Henri Poincaré

‘A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.’ ― Henri Poincaré

The classic biographical book ‘Henri Poincaré: A Scientific Biography’ by Jeremy Gray has recently been published in paperback, making it available to a wider audience. Although its readership will certainly be wider, its subject matter will continue to be of interest to those with a keen interest in the history of scientific ideas in the age of modernism, as well as to those who love mathematics and have wandered in those parts of the Platonic world that Poincaré was so accustomed to explore, leading generation after generation ever since his exciting excursions.

Gray is an erudite and renowned historian of mathematics, notable for his attention to minute detail and historical context. A master of good ‘old fashioned’ academic rigour, Gray’s style is far from dry. On the contrary, his prose brings to life relationships, the rivalries, the passions and even the occasional, refreshing, high-falutin academic gossip.

Prominent mathematicians, physicists, academic philosophers and influential thinkers of his time come alive in their interactions with and influences on the great master of thought, Henri Poincaré, the ‘last universalist,’ as he has come to be known in the annals of science for his overarching bright intellect and scientific originality.

I must confess that I chose this book on the basis of a very personal bias. You see, I came to know, admire and even love Poincaré’s work and personality as one loves and admires a very distant ancestor. As the adviser of the adviser of my thesis adviser, I came to relate to him as one would to a heroic great-grandfather or something like that. A legendary figure who still influences my life and my thinking, that of my Alma Mater, that of my chosen field, but also of science in its broader, true, sense.

It was in my university’s (ULB) celebrations for its 75th anniversary in 1909 where he famously proclaimed:

‘Thinking must never submit itself, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, if not to facts themselves, because, for it, to submit would be to cease to be.’

A true free thinker and major proponent of the spirit of free inquiry or ‘Libre Examen’ Poincaré, although good-hearted and introvert by nature, did not hesitate a second to stand up whenever he felt that social justice was in jeopardy.

During the historic scandal of the Dreyfus Affair, Poincaré defended justice and advocated for the scientific method of evidence assessment. Gray’s book recounts Poincaré’s involvement, highlighting a lesser-known incident where he examined evidence in the unjust arrest of Alfred Dreyfus. As a renowned expert in probability, Poincaré recalculated the incriminating probability and dismissed it as evidence, showcasing his commitment to justice and scientific rigour amidst the scandal.

‘Doubting everything and believing everything are two equally convenient solutions that guard us from having to think.’ ― Henri Poincaré

Another defender of Dreyfus, Jacques Hamadar, a committed Platonist, was a close friend of Poincaré. They pioneered nonlinear analysis and stability problems, and in 1889 discovered what now we call ‘the Butterfly Effect,’ a fundamental concept in chaos theory that shows how infinitesimally small causes can lead to significantly large effects. The book delves into Poincaré’s decisive contributions, though not extensively into his seminal ideas related to chaos and complexity.

It outlines how his ideas continue to inspire current and future investigations in the foundations of mathematics. Poincaré’s famous conjecture, a ‘Millennium Prize Problem,’ exemplifies his enduring influence. His work on logic, the continuum and infinity also laid the foundations for Gödel’s contributions as another inspired Platonist, revealing his everlasting influence in the foundations and nature of mathematical truth.

‘Logic sometimes creates monsters.’ ― Henri Poincaré

One of the topics I particularly enjoyed is the book’s detailed accounts of Poincaré’s philosophy, metaphysics and keen investigation into the nature of truth and self-reflection. He influenced Hadamard and many others to use the results of introspection to study mathematical thought processes. He described his own mathematical thinking as largely wordless, often accompanied by mental pictures representing the whole solution to a problem, and seeing whole solutions with ‘sudden spontaneousness.’

‘The subliminal self is in no way inferior to the conscious self. It knows how to choose and to divine.’ ― Henri Poincaré

Although not a Platonist in the strict academic sense that most classicists subscribe to, the book refers extensively to his belief that mathematical, noetic, entities exist independently of human thought, and that mathematical truths are accessible only by the intellect. While Poincaré acknowledged this other-worldly objective reality, he also emphasised the role of human intuition and pure creativity in the development of mathematical knowledge. As the book recounts, Poincaré’s approach to mathematics was characterised by a deep appreciation of the reality of both its objective and subjective dimensions.

*‘It is through science that we prove, but through intuition that we discover.’

*― Henri Poincaré

*‘Logic remains barren, unless it is fertilised by intuition.’

*― Henri Poincaré, Science and Method

The book touches on the metaphysics of knowledge as witnessed by the genius of Poincaré, but doesn’t go deeply into it. It does, however, provide a wealth of references and detailed annotations to passages in his original texts. It also provides a good glossary and appendices on the mathematical background needed to appreciate these insights more fully.

‘There is no logic and epistemology independent of psychology.’

- Henri Poincaré, Last thoughts

Apart from the warning that ‘here be formulas’ ... as in ‘hic sunt dracones,’ this work offers a detailed portrait of a pivotal figure in science, with a rich exploration of mathematics, physics, and philosophy for anyone interested in the history of mathematics, physics, and the philosophy, both epistemological and ontological, of science. I sincerely hope that you too will come to admire and love such a truly extraordinary thinker as Henri.