# Abstract adventuring

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A review of *Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics (New Histories of Science, Technology, and Medicine)* by amir alexander

The title of Amir Alexander’s new book (his second) and the beautiful unidentified landscape painting on its jacket, refer to an early dawn duel on a deserted street in Paris. On May 30, 1892. Éveraste Galois, a brilliant young mathematician who pioneered the study of groups, a branch of abstract algebra, was killed in a ridiculous pistol duel over a woman. The duel was so little newsworthy that to this day no one knows for sure who shot Galois in the stomach and left him to die. He was twenty. As soon as Galois was buried, a legend formed about him. He became a martyr unjustly scorned by the French establishment, a scorn that contributed to his poverty and early death. This myth found its strongest expression in a flawed chapter on Galois in Eric Temple Bell’s bestseller *Men of Mathematics*.

But as Alexander, a science historian who lives in Los Angeles, makes clear, Galois was a thoroughly obnoxious nerd, suffering from what today would be called a “personality disorder.” His anger was paranoid and unremitting. He insulted friends. His ardent Republicanism, with its hatred of the king, sent him twice to prison. He railed against the French establishment, even though it published many of his papers. “If any person was ultimately to blame for the short and tragic life of this brilliant young mathematician,” Alexander writes, “it was inescapably himself.”

Alexander sees Galois’s death as a turning point in the history of modern mathematics, a point at which math became less a study of nature than a purely abstract realm of its own, uncontaminated by the external world. He skillfully tells the story of this change, weaving it around the often tragic lives of the mathematicians most responsible for the change.

The first chapter of *Duel at Dawn* concerns the French mathematician le Rond d’Alembert. He was the illegitimate son of Claudine de Tencin, the leader of a fashionable Paris salon, notorious for her many love affairs. When d’Alembert was born she left him in a box on the steps of a Paris church. She never acknowledged him as her son, but the father, a French military officer, provided him with a pension on which he lived comfortably. D’Alembert first tried a career as a lawyer, but soon abandoned it for the study of medicine. That, too, was abandoned to pursue his one great love, mathematics. He quickly became a respected mathematician, physicist and philosopher, best known today as the co-editor with Denis Diderot of the famous encyclopedia.

Another pioneer of group theory who rates a chapter in Alexander’s book is the Norwegian genius Niels Henrik Abel. Like Galois he died young (age twenty-six) and in poverty. “Abelian groups” are groups whose elements commute. That is, when two elements are multiplied, the product is the same whether A is multiplied by B or B by A. An old riddle goes: What’s purple and commutes? Answer: An Abelian grape.

Later on, mathematical creativity shifted from France to Germany. The greatest German mathematician of the era was Georg Cantor, the discoverer of transfinite (infinite) numbers. The lowest, called aleph-null, counts the number of integers, aleph-one counts the real numbers, and so on, in an infinite hierarchy of alephs. Cantor died in a mental hospital, not knowing his controversial numbers would eventually be accepted as authentic.

In England, the two most famous mathematicians were G. H. Hardy and his associate Srinivasa Ramanujan, who died at age thirty-three. An autodidact and devout Hindu, Ramanujan’s notebooks are still mined for his thousands of contributions to number theory. Hardy, as Alexander tells it, was so impressed by the beauty of pure mathematics that he had only contempt for colleagues he believed squandered their talents on applied math. He once bragged that none of his discoveries had any use, a statement that turned out to be false.

Of course, Alexander does not overlook Kurt Gödel, the great Austrian logician. He shattered the dream of mathematicians who hoped that, someday, a mathematics would be formulated of such power that all theorems could be proved. To the amazement of most mathematicians, notably David Hilbert and Bertrand Russell, Gödel was able to show that in all formal systems that include arithmetic, even in humble arithmetic, there are theorems undecidable by proofs within the system even though they are true! As Alexander reminds us, Gödel ended his unhappy life at Princeton’s Institute for Advanced Study by starving himself to death.

In Switzerland, Leonard Euler, believed by many historians to have been the world’s greatest mathematician, lived to a comfortable seventy-six, admired everywhere. After he became almost totally blind, his output of papers actually increased. Today Euler is best known for two remarkable formulas. A polyhedron has faces (F), edges (E), and Corners (C). Euler discovered that F - E + C always equals 2. A cube, for example has 6 faces, 12 edges, and 8 corners: 6 - 12 + 8 = 2. Euler’s other famous formula, the most mysterious in all mathematics is e^{iπ} + 1 = 0. This incredible equation links the two most famous transcendental numbers, pi and e, with the imaginary square root of minus-1, a number Euler designated with “*i*.”

Twentieth-century mathematicians covered in Alexander’s marvelous history include Alan Turing, best known for inventing the Turing machine, a greatly simplified computer, and John Nash, the game theorist who is the subject of a movie and book, both titled *A Beautiful Mind.* They tell of his battle, finally won, against mental illness. Turing, depressed over failed efforts to become heterosexual, died after preparing and eating a poisoned apple.

Several pages deal with the curious career of the Russian mathematician Gregory Perelman. Alexander calls him a “modern Galois.” He became world famous in 2009 for solving a notorious conjecture called the Poincaré hypothesis. Imagine a rubber band placed at any spot on the surface of a solid. Can the band always be contracted on the surface until it is a point? If so, Poincaré’s hypothesis says the surface must be topologically equivalent to a sphere.

The conjecture seems obvious, but for more than a century it has been fiendishly difficult to prove. When Perelman succeeded it won him a Fields medal, the mathematical equivalent of a Nobel prize. To show his contempt for the establishment, Perelman refused the award. Later he was offered a million dollars for cracking the Poincaré conjecture. He is expected to turn that down, too! He has recently retired to St. Petersburg where he lives invisibly with his mother.

The penultimate chapter of *Duel at Dawn* impressed me as the highlight of Alexander’s history. It covers the fantastic story of Euclid’s notorious parallel postulate. The simplest way to explain the postulate is to imagine a straight line and close to it a point. The postulate states that, through the point, one and only one line can be drawn parallel to the given line. It looks less like a postulate than like a theorem that could be proved by using Euclid’s other posits. For many decades, geometers tried to find such a proof but in every case the postulate, or a theorem using it, entered the proof. It finally became clear that Euclid did not blunder in calling it an assumption. To the amazement of geometers, the postulate can be violated in two different ways, each resulting is a geometry just as true as Euclid’s!

One non-Euclidean geometry results from assuming *no* line can be drawn through the point parallel to the given line. A different geometry results when it is assumed an *infinite number* of parallel lines can go through the point. Not only did these wonderful geometries open up vast new fields for exploration, but an even greater surprise occurred. It turned out that Mother Nature knew all about non-Euclidean geometry! In Einstein’s general relativity, space itself has a non-Euclidean structure. We think planets circle stars in ellipses, but that’s because our minds are incapable of visualizing a non-Euclidean space. We see a planet’s path as elliptical when actually it is traveling through non-Euclidean space-time along a geodesic, the shortest and straightest possible path. Light also seems bent when it passes close to the sun, but it too is travelling a geodesic we are unable to visualize.

Alexander’s last chapter speculates on changes in mathematical research caused by the computer revolution. Proofs no longer are confined to proofs that can be fully understood by reading them. The four-color map theorem, for example, has been “proved” by hundreds of pages of computer printouts that have to be checked by other computer programs. To many mathematicians, such “proofs” are abominations, not to be trusted. No one thinks them beautiful—indeed, they are actually ugly.

At the front of his book, Alexander quotes these enigmatic lines by Keats:

Beauty is truth, truth beauty, that is all

We know on earth, and all ye need to know.

Alas, the lines are almost meaningless. They are not all we know or need to know. Moreover, there are true mathematical theorems that are ugly, and there are beautiful “proofs” that are false. T. S. Eliot surely spoke for most literary critics when he called Keats’s lines “a serious blemish on a beautiful poem.”

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