One spring morning in 1832, a young revolutionary was mortally wounded in a duel in Paris. The background to the duel is murky - it remains unclear whether its origins were political or romantic - but the youth who died the next day was the brilliant 20-year-old mathematician Evariste Galois. In years to come, it would emerge that he had provided a complete solution to the major mathematical problem of the solution of polynomial equations, and his theory is now the subject of advanced undergraduate courses for pure mathematicians.
Amir Alexander's thesis in this intriguing book is that Galois' death marked a change in how mathematicians saw their subject and themselves: a change that continues to define how mathematics is regarded by many practitioners today. This book's close examination of the lives and posthumous reputations of several mathematicians - including Galois, Jean le Rond d'Alembert, Niels Henrik Abel, Augustin-Louis Cauchy and Janos Bolyai - shows how today's vision of the ideal mathematician came about.
The great mathematicians of the 18th century, such as d'Alembert and Leonhard Euler, saw themselves, Alexander argues, as men whose mathematical insights came from their sensitivity to nature. Johann Bernoulli, in a technical example that Alexander discusses in detail, used arguments derived from the mechanics evident in the real world to prove claims about geometry. For these mathematicians, such evidence was as conclusive as a rigorous abstract mathematical proof. Unlike d'Alembert and Euler, who were recognised figures of the mathematical establishment, Galois was a mathematical outsider as well as a political revolutionary. Despite his mathematical ability, as a teenager he was rejected twice by the Ecole Polytechnique, and when he later submitted work to leading mathematicians it was lost or misunderstood. But Galois was confident in the value of his mathematics: in a letter written on the eve of his fatal duel, he asked his friend Auguste Chevalier to consult the greatest mathematicians of the day on his work - not as to its truth, but as to its importance.
Alexander shows how, following Galois' death, his story was mythologised: the young rebel, rejected during his lifetime by the mathematical world, whose work was appreciated only years after his death. His story has been promulgated by influential writers as different as George Sarton, the historian of science, and the populariser Eric Temple Bell. Galois is often bracketed with Abel, his contemporary, who also made a major contribution to the theory of equations and died young. But while the Galois legend largely matches the terms in which he appears to have seen himself, Alexander argues that Abel would not have recognised the version of his life later established in mathematical folklore. Abel was not a loner ignored by the mathematical world; he and his work were recognised in his lifetime. When he died prematurely he was about to be offered a university chair that would have secured his future. Nevertheless, the myth of Abel's life as a twin to that of Galois remains powerful: two brilliant but doomed youngsters rejected by the mathematical community, their achievements valued only by posterity.
Galois blamed some of his misfortunes on the eminent French mathematician Cauchy, and his legend presents Cauchy as the heart of the establishment that rejected him. But Alexander shows that Cauchy also saw himself as an outsider whose career was hindered by his political views; persecuted for his refusal to conform to traditional teaching methods, and for a time forced into exile. His famous Cours d'analyse, the hugely influential textbook that modernised analysis, brought him into conflict with colleagues who did their best to prevent Cauchy teaching this new approach. In very different ways, the revolutionary Galois and the conservative Cauchy were both rebellious loners.
The Hungarian Bolyai, creator of non-Euclidean geometry, was another misunderstood young man who was denied credit for his discoveries and driven out of mathematics by this rejection, despite the importance we now attach to his ideas. Alexander discusses how Bolyai may have seen himself and how his story has subsequently been presented.
The theme of Duel at Dawn is that, from the time of Galois and Abel, the prototypical mathematician has been a rebellious hero, more at home with abstract ideas than in advancing a career, suffering for his subject and, often, dying young and unacknowledged, only to receive due recognition long after death. He (it is always "he") is driven by a Romantic ideal of abstract truth. We are familiar with the stories of Galois, Abel and Bolyai as illustrations of this theme (greatly surprised though Abel would have been to find his life presented in that way), but Alexander elucidates how even establishment figures such as Cauchy could see themselves similarly.
His argument is that the paradigm of the mathematician as Romantic hero has not only been the predominant model inspiring mathematicians for the past two centuries, but has also determined the kind of mathematics we do. Pure mathematicians see their subject as entirely abstract, and a discipline to be valued for its own sake regardless of any connections it may have with the real world; indeed, a subject whose very lack of application is to be celebrated.
Mathematicians love to quote G.H. Hardy rejoicing in the uselessness of his life's work (even if, in these days when research funding depends on practical applications, it is worth noting that Hardy's work has turned out to be valuable in the real world). Alexander's persuasive technical examples and cogent analysis illustrate this change in the way mathematicians have thought about themselves and their subject, from Galois to the present-day mathematician/outsider Grigory Perelman, who in 2006 declined the Fields Medal.
An interesting chapter uses visual images of mathematicians to support Alexander's thesis. Here he compares portraits of Galois and Abel with images of earlier mathematicians, contemporary poets and painters, and 19-century physicists, concluding that these young mathematicians are presented as Romantics, portrayed in similar style to the literary and artistic figures, whereas the iconography of physical scientists is very different. This is plausible but needs more evidence: a much wider survey would be required to support the conclusion, especially as the portraits of Lord Kelvin and Hermann von Helmholtz at the height of their careers are hardly comparable with those of the youthful Galois and Abel.
This is a fascinating and provocative book. It is also extremely readable: the accounts of Galois, Abel, Cauchy and Bolyai and their posthumous reputations are engaging and entertaining, and along the way we meet many other fascinating personalities, including Guglielmo Libri, the aristocratic revolutionary, mathematician and stealer of rare books. Alexander's arguments are illuminating: I gained many insights into the origins of my own mathematical training, discovering for example that my schoolteacher's disapproval of diagrams comes in a direct line from Joseph-Louis Lagrange's Mecanique analytique.
As the publisher's publicity material claims, Alexander demonstrates here how "popular stories about mathematicians are really morality tales about their craft". This book uses the history of mathematics powerfully to throw light on the attitudes and practices of today's mathematicians.
Tony Mann is head of the department of mathematical sciences, University of Greenwich, and president, British Society for the History of Mathematics.
Amir Alexander, visiting scholar in history at the University of California, Los Angeles, admits that his interest in history and science owes at least something to his family: "I come from a long line of scientists, physicists and mathematicians. And my uncle is a noted Renaissance scholar."
As a youth growing up in Jerusalem, Alexander became involved with Peace Now. Although no longer active in the movement, he continues to support its ideals. However, he says, faith that they can ever be achieved "is missing".
He moved to the US to pursue postgraduate qualifications at Stanford University, and he has stayed in America since.
Alexander likes "hanging out" with his wife and two children and following his favourite sports teams. In addition to supporting the Hapoel Jerusalem soccer club in Israel - a "misfortune", he confesses - he is a fan of Stanford's American football team and the LA Lakers basketball team.
Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics
By Amir Alexander
Harvard University Press
Published 30 April 2010