Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. Peter Pesic. viii + 213 pp. The MIT Press, 2003. $24.95.
Capturing the essence of a problem with a symbolic equation and then solving that equation is an extremely powerful process. One indication of its power is that it has become the dominant way of thinking about mathematical problems. Algebra, for many years considered inferior to her older sister Geometry, has almost completely taken over the mathematical terrain.
The story of equations really begins with the algebraists of medieval Islam. Algebraic equations take the form a_{n}x^{n} + a_{n–1}x^{n–1} + ... + a_{0} = 0 and are classified by n, the degree. If n = 1, the equation is linear; if n = 2, it is quadratic; if n = 3, 4, or 5, it is cubic, quartic or quintic (or of degree 3, 4 or 5), and so on. The early algebraists formulated and knew how to solve quadratic equations, but they found that equations of degree three were much harder. These mathematicians devised methods for approximate solution, but they liked tidiness (as do their modern counterparts) and kept looking for a complete, exact solution.
When algebra came to Italy around 1200 A.D., these questions came with it. The masters of calculation of that era played with the problem, but little progress was made until the 16th century. At that point, a flurry of mathematical activity spanning no more than 50 years yielded complete solutions for equations of degree three and degree four. Such significant success in so little time was cause for much optimism, and in the ensuing decades many mathematicians thought about how to solve equations of higher degree.
What exactly were they looking for? Well, as we learned in high school, a quadratic equation of the form ax^{2} + bx + c = 0 can be solved by using a formula that involves root extraction and the standard operations of arithmetic. The 16th-century solutions for the equations of degree three and degree four were similar. The holy grail, then, was to obtain such formulas for more general equations or, failing that, for the next step: the equation of degree five—the quintic.
It was a hard problem, but still it seems that very few people suspected what the final answer was going to be until late in the 18th century: The formula for an equation of degree five simply does not exist. This does not mean such equations don't have solutions. Rather, it means that a solution cannot be obtained with a formula of the type just described. Although the Italian mathematician Paolo Ruffini was the first to claim to have proved this fact, it is generally agreed that the first correct proof was that of Niels Henrik Abel.
Born in Norway to impoverished parents, Abel spent most of his short life battling difficult financial circumstances and trying to get the attention of mathematicians in France and Germany, then the real centers of mathematical research. Poverty led to illness, but Abel's work did interest many people, and in 1829 August Leopold Crelle obtained a teaching post for Abel in Germany. Alas, Crelle's message arrived in Norway two days after Abel died in April of 1829. He was just 26 years old.
In Abel's Proof, Peter Pesic sets out to tell the story of algebraic equations and their solution, starting in Greek times. Pesic wants not just to present this history but to explain Abel's argument for the unsolvability of the quintic, to explore its implications and to reflect on the very notion of unsolvability. He is quite successful in achieving his first goal, partly successful in giving us access to Abel's proof and mostly unsuccessful in shedding light on "the sources and meaning of mathematical unsolvability."
Pesic's history, which forms the core of the book, is well written and, for the most part, reliable. After a very quick survey of algebraic equations from antiquity to the 18th century, he slows down to discuss the work of Joseph-Louis Lagrange, Ruffini, Abel, Evariste Galois, and their successors and interpreters. The final chapters look at the role of noncommutativity in the unsolvability of some equations and explore the notion of "solving" a problem by proving that it is unsolvable. Throughout, most of the mathematical details are relegated to shaded boxes or appendices where they will not threaten the nontechnical reader. The exception to this is a very nice explanation of the basic ideas of finite group theory.
Writing a book aimed at the general public on the history of a nontrivial bit of mathematics requires some daring. One is liable to be nitpicked to death both by mathematicians and by historians. As someone who is a historian as well as a mathematician, I will acknowledge the temptation and admit that I have many nits to pick. Many pages in my copy are marked with queries, disagreements, corrections and comments to the effect that some things could have been done better. But to indulge this temptation would, in the end, be unfair to a book that does quite a good job of giving the reader access to some fascinating mathematics. Abel's proof that the quintic equation simply could not be solved (at least as that task was originally posed) is a remarkable turning point in the history of mathematical thought, one that should be more widely known. Pesic's book is a good place to begin to learn about this important piece of intellectual history.—Fernando Q. Gouvêa, Mathematics, Colby College, Waterville, Maine