Proof: Demonstration from All Angles

Article excerpt

Pythagoras's 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's 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 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 forebears. 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's mathematical abilities remained modest, Pythagoras's theorem had taught him a new way to reason and to present his conclusions persuasively.

Pythagoras's 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 6th century BC. Ancient Indian texts accompanying the Sutras - from 100 to 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. …