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Chapter 3
This webpage reproduces a chapter of
A History of Philosophy

Frederick Copleston, S. J.

as reprinted by
Image Books — Doubleday
New York • London • Toronto • Sydney • Auckland

The text, and illustrations except as noted,
are in the public domain.


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Chapter 5

Part I: Pre‑Socratic Philosophy

 p29  Chapter IV
The Pythagorean Society

It is important to realise that the Pythagoreans were not merely a crowd of disciples of Pythagoras, more or less independent and isolated from one another: they were members of a religious society or community, which was founded by Pythagoras, a Samian, at Kroton in South Italy in the second half of the sixth century B.C. Pythagoras himself was an Ionian, and the earlier members of the School spoke the Ionic dialect. The origins of the Pythagorean Society, like the life of the founder, are shrouded in obscurity. Iamblichus, in his life of Pythagoras, calls him "leader and father of divine philosophy," "a god, a 'demon' (i.e. a superhuman being) or a divine man." But the Lives of Pythagoras by Iamblichus, Porphyry and Diogenes Laërtius, can hardly be said to afford us reliable testimony, and it is doubtless right to call them romances.1

To found a school was probably not new in the Greek world. Although it cannot be proved definitely, it is highly probable that the early Milesian philosophers had what amounted pretty well to Schools about them. But the Pythagorean School had a distinguishing characteristic, namely, its ascetic and religious character. Towards the end of the Ionian civilisation there took place a religious revival, attempting to supply genuine religious elements, which were catered for neither by the Olympian mythology nor by the Milesian cosmology. Just as in the Roman Empire, a society verging towards its decline, its pristine vigour and freshness lost, we see a movement to scepticism on the one hand and to "mystery religions" on the other hand, so at the close of the rich and commercial Ionian civilisation we find the same tendencies. The Pythagorean Society represents the spirit of this religious revival, which it combined with a strongly marked scientific spirit, this latter of course being the factor which justifies the inclusion of the Pythagoreans in a history of philosophy. There is certainly common ground between Orphicism and Pythagoreanism, though it is not altogether easy to determine the precise relations of the one to the other, and the degree of  p30 influence that the teaching of the Orphic sect may have had on the Pythagoreans. In Orphicism we certainly find an organisation in communities bound together by initiation and fidelity to a common way of life as also the doctrine of the transmigration of souls — a doctrine conspicuous in Pythagorean teaching — and it is hard to think that Pythagoras was uninfluenced by the Orphic beliefs and practices, even if it is with Delos that Pythagoras is to be connected, rather than with the Thracian Dionysian religion.2

The view has been held that the Pythagorean communities were political communities, a view, however, that cannot be maintained, at least in the sense that they were essentially political communities — which they certainly were not. Pythagoras, it is true, had to leave Kroton for Metapontum on the instance of Cylon; but it seems that this can be explained without having to suppose any specifically political activities on the part of Pythagoras in favour of any particular party. The Pythagoreans did, however, obtain political control in Kroton and other cities of Magna Graecia, and Polybius tells us that their "lodges" were burnt down and they themselves subjected to persecution — perhaps about 440‑430 B.C.,​3 though this fact does not necessarily mean that they were an essentially political rather than a religious society. Calvin ruled at Geneva, but he was not primarily a politician.​a Professor Stace remarks: "When the plain citizen of Crotona was told not to eat beans, and that under no circumstances could he eat his own dog, this was too much"​4 (though indeed it is not certain that Pythagoras prohibited beans or even all flesh as articles of food. Aristoxenus affirms the very opposite as regards the beans.​5 Burnet, who is inclined to accept the prohibitions as Pythagorean, nevertheless admits the possibility of Aristoxenus being right about the taboo on beans).​6 The Society revived after some years and continued its activities in Italy, notably at Tarentum, where in the first half of the fourth century B.C. Archytas won for himself a reputation. Philolaus and Eurytus also worked in that city.

As to the religious-ascetic ideas and practices of the Pythagoreans, these centred round the idea of purity and purification, the doctrine of the transmigration of souls naturally leading to the promotion of soul-culture. The practice of silence, the influence  p31 of music and the study of mathematics were all looked on as valuable aids in tending the soul. Yet some of their practices were of a purely external character. If Pythagoras really did forbid the eating of flesh-meat, this may easily have been due to, or at least connected with, the doctrine of metempsychosis; but such purely external rules as are quoted by Diogenes Laërtius as having been observed by the School can by no stretch of the imagination be called philosophical doctrines. For example, to abstain from beans, not to walk in the main street, not to stand upon the parings of your nails, to efface the traces of a pot in the ashes, not to sit down on a bushel, etc. And if this were all that the Pythagorean doctrines contained, they might be of interest to the historian of religion, but would hardly merit serious attention from the historian of philosophy. However, these external rules of observance by no means comprise all that the Pythagoreans had to offer.

(In discussing briefly the theories of the Pythagoreans, we cannot say how much was due to Pythagoras himself, and how much was due to later members of the School, e.g. to Philolaus. And Aristotle in the Metaphysics sparks of the Pythagoreans rather than of Pythagoras himself. So that if the phrase, "Pythagoras held . . ." is used, it should not be understood to refer necessarily to the founder of the School in person.)

In his life of Pythagoras, Diogenes Laërtius tells us of a poem of Xenophanes, in which the latter relates how Pythagoras, seeing somebody beating a dog, told him to stop, since he had recognised the voice of a friend in the yelping of a dog. Whether this tale be true or not, the ascription to Pythagoras of the doctrine of metempsychosis may be accepted. The religious revival had brought to fresh life the old idea of the power of the soul and its continued vigour after death — a contrast to the Homeric conception of the gibbering shades of the departed. In such a doctrine as that of the transmigration of souls, the consciousness of personal identity, self-consciousness, is not held in mind or is not regarded as bound up with soul, for in the words of destroyer Julius Stenzel: ". . . die Seele wandert von Ichzustand zu Ichzustand, oder, was dasselbe ist, von Leib zu Leib; denn die Einsicht, dass zoom Ich der Leib gehort, war dem philosophischen Instinkt der Griechen immer selbstverständlich."​7 The theory of the soul as the harmony of the body, which is proposed by Simmias in Plato's Phaedo and  p32 attacked by Plato, would hardly fit in with the Pythagorean view of the soul as immortal and as undergoing transformation; so the ascription of this view to the Pythagoreans (Macrobius refers expressly to Pythagoras and Philolaus)​8 is at least doubtful. Yet, as Dr. Praechter points out, it is not out of the question if the statement that the soul was harmony of the body, or tout simple a harmony, could be taken to mean that it was the principle of order and life in the body. This would not necessarily compromise the soul's immortality.9

(The similarity in several important points between Orphicism and Pythagoreanism may be due to an influence exercised by the former on the latter; but it is very hard to determine if there actually was any direct influence, and if there was, how far it extended. Orphicism was connected with the worship of Dionysus, a worship that came to Greece from Thrace or Scythia, and was alien to the spirit of the Olympic cult, even if its "enthusiastic" and "ecstatic" character found an echo in the soul of the Greek. But it is not the "enthusiastic" character of the Dionysian religion which connects Orphicism with Pythagoreanism; rather is it the fact that the Orphic initiates, who, be it noted, were organised in communities, were taught the doctrine of the transmigration of souls, so that for them it is the soul, and not the imprisoning body, which is the important part of man; in fact, the soul is the "real" man, and is not the mere shadow-image of the body, as it appears in Homer. Hence the importance of soul-training and soul-purification, which included the observance of such precepts as avoidance of flesh-meat. Orphicism was indeed a religion rather than a philosophy — though it tended towards Pantheism, as may be seen from the famous fragment Ζεὺς κεφαλὴ, Ζεὺς μέσσα, Διὸς δ’ ἐκ πάντα τέτυκται;​10 but, in so far as it can be called a philosophy, it was a way of life and not mere cosmological speculation, and in this respect Pythagoreanism was certainly an inheritor of the Orphic spirit.)

To turn now to the difficult subject of the Pythagorean mathematico-metaphysical philosophy. Aristotle tells us in the Metaphysics that "the Pythagoreans, as they are called, devoted themselves to mathematics, they were the first to advance this study, and having been brought up in it they thought its principles were the principles of all things . . ."​11 They had the enthusiasm  p33 of the early students of an advancing science, and they were struck by the importance of number in the world. All things are numerable, and we can express many things numerically. Thus the relation between two related things may be expressed according to numerical proportion: order between a number of ordered subjects may be numerically expressed, and so on. But what seems to have struck them particularly was the discovery that the musical intervals between the notes on the lyre may be expressed numerically. Pitch may be said to depend on number, in so far as it depends on the lengths, and the intervals on the scale may be expressed by numerical ratios.​12 Just as musical harmony is dependent on number, so it might be thought that the harmony of the universe depends on number. The Milesian cosmologists spoke of a conflict of opposites in the universe, and the musical investigations of the Pythagoreans may easily have suggested to them the idea of a solution to the problem of "conflict" through the concept of number. Aristotle says: "since 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, and the whole heaven to be a musical scale and a number."13

Now Anaximander had produced everything from the Unlimited or Indeterminate, and Pythagoras combined with this notion that of the Limit, or τὸ πέρας, which gives form to the Unlimited. This is exemplified in music (in health too, where the limit is the "tempering," which results in the harmony that is health), in which proportion and harmony are arithmetically expressible. Transferring this to the world at large, the Pythagoreans spoke of the cosmical harmony. But, not content with stressing the important part played by numbers in the universe, they went further and declared that things are numbers.

This is clearly not an easy doctrine to understand, and it is a hard saying that all things are numbers. What did the Pythagoreans mean by this? First of all, what did they mean by numbers, or how did they think of numbers? This is an  p34 important question, for the answer to it suggests one reason why the Pythagoreans said that things are numbers. Now, Aristotle tells us that (the Pythagoreans) hold that the elements of number are the even and the odd, and of these the former is unlimited and the latter limited; and the I proceeds from both of these (for it is both even and odd), and number from the I; and the whole heaven, as has been said, is numbers."​14 Whatever precise period of Pythagorean development Aristotle may be referring to, and whatever be the precise interpretation to be put on his remarks concerning the even and the odd, it seems clear that the Pythagoreans regarded numbers spatially. One is the point, two is the line, three is the surface, four is the solid.​15 To say than that all things are numbers, would mean that "all bodies consist of points or units in space, which when taken together constitute a number."​16 That the Pythagoreans regarded numbers in this way is indicated by the "tetraktys," a figure which they regarded as sacred. FIGURE

This figure shows to the eye that ten is the sum of one, two, three and four; in other words, of the first four integers. Aristotle tells us that Eurytus used to represent numbers by pebbles, and it is in accord with such a method of representation that we get the "square" and the "oblong" numbers.​17 If we start with one and add odd numbers successively in the form of "gnomons," we get square numbers, 2 FIGURES

while if we start with two and add even numbers, we then get oblong numbers.

This use of figured numbers or connection of numbers with geometry clearly makes it easier to understand how the Pythagoreans regarded things as being numbers, and not merely as  p35 being numerable. They transferred their mathematical conceptions to the order of material reality. Thus "by the juxtaposition of several points a line is generated, not merely in the scientific imagination of the mathematician, but in external reality also; in the same way the surface is generated by the juxtaposition of several lines, and finally the body by the combination of several surfaces. Points, lines and surfaces are therefore the real units which compose all bodies in nature, and in this sense all bodies must be regarded as numbers. In fact, every material body is an expression of the number Four (τετρακτύς), since it results, as a fourth term, from three constituent elements (Points, Lines, Surface)."​18 But how far the identification of things with numbers is to be ascribed to the habit of representing numbers by geometrical patterns, and how far to an extension to all reality of Pythagorean discoveries in regard to music, it is extremely difficult to say. Burnet thinks that the original identification of things with numbers was due to an extension of the discovery that musical sounds can be reduced to numbers, and not to an identification of numbers with geometrical figures.​19 Yet if objects are regarded — as the Pythagoreans apparently regarded them — as sums of material quantitative points, and if, at the same time, numbers are regarded geometrically as sums of points, it is easy to see how the further step, that of identifying objects with numbers, could be taken.20

Aristotle, in this above-quoted passage, declares that the Pythagoreans held that "the elements of number are the even and the odd, and of these the former is unlimited and the latter limited." How do the limited and the unlimited come into the picture? For the Pythagoreans the limited cosmos or world is surrounded by the unlimited or boundless cosmos (air) which it "inhales." The objects of the limited cosmos are thus not pure limitation, but have an admixture of the unlimited. From this point of view too, then, it is but an easy step to the identification of numbers with things, the even being identified with the unlimited and the odd with the limited. A contributory explanation may be seen in the fact that the odd gnomons (cf. figures) conserve a fixed quadratic  p36 shape (limited), while the even gnomons present a continually changing rectangular ships (unlimited).21

When it came to assigning definite numbers to definite things, scope was naturally allowed for all manner of arbitrary caprice and fancy. For example, although we may be able to see more or less why justice should be declared to be four, it is not so easy to see why καιρός should be seven or animation six. Five is declared to be marriage, because five is the product of three — the first masculine number, and two — the first feminine number. However, in spite of all these fanciful elements the Pythagoreans made a real contribution to mathematics. A knowledge of "Pythagoras' Theorem" as a geometrical fact is shown in Sumerian computations: the Pythagoreans, however, as Proclus remarked,​22 transcended mere arithmetical and geometrical facts, and digested them into a deductive system, though this was at first, of course, of an elementary nature. "Summing up the Pythagorean geometry, we may say that it covered the bulk of Euclid's Books I, II, IV, VI (and probably III), with the qualification that the Pythagorean theory of proportion was inadequate in that it did not apply to incommensurable magnitudes."​23 The theory which did solve this last arose under Eudoxus in the Academy.

To the Pythagoreans, not only was the earth spherical,​24 but it is not the centre of the universe. The earth and the planets revolve — along with the sun — round the central fire or "hearth of the Universe" (which is identified with the number One). The world inhales air from the boundless mass outside it, and the air is spoken of as the Unlimited. We see here the influence of Anaximenes. (According to Aristotle — De Caelo, 293, a 25‑7 — the Pythagoreans did not deny geocentrism in order to explain phenomena, but from arbitrary reasons of their own.)

The Pythagoreans are of interest to us, not only because of their musical and mathematical investigations; not only because of their character as a religious society; not only because through their doctrine of transmigration of souls and their mathematical metaphysic — at least in so far as they did not "materialise"  p37 numbers​25 — they tended to break away from the de facto materialism of the Milesian cosmologists; but also because of their influence on Plato, who was doubtless influenced by their conception of the soul (he probably borrowed from them the doctrine of the tripartite nature of the soul) and its destiny. The Pythagoreans were certainly impressed by the importance of the soul and its right tendance, and this was one of the most cherished convictions of Plato, to which he clung all his life. Plato was also strongly influenced by the mathematical speculations of the Pythagoreans — even if it is difficult to determine the precise extent of his debt to them in this respect. And to say of the Pythagoreans that they were one of the determining influences in the formation of the thought of Plato, is to pay them no mean tribute.

The Author's Notes:

1 "Ben, invero, possono dirsi romanzi, le loro 'Vite.' " Covotti, I Presocratici, p66.

2 Cf. Diog. Laërt. 8.8.

3 Polybius, II.39 (D. 14, 16).

4 Stace, Critical History of Greek Philosophy, p33.

5 ap. Gell. IV.11.5º (D. 14, 9)

6 E. G. P., p93, note 5.

7 Metaphysik des Altertums, Teil I, p42.

8 Somn. Scip. I.14.19 (D. 44 A 23)

9 Ueberweg-Praechter, p69.

10 D. 21 a.

11 Metaph., 985b25‑6.

12 It seems certain that the Pythagorean acoustic ratios were ratios of lengths and not of frequencies, which the Pythagoreans would hardly be in a position to measure. Thus the longest harpstring was called ἡ ὑπάτη, though it gave our "lowest" note and frequency, and the shortest was called ἡ νεάτη, though it gave our "highest" note and frequency.

13 Metaph., 985b31–986a3.

14 Metaph., 986a17‑21.

15 Cf. art. Pythagoras, Enc. Brit., 14th edit., by Sir Thos. Little Heath.

16 Stöckl, Hist. Phil., I, p48 (trans. by Finlay, 1887).

17 Metaph., 1092b10‑13.

18 Stöckl, Hist. Phil., I, pp43‑9.

19 E. G. P., p107.

20 Philolaus (as we learn from the fragments) insisted that nothing could be known, nothing would be clear or perspicuous, unless it had or was number.

21 Cf. Arist. Physics, 203a10‑15.

22 In Eukleiden, Friedlein, 65, 16‑19.

23 Heath, art. cit.

24 Cf. the words of the Russian philosopher, Leo Chestov: "It has happened more than once that a truth has had to wait for recognition whole centuries after its discovery. So it was with Pythagoras' teaching of the movement of the earth. Everyone thought it false, and for more than 1,500 years men refused to accept this truth. Even after Copernicus savants were obliged to keep this new truth hidden from the champions of tradition and of sound sense." Leo Chestov, In Job's Balances, p168 (trans. by C. Coventry and Macartney).

25 As a matter of fact the Pythagorean mathematisation of the universe cannot really be regarded as an "idealisation" of the universe, since they regarded number geometrically. Their identification of things and numbers is thus not so much an idealisation of things as a materialisation of numbers. On the other hand, in so far as "ideas," such as justice, are identified with numbers, one may perhaps speak with justice of a tendency toward idealism. The same theme recurs in the Platonic idealism.

It must, however, be admitted that the assertion that the Pythagoreans effected a geometrisation of number would scarcely hold good for the later Pythagoreans at least. Thus Archytas of Tarentum, a friend of Plato, was clearly working in the very opposite direction (cf. Diels, B 4), a tendency to which Aristotle, believing in the separation and irreducible character of both geometry and arithmetic, firmly objected. On the whole it might be better perhaps to speak of a Pythagorean discovery (even if incompletely analysed) of isomorphisms between arithmetic and geometry rather than of an interreduction.

Thayer's Note:

a A curious example from a Catholic writer: a much better example would be that the Popes ruled in Rome for over a thousand years, forging international alliances and conducting foreign policy, maintaining armies and conducting wars, but they were not primarily politicians either.

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