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Pythagoras

Pythagoras was one of the most intriguing and puzzling men in history, but very little is known about his life. Born on the island of Samos, and active ca. 550 B.C., he was both the founder of modern mathematics and the leader of a religious cult whose main beliefs were the transmigration of souls and the sinfulness of eating beans. Mathematics, as developed by the Greeks, begins with Pythagoras. His ideas profoundly influenced Plato and Euclid, and therefore the whole evolution of Western philosophy. The heart of Pythagorean mathematics was number. ¡°All things are numbers¡±, he reputedly said, and numbers were considered to have their own personality ¨C masculine or feminine, perfect or incomplete, beautiful or ugly. For example, ten was thought to be the most perfect number because it contained within itself the first four integers (1+2+3+4=10). He was the earliest to understand that vibrating strings produce harmonious sounds when the ratios of their lengths are whole numbers, demonstrating that beneath the beauty of music lay a set of mathematical relationships. Among his most important ideas was the geometric proof, which, as developed by Euclid, begins with a set of ¡°self-evident¡± axioms and through a process of deductive reasoning yields theorems that are very far from self-evident. These theorems were then taken to be true of real space, or as something which could be verified through experience. This led to a spiraling set of assumptions culminating in the belief that the world could be understood through the use of reason alone, a view which dominated European philosophy through the end of the 18th century. When the Declaration of Independence states ¡°We hold these truths to be self-evident¡± it is taking Greek mathematics as its rhetorical model.

Today, as every tenth grader knows, Pythagoras is best known for the Pythagorean Theorem, which states that for a right-angled triangle the square of the hypotenuse is equal to the sum of the squares on the two other sides (c 2 =a 2 +b 2). Central to the Pythagorean system was the belief that all numbers are whole numbers, and consequently behave ¡°rationally¡±. But buried within the Theorem of Pythagoras was a virus that eventually destroyed any possibility of a world based on reason. If the right triangle has two sides with a length of 1, the length of the hypotenuse will equal ¡Ì (12 + 12 ), or ¡Ì 2. The square root of 2, however , turns out to be 1.414213562¡, a never- ending, or ¡°irrational¡± number. The discovery of never-ending numbers, numbers that were not ¡°whole¡±, devastated Pythagoras, for having set out to prove the simplicity of the world, he had accidentally stumbled across its irreducible complexity. But the final irony is that what turned out to be his greatest contribution to modern thought - the discovery of the irrational - became the darkest secret of the Pythagorean cult, the mere mention of which was punishable by death.

Mel Bochner

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Mel Bochner / Alfred Jensen

Hommage to Pythagoras