The authors also bring out the eternally unﬁnished aspect of
math, its open-ended quality. The resolution of any part of math-
ematics invariably modulates the subject into a different key, and
makes a new and deeper set of questions vital. One theorem having
been proved, more further-reaching problems come to prominence.
Fermat’s Last Theorem, posed over 350 years ago, has been proved;
the curious Problem of Catalan, conceived over a century ago to
prove that 8 and 9 are the only two consecutive perfect powers
(8 = 2
), has recently been solved. But you need only glance
at the last chapters of this book to see how, in the wake of the
resolution of older problems, a new, and possibly richer, repertoire
of interesting problems has come to occupy center stage, which
would have astounded ancient Diophantus.
And waiting for future generations are the sweeping expecta-
tions posed by celebrated problems such as the ABC conjecture and
the Riemann Hypothesis.
Fearless Symmetry begins where few math books do, with an
enlightening discussion of what it means for one “thing” to represent
another “thing.” This action—deeming A a “representation” of B—
underlies much mathematics; for example, counting, as when we
say that these two mathematical units “represent” those two cows.
What an extraordinary concept representation is and always has
been. In Leibniz’s essay On the Universal Science: Characteristic
where he sketched his scheme for a universal language that would
reduce ideas “to a kind of alphabet of human thought,” Leibniz
claimed his characters (i.e., the ciphers in his universal language)
to be manipulable representations of ideas.
All that follows rationally from what is given could be found by
a kind of calculus, just as arithmetical or geometrical problems
Nowadays, whole subjects of mathematics are seen as represented
in other subjects, the “represented” subject thereby becoming a
powerful tool for the study of the “representing” subject, and vice
The mathematics of symmetry also has had an astounding
history. It timidly makes an appearance in Euclid’s Elements under