Pythagoras of the Ionian island of Samos (about 572-500 BC) was an influential but obscure figure in history. It is said that Thales was so surprised by the talents of the young man that he recommended that he should go to Egypt to study under the guidance of priests. An equally uncertain story tells that he received learning while a prisoner in Babylonia. At an age of about 40, Pythagoras moved to southern Italy where he and his wife Theano founded a school in the Greek colony of Crotona. The school was actually a religious fraternity, where mathematics, philosophy, and other topics were practised under the leadership of the master.
To the candidates for the first principle Pythagoras added still another entity, number. The cosmos, "ordered universe," is ruled by mathematics. This idea has a far-reaching consequence that we are still feeling in our own science: it is possible for a thinking human being to deduce the structure of the universe, without visiting every corner. The Pythagoreans regarded the Earth as a sphere, as is the starry sky. Planets, among them, the Sun and the Moon, are each attached to their own spheres that revolve around the Earth. Surely there was already evidence for the spheroid of the Earth (e.g., travelers knew that the sky changed when they go from north to south), but likely such empirical aspects just enforced the belief in the primary nature of the complete, beautiful spherical shape.
It is remarkable how one Pythagorean, Philolaus (around 450 BC), taught that the Earth and other cosmic bodies revolve around the fire burning in the center of the world. The fire is not the Sun, so this was not a heliocentric system, but it showed that it was possible to imagine the Earth moving in space even though we do not feel anything of the sort under our feet. Philolaus is said to have theorized that we cannot see the central fire, because the Earth always turns with the same half toward it (like the Moon does relative to the Earth).
Pythagoras founded number theory and proved the famous theorem of Pythagoras about the areas of the squares drawn on the sides of a right-angled triangle. Integer numbers were the basis of the Pythagorean worldview. Those thinkers regarded that integer numbers (or their ratios), which were the only type of numbers known at the time, may measure everything in the world. For example, they thought that a line is formed by a large number of points, like atoms put side by side, and hence the ratio of the lengths of any two line segments would always be rational. It was a shock to find, using the very theorem of Pythagoras, that the ratio of the diagonal and the side of a square (= a/2) cannot be expressed in terms of integers. Along with the old numbers ("rational") one had to accept new ones ("irrational"). In the long run this was necessary for the further development of mathematics.
Irrational numbers served as a healthy reminder that the world is not so simple that first mathematical concepts were sufficient for its description and understanding. Nevertheless, modern scientists view with sympathy the efforts of Pythagoras to grasp the cosmos as a harmonic whole. We also like to believe that the world must be in some deep manner simple and comprehensible.
About 500 BC there was an attack on Crotona, the house of the Pythagoreans was burnt down and several members of the fraternity were killed. Others escaped. Pythagoras himself went to Tarentum (in Italy), but many moved to the mainland of Greece, e.g., to Athens, where the new ideas began to spread.
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