The Equation that Couldn't Be Solved

The Equation that Couldn't be Solved:  How Mathematical Genius Discovered the Language of SYMMETRY.  Mario Livio. 2005, 353 pp., $26.95, cloth.  New York:  Simon & Schuster,   ISBN-13: 978-0-7432-5820-3

Any student who learns the quadratic formula might ask, “Can you find the roots of a cubic or higher-degree polynomial in a similar way?”  A very readable answer lies in this book, which has a much larger agenda, namely, to convince the reader that “Symmetry Rules”—everywhere, it seems.

Chapters 3-5 are the most fascinating.  Here Livio traces the emergence of what we call algebraic equations, and the progress in finding algorithms to solve linear, quadratic, cubic, and quartic equations.  Drama abounds in the sixteenth century—mathematical contests, secrets, jealousies, egos—with del Ferro, Fiore, Tartaglia, Cardano, and Ferrari at center stage. 

The story of the quintic equation jumps to the eighteenth century with important new observations by Euler, Lagrange, and Gauss, but no progress on the general question.  Ruffini first claimed a proof of the unsolvability of the quintic, and he, like Abel and Galois after him, was thwarted over and over by the mathematical titans of the day who ignored, lost, dismissed, or buried his work. Abel, “The poverty-stricken mathematician”, and Galois, “The romantic mathematician”, each have a full chapter recounting the riveting and heartbreaking stories of their short lives.

Every scientific writer who chooses to address a general audience  must tell the truth, but how much of the truth?  Livio chooses to be timid.   He tells us that Abel used a reductio ad adsurdum argument to prove that the general quintic could not be solved by algebraic operations, but we learn nothing more.  The chapter on groups gives a good description of permutations, permutation groups, and even what a solvable group is, in order to state Galois' theorem, “The condition for an equation to be solvable by a formula is that its Galois group should be solvable.”  But then, rather than illustrate with a simple example (even in the appendix), Livio goes off on several tangents (as he often does throughout this book), discussing such topics as: the dating game; lotteries; groups in kinship rules, and others.

So do not expect to find here any real mathematical answers that address the book's title.  Less than half the book really addresses that subject.  Mostly the book is about symmetry, its appearance everywhere, and its manifestations.  “Symmetry represents the stubborn cores of forms, laws, and mathematical objects that remain unchanged under transformations.”  There are many examples drawn from physics—spanning classical Newtonian to string theory illustrating how its laws are intertwined with symmetry.  Despite a few weak caveats about attributing everything to symmetry, occasionally symmetry gets confused with duality.  Still, the author clearly believes there are few aspects of life untouched by symmetry.

For the reader who seeks more information, Livio has included 25-pages of references as well as substantial endnotes  that indicate titles, but, unfortunately, not specific pages of sources.

Doris Schattschneider, Prof. Emerita of Mathematics, Moravian College, Bethlehem, PA

See also the MAA Review by Darren Glass.