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The year 2012 marked the centenary of the death of the great Henri Poincaré, mathematical genius, professor of physics and popular philosopher. In comparison with other tributes, Jeremy Gray’s biography stands out because it is so long, drenched in mathematical and bibliographical detail, and offers several chronologies from diverse disciplinary perspectives. After an initial overview and biographical sketch, each chapter presents Poincaré’s contributions to particular fields, including but not limited to cosmogony, physics, topology and the theory of functions.

Over time, however, Gray’s book may well endure because, paradoxically, it is so short. It is a comprehensive but uncluttered guide to Poincaré’s extensive oeuvres that is technical, even though it omits technicalities, and deep, even though it raises more questions than it answers. In a sublime fit between form and content, the domain-specific chapters typically start in the early 1880s when Poincaré began publishing “in three fields at once”, traces the arc of his contributions to a particular field and then, like a planet in a stable orbit, returns again to the “flood from Caen” with which Poincaré’s prolific career began.

Poincaré was not only a brilliant scientist but a talented writer whose popular essays are still in print in many languages. In 1908, after receiving numerous accolades for his scientific and mathematical achievements, he became “an immortal” when elected to the Académie française, an honour traditionally bestowed on literary giants, even though in this case, “the achievements of this best-selling author were hard for most writers to convey intelligibly and in a few words”. Poincaré’s style is aphoristic and elliptical. Contemporaries noted, often in frustration, that he could see more than he could say. When mathematicians asked him to fill in gaps, the genius would simply reply, “But that’s the way it is”. Gray explains how Poincaré worked, from an invitation to lecture, to an initial publication in what was often a new journal. Then, after corresponding with critics, he would publish the final form of his essay in an anthology - except that the “final form” of an essay could remain elusive, as each project waxed and waned in accordance with the feedback received and the intended audience. In sorting out the tangled stories behind these multiply instantiated essays, Gray places them in context and shows how they relate. He argues throughout that “what holds [Poincaré’s] life’s work together to a remarkable degree is the tight hold his epistemology had on his ideas of ontology”.

For Poincaré, what exists cannot be understood apart from what we know, and what we know best are the enduring relations between hypothetical entities. What mathematics is about cannot be separated from what mathematicians do. Gray reads Poincaré’s essay, “On the foundations of geometry” as “an early example of cognitive science”.

Our accounts of space reflect our experiences as rigid bodies in the world, both as individuals and as human beings. According to Poincaré, space “is in reality amorphous, and it is only the things in it that give it form”. But if Poincaré was one of the architects of relativity, why did he not take up residence in Minkowski spacetime, so quickly inhabited by physicists, rather than choose, as Gray puts it, “to be a Galilean to the end”?

The short answer is that Poincaré “was at times more modern and at times more traditional than his opponents”. The long answer Gray offers requires a patient reader. Even so, what Poincaré said about his Acta Mathematica papers may be applied to Gray’s book also; namely, that it only skims a vast subject that will doubtless furnish future scholars with the occasion for numerous important discoveries.