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heaven and hell (and even into American presidential statements such as “You are either with us, or you are with the terrorists”). More generally, it has always been true that the meaning of life has been illuminated by death, and of knowledge by comparing it to ignorance.
    Not all the Pythagorean teachings had to do directly with numbers. The lifestyle of the tightly knit Pythagorean society was also based on vegetarianism, a strong belief in metempsychosis—the immortality and transmigration of souls—and a somewhat mysterious ban on eating beans. Several explanations have been suggested for the bean-eating prohibition. They range from the resemblance of beans to genitals to bean eating being compared to eating a living soul. The latter interpretation regarded the wind breaking that often follows the eating of beans as proof of an extinguished breath. The book Philosophy for Dummies summarized the Pythagorean doctrine this way: “Everything is made of numbers, and don’t eat beans because they’ll do a number on you.”
    The oldest surviving story about Pythagoras is related to the belief in the reincarnation of the soul into other beings. This almost poetic tale comes from the sixth century BC poet Xenophanes of Colophon: “They say that once he [Pythagoras] passed by as a dog was being beaten, and pitying it spoke as follows, ‘Stop, and beat it not; for the soul is that of a friend; I know it, for I heard it speak.’”
    Pythagoras’s unmistakable fingerprints can be found not only in the teachings of the Greek philosophers that immediately succeeded him, but all the way into the curricula of the medieval universities. The seven subjects taught in those universities were divided into the trivium, which included dialectic, grammar, and rhetoric, and the quadrivium, which included the favorite topics of the Pythagoreans—geometry, arithmetic, astronomy, and music. The celestial “harmony of the spheres”—the music supposedly performed by the planets in their orbits, which, according to his disciples, only Pythagoras could hear—has inspired poets and scientists alike. The famous astronomer Johannes Kepler (1571–1630), who discoveredthe laws of planetary motion, chose the title of Harmonice Mundi ( Harmony of the World ) for one of his most seminal works. In the Pythagorean spirit, he even developed little musical “tunes” for the different planets (as did the composer Gustav Holst three centuries later).
    From the perspective of the questions that are at the focus of the present book, once we strip the Pythagorean philosophy of its mystical clothing, the skeleton that remains is still a powerful statement about mathematics, its nature, and its relation to both the physical world and the human mind. Pythagoras and the Pythagoreans were the forefathers of the search for cosmic order. They can be regarded as the founders of pure mathematics in that unlike their predecessors—the Babylonians and the Egyptians—they engaged in mathematics as an abstract field, divorced from all practical purposes. The question of whether the Pythagoreans also established mathematics as a tool for science is a trickier one. While the Pythagoreans certainly associated all phenomena with numbers, the numbers themselves—not the phenomena or their causes—became the focus of study. This was not a particularly fruitful direction for scientific research to take. Still, fundamental to the Pythagorean doctrine was the implicit belief in the existence of general, natural laws. This belief, which has become the central pillar of modern science, may have had its roots in the concept of Fate in Greek tragedy. As late as the Renaissance, this bold faith in the reality of a body of laws that can explain all phenomena was still progressing far in advance of any concrete evidence, and only Galileo, Descartes, and Newton turned it into a proposition defendable on inductive grounds.
    Another major contribution attributed to the