The Pythagorean society at Croton soon evolved into a moral, religious, and political movement of reform, influenced the city's politics, elevated Croton to a commanding position among the Greek cities in southern Italy, organized branches in other such cities, whose politics the Pythagoreans purified, and created a loose political and economic alliance among the communities which the Pythagoreans controlled by placing their members in prominent magistracies. But the rising tide of democracy destroyed the Pythagorean brotherhood in southern Italy, probably during 450- 440 B.C., when the democrats persecuted the aristocratic Pythagoreans, burned their several headquarters. killed some and exiled other members.
But during 400-390 B.C., when Dionysius the Elder, tyrant of Syracuse, was threatening the independence of Greek cities in southern Italy, the survivors of the Pythagorean order returned and resumed a prominent part in politics by constructing in defense of liberty a coalition of aristocratic and democratic forces against autocracy. The final conquest of most of the larger Greek city-states in that area by Dionysius during 390-379 B.C. caused the second catastrophe of political Pythagoreanism. This time most of the members were exiled. Some of the society turned to Tarentum (Taranto), where they exerted considerable influence, especially during 370- 360 B.C., when Pythagoreans officially operated the government.
Others of the order traveled to Greece and wandered from city to city as itinerant missionaries of their doctrine. But by 350 B.C. the brotherhood as an organization disappeared from the political scene everywhere.
Although much of the tradition about Pythagorean philosophy is confused because of dissensions within the school and on account of intermixture of later speculation with earlier doctrine, yet some of the chief principles are quite clear. Pythagoras' discoveries in musical theory, such as that the basic musical harmonies depend on very simple numerical ratios between the dimensions of the instruments (such as strings, pipes, disks) producing them, led him to interpret the world as a whole through numbers.
This discovery was the basis for the Pythagorean theory of numbers, of which the systematic study induced the intense Pythagorean devotion to mathematics and the subsequent development of this science by Greek scientists. Pythagoras taught that since all things can be counted, one cannot imagine a universe wherein is no number and that number is a fundamental part of the world's framework. According to his theory the dominant notes of the universe are proportion, order, and harmony. All three are expressible by numerical relations. Pythagoreans thus considered that the universe's essential character is number, but they went beyond this by asserting that the world is made of numbers-a doctrine that is the core of Pythagorean philosophy. In preaching this principle the Pythagoreans both propounded several semi-mystical speculations and discovered some scientific truths.
On the speculative side occurs the celebrated Pythagorean table of opposites, derived from their proposition that the universe is composed of pairs of contradictories. The pairs are 10 in number: (1) limited and unlimited; (2) odd and even; (3) one and many; (4) right and left; (5) masculine and feminine; (6) rest and motion; (7) straight and crooked; (8) light and darkness; (9) good and evil; (10) square and oblong. Though this theory may not be so fantastic as it appears, the Pythagorean development of numbers was quite arbitrary in the following proposition. The number 1 is the point, 2 is the line, 3 is the plane, 4 is the solid,S is physical qualities, 6 is animation, 7 is intelligence and health, 8 is love, friendship, wisdom. Identification of different numbers with different things exemplifies no principle. The Pythagoreans themselves disagreed on what number should be assigned to what thing. Thus. since justice is that which returns equal for equal, the only numbers which do this are square numbers; thus 4 equals 2 into 2 and so returns equal for equal; thus 4 must be justice. But, since 9 is equally the square of 3, 9 also can represent justice. Such speculation seems sterile, save to numerologists.
Among the Pythagorean achievements in science were:
(1) The Pythagorean theorem, reliably reported to have been discovered by Pythagoras, to whose speculation was owed also, quite probably, most of the first book of Euclid's Stoicheia (Elements) on geometry.
(2) By 500 B.C. the earth's sphericity was proclaimed by Pythagoreans, who were among the first, if not the first, to teach it.
(3) Hippasus (450 B.C.) discovered incommensurability and elaborated a theory of proportions applicable to incommensurables.
(4) By 400 B.C. the Pythagoreans taught the theory that the earth, sun, moon, planets. and fixed stars revolve around a central fire-a denial of the earlier and later geocentric view of the universe and an anticipation of Nicolaus Copernicus' heliocentric hypothesis announced in 1543. From this theory they developed the doctrine of the music of the spheres, which lasted into modern times.
(5) Archytas of Tarentum (360 B.C.) developed a very advanced theory of acoustics and founded mechanics.
(6) At an undetermined date Pythagoreans developed the theory of mathematical "means" and they also invented the theory of polygonal numbers.
Pythagorean ethics consisted in ascetic practice. Happiness was the perfection of the soul's virtue, which was a kind of harmony. The process of purification of the soul was accomplished by metempsychosis, the transmigration of the soul, a theory imported by Pythagoreans from the Orient and one of their most characteristic dogmas.