Borges and Mathematics

6. Euclid, or the Aesthetics of Mathematical Reasoning

6 Euclid, or the Aesthetics of Mathematical Reasoning At the end of the 1930s, a diminutive man with a fragile demeanor and a broad forehead arrived at the Universidad Nacional del Litoral, persecuted by Mussolini. He was Beppo Levi, among the most important mathematicians of the twentieth century. He had been hired as a researcher at one of the first specialized institutes in Argentina, but, due to a typicalArgentine paradox, there was a sudden, devastating intervention, and Levi ended up teaching ordinary classes in mathematical analysis to first-year students. It was also in the city of Rosario inArgentina where his Leyendo a Euclides (Reading Euclid) was first published. Nearly fifty years later, a group of his academic disciples issued a new edition of this detective-like incursion into Socratic thought. In order to understand the importance of this book, it’s important to keep in mind that Euclid’s geometrical axioms not only were— and still are, to a great extent—the paradigm of the way mathematical reasoning operates, but that they also forged a profound and almost imperative aesthetic for that reasoning, with multiple philo-| 81 | | 82 | BORGES AND MATHEMATICS sophical implications that endure to this day. That aesthetic is the delicate balance between simplicity and scope, between the minimum number of assumptions and the maximum number of consequences that can be derived from those assumptions. In effect, the beauty and seductiveness of the Euclidean model lies in that fact that, by using basic concepts such as point, straight line, circle, and only five axioms to connect those concepts in a fairly obvious way, proceeding from theorem to theorem, all of classical geometry can be derived: that is, the sum total of geometry known to humanity until not very long ago, a geometry that Kant believed to be the only one possible. It is the geometry that corresponds to the way in which we see the world, and the one that mapmakers, architects , and surveyors employ for all their daily needs. This age-old influence of the axiomatic approach to philosophy can be found in Spinoza’s Ethics, whose subtitle is Demonstrated in Geometric Order, as well as in Descartes’search for a truth “beyond all reasonable doubt,” one that might serve as first principle and foundation on which to build an impregnable system of thought through purely logical steps. But perhaps the best-known story about Euclidean geometry is the one having to do with the fifth postulate: Given a straight line and a point outside of it, there can be only one straight line parallel to the given line that passes through that point. Of the five axioms, this last one was the least obvious, even to Euclid himself, and he tries to use it in his proofs only when strictly necessary. For two thousand years it was thought that it might be possible to prove this fifth axiom by using the four previous ones, like just one more theorem. To find that elusive proof became geometricians ’ primary unresolved problem. At last, in 1926, a Russian student named Nikolai Lobachevsky discovered that it was completely possible to develop a new geometry in which the first four axioms were valid, but not the fifth. Later, Hungarian mathematician János Bolyai proved something even more curious: that the new geometry , strange as it might have seemed intuitively, was as legitimate and solid as Euclidean geometry in the sense that if it happened to EUCLID | 83 | lead to a logical contradiction, the “fault” of that contradiction could not be attributed to the negation of the fifth postulate, but rather to the four previous ones, which are shared with classical geometry. German mathematician Carl Friedrich Gauss, who had arrived at the same conclusions on his own, was one of the first to observe that the existence of a non-Euclidean geometry threatened the Kantian idea of an a priori notion of space. This was one of the harshest blows to Kant’s philosophy, later compounded by experiments in the geometry of visual perception, also not wholly Euclidean, by German physicist Hermann von Helmholtz. Hilbert’s Program and Incompleteness Euclid’s spirit was revived with special vigor in early 1900 with Hilbert’s program for laying the foundations of mathematics. Certain logical paradoxes, as pointed out by Russell in set theory, had caused the venerable edifice of mathematics to creak for the first time, revealing the need to look...