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An ethical, religious, and mystical system of teaching founded by
Pythagoras in the sixth century BC. Those holding to such teaching were called Pythagoreans. Their first society or brotherhood was established in Croton about 530 BC. The teaching exerted political influence in Croton and in other city-states throughout the region. By the fifth century BC Pythagorean societies in southern Italy had became involved in the fierce fighting between the aristocracy and the democratic forces of government. When the democratic party gained control it fiercely turned on the Pythagoreans in their settlements and burned them. Those that survived fled back to the Greek mainland and settled around Thebes and Phlius.

About this time in the fifth century BC the Pythagoreans separated into two distinct groups called the Acusmatici (from akousma, meaning "oral precept") whose members emphasized the observation of the special Pythagorean way of life taught by the master himself. The second group was the Mathematici (meaning "students of theoretical subjects"), who prsued interests in arithmetic, the theory of music, astronomy, and cosmology.

The Mathematici group eventually became closely associated with the doctrine of Plato and his followers who established the Platonic Academy. The Acusmatici became wandering ascetics finally joining the Cynics of the fourth century BC.

There is speculation as to the reasons for the separation of the Pythagoreans. It should be understood that the Mathematici group did not abandon the ethical rigorism of the master which the Acusmatici maintained, but developed a Pythagorean world-view which served as a framework and justification for the older themes. Most probably the two factors which separated the groups the most was the function of number symbolism in ritual, and the very obsessive role of ideas of order, measure, and organization in the Pythagorean way of life.

From the end of the third century BC to the Christian era both Pythagorean groups were in recession. Although there is a large body of Pythagorean literature, most of it is thought to have been borrowed or copied from Platonic, Aristotelian, and Stoic material. Within the first century BC there was a revival of the teaching, called Neo-Pythagoreanism, as a school and way of life centered around Rome and Alexandria.

The seven cardinal doctrines overlap the doctrines of
Platonism so to appear that Pythagoreanism joined with, or was derived from the latter. The doctrines: 1. The fundamental realities of the world are structural and mathematical. 2. These structures constitute not only what is more fundamental but also what is normatively better, what is aesthetically more beautiful, and what displays greater simplicity, regularity, and coherence in its mathematical proportions or parameters. 3. Structures in superficially dissimilar contexts can be basically the same. Indeed, there is a pervasive affinity or sympathy between the inanimate and the animate, between man's psyche and the whole cosmos. 4. This cosmic sympathy affords the possibility of moral improvement through a patterning of the individual psyche on the cosmos. 5. Beyond moral improvement, the cosmic sympathy affords the prospect of ascent to a trans-human level of existence, even to immortality, through a process of purification. Correlatively, it also poses the threat of descent into an infra-level of existence. 6. Knowledge or understanding is inherently mystical and can be attained only by the elite. 7. The study of mathematics is the indispensable basis for all intellectual and spiritual progress.

Ultimately, Pythagoreanism has been a dynamic force on Western culture. It has creatively influenced philosophers, theologians, mathematicians, and astronomers (notably Copernicus and Kepler), as well as musicians, composers, poets, and especially architects of the Middle Ages.

As previously stated, much of Pythagoreanism seems to overlap Platonism. Although the Mathematici groups have been give much credit for the development of mathematics there is an incident which lends credence to the overlapping concept. In the fourth century there existed a friendship between a leading Pythagorean and Archytas of Tarentum, a statesman and brilliant mathematician, whose was also a friend of Plato. It is assumed some of the Pythagorean "number lore" was attributed by Archytas.

Sources: 1, (Alexander P.D. Mourelatos, The University of Texas in Austin) 61.

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