“The soul of the fact”—Poincaré and proof

Henri Poincaré acquired a reputation in his lifetime for being difficult to read. It was said that he missed out important steps in his arguments, assumed the truth of claims that would be difficult if not impossible to prove, and in short that he lacked rigour. In the years after his death this view coalesced into an exaggerated claim that his work was simply too vague, and has become a cliché. This paper argues that Poincaré was far from indifferent to rigour, and that what characterises his work is an attempt to convey a particular sense of what it is to understand a topic. Throughout his working life Poincaré was concerned to promote the understanding of many domains of mathematics and physics. This is as apparent in his views about geometry, his conventionalism, and his theory of knowledge, as it is in his work on electricity and optics, on number theory, and function theory. It is one of the ways Poincaré discharged his responsibilities as a scientist, and that it accounts not only for a surprising degree of unity in his work but also gives it its distinctive character—at once profound and elusive.