematical logic, and limits to what we can achieve in im-
plementing democracy. The most well-known of the three is Werner
Heisenberg’s uncertainty principle, discovered in 1927. The uncertainty
principle shows that not even an individual possessed of omniscience
could have supplied Laplace with the positions and velocities of all the
objects in the universe, because the positions and velocities of those ob-
jects cannot be simultaneously determined. Kurt Gödel’s incompleteness
theorem, proved a decade later, reveals the inadequacy of logic to deter-
mine mathematical truth. Roughly fifteen years after Gödel established
the incompleteness theorem, Kenneth Arrow showed that there is no
method of tabulating votes that can satisfactorily translate the preferences
of the individual voters into the preferences of the society to which those
voters belong. The second half of the twentieth century witnessed a pro-
fusion of results in a number of areas, demonstrating how our ability to
know and to do is limited, but these are unquestionably the Big Three.
There are a number of common elements to these three results. The
first is that they are all mathematical results, whose validity has been es-
tablished by mathematical proof.
It is certainly not surprising that Gödel’s incompleteness theorem,
which is obviously a result about mathematics, was established through
mathematical argument. It is also not surprising that Heisenberg’s un-
certainty principle is the result of mathematics—we have been taught
since grade school that mathematics is one of the most important tools of
science, and physics is a discipline that relies heavily on mathematics.
However, when we think of the social sciences, we do not usually think of
mathematics. Nonetheless, Arrow’s theorem is completely mathematical,
in a sense even more so than Heisenberg’s uncertainty principle, which
is a mathematical result derived from hypotheses about the physical
world.
Arrow’s theorem is as “pure” as the “purest” of mathematics—it deals
with functions, one of the most important mathematical concepts. Math-
ematicians study all types of functions, but the properties of the func-
tions studied are sometimes dictated by specific situations. For instance,
a surveyor would be interested in the properties of trigonometric func-
tions, and might embark upon a study of those functions realizing that
Introduction xii