Henri Poincaré: A Scientific Biography

  • Jeremy Gray
Princeton Univ. Press: 2012. 616 pp. $35 9780691152714 | ISBN: 978-0-6911-5271-4

Were it not for the Poincaré conjecture, it is doubtful whether many non-mathematicians today would know of Henri Poincaré. His vexed question in topology was solved only in 2003 — nearly a century after it was published and some years after its conqueror, Russian mathematician Grigori Perelman, began to unpick it. Perelman has vanished from public view. Poincaré remains a household name.

Henri Poincaré posed a puzzle that remained unsolved for 99 years. Credit: DORNAC (PAUL FRANCOIS ARNOLD CARDON)/ARCHIVES LAROUSSE, PARIS/GIRAUDON/BRIDGEMAN

He was hardly unknown in his day. As John Gray recounts in his masterly Henri Poincaré: A Scientific Biography, Poincaré was one of France's great intellectuals in the late nineteenth and early twentieth centuries. When he died at just 58 in 1912, the French Minister of Education called him “a kind of poet of the infinite, a kind of bard of science”, and his funeral cortège was a veritable who's who of the world's intellectual elite.

Poincaré was also a prodigiously versatile thinker. A brilliant mathematician, equalled in his time only by David Hilbert in Göttingen, Germany, Poincaré was also rightly considered a physicist and philosopher of science of the first order. Gray encapsulates Poincaré's multiple dimensions; his intellectual biography is both a tour de force and a triumph of readability. He leads us through Poincaré's life, and the vast array of subjects he touched on, covering practically the entire corpus of what interested mathematicians and physicists at the turn of the twentieth century — from topology and algebraic geometry to Lie groups.

The field that Poincaré spawned is algebraic topology, which explores surfaces in higher-dimensional spaces using techniques from abstract algebra, the discipline concerned with mathematical structures. And, within topology, he formulated his conjecture.

Poincaré in fact posed a version of the conjecture four years before the one for which he is remembered: a theorem he decided to publish to “avoid making this work too prolonged”, as he put it, with the promise of a proof to follow. Instead, he proved himself wrong by providing a counterexample. In 1904 he was much more cautious, and published the puzzle as a question. In essence, he asked whether a three-dimensional surface is equivalent to a three-dimensional sphere if rubber bands, wound around it, can be contracted, lasso-like, to a single point. He ends with the ominous words, “However, this question would carry us too far.”

Mathematicians laboured over the problem for 99 years. Most tried to solve it in the affirmative; some attempted to find counterexamples. It was left to Perelman to prove that the answer to Poincaré's question was “Yes” — after which he refused both the US$1-million Millennium Prize from the Clay Mathematics Institute and the Fields Medal.

Like many, Gray wonders why the Nobel Prize in Physics was never awarded to Poincaré for his work in, say, electromagnetism, optics or thermodynamics. As Gray tells us, Poincaré was ahead of Albert Einstein in speculating about a truly relativistic theory of gravity. Quoting Maurice de Broglie, who pioneered X-ray spectroscopy, Gray writes that the reason Poincaré did not take the decisive steps that Einstein did may have been his “too hypercritical turn of mind, due perhaps to his having first been a pure mathematician.” Another reason that Poincaré did not win the prize, Gray suggests, was that he was a theorist in mathematical physics, and Nobel prizes at the time were awarded mainly for experimental discoveries. After all, even Einstein was awarded the Nobel prize only in 1921 — for the discovery of the law of the photoelectric effect.

It would be petty to find faults in a work of this calibre, but some reference to Louis Bachelier would have been welcome. He was the visionary of the Black–Scholes options pricing formula of modern financial theory — which gives the correct price of financial derivatives — and one of Poincaré's handful of doctoral students. And the bibliography seems to have omitted Perelman's postings to the Internet.

On the whole, however, this book is an achievement in its own right. Gray keeps the tone light and embeds each of the equations in explanatory text.

Fortunately, Gray also tells it like it was, warts and all. Poincaré's work could contain errors, and often lacked rigour. Aside from his initial, incorrect attempt at his conjecture, his first stab at a prize question — posed to honour the 60th birthday of Sweden's King Oscar II — contained a serious flaw. All copies of the journal that published it were pulped. But most of Poincaré's fumbles are there for all to see; and by studying such blunders, we may observe the meanderings of science as it advances by trial and error. Presenting only the finished product, as Isaac Newton did when he concealed his discovery of calculus, does injustice to the scientific process.