This *Standpoint* column gives scientists a forum to express their ideas publicly, though doing so is not easy. Among those who were good at it was the great French scientist Henri Poincaré — not to be confused with his cousin Raymond Poincaré, who was President of France during the First World War. Henri died in 1912 before that war, and if I could reincarnate one figure from the past to comment on modern scientific ideas and breakthroughs, it would be him.

Poincaré arrived at the main ideas of Relativity Theory before Einstein, foreseeing the gravitational implications that emerged later, and a putative 1970s textbook on relativity by a top Princeton mathematician was never published when it was seen to downplay Einstein’s role relative to Poincaré’s. In fact the two men’s work was independent and as the French physicist Louis de Broglie wrote, “he left to Einstein the glory of seeing all the consequences of relativity and, in particular . . . the true physical character of the relationship the principle of relativity establishes between space and time”. Why? De Broglie gives an answer in a 1954 address, available on YouTube: it was “without doubt his a little too hypercritical turn of mind, due to his having first been a pure mathematician, that was the cause”.

These quotations appear in *Henri Poincaré: A Scientific Biography* by Jeremy Gray (Princeton, £24.95). It is full of the mathematical, physical and metaphysical ideas of a man who was not only a dispassionate observer of the world around us, but of our way of understanding it. Epistemology was a vital subject for him and one of his favourite questions was, “How do we know that?” As a disinterested observer he was even called in to examine the Dreyfus case, and completely demolished the “proof” that the handwriting of the main incriminating document was that of the accused. It wasn’t.

Henri Poincaré, born in 1854, could already talk at nine months, and during his fourth year at the Lycée (13-14 years) his teacher called at the Poincaré house to say that the boy would be a mathematician. When his mother, who rather liked mathematics, appeared not unduly surprised, the teacher added, “I mean to say, a great mathematician.”

This was the man whose name became attached to one of the most famous conjectures in mathematics, only recently solved. It was originally a question Poincaré posed about three-dimensional spaces in which distance is not present. Geometry without distance is called topology, and Poincaré himself was the first to use algebraic methods in studying it. It was a subject dear to his heart because he rejected the idea of absolute distance in physics. Distance is merely what we measure, but how do we measure it, how do we know? He was absolutely right, and in the theory of relativity it turned out that physics admits no distance in the usual sense. It depends on the observer, and lengths appear foreshortened when travelling at great speed, leading to paradoxes like the one about a train on a railway track heading for a gap in the rails. Travelling at close to the speed of light, the train driver sees a shrunken gap that will not disturb the smooth ride, but an observer near the track sees a shrunken train smaller than the gap and heading for disaster. Who is right? Fortunately it’s the train driver — the nearby observer has not understood that from the train’s point of view the front and back cannot be over the gap at the same time.

Poincaré’s research on the ideas underlying relativity theory emerged from his study of the new electromagnetism developed by the Scottish physicist James Clerk Maxwell, on which he became the pioneering French expert.

He had already made a huge mathematical breakthrough on one of the great problems of celestial mechanics: how to find a formula describing the motion of three bodies under their mutual gravitational attraction. Two bodies had been successfully tackled by Newton, and Poincaré’s answer proved, rather surprisingly, that it was impossible for three. He showed that although most orbits would be stable, a tiny change, as happens in the real world, could yield unexpected instability. In modern terms this is like the proverbial flap of a butterfly’s wings in Siberia creating unexpected changes to the weather in London.

The scope and depth of Poincaré’s mathematical ability was surely greater than any physicist who followed him, but was he indeed a physicist or a mathematician? As a young graduate of the Ecole Polytechnique he joined the Corps de Mines, making important reports and recommendations on mining disasters, and remained a member of the corps all his life, rising to ever higher positions. He held academic chairs in astronomy, mathematics, physics, experimental mechanics, and even electrical theory after taking an interest in wireless telegraphy. He was at various times president of the main French scholarly societies in mathematics, physics, and astronomy, and was the man you wanted on important committees. No wonder they brought him in to the Dreyfus case, and he was a leading proponent of the unsuccessful French attempt to decimalise circular measure. Had there been a Nobel Prize in mathematics he would have won it hands down, and he very nearly got the physics prize in 1910, despite opposition from the experimentalists, particularly in Britain.

Not that Poincaré had any objection to experimental work. On the contrary, he believed in experimental results and mathematical rigour. “Explanations are what we lack the least,” he once said. What would he have made of String Theory? The Higgs boson? Dark matter? We still need his epistemological scepticism, combined with huge mathematical ability. He believed the universe to be truly intelligible, but also that we can never know what it is “truly” like. He was sceptical of what we call space, or even space-time, and this was before quantum theory, before we fully realised that space-time cannot be infinitely divisible. He even talked of an “atom of time”, and believed in the primacy of processes and relationships, rather than some nebulous space in which they take place, and used the concept of symmetry (mathematically speaking: group theory) in preference to that of space. How interesting, then, that symmetry has been the guiding hand in studying interactions unknown in Poincaré’s time, the strong and weak nuclear forces, which along with electromagnetism have given us the three quantum forces of the so-called Standard Model of elementary particles.

If space is an illusion, where then does it come from? Poincaré located our intuition of space in our nervous impulses. He would have loved modern brain-imaging techniques, partical accelerators, computers and on the centenary of his death last year some writers referred to this great polymath as “the last universalist”. I fervently hope not, but where do we find another like him?

The ravenous longing for the infinite possibilities of “otherwhere”

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