Mathematics and history

Abstract

I begin with a caricature---indeed, a travesty--of the typical history of mathematics textbook. This caricature will draw attention to some of the philosophical presuppositions which often underlie these books. I shall then discuss alternatives to these presuppositions. Mathematics epitomises Reason. It began in Egypt and Mesopotamia. However, it really began in Greece, because that is where pure mathematics began, and pure mathematics is better than applied mathematics, because pure Reason is better than impure Reason. The greatest Greek mathematicians were Eudoxus, Apollonius, Archimedes, and Hypatia. Hypatia did very little compared to Archimedes, but Hypatia was the only woman who did mathematics, so we can be sure that her true greatness was hidden by male chauvinism. In spite of their preference for geometry, and their rejection of motion in mathematics, the Greeks were enormously wonderful. Unfortunately, superstition and ignorance made a comeback when Cyril, the Christian bishop, had Hypatia murdered (in 415 AD), and, for one thousand years, no one in Western Europe did any mathematics. Meanwhile, the Arabs were developing algebra. Although he could not prove the Theorem of Pythagoras for non-isosceles right triangles, AI-Khwarizmi was a magnificent algebraist. He once found two solutions to a quadratic equation, and he used three different values for pi. In the sixteenth century, Europe rebelled against the Church, and Reason (and happiness) returned. Taking up where Hypatia had left off, Newton and Leibniz invented calculus, and introduced motion into mathematics. The odd thing was that Newton and Leibniz worked independently. Therefore both of them deserve the praise and glory for creating calculus. In the nineteenth century, Reason really came into her own. Before that, calculus was not rigorous (partly because it dealt with motion). Today, however, calculus is the most rigorous, most reasonable thing possible. Unfortunately, we cannot tell you much about contemporary mathematics because it would take all our time and energy just to find out ourselves what is happening. We do know, however, that it is wonderful.