The Equation That Couldn't be Solved: How Mathematical Genius Discovered the Language of Symmetry [Hardcover]

Mario Livio's title suggests an exploration of unsolvable equations, in particular the drama enshrouding the mathematical conundrum of solving general, fifth degree polynomial equations, known as quintics. His subtitle, "How Mathematical Genius Discovered the Language of Symmetry," indicates that his work will also explore the role of symmetry in ultimately resolving the question of whether such polynomials could be solved by a formulas using nothing more than addition, subtraction, multiplication, division, and nth roots. These two subjects portend an interesting discussion on the solvability of equations and the peculiar mathematical race in Renaissance Europe to "discover" the magical formulas for solving cubics and quartics.

One could reasonably expect that the groundbreaking work of Tartaglia, Cardano. Ferraro, Galois, Abel, Kronecker, Hermite, and Klein would be encompassed in this survey, and indeed they are. However, purchasers of this book are given no indication that they will spend well over half their reading time on rehashes of Abel's tragic life story and the mythology of Evariste Galois's foolish death, Emmy Noether's challenges as a woman mathematician in Germany, a history of group theory, Einstein's theory of relativity, the place of string theory in modern cosmology, the survival benefits of symmetry in evolution, Daniel Gorenstein's 30-year proof that "every finite simple group is either a member of one of the eighteen families or is one of the twenty-six sporadic groups," a trite and unnecessary diversion on human creativity, and finally, an even more outlandish (and utterly inconclusive) "comparison" of Galois's brain with that of Albert Einstein. The persevering reader can only conclude that anything and everything that remotely touches upon the quintic and Galois's work was given a chapter of its own, a mathematical version of "everything but the kitchen sink." The end result is an unfortunate mishmash, a sort of treetop skimming of modern mathematics, post-Newtonian physics, and cognitive theory.

Sadly, Mr. Livio misses a number of opportunities to enlighten his readers on the theory of polynomials, the nature of their roots, and the curious symmetries one encounters. For example, he makes no effort to discuss the nature of polynomial roots beyond a short Appendix, and he passes on the chance to detail the marvelous symmetry of imaginary roots in equations such as x^6 = 1. While he outlines the general thrust of Galois's approach to the unsolvability of quintics, Livio also mentions that Hermite found a method to solve the general quintic using elliptic functions, but we are not told how such a solution is discovered. What about sixth degree polynomials and beyond? Mr. Livio doesn't tell us - he's too busy worrying over the fairness of the first draft lottery in 1970. There is also the small matter of the author's style of explication. At times, such as his introduction to symmetry, he writes for a general, non-mathematical audience. Later, he tosses out references to elliptic functions without explanation and culminates his group theory discussion with sentences like, "We can use the family tree of these subgroups to create a sequence of composition factors (order of the parent group divided by that of the maximal normal subgroup)."

What THE EQUATION THAT COULDN'T BE SOLVED really needed was a good editor to bring these widespread ramblings into focus. A bit of truth in advertising might have been appropriate as well, but a book entitled "The Role of Group Theory in Modern Mathematics and Science" (primarily what this book is about, along with the author's peculiar obsession with Evariste Galois's death by duel) wouldn't tap well into the market developed by Keith Devlin, John Allen Paulos, Ian Stewart, Eli Maor, Simon Singh, and other popularizers of mathematics for mass market audiences. In the end, this book falls short of its companions for its sheer lack of focus and somewhat misleading cover presentation. At times, the book is interesting; at others, regrettably, it's simply too much of a superficial slog through too many loosely connected disciplines.

I became interested in this book for several reasons. The first is that I find Livio to be an entertaining writer. I read his book on phi and its relationship to beauty and found it interesting and enlightening. I have reviewed that book on amazon earlier. I met Livio in Princeton a little over a month ago when he gave a lecture on symmetry at the Princeton Plasma Physics Laboratory in one of a series of lectures intended for high school students. It was a fascinating presentation and he briefly discussed the book, mentioning how his research into the death of Galois led him to a new theory about how he died in the duel and who killed him. I found this very intriguing and I wanted to read about it.

As a college undergraduate I majored in mathematics and modern algebra was my favorite subject. The course I took on Galois theory was the most fascinating to me and I marveled over the fact that a teenage boy had developed a branch of group theory that answered questions that had stumped the greatest mathematicians for centuries.

So I bought the book and read it with very high expectations. I preface my remarks this way because I was somewhat disappointed in the book and my disappointment leads to my criticism here. But I don't want the critcism to detract from the fact that it is a well written and researched book and written in a style that like his other books makes it accessible to the general public and even the highly motivated high school students.

First of all the title leads you to believe that it is completely about the solving of the problem for which polynomials can be solved by radicals (i.e. equations that only involve basic arithmetical operations a roots, e.g. square cube roots etc,)and which ones cannotbe so solved. The book starts out discussing symmetry and to some extent how it is connected to group theory and polynomial equations. Livio takes us through the competitions that existed as mathematicians struggled to solve the general cubic and quartic equations. He even goes into the many attempts and "near proofs" of the solution to the fifth degree equation. Here this leads into a natural discussion of the lives of Abel and Galois, the two young mathematical geniuses who both died in their twenties but did the amazing work of showing that the general fifth degree polynomial could not be solved by radicals.

Livio slips into vague terminology, at times referring to the result as simply the equation that could not be solved as he does in the title and at other times he calls it "the equation that could not be solved by formula." Both expressions are vague and a little misleading. All polynomial equations with real coefficients have solutions in the complex plane. In fact there are n solutions to a nth degree polynomial, some may be real, some complex and some may occur as multiples but every nth degree polynomial with real coeefficients can be factored into the product of n first degree polynomials with each term of the form x-a where a is a complex number.

Galois in his writings refers to this question as one of solvability by radicals which is the precise accepted mathematical term that I think should have been used in this book.

Although another reviewer, Steve Koss, criticizes Livio for his discussion of the lives of Galois and Abel I can see justification for this. I was actually most interested in learning more about these two geniuses and the circumstances that led to their early deaths. For this I give Livio high marks. I was a little disappointed in the Galois story because I had been led to believe that Livio had a new convincing theory about the duel that led to Galois' demise at the age of only 20. But the theory was not completely new and not as convincing as I had expected.

With regard to Galois, Livio refers to his discovery as the creation of group theory. I did not know this and I am a little skeptical because in my studies of modern algebra I have never seen a reference to Galois as the founder of group theory. What I do know about Galois is that the theory he developed solved not only the impossibility of solving the general polynomials of fifth degree and higher but also the trisection of an angle with only a ruler and compass and several other questions that had stumped the Greek mathematicians many centuries earlier. These problems of the Greeks were mentioned to us when we were in high school but often the fact that they cannot be solved is avoided because the mathematics that proves it is too advanced for high school. Unfortunately it has held many a student to try to come up with a construction and some have been convincing even though they are flawed.

This theory, that concerns itself with special groups and fields that are called Galois groups and Galois fields is highly focussed on the aspects of modern algebra that address the problem of solvability of polynomials by radicals which in turn leads to the results about the Greek constructions as well. I was disappointed that Livio missed the opportunity to point this out. This theory is rightfully called Galois theory and it encompasses groups, fields and isomorphisms. It does not cover all aspects of group theory but it does branch out into other areas of modern algebra. I agree with Koss that the theory of solvability was shortchanged in the book.

The latter part of the book is not really at all about polynomial equations. Rather, the author has taken the liberty of moving on into discussions of mathematical symmetries and there relationships to physics, non-euclidean geometry, human psychology and other fields. Like Steve Koss I found this part much less focussed and somewhat disorganized. Also, I think the book title is misleading. The book is about symmetry as a part of group theory and mathematics and other disciplines. The solvability of polynomials is only a part of it. I didn't enjoy the latter chapters nearly as much as the earlier ones.

Inspite of these shortcomings this book is well worth reading especially if you have an interest in abstract mathematics and life's connections to symmetry. What Abel and Galois accomplished as very young mathematicians is the most difficult thing to do in mathematical research. They showed that something that people thought was possible to prove was actually impossible. It is very difficult to accept when doing research that failure to prove something is not your own personal shortcoming but rather the fact that the endeavor is futile. I have often heard the expression "How can you prove a negative?" My most recent recollection was Roger Clemens saying this in front of a congressional committee investigating the use of performance enhancing drugs in baseball. Well, as we see here in mathematics it is possible to prove a negative!

I picked up this book not knowing anything about symmetry and, frankly, not being too interested in it. What I discovered was a brilliant, cerebral yet entertaining examination of both the mathematical foundations of this concept and its artistic, cultural, and social significance. Perfectly mixing mathematical analyses with fascinating biographical, historic and artistic information (as well as the occasional amusing anecdote), Livio's incredibly well-researched book is as illuminating as a great work of philosophy and as thrilling as a Sherlock Holmes mystery. Those with absolutely no knowledge of mathematics (like me) should not be deterred, because the author inventively elucidates any difficult concepts, leaving nothing unexplained yet never digressing unnecessarily from the central narrative. Above all, the haunting character of Evariste Galois will remain with readers for a long time after they have completed reading this masterful account.