Pythagoras founded* a school of philosophy in Croton in southern Italy that was renowned throughout the ancient Mediterranean world. As much a religious brotherhood as a scholastic academy teaching music, astronomy, ethics, arithmetic and geometry, its students were taught mathematics not — as it was considered by Plato — as a preparation for an otherworldly contemplation of divine principles but in order to contemplate the divine immanent in nature. The mathematical doctrines of Pythagoras cannot be properly understood without recognising that the modern dichotomy of science and religion did not exist for the Pythagoreans, who believed that man can realise his divine nature by knowing the universal principle which governs the cosmos (a word coined by Pythagoras himself, meaning “world-order,” a world ordered in a state of mathematical harmony). This principle is Number, which is “the principle, the source and the root of all things” (1). For the Pythagoreans, the spiritual and scientific dimensions of number were complementary and could not be separated.</p>Pythagoras was not only the first to call himself a philosopher but also a priest -initiate of a mystery religion influenced heavily by Orphism, which taught that the essence of the gods is defined by number. Numbers, indeed, expressed the essence of all created things. According to the Pythagorean Philolaus: “All things which can be known have numbers, for it is not possible that without number anything can either be conceived or known” (2). The Pythagoreans were the first to assert that natural phenomena conformed to mathematical principles and so could be understood by means of mathematics. In this sense, they may be considered the first physicists. But their doctrine gradually became distorted into the proposition that not only does number express the essence of things but also that, ultimately, all things are numbers. Unconvinced by the peculiar emphasis Pythagoreans gave to numbers because he was not privy to the secrets of their teachings, Aristotle said of them: “These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent state” (3). The Pythagoreans thought that numbers had metaphysical characters, which expressed the nature of the__________________________________________* Pythagoras’ early biographers provide an unreliable, inconsistent chronology. But, as they agree that he left his home Samos for Italy during the rule of its dictator Polycrates (528–522 BCE) and was deported from Egypt after its invasion by Cambyses in 525 BCE, he must have started his school at Croton between 525 BCE and 522 BCE. gods. The number one (the Monad) represented the principle of unity — the undifferentiated source of all created things. The Pythagoreans did not even regard it as a number because for them it was the ultimate principle underlying all numbers. The number 2 (Dyad) represented duality — the beginning of multiplicity, but not yet the possibility of logos, the principle relating one thing to another. The number 3 (Triad) was called “harmony” because it created a relation or harmonia (“joining together”) between the polar extremes of the undifferentiated Monad and the unlimited differentiation of Dyad.</p>Pythagoras (Fig. 1) was the first to use geometrical diagrams as models of cosmic wholeness and the celestial order. Numbers themselves were represented by geometrical shapes: triangles, squares, pentagons, etc. For example, a ‘triangular number’ is any number that is equal to the number of dots forming a triangular array and a ‘square number’ is one that can be represented by a square array of dots (Fig. 2). The Greeks generalised such ‘figurative numbers’ by considering nests of n regular polygons nested inside one another so that they
