Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through them, all of Western philosophy.
Historians from the Stanford Encyclopedia of Philosophy pointed out
Aristotle makes clear that there are several groups of people included under the heading “so-called Pythagoreans,” by explicitly distinguishing those Pythagoreans who posited the table of opposites from the main Pythagorean system.
According to tradition, pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi (μαθηματικοί, Greek for "Teachers") and the akousmatikoi (ἀκουσματικοί, Greek for "listeners"). There is the inner and outer circle
John Burnet (1892) noted
Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, and Anaximenes, for instance, can only be understood as the elaboration of a single idea in a school with a continuous tradition.
According to Iamblichus (ca. 250-330 AD, 1918 translation) in The life of Pythagoras, by Thomas Taylor
There were also two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras but from Hippasus. The philosophy of the Acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process; because it merely ordered a thing to be done in a certain way and that they should endeavor to preserve such other things as were said by him, as divine dogmas. Memory was the most valued faculty. All these auditions were of three kinds; some signifying what a thing is; others what it especially is, others what ought or ought not to be done. (p. 62)
- It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth.
- Musical principles played almost as important a part in the Pythagorean system as mathematical or numerical ideas. The opposite principia of the unlimited and the limiting are, as Philolaus expresses it neither alike, nor of the same race, and so it would have been impossible for them to unite, had not harmony stepped in."This harmony, again, was, in the conception of Philolaus, neither more nor less than the octave (Brandis, /. c. p. 456). On the investigation of the various harmonical relations of the octave, and their connection with weight, as the measure of tension, Philolaus bestowed considerable attention, and some important fragments of his on this subject have been preserved. We find running through the entire Pythagorean system the idea that order, or harmony of relation, is the regulating principle of the whole universe.
...the hammers beating out a piece of iron on the anvil and producing sounds that accorded with each other, one combination only excepted. In these sounds he recognized the diapason, the diapente and the diatessaron, harmony. And the sound that was between the diatessaron and the diapente was by itself dissonant, yet gave completion to that which was the greater sound among them. (p.49)
- The chorus of the Muses was always one and the same, and they had charge of unison, harmony and rhythm, all that goes to make up concord.
Thus this is what it is to use anything: if the capacity is for a single thing, when someone is doing this very thing, and if the capacity is for a number of things, when he is doing the best of them; for example, with flutes, one uses them either only when playing the flute, or most of all then, as its other uses are presumably also for the sake of this. Thus one should say that someone who uses a thing correctly is using it more, for the natural objective and mode of use belong to someone who uses a thing in a beautiful and precise way. And now the only function of the soul, or else the greatest one of all, is thinking and reasoning. Therefore it is now simple and easy for anyone to draw the conclusion that he who thinks correctly is more alive, and he who most attains truth lives most, and this is the one who is intelligent and observing according to the most precise knowledge; and it is then and to those that living perfectly, surely, should be attributed, to those who are using their intelligence, i.e. to the intelligent. (p. 56)
The Neoplatonists were quite justified in regarding themselves as the spiritual heirs of Pythagoras; and, in their hands, philosophy ceased to exist as such, and became theology. And this tendency was at work all along; hardly a single Greek philosopher was wholly uninfluenced by it. Perhaps Aristotle might seem to be an exception; but it is probable that, if we still possessed a few such "exoteric" works as the Protreptikos in their entirety, we should find that the enthusiastic words in which he speaks of the "blessed life" in the Metaphysics and in the Ethics (Nicomachean Ethics) were less isolated outbursts of feeling than they appear now. In later days, Apollonios of Tyana showed in practice what this sort of thing must ultimately lead to. The theurgy and thaumaturgy of the late Greek schools were only the fruit of the seed sown by the generation which immediately preceded the Persian War.
- The Pythagorean idea that whole numbers and harmonic (euphonic) sounds are intimately connected in music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantified and described. This may have paved the way to his son's crucial insight that all physical phenomena may be described quantitatively in mathematical language (as physical "laws"), thus beginning and defining the era of modern physics.
- Pythagoreanism has been regarded by some to have had a clear influence on the texts found in the hermetica and thus to have flown over into hermeticism, gnosticism and alchemy.
- The Pythagorean cosmology also inspired the Arab gnostic Monoimus to combine this system with monism and other things to form his own cosmology.
- The pentagram (five-pointed star) has been thought to be an important religious symbol used by the Pythagoreans.
- The Pythagorean school doubtless had a monumental impact on the development of numerology and number mysticism, an influence that still resonates today. For example, it is from the Pythagoreans that the number 3 acquires its modern reputation as the noblest of all digits.
- The Pythagoreans were advised to "speak the truth in all situations," which Pythagoras said he learned from the Magi of Babylon.
- The Pythagorean theory of harmonic ratios is the basis of studies on music theory in the Islamic world, for example al-Farabi's Kitab al-Musiqa al-kabir.
- Pythagorean philosophy had a marked impact on the thoughts of early modern scholars involved within the Scientific Revolution. Of particular interest is the focus applied to the Platonic Solids derived from the Pythagorean theories of geometry and numbers by Plato. Within the work of Leonardo fascination can be found within manuscripts describing the Platonic Solids, and also within the work of Kepler who supported the Copernican theory of heliocentrism and attempted a theory of the universe based on musical, geometrical harmony.
- Cornelli, G.; McKirahan, R.; Macris, C. (eds.), On Pythagoreanism, Berlin, Walter de Gruyter, 2013.
- Cerqueiro, Daniel. Evohé (Pythagoras doxography).Buenos Aires 2004: Ed. Peq. Ven.
- Jacob, Frank Die Pythagoreer: Wissenschaftliche Schule, religiöse Sekte oder politische Geheimgesellschaft?, in: Jacob, Frank (Hg.): Geheimgesellschaften: Kulturhistorische Sozialstudien/ Secret Societies: Comparative Studies in Culture, Society and History, Globalhistorische Komparativstudien Bd.1, Comparative Studies from a Global Perspective Vol. 1, Königshausen&Neumann, Würzburg 2013, S.17-34.
- O'Meara, Dominic J. Pythagoras Revived: Mathematics and Philosophy in Late Antiquity , Clarendon Press, Oxford, 1989.
- Riedweg, Christoph Pythagoras: his life, teaching, and influence ; translated by Steven Rendall in collaboration with Christoph Riedweg and Andreas Schatzmann, Ithaca : Cornell University Press, (2005),