Show description

We have only the faintest echoes of the story, in late and unreliable sources at that, since secrecy obsessed the Pythagoreans generally, but at this moment most of all. A Pythagorean named Hippasus, they say, from Metapontum, used that great theorem to prove there was a magnitude which, when compared to the unit length, couldn’t make a ratio of two natural numbers. But if this were so, where would the music of the spheres and the harmony of things be? Where the whole, the tetractys, the moral foundations of life?

So if an operation made sense for the natural numbers, it must—by Peacock’s Principle—make sense for any kind of number. His Principle never stooped to ask why this should be so, and in fact (as we shall see on page 93), led to nonsense. Hidden in the neutral word “form”, however, was the embryo of an abstractly formalist point of view that would utterly shift our understanding of mathematics. This changing way of looking was part of the broader Romantic rebellion against Enlightenment ideals.

Prove that it is true for k’s successor, k + 1. You will probably also want to use the fact that the kth odd number has the form 2k – 1 and the next one 2k + 1 (you can check your proof against that in the Appendix). It tells you how risky this new kind of proof must have seemed to its inventor that he actually checked the statement not only for 1 but for 3 and 5. It is as if we were witnessing scientific induction turning into mathematical induction. History isn’t inductive, since there never seems to have been a definitive first instance of any notion you can name.