The wonder of proof

PYTHAGORAS’ theorem changed the life of the British philosopher Thomas Hobbes (1588-1679). Until he was 40, Hobbes was a talented scholar exhibiting modest originality. Versed in the humanities, he was dissatisfied with his erudition and had little exposure to the exciting breakthroughs achieved by Galileo, Johannes Kepler and other scientists who were revolutionizing the scholarly world.

One day, in a library, Hobbes saw a display copy of Euclid’s “Elements� opened to Book I, Proposition 47 -- Pythagoras’ theorem. He was astounded, exclaiming, “This is impossible!� He read on, intrigued. The demonstration referred him to other propositions, and he was soon convinced that the startling theorem was true.

Hobbes was transformed. He began obsessively drawing figures and writing calculations on bedsheets and even on his thigh. His approach to scholarship changed. He began to chastise philosophers of the day for their lack of rigor and for being unduly impressed by their forbearers. He compared other philosophers unfavorably with mathematicians, who proceeded slowly but surely from “low and humble principles� that everyone understood.

In books such as “Leviathan,� Hobbes reconstructed political philosophy by establishing clear definitions of terms, then working out implications in an orderly fashion. Though Hobbes’ mathematical abilities remained modest, Pythagoras’ theorem had taught him a new way to reason and to present his conclusions persuasively.

Pythagoras’ theorem is important for its content as well as for its proof. But the fact that lines of specific lengths create a right-angled triangle was discovered in different lands long before Pythagoras. Another pre-Pythagoras discovery was the rule for calculating the length of the long side of a right triangle (c) knowing the lengths of the other sides (a and b): c2 = a2 + b2.

Indeed, a Babylonian tablet from about 1800 BC shows that this rule was known in ancient Iraq more than 1,000 years before Pythagoras, who lived in the sixth century BC. Ancient Indian texts accompanying the Sutras -- from between 100 and 500 BC, but clearly passing on information of much earlier times -- also show a knowledge of this rule. An early Chinese work suggests that scholars there used the calculation at about the same time as Pythagoras, if not before.

But what we do not find in these works are proofs -- demonstrations of the general validity of a result based on first principles and without regard for practical application. “Proof� was itself a concept that had to be discovered. In Euclid’s “Elements,� we find the first attempt to present a more or less complete body of knowledge explicitly via proofs. Euclid does not mention Pythagoras, who lived about 200 years earlier, in connection with Proposition 47. We credit it to Pythagoras on the authority of several Greek and Latin authors, including Plutarch and Cicero.

Pythagoras’ theorem is unique for the peculiar way in which it has become a challenge to devise new proofs for it. These proofs are not necessarily any better; most rely on the same axioms but follow different paths to the result. The Guinness Book of World Records website names someone who, it is claimed, has discovered 520 proofs.

What is gained by proving a theorem over and over in different ways? The answer lies in our desire not merely to discover but to view a discovery from as many angles as possible. But what is it that is so fascinating about Pythagoras’ theorem in particular?

First, the theorem is important. It helps to describe the space around us and is essential not only in construction but also, in modified form, in equations of thermodynamics and general relativity. Second, it is simple. The Hindu mathematician Bhaskara II constructed a proof using a simple diagram -- and instead of an explanation wrote a single word of instruction: “See!�

Third, it makes the visceral thrill of discovery easily accessible. In an essay, Albert Einstein wrote of the “wonder� and “indescribable impression� left by his first encounter with Euclidean plane geometry as a child, when he proved Pythagoras’ theorem for himself.

In Plato’s “Meno,� Socrates teaches it to an illiterate slave boy. And in his memoirs, novelist George MacDonald Fraser writes of proving it to a war-weary but delighted comrade, tracing outlines in the sand with a bayonet, in a break during the fighting against the Japanese in World War II.

For Hobbes, Socrates, Fraser and countless others, Pythagoras’ theorem was far more than a means to compute the length of hypotenuses. It shows something more -- the idea of proof itself. It reveals more than a bare content but a structure of reasoning. It is a proof that demonstrates proof.