July 2010

Leigh Arber


Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics by Amir Alexander

Those who have studied some math and science might remember that D’Alembert has a principle, Bernoulli has an equation, and Varignon has a theorem. But not many of us know about these mathematicians as people; we remember math class as the place where personality goes to die. However, in Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics, Amir Alexander presents the lives and work of the people behind the equations. His perspective on the history of mathematics is an unusual one: he is as interested in cultural notions of mathematicians as the mathematicians themselves.

Duel at Dawn provides a biographical account of prominent European mathematicians of the nineteenth and twentieth centuries. The title refers to the famed 1832 duel between Évariste Galois and his unknown opponent, an event which Alexander regards as a major pivot in the history of mathematics. Prior to 1832, mathematics was seen as an extension of physical reality, and dependent on its laws. It was connected to the natural word and relevant to science, in keeping with the principles of the Enlightenment. After Galois, however, as the Romantic movement progressed, mathematics became a self-contained, separate field which existed only within its defined boundaries, following only its own rules of logic. Most of the mathematicians presented in the book fit the Galois mold; Alexander holds that this notion still endures.

This shift in the general conception of mathematics, marked by the famous duel, was accompanied by a shift in the personalities of mathematicians. Before Galois’s death in the duel at the age of 20, mathematicians were well-adjusted and successful lives. These Enlightenment mathematicians, or “natural men,” were a part of the real world; so was the mathematics they studied.

D’Alembert, Euler, and Lagrange served in the Academies, wrote books and treatises, and founded major learning centers. However, beginning with Galois, mathematicians became fragile, innocent heroes who were unsuited to this world and died early deaths. Math had become a pure, disconnected discipline, and mathematicians were pure and disconnected too. Alexander eloquently describes Galois as a mathematician who was “interested in the deep structure of equations, not in the fortunes of any particular one.”

Alexander does more than relay facts; he also analyzes how those facts appear to the imagination. Biographers often embellished the tragic elements of their subjects, especially when they wanted to accuse the establishment of excluding other mathematicians from the powerful academies. But Galois himself seemed to relish such a self-image, and he admitted that he was unable to let anyone else have the last word, even on an issue as trivial as whether or not he was drunk. He predicted that he would not live long. And Alexander acknowledges the psychological reasons for dramatizing such a figure: “who among us… in a world of compromise and shades of gray, does not wish that for a single moment they could live life as brightly and fully as Galois did in his twenty years?”

The book is at its best when Alexander relies on direct quotes from the mathematicians or odd details about their lives. He points out that Cauchy prefaced his mathematical writing with the statement that he had made “incertidue disappear,” and that the scientist and statesman Francois-Vincent Raspail is currently “honored with a wide Paris boulevard and a metro stop on the Left Bank.” He also includes excerpts from mathematicians’ letters: Niels Hendrik Abel wrote, simply, “I am as poor as a church rat,”and the writer Stendhal described Cauchy as “a veritable Jesuit in short frock.” In the twentieth century, Alan Turing, after breaking the German Enigma code during World War II and being persecuted for homosexuality, dies “mysteriously” in his lab from an apple coated with potassium cyanide. These stories do more justice to mathematicians than the sanitized biographies in textbooks; Alexander has preserved for us the extraordinary details and firsthand accounts of mathematicians’ lives and letters.

If Duel at Dawn is read as a history text, it may seem inverted; mathematicians occupy the foreground while the major events of modern European history unfold in the distance. As Cauchy considers his career options -- mathematician, lawyer, or priest -- Napoleon’s empire crumbles. Galois was not always studying; in 1831 he led a march on Bastille Day to plant a liberty tree in front of City Hall, and he also spent time in prison. It is unusual and interesting to read about these events as mere factors in the lives of mathematicians rather than as the main subject of the text.

Because it is such an engrossing story, it’s easy to forget that the book’s purpose also is to educate. Alexander conveys a general sense of who mathematicians were and how they fit in with society, but he’s less successful at summarizing the lives and work of each of the mathematicians presented. The dates and the facts are all there, but the book is not always chronological, and recognizable figures of history show up without notice. Alexander’s aim is to track changes and reveal trends; therefore, he follows the arc of the story and sometimes loses his grip on each individual segment. I wished the book included a brief timeline or a list of characters. This would have helped readers come away with specific knowledge in addition to a general impression.

It is also unfortunate for readers that the book is confined to Europe. The history of mathematics is long, and Alexander’s book is fairly comprehensive. Nevertheless, I would have liked to read more about what mathematics was like in other parts of the world. Alexander does make reference to the Indian mathematician Srinivasa Ramanujan, and the fact that Euclid’s fifth postulate was kept alive by mathematicians in the Arab world, but these brief comments could have been expanded. In a book which emphasizes the importance of popular imagination and cultural climate, the absence of non-European mathematics and historical legacy seems like a serious omission.

Alexander argues that the view of modern mathematics which began with Galois still persists today. I was happy to discover that Alexander extends his story into the present. Mathematics already seems so far away, and a book which only covers the past would have made it seem even more distant. Alexander discusses a 1996 New Yorker article about Grigory Perelman, Shing-Tung Yau, and the Poincaré conjecture. He criticizes the authors of the article for painting Perelman as a tragic, Galois-like figure when the truth is that Perelman was offered prestigious positions and a Fields Medal, which he turned down. This shows that, in the eyes of the onlookers, the Galois image still exists today, and it remains half-truth, half-myth.

Fortunately, Amir Alexander is able to make the important questions of mathematics accessible to non-mathematicians. Unfortunately, though, the book might lead readers to oversimplify mathematics. As we read about one tragic mathematician after another, we may think of our tortured lives as every bit as doomed and dazzling as theirs were. We are almost fooled into believing that a comprehension of math, too, is within our grasp. But it’s not, of course; we may have brooded in a one-roomed garret with the best of them, but we didn’t, like Georg Cantor, discover transdefinite numbers along the way. Without understanding what they achieved, we probably can’t fully understand who they were. This awe of math should not diminish as we grow more familiar with the facts of their lives.

My sense of awe was renewed when Alexander launches unexpectedly into a technical discussion of mathematics. At these points, it becomes clear that it’s not so easy to understand modern mathematics. I certainly didn’t follow all of it, and the description of the catenary shape of a hanging cable was more complicated than other versions I’ve read. But the technical discussions are asides in an otherwise accessible story, and Alexander gives the reader permission to skip them.

I was puzzled by the end of the book, when Alexander expresses his disdain for computer methods in mathematics; he seems to prefer a pure and detached form of math. According to Alexander, who sees no role for computers in mathematics, the computer wiz lives in his “parents’ basement,” keeps his sights “focused firmly on our own material world,” and is busy “nursing all the injuries done to him by fellow humans and secretly plotting his revenge.” I studied engineering and came into contact with many a “computer wiz,” none of whom seemed to be bitter and spiteful. And furthermore, Alexander’s contempt relies on a contradiction; someone who has his gaze on the material world is likely to have a successful career and therefore live somewhere other than his parents’ basement. Alexander’s disdain for the material world is difficult to understand.

If his goal is to elevate the status of mathematicians, Alexander should not have targeted the “computer wiz.” A better conclusion might have appealed to the general reader with the argument that mathematicians have already earned their place in history, even though they receive little glory. Earlier in the book, he does make such a plea: “In their lives and in their works, Galois, Abel, Cauchy, and Bolyai were as much figures of the Romantic age as Byron, Shelley, Delacroix, and Beethoven.”

Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics
by Amir Alexander
Harvard University Press
ISBN: 0674046617
320 Pages