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Work in the history of mathematics can, and often should, involve understanding some quite hard mathematics, as it does in questions like these.

By the 1870s, geometry was beginning to be a well-understood subject, with clear research programmes. The study of algebraic curves could be undertaken in several ways, projective geometry was acquiring its status as a fundamental branch of geometry, devoted to the most basic properties of figures (those deeper than the metrical ones), non-Euclidean geometry was acceptable to, and accepted by, most mathematicians (but probably not philosophers). This is indeed the perception of most mathematicians at the time, and most historians of mathematics since, but, as we shall see later, there is another aspect to the story too. But first let us document that projective geometry had become central to many mathematicians perception of geometry.

David Hilbert, who came to dominate German mathematics between 1890 and the 1920s, and who many would say was the leading mathematician in the world of his generation, was born in Königsberg on 23 January 1862. Königsberg was a small town in East Prussia, best known for being the home town of the philosopher Immanuel Kant, and Hilbert’s schooling there was unremarkable. He later said that “I did not particularly concern myself with mathematics at school because I knew that I would turn to it later”. He then went to the university in Königsberg in 1880. Although small, it had a strong tradition in mathematics and physics, beginning with Carl Jacobi and the physicist Franz Neumann, who pioneered the teaching of experimental physics. A steady stream of mathematicians and scientists came to or graduated from Königsberg: the physicist Gustav Kirchhoff, the geometer Otto Hesse and Heinrich Weber, who held the chair in mathematics at Königsberg from 1875 to 1883 and was succeeded by Ferdinand Lindemann, who had just become famous for his proof that is transcendental.

When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. Recent historians of mathematics have shown that, in Italy at least, Fano’s point of view on the nature of geometrical entities had been a generally accepted theory for at least a decade, but it was not in fact axiomatic in Hilbert’s manner, and that other Italian mathematicians were, however, ahead of Hilbert in this manner. How did this come about, what did they do, and why did they lose out?

Jules Henri Poincaré was born at Nancy, a town in Lorraine in the East of France, on 29 April 1854. His father was professor of medicine at the university there, his mother, a very active and intelligent women, consistently encouraged him intellectually, and his childhood seems to have been very happy, at least until the war intervened. The town was surrendered to the Germans as part of the settlement of the Franco-Prussian war 1870—1871, and Poincaré remembered seeing German troops occupying it. This may have been one reason for his choosing the military school, the École Polytechnique, over the increasingly popular civilian École Normale, when the time came. As a child he did not at first show an exceptional aptitude for mathematics, but towards the end of his school career his brilliance became apparent, and he entered the École Polytechnique at the top of his class. Even then he displayed what were to be life-long characteristics: a capacity to immerse himself completely in abstract thought, seldom bothering to resort to pen and paper, a great clarity of ideas, a dislike for taking notes so that he gave the impression of taking ideas in directly, and a perfect memory for details of all kinds. When asked to solved a problem he could reply, it was said, with the swiftness of an arrow. He had a slight stoop, he could not draw at all, which was a problem more for his examiners than for him, and he was totally incompetent in physical exercises.

From 1870 to 1914, public interest grew in the idea that space might not be Euclidean. This interest was inextricably linked with the idea that space might be four-dimensional, which was also mixed up with the idea that time could be considered as a dimension. All these ideas are separate, as matters of mathematical fact, but they attracted a tremendous amount of interest, not least because of the prospect that one could travel in the fourth dimension, or in time, in ways denied to us hitherto.

Such have been the transformations in geometry between the 1840s, when we last took stock, and 1900 that it seems advisable to pass it briefly under review before moving to some final considerations. In Chapter 13, we looked at Möbius’s algebraic version of projective geometry and his introduction of barycentric coordinates. All of projective geometry was thereby turned into algebra, including the line at infinity in the projective plane, and projective transformations.

We have seen a number of types of argument used in geometry: algebraic, differential geometric, projective and, especially, axiomatic arguments. An interesting paper published by the American historian and philosopher of mathematics Ernest Nagel, in 1939, made the provocative suggestion that the principle of duality in plane projective geometry was an important stimulus to thinking of geometry in a purely logical way, independent of any appeal to intuition [171].

Although we shall not discuss the point in any detail, you might ask if non-Euclidean geometry passes or fails this test: can there be a mechanics in such a space? Can one set up such things as an inverse square law of gravity, or conservation of energy and momentum? Could one recover, doubtless in an altered form, everything that physicists had discovered about Euclidean space if it turned out that space obeyed non-Euclidean geometry? The answer is, as you might suspect, yes, and this was shown by Lipschitz, a follower of Riemann, in the 1870s.

It is oddly hard to think of true statements once you allow doubt to play a role. What could another person say to you that you would absolutely have to believe? Statements people make about themselves are surely not acceptable (any follower of detective fiction will know that that is hopeless). The weather? The date? Are you sure you haven’t been hoodwinked, kidnapped, drugged? If we decide to cut these speculations short of paranoia and allow a reasonable degree of interaction with the world in good faith, we enter an old and vexed distinction between certain and merely probable knowledge. Merely probable knowledge was once known as scientia, the word from which science derives, and it had an aspect of unreliability about it. Today, statements of science are among the most highly regarded (and may well figure in any answer to the opening question).