At some point in his or her life every working mathematician has to

explain to someone, usually a relative, that mathematics is hardly

a ﬁnished project. Mathematicians know, of course, that it is far

too soon to put the glorious achievements of their trade into a big

museum and just become happy curators. In many respects, the

study of mathematics has hardly begun. But, at least in the past,

this has not always been universally acknowledged.

Recent successes (most prominently the proof of Fermat’s Last

Theorem) have advertised to a wide audience that math remains

humanity’s grand “work-in-progress,” where mysteries abound and

profound discoveries are yet to be made. Along with this has come

a demand from a larger public for genuinely expository, but serious,

accounts of currently exciting themes in mathematics.

It is a hard balancing act: to explain important and beautiful

mathematical ideas—to truly explain them—to people with a gen-

eral cultural background but no technical training in math, and yet

not to slip away from the full seriousness and ambitious goals of the

subject being explained.

Avner Ash and Robert Gross do a wonderful job with this

balancing act in Fearless Symmetry. On the one hand the substance

of their book is honestly—fearlessly, even—faithful to the great

underlying ideas of the mathematical story that they tell. On

the other hand, the authors are keenly sensitive to the basic,

almost premathematical, issues that would occur to, and perhaps

challenge, a newcomer to these ideas, and they treat these issues

with an exemplary level of thoughtfulness.

xvi FOREWORD

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

3

;9= 3

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

are solved.

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

versa.

The mathematics of symmetry also has had an astounding

history. It timidly makes an appearance in Euclid’s Elements under

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