O'Reilly logo

Fearless Symmetry by Robert Gross, Avner Ash

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required