Birth of a Theorem: A Mathematical Adventure

  • Cédric Villani
Bodley Head/Faber and Faber: 2015. 9781847922526 | ISBN: 978-1-8479-2252-6

Mathematics Without Apologies: Portrait of a Problematic Vocation

  • Michael Harris
Princeton Univ. Press: 2015. 9780691154237 | ISBN: 978-0-6911-5423-7

One evening in 2009, French mathematician Cédric Villani stepped into his children's room. He locked the door, turned off the light and began to pace, pondering the statistical properties of plasma. A few metres away, his wife, Claire, was in the kitchen, cooking dinner for the family. The contrast, Villani concedes in his engaging Birth of a Theorem, was “a bit much”, yet immediately after dinner he returned to the dark room to grapple for hours with his elusive proof.

Anyone reading that anecdote will feel sympathy for Claire as she tries to preserve family normality. Anyone who has seriously engaged with mathematics will also understand her husband. Perhaps more than any other field, mathematics pulls the practitioner away from the 'normal' world of things and people into a strange alternate universe, in which we catch glimpses of beauty and coherence, but spend most of our time groping in the dark. In Birth of a Theorem, Villani offers one way of straddling that divide; in Mathematics Without Apologies, fellow mathematician Michael Harris presents a very different one. Together, they provide an unmatched perspective on life in this “problematic vocation” by two of its leading practitioners.

Cédric Villani studies the maths of plasmas. Credit: Joel Saget/AFP/Getty

Birth of a Theorem is the story of Villani's quest to give a full mathematical account of Landau damping. Whereas gas becomes increasingly disordered over time as entropy increases, plasma spontaneously stabilizes, with no increase in entropy. Soviet physicist Lev Landau was the first to mathematically describe this improbable phenomenon, but he used a simplified model that left many unconvinced. Working closely for several years with mathematician Clément Mohout, Villani succeeded, and he was awarded a Fields Medal in 2010.

Villani's quest takes him across the world, from Lyons, France, to Princeton, back to Paris and on to Hyderabad in India. At every stop, he talks to local mathematicians, demonstrating that, for all its abstractness, mathematics can be an intensely social activity. The book is sprinkled with brief, telling portrayals of mathematicians and physicists past and present. The grumpy, grey-haired Étienne Ghys of Lyons and Chinese expat Alice Chang of Princeton alternate with the autocratic Lev Landau in 1960s Moscow and the ever-present shadow of Albert Einstein at the Institute for Advanced Study in Princeton. Much of Villani's e-mail correspondence with Mohout is reproduced, chronicling moments of triumph and despair.

Charismatic and flamboyantly dressed, Villani is the opposite of the 'mathematical hermit' and annoyed by the stereotype. He attends a recital by one of his children, joins his family at the American Museum of Natural History in New York and travels a long way to attend a concert by rock band Têtes Raides. Yet he studies the mathematics of galaxy formation during the recital, works out a step in his proof on the bus from the museum, and explains his research to a stranger who drives him back from the concert. The mathematical life, in his telling, is a delicate dance between the demands of the 'real' world and the allure of the mathematical one.

If Birth of a Theorem is the personal record of a single-minded quest, Mathematics Without Apologies is a kaleidoscope of philosophical, sociological, historical and literary perspectives on what mathematicians do, and why. Do they pursue their work for the public good? Harris dismisses that as a pose, useful for grant applications and little else. Is it the absolute truth of mathematical demonstrations that drives the field? That, Harris contends, is a conceit of philosophers: practising mathematicians seek insight, not certainty. What about the lauded beauty of mathematics? Perhaps, Harris concedes, but when mathematicians talk about beauty, what they mean is pleasure. A 2012 sociological survey found that 91% of pure mathematicians cited it as a key attribute of the field. Mathematicians, Harris concludes, do what they do because of the enormous pleasure it brings them.

Pleasure is not an explanation likely to satisfy funding agencies. Yet Harris makes no apologies. He is concerned that the field has made a Faustian bargain with the institutions that provide the material conditions for mathematical research. Leading mathematics departments have trained an army of quantitative analysts, or quants, to implement the algorithms that govern financial trading practices. The impersonal, unchallengeable equations of higher mathematics, he worries, contribute to a moral vacuum at the heart of high finance.

Harris's insider view reveals a community in which each mathematician is placed in an informal but strict hierarchy, depending on acknowledged brilliance and accomplishments. He takes a playful detour, arguing that each of US writer Thomas Pynchon's 'non-linear' novels is organized around a different conic section, such as a parabola. Throughout the book, he verbally spars with an imaginary “performing artist” while trying to explain the mysteries of number theory.

But, like Villani, Harris returns repeatedly to the chasm between the human world and the mathematical one — a tension that in his own life has proved fruitful. Stuck in a professional cul-de-sac in the 1990s, Harris experienced a revelation: a dream showed him a new mathematical path, which led to his ascent up the mathematical hierarchy and transformed his life.

For him as for Villani, mathematical insight at its deepest core remains an irreducible personal experience.