The Eclectic Pythagorean

“There is geometry in the humming of the strings, there is music in the spacing of the spheres. ” -Pythagoras

Pythagoras and the Pythagorean School

Posted by The Eclectic Pythagorean on October 4, 2008


In developing a way of life which distinguishes the Pythagoreans from the rest of the world, Pythagoras clearly distinguishes himself from the Milesians and their nature philosophy, as K&R point out: but the distinction is not quite so clear-cut — Heraclitus too begins philosophy as a way of life, insofar as he includes a concern with a certain kind of wisdom.

In any case, Marias’ comment is also helpful here:

We have in the Pythagorean school a first clear example of philosophy understood as a way of life. The problem of the self-sufficient life issues in a special discipline consisting in contemplation. There appears here the theme of freedom, of self-reliance.

“The Mystical Side of Pythagoras’ Teaching”

Includes the notion of the transmigration of the soul; story of Pythagoras recognizing a soul in a whipped dog.

More generally establishes the kinship of all living things. This kinship follows from the notion that souls qua eternal can be reincarnated in a variety of living things: the suggestion is that the process is cyclical (–> source of Nietzsche‘s notion of eternal return). To put it still differently: the kinship of all living things is a “biological” expression of an emphasis on unity apparent in physical matters.

P. further established rules of abstinence and prohibitions, including a “philosophically-inspired vegetarianism.”

A Pythagorean vocabulary:

theoria (contemplation)kosmos (an orderliness found in the arrangement of the universe)

katharsis (purification)


schole (“leisure” –> “school”)

mania (“orgy”) –> “enthusiasm” (en-theos) –> sophos

philosophia (includes notions of freedom and self-reliance)

mathematikos (fond of learning)

bios theoretikos (the theoretical life) — see above]

“By contemplating the principle of order revealed in the universe — and especially in the regular movements of the heavenly bodies — and by assimilating himself to that orderliness, man himself was progressively purified until he eventually escaped from the cycle of birth and attained immortality.”

In other words, what we might characterize as a religious sort of salvation is centrally dependent upon a “scientific” understanding of the central order of nature, and an “ethic” (from ethos, habit, rule or pattern of behavior) based on that “scientific” understanding which seeks to replicate the cosmic order in the life of the individual.

(This complementary attitude towards what we might call “faith” and “reason” will reappear in the early Middle Ages, and make possible both the recovery of the ancient Greek and Roman developments in science, mathematics, technology, and philosophy , especially as expanded and refined in the Muslim world – and the development of these knowledges and disciplines into the foundations of modern natural science.  The sense of opposition between “faith” and “reason” emerges in the West primarily post-Augustine (4th ct. C.E.) through the Dark Ages, and again with Cartesian dualism (17th ct. C.E.) and some strands of modern Protestantism [especially 19th ct. North American Fundamentalism].)

“Scientific” achievements:

1) establishes an ultimate dualism between Limit and the Unlimited;2) establishes the equation/identity of things with numbers.

More specifically, it is probable that Pythagoras discovered that the chief musical intervals are expressible in simple numerical ratios of the first four integers, i.e.:

Octave — 2:1Fifth — 3:2

Fourth — 4/3

This discovery, coupled with the discovery/invention of a mathematical order to the universe itself (familiar to us since Anaximander), leads to the venerable notion of “the harmony of the spheres.” As Julian Marias paraphrases it: since the distances of the planets correspond approximately to the musical intervals — then every/ star emits a note, all the notes together comprise the harmony of the spheres, a celestial music. We do not hear it because it is constant and without variation.

While we may be tempted to dismiss such a notion, note that this vision provided a foundation for such “modern” figures as:

a) Copernicus (who follows the Pythagorean astronomer Ecphantus in affirming the rotation of the earth), andb) Kepler (who diligently searched for over 10 years to find the Pythagorean harmonies — discovering the three laws of planetary motion in the process).  [We will also hear the computer realization of Kepler’s version of the Harmonia Mundi when we explore modern philosophy and natural science.]

More broadly, as K&R put it:

If the musical scale depends simply upon the imposition of definite proportions on the indefinite continuum of sound between high and low, might not the same principles, Limit and the Unlimited, underlie the whole universe? If numbers alone are sufficient to explain the “consonances,” might not everything else be likewise expressible as a number of a proportion?

Moreover, since the first four integers contain the whole secret of the musical scale, their sum, the number 10 or the Decad, might well “seem to embrace,” as Aristotle puts it, “the whole nature of number,” and so come to be regarded, as it certainly was, with veneration. As well, the first four integers generate the three maior figures beyond the point (cf. Speusippus, Kirk & Raven, pp. 253ff.).

Also attributed to Pythagoras – the Pythagorean theorem, with its corrollary, the incommensurability of the diagonal and the side of a square. Revealing this secret cost one poor student his life, it is said.

[For those who are really with it: the experience and conception of a harmony (=connection in the face of difference) avoids the conflict implicitly raised by Anaximander (dualism) and Anaximenes (monism) — and between Parmenides (dualism) and Heraclitus (monism)]

The Pythagoreans

Because of the Persian domination, philosophy moves from Ionia to the coasts of Magna Graeca, southern Italy and Sicily — to form what Aristotle calls the Italian school.

Pythagoras is a highly obscure figure. He apparently came from the island of Samos, settled in Croton (Magna Graeca). Several journeys are attributed to him, including one to Persia where he is said to have met the Magus Zaratas [= Zoroaster/Zarathustra].

He is further associated with the Orphics and the revival of the worship of Dionysus.

[CE] Indeed, it should be emphasized that the insights gained in this “philosophy” qua “theory” of the physical order are not designed so much for manipulating the environment as for saving one’s soul.

The Pythagoreans settled in a number of cities on the Italian mainland and Sicily, and from thence to Greece proper.

They formed a league or a sect. They did not eat meat or beans; the could not wear clothes made of wool; the could not pick up anything that had fallen, stir a fire with iron, etc.

The sect was divided between the akousmatikoi (hearers) and the mathematikoi (learned). The local democrats frowned on this aristocracy, if not on the sect as such, and many were killed.

The Pythagoreans formed the first “school” (from schole, “leisure”), defined as a way of life. And, perhaps because of their situation as foreigners, they understood themselves as following the spectator’s way of life (in contrast with those who buy and sell, and those who run in the stadium). This is the bios theoretikos, the contemplative or theoretic life.

The main difficulty to overcome: the body and its necessities which subdue man. It is necessary to free oneself from these. The body is a tomb — one must triumph over it, but not lose it. To so so requires that one attain the state of enthusiasm (en- theos). (This seems to suggest a connection with the Orphics and their rites, founded on mania, “orgy” — though the Pythagoreans apparently moderated this somewhat.)

In this way, one attains a self-sufficient, theoretic life — a life not tied to the necessities of the body, a divine life.

Such a man is a wise man, a sophos.

(The term philosophia, “love of wisdom,” is first used in Pythagorean circles.)


Greek mathematics began in the Milesian school (cf. Thales, Anaximander), inheriting the knowledge of Egypt and Asia Minor (Babylonia). The Pythagoreans transform it into an autonomous and rigorous science.

In mathematics, the Pythagoreans discovered a type of entity — numbers and geometric figures — which is not corporeal, but which seems to have non-arbitrary features of its own (in contrast with the arbitrary, changing whim of fancy, imagination, dream). Marias suggests that this discovery perhaps leads to the further claim that Being is not simply corporeal, material being — in which case, we would now have a problem. A development of the concept of being is called for[?].

In any case, for the Pythagoreans, Being means the being of mathematical objects:

Numbers and figures are the essence of things;Entities which exist are imitations of mathematical forms

[anticipates Plato’s alleged theory of forms]

Pythagorean mathematics is not an operative technique: it is the discovery and construction of new entities, which are changeless, eternal — in contrast with things which are variable and transitory.

Aristotle gives this account – and critique – of the Pythagoreans:

Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being – more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity – and similarly almost all other things being numerically expressible); since again they saw that the attributes and the ratios of the musical scales were expressible in numbers; since, then, all other things seemed in their whole nature to be modelled after numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangment of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth the ‘counter-earth.’ (Metaphysics A5, 985b23)

PreParmenidean Pythagoreanism

Beyond the insight first articulated by Pythagoras — that the universe, on analogy with the lyre, is built out of numbers and a harmony expressible in numbers — the dualism of Limit and Unlimited is expanded:

Limit Unlimited
odd even
one plurality
right left
male female
resting moving
straight curved
light darkness
good bad
square oblong

(see Jones)

Aristotle further reports that the Pythagoreans — evidently in contrast with all other Greeks — regarded the unit to have spatial magnitude (thus confusing “the point of geometry with the unit of arithmetic.”)

It is against such an assumption that Zeno’s paradoxes have their greatest force.

These unit-points functioned also as the basis of physical matter: they were regarded in fact as a primitive form of atom. Concrete objects are literally composed of aggregations of unit-point-atoms

Hence Aristotle:

Further, how are we to combine the belief that the modifications of number, and number itself, are causes of what exists and happens in the heavens both from the beginning and now, and that there is no other number than this number out of which the world is composed? (Met. A8, 990al8)

But the Pythagoreans, because they saw many attributes of numbers belonging to sensible bodies, supposed real things to be numbers – not separable numbers, however, but numbers of which real things consist. (Met.. N3, lO9Oa20)

While this may seem bizarre to us, Plato seems to have been the first Greek to have consciouslv thought that anything could exist otherwise than in space, and he was followed in this respect by Aristotle.


“When the one had been constructed, either out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by limit.”

Cf. modern accounts of the “Big Bang,” etc.

This is apparently a biologically-based conception, one which

(a) further recalls the basic similarity between the mythopoetic and philosophical/scientific structures of explanation, and(b) jibes with placing the male principle under limit and the female under unlimited in the table of opposites:

“The early Pythagoreans may well, therefore, have initiated the cosmogonical process by representing the male principle of Limit as somehow implanting in the midst of the surrounding Unlimited the seed which, by progressive growth, was to develop into the visible universe.” (K&R 251)

In this process, the void exists and functions to differentiate things:

Apparently the first unit, like other living things, began at once to grow, and somehow as the result of its growth burst asunder into two; whereupon the void, fulfilling its proper function, keeps the two units apart, and thus, owing to the confusion of the units of arithmetic with the points of geometry, brings into existence not only the number 2 but also the line. So the process is begun which, continuing indefinitely, is to result in the visible universe as we know it.

Notice here as well that the leap from the lyre to the conception of the entire universe as number and the harmony of the spheres rests on the Milesian tendency to draw analogies between the human and the natural — a tendency in keeping with the attempt to uncover an original unity which accounts for both the human and the physical orders.

Indeed, as already noted above, Aristotle chastizes the Pythagoreans in regard to their theory of a counter-earth (so as to complete the nine bodies in the heavens with a perfect 10th) this way “In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accomodate them to certain theories and opinions of their own.” (see K&R, pp. 257ff.)

Student Comments on the Pythagoreans

The pythagoreans “… infused all nature with mathematical concepts.” Mathematics became abstract and deductive. “The early Pythagorean community was a mystical, religious group; researches into the science of mathematics were part of a larger philosophy.” An idea that was very important to the Pythagoreans was an idea of a natural harmony in the universe. Janet L.

I found that Alioto had an interesting quote about the Pythagoreans: “In other words, with Thales, mathematics became deductive and therfore abstract. The Pythagoreans extended this process of abstraction and in turn infused all of nature with mathematical concepts. It seems that they were the first to stress the idea of number and geometry underlying diverse natural phenomena. The result, adapted and enshrined in Plato’s later philosophy along with an ethical, transcendental corollary, was the important recognition that numbers are abstractions, mental concepts, suggested by material things but independent of them. For the early Pythagoreans, however, the physical world was actually constructed from numbers.” (36). This, I believe, was a major step for philosophy because of the use of abstraction to relate to reality.  – Robert


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