Lore and Science in Ancient Pythagoreanism

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Burkert, Walter 1972. Lore and Science in Ancient Pythagoreanism. Translated by Edwin L. Minar, Jr. Cambridge, Massachusetts: Harvard University Press. [Google Books]

The "Pythagorean question" has sometimes been compared with the Homeric question. Not that the details of the problem would especially suggest this; what does remind one of that most famous of philological controversies is the difficulty of the argument and the lack of agreement on methodology, as well as the multiplicity and contradictory character of the solutions advanced. Another similarity, and not the least striking, lies in the tremendous importance of the questions about the life, activity, and influence of Pythagoras of Samos. Over the origins of Greek philosophy and science, as over the beginning of Greek literature, lies the shadow of a great traditional name. The attempts of scholarship to grasp the underlying historical reality keep getting entangled in contradictions; where some think they discern the figure of a world-historical genius, others find little more than empty nothingness. (Burkert 1972: 1)

An important figure, about whom very little is known for certain. Yet, as Charles Peirce pointed out, there is more historical material, and from more respectable authorities (e.g. Aristotle), for Pythagoras' golden thigh than there is for the life of one Jesus of Nazareth.

Pythagoras' influence was a lasting one. The ancient tradition of the history of philosophy made him the ancestor of the "Italian School" and therefore, after Thales, the second, and more important, originator of philosophia - in fact, the inventor of the word. The doctrine transmitted under his name, that numbers are the principles of what exists, that the "One" is its primal ground, became part of the amalgam of Neoplatonism. In the trend set by Iamblichus, Pythagoras was the high priest, par excellence, of the divine wisdom. He then became, in the trivializing school tradition of the Middle Ages, the master of the quadrivium, and in particular the inventor of arithmetic. The early modern period discovered Pythagoras as the creator of natural science, which was just then being reborn; what Copernicus and Galileo taught was regarded by their contemporaries as a revival of Pythagorean science. (Burkert 1972: 1)

All already familiar themes: the first endonymic philosopher or lover of wisdom, "All is number", everything is an emanation from a singular oneness, the Monad. The most divine Iamblichus regarded Pythagoras as the most divine, etc.

E.g., Campanella in a letter to Galileo of Jan. 13, 1611 (T. Campanella, Lettere, ed. V. Spampanato [Bari, 1927] 165). Further literature in Capparelli, I 29ff; see below, ch. IV 3, nn. 1-2. (Burkert 1972: 1, footnote 1)

Thus, my finding pythagorean strands in Campanella is not completely baseless. Perhaps I should one day turn to Campanella's own idiosyncratic philosophy and examine its pythagorean-ness more closely.

[...] the traditional picture of Pythagoras, imposing though also vague in outline, inevitably gave way more and more before criticism. In the scholarly controversy that followed scarcely a single fact remained undisputed, save that in Plato's day and then later, in the first century B.C., there were Pythagoreioi. (Burkert 1972: 2)

For a long time, the only thing known for certain was that Plato had dealings with "pythagoreans" (e.g. Archytas, Philolaos, Timaeus, etc.). It goes: 1) death, 2) taxes, 3) Plato met pythagoreans.

When we set out to survey the most important attitudes and trends in modern Pythagorean scholarship, the point of departure must be the work of Eduard Zeller. In it the material is not only collected, with a completeness scarcely to be surpassed, but sifted with uncommon methodological rigor. (Burkert 1972: 2)

This must be Zeller's chapter on "The pythagoreans" (1881: 306-533) in his A history of Greek philosophy from the earliest period to the time of Socrates. If Huffman says to begin with Burkert and Burkert says to begin with Zeller, then that's what I'll do. What's an extra 228 page chapter to their 500 pages each. Oh god.

The most important source, nearly the only one which is left, is in the reports of Aristotle, in his surviving treatises. A second primary source is found in the fragments of Philolaus, in which August Boeckh, in his day, claimed to have found a firm foothold amidst the bog of Pythagorean pseudepigrapha. (Burkert 1972: 2)

Specifically, the 5th chapter of the 1st book of Aristotle's Metaphysics (that is, 986a).

Pythagoras is still recognized as the "founder of a religious society" and teacher of transmigration (411), but all philosophical significance is denied his ethical and religious doctrines (557ff). Alongside this stands, without connection, the number philosophy of the Pythagoreans. This can be reconstructed from Aristotle and is confirmed by Philolaus, though aside from him the Pythagoreans remain anonymous and scarcely datable. (Burkert 1972: 2)

According to Zeller, Philolaus and Aristotle are indeed the only reliable sources, which does not bode very well for decyphering the intricacies of the pythagorean number symbolism.

Zeller's solution, however, left a number of problems unsolved, and later research entered in with supplement, modification, and criticism. Above all, a gap had opened between Pythagoras the religious founder and the number philosophy of anonymous Pythagoreans; to connect these disparate elements and to show their original unity was bound to be an extremely enticing challenge. (Burkert 1972: 3)

"It was unavoidable from the capricious irregularity of this whole procedure, that among all these comparisons there should be numerous contradictions; that the same number or figure should receive various significations" (Zeller 1881: 420).

The direction was set by August Döring (1892); his thesis became most influential because he was followed by John Burnet in the later editions of his Early Greek Philosophy. (Burkert 1972: 3)

Cf. ch. 7, "The Pythagoreans" in the second (1908), third (1920), and editions of John Burnet's Early Greek Philosophy (the fourth edition (1930) can be borrowed and looks like a good scan, but the text appears to be the same as in the third).

The most significant contribution, after Burnet, was F. M. Cornford's article "Mysticism and Science in the Pythagorean Tradition"; but his conclusions were importantly modified by his own pupil J. E. Raven. Here should be mentioned also the independent accounts of Léon Robin, Abel Rey, and Pierre-Maxime Schuhl. The fact that each scholar had to erect a whole new superstructure shows the weakness of the foundation. (Burkert 1972: 4)

This is "Mysticism and Science in the Pythagorean Tradition" (Cornford 1922). F. M. Cornford also wrote a chapter on "The Italian Philosophy", mostly about Pythagoras, in his Greek Religious Thought... (1923). J. E. Raven's Pythagoreans and Eleatics. An account of the interaction between the two opposed schools during the fifth and early fourth centuries B.C. (1948) is available on lg (a tolerable scan); Burkert also cites his "Polyclitus and Pythagoreanism" (Raven 1951). Rey (1933), Schuhl (1934), and Robin (1928) are in French, but the last has a chapter on "Pythagoreanism and the Italian school" in a translation.

Erwin Rohde initiated the careful analysis of the sources of the lives of Pythagoras by Porphyry and Iamblichus, and this led back, for substantial portions, to authors of the fourth century B.C. - Aristoxenus, Dicaearchus, Heraclides Ponticus, then Timaeus. With growing optimism others followed along the path Rohde had pioneered. It was not only in the realm of political history that many new insights could thus be attained; the problem of the philosophy and science of Pythagoras also began to appear in a new light. (Burkert 1972: 4)

Overcoming Zeller's limitations. Only thing available in English is Rohde's Psyche: the cult of souls and belief in immortality among Greeks (1925[1894]).

There was also an attempt to discover indirect sources; one sought to find Pythagorean material reflected in other pre-Socratic thinkers, whether by way of influence or of polemic. The most important step in this direction was the thesis of Paul Tannery, that Parmenides took the cosmology of the doxa section of his poem from the Pythagoreans and that Zeno's polemic was directed against their number theory. This brought some very ancient evidence into the field, which could help to classify, to supplement, and even to test the reports of Aristotle. (Burkert 1972: 5)

There are hints of Parmenides' pythagorean connections in Ritter and Zeller (cf. 1881: 511-512)

The theory of Burnet and Taylor, which takes every portrayal in the dialogues as historical fact, not only transforms the entire Timaeus to a Pythagorean document of the fifth century B.C., but presents us with a Socrates who is an advocate of the theory of ideas and an adept of Pythagorean wisdom. Though this radical solution has attracted no following, the Pythagorean origin of the theory of ideas, and especially of the [|] doctrine of recollection, is still being discussed; and it is generally taken as proved that Plato owed his scientific knowledge to the Pythagoreans, especially in the realm of astronomy. (Burkert 1972: 5-6)

"According to Pythagoras, and after him Plato, idea is the common name of Types eternally existent in the mind of God, - types conformably to which all contingent things were made." (Clay 1882: 20-22)

Research in the religious history of late antiquity led in a similar direction. Franz Cumont and Jérôme Carcopino made extensive use of Pythagorean tradition in the interpretation of funerary symbolism of the imperial period. From this point of view there was no difference discernible between early and late Pythagoreanism; it was rather as though a powerful and continuous stream flowed from an ancient source. The numerous studies of Pierre Boyancé also follow this tendency; their aim is to grasp the "origine pythagoricienne" behind late material. (Burkert 1972: 6)

Zeller indeed draws a deep chasm between the 4th c. B. C. scientific mathematicians and the 1st c. A. D. religious acousmaticians.

Over against all these attempts to achieve a more positive view than Zeller are energetic movements of scholarly criticism which have even called into question testimony accepted by that scholar. The genuineness of the Philolaus fragments was attacked by Carl Schaarschmidt in 1864, and in 1868 by Ingrab Bywater. While Zeller's authority held up for a while in Germany, Burnet followed Bywater and therewith ensured the predominance of the negative verdict on the Philolaus fragments which still holds in the English-speaking world. (Burkert 1972: 7)

So Burnet ain't no saint. Bywater's "On the fragments attributed to Philolaus the Pythagorean" is found here.

Rejection of these Philolaus fragments is an essential element in the thesis of Erich Frank, whose book Plato und die sogenannten Pythagoreer (1923) towers over everything else that has appeared since Zeller on the history of Pythagoreanism, in the qualities of critical vigor, penetration, and firmness of judgment. To be sure, its merits are counterbalanced by one-sidedness and obvious perversities. Frank's methodological contribution was that he consistently held to the history of the natural sciences - mathematics, music, and astronomy - as basis for the reconstruction of Pythagoreanism. The first result was to date the development much later; all Pythagorean science, he thought, had come into existence in the circle of Archytas, about 400 B.C., influenced by the fully developed atomism of Democritus. The philosophy of the "so-called Pythagoreans," however, the number theory, was dependent on the late Plato, and was basically a creation of Speusippus, who had also himself forged the book attributed to Philolaus. (Burkert 1972: 7)

Absolutlely wild.

Nevertheless, the book still has importance, above all because of the extreme way in which the problem is put: "Plato and the Pythagoreans" - their mutual relationship is in fact the central problem of any historical investigation of Pythagoreanism, and Frank was right in perceiving that the influence did not go entirely in one direction. There is Platonic material which at a later date was wrongly labeled Pythagorean, and the generation of Plato's immediate disciples - Speusippus, Xenocrates, and Heraclides - played the decisive role in this development. (Burkert 1972: 8)

Especially with respect to Plato's later dialogues, which were reportedly mainly held with various pythagoreans.

There is no longer even any discussion of their authenticity, except for the Philolaus fragments and some of the Archytas material. Of course, "forgery" has its own importance in intellectual history; but in the discussion of early Greek thought there is no place for that which - like the book of Ocellus or of Timaeus of Locri - is obvious imitation of Platonic and Aristotelian material. (Burkert 1972: 9)

"Modern criticism has clearly shewn that the works attributed to Timæus and Archytas are spurious; and that the treatise on the nature of the All, which has been assigned to Ocellus Lucanus, cannot even have been written by a Pythagorean" (Ritter 1836: 346-347).

Just as a city which was continuously inhabited over a period of time, by changing populations, presents to the archaeological investigator far more complicated problems than a site destroyed by a single catastrophe and then abandoned, the special difficulty in the study of Pythagoreanism comes from the fact that it was never so dead as, for example, the system of Anaxagoras or even that of Parmenides. When their systems had been superseded and lost all but their philological [|] and historical interest, there still seemed to be in the spell of Pythagoras' name an invitation to further adaptation, reinterpretation, and extension. And at the source of this continuously changing stream lay not a book, an authoritative text which might be reconstructed and interpreted, nor authenticated acts of a historical person which might be put down as historical facts. There is less, and there is more: a "name," which somehow responds to the persistent human longing for something which will serve to combine the hypnotic spell of the religious with the certainty of exact knowledge - an ideal which appeals, in ever changing forms, to each successive generation. (Burkert 1972: 10-11)

Name-magic. "Pythagoras" invokes a mystical scientist, the first endonymic philosopher.

Study of the oldest, pre-Platonic tradition can thus be supplemented by those pieces of evidence which stand outside the Platonic influence, and were not affected by the reinterpretations mentioned above. Once more the reports of Aristotle becomes especially important, the fragments of his lost monograph on the Pythagoreans. The Pythagoras story, which used to be, for the most part, written off as the unfortunate product of the obfuscation of historical facts, may be understood as the expression, precisely, of a definite historical reality. Pioneers of this line of interpretation were Karl Meuli and E. R. Dodds. (Burkert 1972: 13)

Meuli is in German. From Dodds the bibliography includes "The Parmenides of Plato and the Origin of the Neoplatonic 'One'" (Dodds 1928), his revised Greek edition of Plato's Gorgias (1959), and his The Greeks and the Irrational (1951), which are available on lg.

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1. Platonic and Pythagorean Number Theory

Πλάτων πυθαγορίζει - from the time of Aristotle, this finding has often been repeated, but there is little clarity as to the extent and the manner in which Plato borrowed Pythagorean doctrine, or as to what Pythagoreanism was like before Plato. This applies especially to those attempts to derive the Ideas from numbers, to equate them with numbers, or even to replace Ideas with numbers, which Aristotle and others attribute to Plato and his pupils, in particular to Speusippus and Xenocrates. (Burkert 1972: 15)

Calls to mind Peirce's doctrine of signs, which - what with him being primarily a mathematician and logician - were probably intended to be either numbers and arithmetical symbols or propositions of the syllogism.

The foundation for the study is the evidence of Aristotle; he alone sets up Platonic and Pythagorean doctrines side by side, specifying [|] their points of agreement and - what is almost more important - their differences. (Burkert 1972: 15-16)

Somewhat ironic, as we today generally set Plato and Aristotle side by side.

There occurs a curious reticence when the conversation touches upon the most essential questions, even in early dialogues. In the Timaeus, Plato gives more definite indication that it is the question of the πάντων ἀρχή (the first principle of all things) that is being bracketed out (48c): Plato has reduced the multiplicity of the world to the four elements, the elements to regular solids, and these to triangular surfaces; "but the principles that are still prior to these god knows, and he among men who is dear to him" (53d). The Republic introduced the Good as the Highest, the Sun in the realm of ideas, "beyond being" (509b). The opposite to the good, in all the later dialogues, is described as an indefinite oscillation in two directions, toward "great and small," "more and less," "stronger and weaker," and it is the Good that constitutes measure and definiteness in this continuum. (Burkert 1972: 17)

(1) the Good, "beyond things"; (2) duality, oscillation between opposites; (3) triangular surfaces; (4) regular solids.

In reporting on it, Aristotle's attitude varies according to the subject he is dealing with. The ideal numbers are not treated in Περὶ ἰδεῶν, whereas in [|] his discussions of "first philosophy," he has a tendency to put a disproportionate emphasis on number theory: "If the Ideas are not numbers, it is impossible for them to exist at all; for from what kind of principles will the Ideas come?" (1081a12f). (Burkert 1972: 18-19)

Relations? Between what, then, if not numbers? Perhaps metaphorical numbers, i.e. firstness, secondness, and thirdness. Cf. 1081a.

As the audience was startled that "the Good" should concern mathematics and astronomy, the letter binds up "philosophy of nature" with "virtue and vice." Plato states however, "There is not any writing of mine on these matters, nor will there ever be, for this is a thing which cannot be put into words like other doctrines" (341c) - a sentence as famous as it is controversial in its interpretation. While Cherniss believes that anyone who takes these words as genuine and [|] relates them to the On the Good must in consistency give up trying ever to understand Plato, Krämer emphasizes that these "ultimate matters" are not "inexpressible" in an absolute sense, but only for the great mass of mankind. (Burkert 1972: 19-20)

To me, this certainly sounds like what is hinted is the pythagorean number symbolism, which does indeed tie together philosophy of nature with virtue and vice. But I take it that these doctrines are not so much "inexpressible" as much as they would simply sound silly if extolled very briefly. Hence the lengthy course of study, in silence, in the pythagorean school.

Yet Plato himself shows how serious he is about this "inexpressibility" by tracing its cause, in an excursus, to the relation between Being and the means of knowing. There are four means or steps of "knowing" an object: by name, by definition, by image, or by knowledge to which mind and right opinion are added (ἐπιστήμη καὶ νοῡς ἀληθής τε δόξα, 342c). But the fifth, Being itself, stands apart as "that which is the object of knowledge and truly exists" (342b). Mind (nous) comes closest to this (342d) but even mind does not grasp it completely and unambiguously. Each of the four kinds of knowing comprehends a qualitative aspect (poion ti) as much as the Being which the soul is seeking (342e): "There are two things, the Being and the qualitative aspect, and it is not the 'what kind' but the 'what' that the soul seeks to know. But each of these four proffers to the soul the thing that is not being sought, and thus fills every man's mind with puzzlement and unclarity" (343b-c). Because of this inadequacy - even on the part of mind and knowledge - it is easy to contradict and refute where the "fifth" is concerned: "When we are under the necessity of separating out and revealing the fifth element, anyone who likes to do so has the means of confuting us" (343d). (Burkert 1972: 20)

I wonder if this fifth corresponds to the pythagorean five, e.g. "Philolaus [...] derived physical qualities from five" (Zeller 1881: 475). The problem being that the first four correspond to "geometrical determinations (the point, the line, the surface, the solid)" (ibid, 475), but 5 is "Being itself" meaning, imo, physical reality, after which the series gets increasingly abstract: soul; reason, health, and light; love, friendship, prudence, and inventive faculty. Nine is a question-mark and ten is perfect.

The highest principle of Platonic ontology is the One; alongside it stands the Indefinite Dyad, a principle that is also described as great-and-small, many-and-few, exceeding-and-exceeded, and unequal. [.fn.] Becker (ZwU 18) by "logical extension," though Plato wants us to think of this "extension" as a duality, a deviation in two directions from the center, the measure, the One. Krämer has shown that this pattern of thought is discernible in ethical discussions of the late dialogues (146ff, 244ff). (Burkert 1972: 21)

Sounds surprisingly reasonable. The line as if extends in either direction from its center point.

The main concern of these philosophers, however, is not to lay the foundations of mathematics, but to explain the world by means of its principles. The ideal numbers are not only the ideas of particular numbers - "twoness," "threeness," etc., but somehow govern the structure of reality: they are ideas themselves. It is not clear how this connection [|] of ideas and numbers is to be understood, in detail. While Aristotle says simply that the ideas are numbers, Theophrastus speaks of an "attaching" (anaptein) of the ideas to certain numbers, and thus allows us to imagine a looser relationship. (Burkert 1972: 22-23)

Made more difficult when the ideas of numbers are projected as a sequence: firstness, secondness, and thirdness.

Aristotle further explicates the doctrine of ideal numbers in connection with pythagorical theory: Plato is said to have formed the soul out of "elements," in the Timaeus, in pursuance of the thought that like is known by like; [|]

and in the same way it was laid down in the work entitled On Philosophy that the Animal-itself is composed of the Idea itself of the One, along with the primary length and breadth and thickness, and the rest in a similar manner. Again, putting it differently, mind was said to be the One, knowledge two (because it goes in a straight line to the One), whereas the number of the plane figure is opinion and that of the solid, sensation [...] And, since the soul seemed to be productive of both motion and knowledge, some have compounded it of both [...] (De anima 404b18ff).

It is a doctrine of Plato or of Xenocrates that is reported here? (Burkert 1972: 24-25)

This famous passage is contained in the second chapter of Aristotle's On the Soul, e.g. here.

The identification of the One with mind (νοῡς) is attested for Xenocrates, as well as the series knowledge-opinion-sensation, and he knows the derivation-series line-plane-solid (above, n. 39). [.fn.] Fragments 15 and 5. Cf. Heinze 2ff, Cherniss, Plato 570f. The series νοῡς-ἐπιστήμη-δόξα-ἀἵσθησις is familiar to Aristotle himself (De an. 428a4, Met. 1074b35). (Burkert 1972: 26)

Knowledge - thirdness; opinion - secondness; sensation - firstness.

The close connection of all this with Plato is shown by a remarkable passage in the Laws (894a):

The origin of each thing takes place [...] when a first principle, taking on increment [the line], passes into its second transformation [the plane] and from this to its neighbor [the solid], and having made three transformations makes perception possible to those who perceive it.

Plato is dealing with the relation of the soul to the physical world, the priority of soul over matter. Thus there are present in Plato both the series line-plane-solid and the equation of the last stage with perception, though to be sure there is nothing about the application of numbers. (Burkert 1972: 26)

Epic! Note that the three transformations stand between 1-2 (the first principle takes on an increment, i.e. becomes a line), 2-3 (the line becomes a plane, i.e. sensation becomes an opinion), and 3-4 (the plane becomes a geometrical body, i.e. opinion passes into knowledge).

The commentators on Aristotle, from Alexander of Aphrodisias on, are unanimous that the doctrines develope in On the Good were Pythagorean, and Aristotle also says that in his theory of the first principles Plato "mostly" followed the Pythagoreans, though he did have "something of his own" to add. The seventh Letter, in the passage on these doctrines (338c), makes reference to Archytas: from his circle, it is suggested, Dionysius might have had knowledge of doctrine which Plato, himself, had not imparted to him. (Burkert 1972: 27)

The obvious conclusion being that Plato's "esoteric" knowledge, which he did not write down, were pythagorean in nature, and possibly derived from his acquaintance with Archytas.

In addition, Aristotle devoted two special books to the exposition and criticism of Pythagorean doctrines, and also wrote on Pythagoras. He himself alludes once (Met. 986a12) to his "more exact" discussions. Plutarch, Alexander of Aphrodisias, Aelian, and especially Iamblichus have preserved important material from these books, which supplemets the reports of the didactic treatises. (Burkert 1972: 29)

For the time-travelling bibliophiles to cop: Πρὀς τοὐς Πυθαγορείους and Περἰ τῶν Πυθαγορείων.

Plato and the Pythagoreans both accepted numbers as the principles (987b24) - number not as the number of other assumed objects, but as an independent entity, οὐσία. In this sense Pythagoreans and Plato are regularly mentioned in conjunction, and are set over against all other thinkers earlier than Aristotle, and in this conjunction Aristotle regularly attacks them. (Burkert 1972: 31)

Not surprising in the least. Noted for whenever "principles" come up in relation with Plato.

"Elements" of the numbers are, according to Pythagorean doctrine, the "even" and the "odd"; the "odd" is at the same time "limit," the "even" is "unlimited." In the pair limit-unlimited we have a primeval cosmic opposition lying behind the number which is the world. To Greek linguistic eeling, "limit" is the positive principle; it is conceived at the same time as [|] masculine, the "unlimited" as feminine, and correspondingly the odd number is also masculine, and the even feminine. Aristotle gives a complicated explanation of the correspondence of odd and limit, even and unlimited, at the basis of which lies the representation of numbers by arrangements of pebbles. (Burkert 1972: 32-33)

Pütaagorlaste põhiliste vastandite hulgas on "piir ja piiritu, [...] meheline ja naiseline" (Kulpa 1947: 53).

What the exegetes mean, however, can be deduced from Porphyry ap. Simpl. Phys. 453.25ff: the number 2 is the principle of division which proceeds to infinity, and only to this extent is there a connection between even numbers and the principle of infinity. These considerations derive from the thought of the Indefinite Dyad, and are therefore not early Pythagorean. (Burkert 1972: 33, footnote 27)

Absolutely wild speculation, but what if odd signifies addition and even signifies division? The female principle (2) is hence divisible (women can bear children), whereas the male principle (3) is indivisible, but additive (women can bear children for the man, in a patriarchial sense increasing him as a collective-familial person).

All the detailed accounts ascribe to Pythagoras the unequivocally Platonic concept of the Indefinite Dyad, καθ' ύπερβολἠν καἰ ἔλλειψιν (see below, ch. I 3), but Aristotle speaks in this passage of ἄπειρον, not of ἀόριστον, and nowhere else makes any mention of an important role played by "twoness" in Pythagoreanism. (Burkert 1972: 33, fn 27)

Probably not Aristotle, then, from whom was derived that "in their arrangement of the mundane system, the first body, reckoning from the central fire, was the counter-earth, and earth the second, this must have furnished to the Pythagoreans a complete and convincing proof that imperfection is the exclusive portion of earth" (Ritter 1836: 401, emphasis mine).

Plutarch, what is more, in a passage where at least for part of the way he is following Aristotle, gives a perspicuous explanation, also from the viewpoint of the pebble figure, of the masculine character of the odd number and the feminine of the even. The even number, he says, has at its middle an empty space, capable of reception, whereas the odd number has a middle member with procreative power. This direct symbolism must be regarded as old, and not only because of its attestation; it has a connection, at least subliminally, with the general Greek association of masculinity with the word περαίνειν. (Burkert 1972: 34)

This scheme and explanation makes a lot more sense than the previous one.

The One has a share in each of the opposite forces; it is "even" and "odd" at the same time, ἀρτιοπέριττον; it is perfectly in keeping that it should be, as late sources say, bisexual, ἀρσενόθηλυ. This One comes into being and develops further - it is nothing else thn the world before its further evolution. The "first principles," Limit and Unlimited are, then, what was there before the world came into being. To the Pythagoreans, number philosophy is cosmogony: κοσμοποιοῦσιν (1091a18). The further development of number is also cosmogonic:

They say clearly that when the One had been constructed - whether of planes or surface or seed or something they cannot express, then immediately the nearest part of the Unlimited began "to be drawn and limited by the Limit."

The One becomes a Two as the Unlimited penetrates it. Here is one [|] of the most widespread cosmogonic themes, "the separation of Heaven and Earth":

ώς οὐρανός τε γαῖά τ ἦν μορφἠ μία.
ἐπεἐ δ' ἐχωρίσθησαν ἀλλήλων δίχα...

The process was modified by the Pythagoreans, with their ideas of the earth in motion and the central fire. But in these very ideas is apparent a complete equivalence of the things separated: the "HeartH" of the universe and its fiery envelope, "Zeus's castle" and Olympus. This separation has happened in the past: when ARistotle quotes exactly, verbs in past tenses suddenly appear. (Burkert 1972: 36-37)

A mathematical creation story.

Cherniss (Pres. 39ff, 44ff, 224f, 387f) tries to show that the Pythagoreans had no doctrine of the origin of number, and that Aristotle only produces this impression by his projection of Platonic ways of thinking. To establish this he must, at Met. 1091a13ff, make a radical separation of the cosmic One ("the universe itself") and the "numerical unit": "Aristotle is confusing [...] the cosmogony with the number-theory" (p. 39). But Aristotle says unequivocally that the Pythagoreans knew only one kind of nuber, the cosmic (990a21), that is, that they thought of number theory as cosmogony, of cosmogony as the development of arithmetic. (Burkert 1972: 38, fn 50)

Perhaps I should call the pythagorean "numerical symbolism" something like "arithmological cosmogony" instead.

Perhaps a quite specific mythical cosmogony forms the background of the Pythagorean number theory. There are striking similarities of detail in the Orphic cosmogony which in the romance of Pseudo-Clement is given by Apion as an example of pagan theology. The problems of transmission are exceedingly complicated, but the basis is unquestionably a hexameter poem ascibed to Orpheus. Allegorical interpretation of Orphic poems, from a philosophical point of view, goes back at least to the fourth century B.C., as the papyrus from Dherveni has proven; so it is quite possible that in the tradition of philosophical exegesis ancient material has been preserved. In specific [|] details, this "Orphic" text gives an impression of antiquty, and it parallels to a surprising extent the first stages of the Pythagorean number theory. In the beginning was an abyss, a "boundless sea," a limited chaos. In it there came to be, by and by, and for no particular reason, a "bubble," which began to grow and become firmer. It sucked in the surrounding πνεῦμα, its "skin" became hard, and soon there floated on the sea of boundlessness a glittering sphere: the world egg. In this there developed a living creature, like the sphere in shape, winged, bisexual. It broke the egg and "appeared" in radiant brilliance: Phanes! Then the two halves of the broken shell fitted themselves together "harmoniously," while Phanes took position at the utmost boundaries of the heavens, a secret, spiritual light; and from the "procreative" content of the egg arose the realms of the world. (Burkert 1972: 38-39)

Exceedingly interesting stuff. "Now, although he is supported by the Peripatetic commentators, I consider that it admits of still another interpretation, which would make the infinite πνεῦμα indicate only the infinite breath of the world, by which the void is inbreathed" (Ritter 1838: 386, fn 4).

All the same, the specias position of these items of "knowledge," as compared with "speculations," does not seem to have been grasped. Rather, we find even more bizarre statements: 1 is mind (νοῦς), 2 opinion, 3 the number of the Whole, 4 (or 9) "justice," 5 "marriage," 7 "right time" (καιρός), and 10 is "perfect." Aristotle tries to understand all this as attempts at scientific definition, though he also complains of "superficiality." It is even more amazing, though also consistent, that these abstract concepts (or perhaps more correctly: these powers) each takes its specific position in the cosmos - for the cosmos itself does consist of numbers. (Burkert 1972: 40)

Once again a painful omission - 8 and 9 are left out, left unnamed.

990a18ff; on this, Alex. Met. 75.15, with the all-too-brief remark that in his second book On the Pythagoreans Aristotle dealt with the arrangement of the numbers in the heaven. Alexander says here that the One had the mid-position, and thus must have been counting from the middle out; in 38.21ff = Arist. fr. 203, however, where we learn that the sun is 7, the enumeration goes in the other direction. One would not expect purely arbitrary improvisations in Alexander, so that the irrational ambiguity must be Pythagorean. (Burkert 1972: 40, fn 64)

Source for Aristotle's claim that "the cosmos itself does consist of numbers" for the pythagoreans. That the pythagoreans started counting from a mid-position is becoming more and more forcefully obvious - if the central fire is 1, it only makes sense that every other number / heavenly body should revolve around it.

From our perspective, we may distinguish at least four quite different functions of number on which the Pythagorean doctrines are built: number as the symbol of certain concepts or powers of ordering; number as the designator of order, position, or rank; number as determiner of spatial extent (the pebble figures); and finally number as ratio and mathematical formula, as natural law. Aristotle already was [|] complaining, in his day, that very different things were being equated here; no wonder that historians of philosophy, with so many different points of departure, have arrived at quite disparate results. Some interpret the Pythagorean number theory as a radical materialization of number and find it a kind of atomism, while others see in it a philosophy of mathematical form, an idealism closely akin, if not identical to, the Platonic theory of ideas. (Burkert 1972: 40-41)

Previously there were three varieties: "Now, we find yet many other and similar formularies expressive of the Pythagorean doctrine of numbers; and it is above measure necessary to notice, that in some numbers, in others the number, or the elements of number, are made to be the principles of things" (Ritter 1836: 360-361). My own count yields at least two distinct series: (a) geometrical forms/beings; and (b) heavenly bodies.

The "number atomism" interpretation goes back to Cornford. In his account of Pythagorean doctrine, Aristotle speaks of a plurality of extended monads, and he often alludes to the definition of the point as a "monad having position." If we interpret this as a comprehensive key idea, to be taken along with the pebble figure, the "star pictures" (constellations), and the procedure of Eurythus, who would determine the "number" of man or horse by making an outline picture with pebbles - the result is the thesis that the Pythagoreans understood the materialized point as a kind of atom. They thought of all bodies as consisting of such point-atoms, and therefore things "are" numbers in the most literal sense; that is, they are the number of atom-point-units which they at any given moment contain. Does not Aristotle himself say that the Atomists "in a way" claim that things "are numbers or composed of numbers" (Cael. 303a8)? (Burkert 1972: 41)

Not even out of the question, seeing as the first series (geometrical forms / beings - all life, plants, animals, humans, etc.) would indeed enable one to say, according to the pythagorean system, that a horse is 7 and a human is 8... This is probably not what is meant here, though.

According to all the sources, the fundamental th for the Pythagoreans was the numbers from 1 to 10, not myriads; the important thing is form, not statistics, and this includes the method of Eurythus, who looked for the significant points, the specific criteria of the shape. (Burkert 1972: 42)

So, more like a "superficial" classification, rather than a numerical identification (like a MD5 hash).

To be sure, the underlying consideration, that there can be a surface without a body, but no body without surface, line, and point - this "conceptual experiment" is fundamental for the system of derivation found in Aristotle's On the Good. But for the Pythagoreans, who know of nothing else than what is sensually perceptible, there can scarcely be a "limit" without a body (ἄνευ σώματος, 1002a6). In fact the limit doctrine appears once where the point of view is that of χωρισμός, and this shows it cannot be Pythagorean. (Burkert 1972: 43)

Which came first: the geometrical body or the physical body?

For the Pythagoreans even the primary One is three-dimensional, or corporeal. The reduction of the physical world in the schema of body-surface-line-(point) belongs to the Platonists, not to the Pythagoreans, who knew only the one world of the sensible. (Burkert 1972: 43)

Exactly the reason why one should be weary of the all too sensible series of sensation-opinion-knowledge-mind: it may very well be a later Platonic interpretation; in all likelihood the original (early) pythagorean theory is more simple or primitive.

[...] by Jaeger, Paideia I 163 (Eng. tr.), for whom, in the other reports, "he is no doubt making the mistake of translating into material terms their theoretical identification of numberness and existence"; [...] (Burkert 1972: 44, fn 85)

Arvulisus.

Evidently Aristotle knew the Pytagoreans had used the word μίμησις. He surely must have intended to belittle Plato's originality in the doctrines which Aristotle himself disliked. Thus the thought that this Pythagorean μίμησις was the same as μέθξις, and therefore implied a theory of ideas, will be seen to be interpretation. Aristotle does give some clues as to how imitation can be understood without a theory of ideas. (Burkert 1972: 44)

"Mimesis" is one of those scientific technical terms very much still in use but which seem to have a very ancient origin - as it turns out here - in the polemics between Plato and Aristotle.

Again and again it becomes clear that the Pythagorean doctrine cannot be expressed in Aristotle's terminology. Their numbers are "mathematical" and yet, in view of their spatial, concrete nature, they are not. They "seem" to be conceived as matter (ὕλη) and yet they are something like form (εἶδος). They are, in themselves, being [|] (οὐσία), and yet are not quite so. They cannot be expressed in the Aristotelian framework of the four principles, or in the categories of form and matter; great as the temptation has been, both for Aristotle and for modern scholars, to understand the opposition of Limit and Unlimited as identical to that of form and mater, the explicit statements that the One partakes of both Limit and Unlimited, and that number is a kind of material (ὕλη), stand in the way. Missing are the impact of the theory of ideas and the dialectic, the classification of Being into stages of differing reality (οὐσία), the reduction of the sensible world to immaterial principles. Neither the system of Aristotle nor the conceptual framework developed by the Academy forms any part of the background of these Pythagorean doctrines; rather, they obstruct our access to them and impede our understanding of them. When one puts these observations alongside the traces of cosmogonic myths which dominate the apparently abstract pattern of the genesis of the "numbers," there can remain no doubt. What Aristotle presents as the philosophy of the Pythagoreans is truly pre-Socratic, unaffected by the achievements of Socratic-Platonic dialectic, and not to be measured by their standards. Οί γἀρ πρότεροι διαλεκτῆς οὐ μετεῖχον (Arist. Met. 987b32). (Burkert 1972: 45-46)

I've said it before and I'll say it again: truly pythagorean philosophy may be so primitive as to be completely incomprehensible to us moderns.

A striking fact about the examples that Aristotle gives for the Pythagoren equation of numbers and things is that it is never a question of the relation of individual things and individual number - aside from the isolated fooleries of Eurythus - but of the correspondence of a plurality of things to the system of numbers, and in particular the correspondence of alterations in things to alterations in the number series. (Burkert 1972: 48)

A horse and a pig are as much 7 as you and I are 8.

Finally we come to the "table of ten opposites," which Aristotle sets apart from the rest of the Pythagorean number theory which he treats: "Other members of this same school say there are ten principles, which they arrange in two columns of cognates" (986a22, Ross tr.):

(1) limit (τέρας)	: unlimited (ἄπειρον)
(2) odd (περιττόν)	: even (ἄρτιον)
(3) one (ἕν)	: plurality (πλῆθος)
(4) right (δεξιόν)	: left (ἀριστερόν)
(5) male (ἄρρεν)	: female (θῆλυ)
(6) resting (ἠρεμοῦν)	: moving (κινούμενον)
(7) straight (εὐθύ)	: crooked (καμπύλον)
(8) light (φῶς)	: darkness (σκότος)
(9) good (ἀγαθόν)	: bad (κακόν)
(10) square (τετράγωνον)	: oblong (έτερόμηκες)

Though the good has the second-to-last position, the arrangement is clearly made from a normative point of view. (Burkert 1972: 51)

Thus far the best formatting I've seen. It helps that the original Greek items are also listed.

In the Nicomachean Ethics Aristotle speaks explicitly of a "column of goods"; there, too, he attributes the system to the Pythagoreans, but adds, "and Speusippus, too, seems to have followed them" (1096b6). The inclusion of movement in the same column as the unlimited or indefinite (ἄπειρον, ἀόριστον) is alluded to by Aristotle in the general remark that "the principles in the second column, because they are negative (στερητικαί) are indefinite (ἀόριστοι). (Burkert 1972: 51)

There is one way to be straight, but infinite varieties of crookedness.

Hermodorus' report of the On the Good reveals a similar train of thought: an initial threefold division of Being is traced back to a [|] twofold division, in which one side has "equal, abiding, harmonized," (ἴσον, μένον, ἡρμοσμένον) and the other "unequal, moving, unharmonized) (ἄνισον, κινούμενον, ἀνάρμοστον). The connection with Speusippus is particularly close. Therefore it is not surprising if later Platonists, and also pseudo-Pythagorean works, keep introducing similar "tables of opposites." [.fn.] The following (a and b) are probably based on Aristotle: (a) Plut. De Is. et Os. 48.370e (the order, compared with Aristotle's, 9 3 1 6 7 2 10 4 8; the "good" is at the beginning; instead of "plurality" he has "dyad"). (b) Por. VP 38 (from Diogenes Antonius?): monad-dyad, light-darkness, right-left, equal-unequal, abiding-moving, straight-circular. (c) Eudorus ap. Simpl. Phys. 181.22ff: ordered-disorderly, definite-indefinite, known-unknown, male-female, odd-even, right-left, light-darkness. (d) "Pythagoras" in Varro Ling. 5.11: finitum-infinitum, bonum-malum, vitam-mortem, diem-noctem, status-motus. (e) Ps.-Archytas p. 19, 5-13 Thesleff: ordered-disorderly, limited-unlimited, speakable-ineffable, rational-irrational, binding-bondspoiling, etc. (f) "Eurysus," Stob. 1.6.19: speakable-ineffable, ordered-disorderly, rational-irrational. (g) Philo Qu. in Exod. 2.33: odd and god-even and mortal, equality-inequality, similarity-dissimilarity, same-different, unification-dissolution, better-worse.Cf. also Tim. Locr. 1. (Burkert 1972: 51-52)

Curious additions are listed here. Especially ordered-disorderly, speakable-ineffable, and binding-bondspoiling. The latter is an especially interesting term.

The most important of the later sources for Pythagorean philosophy are the Pythagorean Memoirs (Πυθαγορικὰ ὑπομνήματα) excerpted by Alexander Polyhistor, the Life of Pythagoras (Πυθαγόρου βίος) excerpted by Photius, the reports of Aëtius, and (most extensive of all) those of Sextus Empiricus. (Burkert 1972: 53)

The only one I'm even slightly familiar with is Sextus Empiricus.

Philo's mention (50) of the game of putting a nut on top of three others to make a pyramid (καρυατίζειν, cf. Anat. 32.3ff), provides an explanation for the word ἐπαιωρήσωμεν in Sext. Emp. Math. 7.100. (Burkert 1972: 55, fn 12)

Three nuts placed together represents a triangular surface, but place a fourth nut on top and it becomes a pyramidal body.

Now since in a later passage, which is certainly Posidonian, there are mentioned precisely the "incorporeal ideas [...] consisting in the borders of bodies," the present passage, too, will have to be attributed to Posidonius. (Burkert 1972: 56)

Isn't the four-nut pyramid also representative of incorporeal ideas? Surely one cannot build real pyramids by placing nuts one atop another.

If Posidonius, à propos of the Timaeus, wished to speak about the recognition of like through like, he could not ignore the fact that in the Timaeus the soul, which recognizes, is created as a number pattern, as the physical world is made up of mathematically determined triangles. Obviously, Posidonius is in part dependent on Aristotle, who, following a similar line of thought, brought the Timaeus and the theory of ideal numbers into connection, and also treated of this matter in the dialogue On Philosophy. (Burkert 1972: 56)

The world is a polygonal simulation. Footnote cites Frank Egleston Robbins' "Posidonius and the Sources of Pythagorean Arithmology" (1920) and "The Tradition of Greek Arithmology" (1921).

Now, we immediately notice a fact of great importance: the majority of the reports about Πυθαγόρειοι can be confidently referred to Aristotle as source, but this is not so of even one of the reports about Πυθαγόρας. Pythagoras is frequently named in the same breath as Plato, Pythagoreans' never. Thus from the external form of the tradition itself, it is clear that alongside the Aristotelian tradition about the Pythagorean philosophy there was another, which dared to name Pythagoras himself and connects him closely with Platonism. (Burkert 1972: 57)

Interesting indeed. In the 19th century histories of philosophy, there are variations in chapter headings, "Pythagoras", "The Pythagorean philosophy", and several iterations of combination, i.e. "Pythagoras and the Pythagoreans".

Further on the Monad is also called mind, god, and good, and the Dyad is called divinity (δαίμων) and evil. The long passage in Sextus builds on the same foundation: "Pythagoras said that the first principle of existing things is the Monad [...] and being added to itself [...] it produces the so-called Indefinite Dyad [...]" (Math. 10.261; cf. 276) "[...] and the rest of the numbers were produced from these [...] The Monad had the position of the active cause, and the Dyad that of the passive matter" (277). (Burkert 1972: 58)

The beginning of the Golden Verses: first respect gods above, and then the demons below, after them your family and everyone else. The implication being, as if, that the gods made humans from demons.

In fact, people did realize the inconsistency with the reports of Aristotle. This appears not only from the polemics of the neo-Platonists, but from a curious document, attributed to Theano, who was usually known as the wife of Pythagoras;

I have learned that many of the Greeks suppose Pythagoras said that everything came to be from number. This statement, however, involves a difficulty - how something that does not even exist is even thought to beget things. But he did not say that things came to be from number, but according to number. For in number is the primary ordering, by virtue of whose presence, in the realm of things that can be counted, too, something takes its place as first, something as second, and the rest follo in order.

Here we have the neat separation of ἀριθμός and ἀριθμητόν, the concept of μετουσία ("presence" above), in almost verbal agreement with formulations in Sextus and Theo, and in addition direct polemic against a tradition according to which things come to be "from numbers." "Many of the Greeks" have learned this false representation of the Pythagorean doctrines; here there is unmistakable polemic against Aristotle. The method is subtly indirect: an "orginal" document is witness against his interpretation. And who is qualified to offer authentic exegesis, if not Pythagoras' wife and student herself? (Burkert 1972: 61)

"All is number" = everything is measurable.

To be sure, Syrianus was in a still more favorable situation; he read the 'Ιερὀς λόγος of Pythagoras himself, which named Πρατεύσ and Δυάς [|] as first principles and made number, the "ruler of shapes and forms," the basis of the "origin of all things." Aristotle, accordingly, must have been mistaken... This is how the gap between the different traditions created the demand for apocrypa. (Burkert 1972: 61-62)

Aristotle's misrepresentation of Pythagorean philosophy created the demand for pseudo-Pythagorean writings.

The line of thought is so close to that of Theophrastus, who was a generation younger, that we can no longer harbour any doubt as to the source from which the latter acquired his non-Aristotelian conception of Pythagorean doctrine. Plato's nephew and successor claimed that the basic thought of the Platonic doctrine of ideal numbers was Pythagorean. (Burkert 1972: 64)

This is becoming more and more obvious.

Furthermore, Xenocrates developed from the Timaeus his definition of soul as "number moving itself," and precisely this definition of soul is ascribed by the doxographers to Pythagoras, as well as the doctrine of the One and the Indefinite Dyad. The later tradition about Pythagoras is largely based on the exegesis of the Timaeus by Xenocrates, who understood the ideas contained in Plato's dialogue as the teaching of Pythagoras. (Burkert 1972: 65)

Hing on iseennast liigutav arv.

A third writer deserves mention in this context, Heraclides Ponticus. He attributed to Pythagoras the invention and definition of the word φιλοσοφία, and this ascription made its way, via the doxographers, into all the ancient handbooks. Yet it is disproved by the semantic history of the word. It had meant close acquaintance and familiarity with σοφία; and Plato was first to define it as in insatiable striving, and set it in opposition to the possession of wisdom. This was after the Sophists and their claims had roused popular animosity. Heraclides probably combined with this a number theory that took its direction from the Timaeus; at least he ascribed to Pythagoras the sentence, "The knowledge of the perfection of the numbers of the soul is happiness." (Burkert 1972: 65)

Diogenes Laertios' story about Pythagoras' testimony, the self-description as a "philosopher" may be bull.

As a rule the number 2 is assigned to the line, 3 to the plane, and 4 to the solid. Sextus emphasizes the difference from another derivation, according to which the line comes to be through a continuous movement (ῥύσις) of the point, the plane through movement of the line, and the solid through movement of the plane. Here there is no need for any other "first principle" than the One. Yet the two ideas get mixed together. (Burkert 1972: 66)

The "transformation" above mentioned is simply movement.

In addition to such beliefs about souls and daimones, the doxographical tradition assigns to Pythagoras or the Pythagoreans the threefold division of the soul into νοῦς, θυμός, ἐπιθυμία, and the authority of Posidonius is cited for this. But it is apparent by now that this is no isolated phenomenon; in other respects as well Posidonius appears as a link in the Pythagoras tradition. Now, this tripartition of the soul is closely related to the theory of the "three ways of life", which Heraclides Ponticus cites as coming from Pythagoras. It is possible that the soul doctrine insinuated its way into the tradition by the same route. Anyway, here too Platonic and Pythagorean material is seen as a unity, and Aristotle's reports lead in a different direction. (Burkert 1972: 74)

Unsurprisingly, this, too, is heavily mixed with Platonism.

The Hypomnemata present a somewhat different division of the psyche (30) "into νοῦς and φρένες and θυμός. νοῦς and θυμός are also present in the other animals, but φρένες only in human beings." Von Fritz has tried to show, on the basis of a thorough semantic investigation, that in this respect the author is reproducing a completely un-Platonic and therefore pre-Platonic doctrine. (Burkert 1972: 74)

There may be a pre-Platonic (Pythagorean) core to this tripartite division yet.

Not the least important cause of this is a certain resignation, for it is quite impossible to determine and delimit, from the study of Plato alone, all of his "sources." In Plato, every thesis or argument derives its importance from its truth value, not from its origin in one source or another. Foreign material is no longer foreign, but an integral part of the Platonic structure. This is why the question of the nature of historical Pythagoreanism is perhaps hardest of all to answer, from Plato alone. (Burkert 1972: 83)

Who would have guessed that the successors of an illiterate philosopher (Socrates) would skip citing their sources.

Pythagoras is named in a single passage in the Republic, in the final reckoning with Homer. Has Homer, Socrates asks,earned the laurels of a lawgiver like Lycurgus, Charondas, or Solon, or even [|] those of a practical adviser like Thales or Anacharsis? Or (600 a-b),

if not in public life, is Homer said in private life to have been, during his lifetime, influential in the education of any persons who cherished him and his association with them and passd on to their successors some kind of "Homeric way of life," as did Pythagoras? He was greatly loved, in this way, and his followers even to the present day speak of a "Pythagorean way of life," and seem in some way to stand out from the rest of mankind.

What gives this passage its importance is its connection with Plato's own lot. For him, too, the most desirable career would have been great influence in the polis; after this was denied him, he decided to be, at least "in private life [...] influential in the education" of individuals: he founded the Academy. [.fn.] The relationship between the Academy and the Pythagorean society has often been emphasized. Boyancé brought out the significance of their common cult of the Muses (Muses 249ff). Cf. also Morrison, CQ 1958, 211f. (Burkert 1972: 84-84)

Those who can, do; those who can't, teach.

Socrates-Plato attacks the problem of ἡδονή by relating it to the more general problem of the "one and many," unity, plurality, and their mutual interpenetration, not only in the objects of experience but in the realm of ideas itself (14d et seq.). An ancient tradition, he says, shows us the way toward solution of the problem:

There is a gift of the gods - so at least it seems evident to me - which they let fall from their abode; and it was through Prometheus, or [|] one like him, that it reached mankind, together with a fire exceeding bright. The men of old, who were better than ourselves and dwelt nearer the gods, passed on this gift in the orm of a saying: all things (so it ran) that are ever said to be consist of a one and a many, and have in their nature a conjunction of Limit and Unlimitedness [...] (16c; tr. Hackfort).

Our task is, he says, not to proceed immediately from the One to the Many but to comprehend, stage by stage, the numerical structure that lies between the One and the Unlimited - and herein lies the difference between dialectics and eristic - just as the grammarian knows the number and nature of the sounds which, in the unitary realm of language, determine the multiplicity of linguistic expression, or as the musician becomes master of the infinite realm of tones by his knowledge of the limited number of the intervals. (Burkert 1972: 85-86)

To my simplistic mind this reads as if everything is a whole singular One, and everything within is a limited part of it, of infinite variety of measures. This is naturally not a good interpretation.

Rhythms and meters are measured "by numbers" (17d); and coming back to this theme later (25d-e), Plato states unmistakably that musical harmony depends on numerical propositions. Thus it is natural to suppose that from the beginning Plato was thinking of the same musical theorists; from what Plato himself says it emerges that the ontology of the Philebus has its roots in Pythagoreanism. (Burkert 1972: 87)

Hmm: "It has been proposed that the [Philebus] is among the last of the late dialogues of Plato, many of which do not figure Socrates as the main speaking character." (Wiki) - The page contains a section on "the major ontological themes of the work", where it looks like "The limitless" is qualitative, and "The limited" is quantitative.

So we come into possession of a piece of pre-Aristotelian evidence for a Pythagorean philosophy of some scope, musically oriented. A prime necessity, however, is to distinguish precisely between what Plato inherited and what it became in the alembic of his own mind. (Burkert 1972: 87)

Good phraseology.

The fundamental problem, that all being is at the same time one and many, had been formulated long ago (14c). It is also discussed in other dialogues and brought into connection with the problems raised by the Eleatics. Empirical things (γιγνόμενα) had already been designated, in passing, as unlimited (ἄπειρα, 15b), in contrast to the "unities" represented by the ideas. What is new in the pronouncements credited to "the ancients" (16c) is, first of all, the antithesis of πέρας and ἀπειρία. (Burkert 1972: 88)

Empirical things are many, ideas are few.

Even in the Life of Pythagoras which Photius excerpted, Pythagoreanism, Academy, and Peripatos are seen as making an unbroken unity: the ninth successor of Pythagoras is Plato, the tenth Aristotle. (Burkert 1972: 93)

An English translation (four A4 pages, by Ɔ. Martiana) is available online.

One cause for this lies in the fact that tradition about Plato's oral teaching lost in importance, in comparison with the steady influence of the Platonic dialogues. The doctrines formulated in the latter were well known, while the theory of ideal numbers sank into desuetude. (Burkert 1972: 94)

The "esoteric" Plato. Cf. "Secret Teachings of Plato & Theology of Arithmetic - Pythagorean Origins of Sacred Geometry" (Justin Sledge - Youtube).

There were necessarily two aspects to this breach. Insofar as the Academics following the Socratic-skeptical trend still felt themselves to be Plato's successors, they had to push diligently aside everything "dogmatic," and especially the mathematical-scientific and metaphysical teachings of the school, as not genuinely Platonic. They found another origin for it: Pythagoras. On the other hand, what this process discarded retained, even for the rationalistic Hellenistic world, a certain fascination. Those who were attracted by it could no longer attribute it to Plato, against the authority of the Middle Academy, but found it necessary to reach back for the authority of "Pythagoras." And when he took the limelight, Plato and his pupils were stigmatized as plagiarists. (Burkert 1972: 94)

A part of Plato's teachings were dumped on Pythagoras' shoulders afterwards.

The interpretation of Pythagoreanism that Speusippus, Xenocrates, and Heraclides had given is hypostasized in this revival, and the purported originals make the intermediaries superfluous. Thus the criticism of Pythagoreans themselves is turned against the Platonists:

[...] that Plato and Aristotle, Speusippus and Aristoxenus and Xenocrates, as the Pythagoreans say, appropriated what was fruitful, with slight modification, but collected some superficial or inconsequential things, whatever is brought forward by those later malicious slanderers in an effort to refute and mock the school, and put thee down as the special doctrines of the sect [...]

Thus even these Pythagoreans admit, implicitly, that what was essential, or "fruitful," in their doctrines agrees with Plato and Aristotle, and that they could make no good use of what historical tradition had to offer as the ἴδια of Pythagoreanism. Later Pythagoreanism is stamped [|] so deeply with Platonism that it has no longer any conception of its real origin. (Burkert 1972: 95-96)

"This is betrayed by the author when he says that Plato and Aristotle collected all that they could not adopt, and omitted the remainder, called that the whole of the Pythagorean doctrine" (Zeller 1881: 309, fn2).

---

2. Pythagoras in the Earliest Tradition

Most of our material on the life and activities of Pythagoras is collected in the eight book of Diogenes Laertius, in Porphyry's Life of Pythagoras, and especially in Iamblichus. We may add the tenth book of Diodorus, of which only fragments are preserved, and the very short sketch in Justin. Photius' excerpt from an anonymous life of Pythagoras has very little that pertains to history or biography. (Burkert 1972: 97)

The plan to go through 19th century chapters on Pythagoras should begin with a revisitation of Diogenes Laertios' chapter, from whence most information about him seems to have come from.

Our first question must be as to the direct sources of Iamblichus. Here analysis is easiest in the case of the Theologumena arithmeticae; one source, Anatolius On the First Ten Numbers, has been recovered, and the second, Nicomachus 'Αριθμητικῶν θεολογυμένων βιβλία β', we have in an excerpt by Photius (Bibl. 187). (Burkert 1972: 98)

Not sure yet where to get a hold of it, but Charles Peirce wrote about it, so the search results feature a surprising measure of semiotics, even Sign Systems Studies.

The identity of this Apollonius with the wonder-worker of Tyana was contested, after Wyttenbach, by Méautis (91), but has probability on its side. Apollonius was a conscious and enthusiastic Pythagorean. (Burkert 1972: 100, fn 10)

Wording.

For the earlier stages of the tradition Diogenes Laertius is particularly important, though his work is hard to analyze. He has woven together material from various handbooks, and his "card-file" method makes it almost impossible to discern connections of any larger elements. (Burkert 1972: 101)

Analogous evaluations may await this blog.

Callimachus' student Hermippus wrote several influential books on Pythagoras; Josephus calls him the "most distinguished" of the biographers of Pythagoras. The fragments we have contain the most [|] eccentric material in the whole Pythagorean tradition; Rohde considered the book "a malicious satire on Pythagoras." [.fn.] Q III. Especially bizarre is the report that, in a war between Acragas and Syracuse, Pythagoras lost his life in the course of a retreat, because he refused to run through a field of beans. (The same motif in Neanthes FGrHist 84F31 = Iam. VP 189ff, in the story of Myllias and Timycha.) (Burkert 1972: 102-103)

"According to Hermippus and others, ap. Diog. 39 sq., Pythagoras was slain in his flight, because he would not escape over a bean field. Neanthes (ap. Iambl. 189 sqq.) relates the same of Pythagoreans in the time of Dionysius the elder." (Zeller 1881: 344, fn4)

On the whole, the "later" tradition seems to be not so much the result of unscrupulous falsification as of simple-minded, naive compilation and transmission of whatever could be found, contradictions and all. (Burkert 1972: 105)

I've been considering writing such a naive compilation on the basis of anglophone sources (up to early 20th century) alone. I'm certain that there are idiosyncrasies and outright inventions in that corpus.

Dicaearchus said that "soul" is a mere word (frr. 7ff), and there is an unmistakable irony in his account of Pythagoras' doctrine of metempsychosis. Pythagoras was once, he says, a beautiful courtesan. Dicaerchus has Pythagoras, at the end, stumbling from one catastrophe to another; and the Locrians, who were famous for their εὐνομία, denied him admittance to their city. (Burkert 1972: 106)

I can concur with the bold statement.

Aristoxenus dealt most fully with Pythagoras and his pupils. He was [|] a native of Tarentum, and cited in evidence the acquaintance of his father with Archytas, as well as his own acquaintance with the "last" of the Pythagoreans. Clearly, he put himself forward as an expect in Pythagorean matters, just as he was, at the same time, an authority in musical theory. [.fn.] Aristoxenus is the fullest of the ancient sources for Pythagoreanism, and therefore the question of his credibility is especially important and much discussed. The very rationality which characterises Aristoxenus' Pythagoreans, far from mysticism and magic, seems a favorable sign to some scholars, suspicious to others. (Burkert 1972: 106-107)

E.g. "In regard to beans, Aristoxenus [...] maintains that Pythagoras, far from prohibiting them, particularly recommended this vegetable" (Zeller 1881: 364, fn5).

It is customary to complain about the sparseness of early testimonies. Still, whereas we have no explicit reference to Anaximander or Parmenides by any fifth-century author, there is a quite imposing array of references to Pythagoras. Interestingly, most of these had already been assembled in ancient times; we hve them as quotations in the lives of Pythagoras. The history of Pythagoreanism was already, at that time, the laborious reconstruction of something lost and gone. (Burkert 1972: 109)

Probably due to the sweetness of the forbidden fruit - replace "forbidden" with "inaccessible", and voila.

It is remarkable that only Croton and Metapontum, among the south italian cities, are noted for the worship of Apollo. There seem to have been quite ancient Apollo cults in metapum, as well as in Macalla, which belonged to Croton. The facts that Croton used the tripod of Apollo on its oldest couns (about 550 B.C.), that Caulonia, [|] a colony of Croton, showed on its coins Apollo καθαρτής with a stag, that in about 470 Metapontum issued Apollo coins - these facts doubtless have implications about the soil in which Pythagoras' doctrine took root. His unique success had a quite individual kind of background, which combined the piety of chthonic mysteries with worship of Apollo the "purifier." (Burkert 1972: 113-114)

The Hyperborean Apollo found a fruitful ground, probably pre-tilled.

After the defeat at the Sagras, according to Justin's account, he succeeded in reforming the Crotoniates' character to such a degree, by "daily praise of virtue," that the victory over Sybaris became possible. Then, according to Timaeus, directly after this victory, they fell into luxurious living (τρυφή). According to a detiled account in diodorus, Pythagoras, functioning as the embodied conscience of Croton, had a part in bringing about the war against Sybaris. A number of factors conspire, however, to make this account seem very suspicious: the quite different account of Herodotus, the inherent improbabilities, and, most of all, the structure of Diodorus' exposition, which is very like the plot of a tragedy. The destruction of Sybaris was the worst atrocity wrought by Greeks against a Greek city in that era; the attempt to make the unheard-of comprehensible [|] was bound to give rise to legends, and the contrast of "Sybaritic" luxury with Pythagorean sobriety was a strong stimulus to the creation of moralistic and edifying fiction. (Burkert 1972: 116-117)

Pythagoras the lawgiver and ethical teacher, going around the city, exhorting the people to behave themselves.

When Plato contrasts Pythagoras with the lawgivers as one who became, ἰδίᾳ τισὶν, a guide to the good life, he is not thinking of him as a political figure. In some respects, Pythagoras is connected with [|] Metapontum more closely than with Croton, while the political activity and the burning-episode belong to Croton. Some modern scholars have gone so far in a skeptical direction as to assert that Pythagorean political activity in Croton is an invention - Aristoxenus and Dicaerchus, they suppose, projected upon the Pythagoreans their own ideal of the βίος πρακτικός, and thus invnted Pythagorean politics together with an appropriate historical background. (Burkert 1972: 117-118)

"Pythagorean communism" (cf. Minar 1944) may have been a fabrication.

There is, however, an often forgotten testimony of Theopompus, in the midst of a fragment of Posidonius about the tyrant Athenion of Athens. Athenion had been a member of the Peripatetic school, but at the first opportunity he cast aside the mask of philosophy and became a tyrant, thus illustrating "the Pythagorean doctrine regarding treachery, and the meaning of that philosophical system which the noble Pythagoras introduced, as recorded by Theopompus in the eight book of his History of Philip, and by Hermippus the disciple of Callimachus." Theopompus must have said that the secret, but genuine, goal of the philosophy introduced by "the excellent Pythagoras" (ὁ καλὀς Πυθαγόρας) was tyranny. (Burkert 1972: 118)

Sounds like ancient propaganda.

There is no inconsistency between this and the religious and ritual side of Pythagoreanism. In fact, cult society and political club are in origin virtually identical. Every organized group expresses itself in terms of a common worship, and every cult society is active politically asa ἑταιρία. Pythagoreanism fits into this picture and can be seen to have firm rootage in the social and political conditions of the time. (Burkert 1972: 119)

No separation of church and state 500 years B.C.

An allusion of Aristotle is equally explicit. He complains of his predecessors that in their theories about the soul they paid far too little attention to the necessary presuppositions about the body: "They try to say what kind of thing the soul is, but do not go on to specify about the body which is to receive the soul, as though it were possible, as in the tales of the Pythagoreans, for just any soul to clothe itself in just any body." In his critique of various philosophers, he introduces this ironical comparison with the Pythagoreans' "myths" as though they were something well known. What he cares about here is not distinctions among individuals, but among species. The soul "clothes itself" (a common expression in the doctrine of metempsychosis) in any kind of body, human or animal; and this failure to distinguish between human and animal gives rise to the same scandalized tone that we hear in Xenophanes. Theophrastus, on the other hand, in arguing against sacrificing animals, tries to bring animals and human beings closer together. The former, too, have souls, quite like those of human [|] beings, or perhaps even identical with them, "as Pythagoras taught." (Burkert 1972: 121)

Cue all the comparisons with Indian religious motives, such as karma.

Porphyry gives a description, taken from Dicaerchus, of the arrival of Pythagoras in Croton: Pythagoras had great success, and was invited to give lectures before the civic leaders, the young men, the boys, and the women.

As a result of these events, a great reputation grew up about him, and he won many disciples from the city itself; not only men but women too [...] [including Theano], as well as many from the non-Greek territory nearby, kings and nobles. Now the content of his teaching to his associates no one can describe realiably, for the secrecy [σιωπή] they maintain was quite exceptional. But the doctrines that became best known to the public were, first, that the soul is immortal, then that it migrates into other species of animals, in addition that at certain intervas what has once happened happens again, so that nothing is really new, and finally that we ought to regard all living things as akin. Pythagoras is said to have been the first to introduce these opinions into Greece.

Whether all this is from Diraerchus is controversial and cannot be definitely decided by philological means. (Burkert 1972: 122)

"You are here because Zion is about to be destroyed – its every living inhabitant terminated, its entire existence eradicated. Denial is the most predictable of all human responses, but rest assured, this will be the sixth time we have destroyed it, and we have become exceedingly efficient at it. The function of the One is now to return to the Source, allowing a temporary dissemination of the code you carry, reinserting the prime program."

The Memoirs have no place for palingenesis; souls that are pure rise to the "highe," and the impure are given over to the Erinyes. [.fn.] 31; see above, ch. I 3. In the Memoirs, as in Aëtius, only the highest part of the soul is immortal. The neo-Pythagorean Alexander of Abonuteichus rejected metempsychosis in somewhat similar terms: the "soul" develops and then perishes, but the important thing si the "spirit" (φρήν) which emanates from the mind of "Zeus." (Lucian Alex. 40; in the background is Pl. Tim. 30b; cf. Plut. De fac. 944e.) (Burkert 1972: 124)

What, indeed, goes to the Islands of the Blessed?

The doctrine of metempsychosis is set forth, with some detail, in works of Pindar, Empedocles, Herodotus (2.123), and Plato. The question how much of this can be accepted as testimony on Pythagoreanism depends on one's judgment of the complicated and much discussed phenomenon of "Orphism." Scholars' conceptions of Pythagoreanism and of Orphism are inevitably as interdependent as he pans of a balance. A "minimalist" attitude to the Orphic tradition rapidly raises the importance of Pythagoreanism, while hypercriticism toward Pythagoreanism peoples Greece with Orphetelestae. There is no such thing as a communis opinio on Orphism, especially since the sensational discovery of the papyrus of Derveni has shaken many established views. (Burkert 1972: 125)

I still know next to nothing about Orphism.

Metempsychosis is not attested for Orphism in any ancient source - only the preexistence of the soul. It is undergoing punishment in its confinement to the body, which is both a prison and a protection. Souls are borne by the wind into the body; while it is an almost unavoidable supplement to suppose that oth living creatures, at their death, have "breathed out" these same souls. (Burkert 1972: 126)

"If the soul originally floats in the air, and enters the body of the newly-born with the first breath, it escapes equally from the body of the dying with the last; and if it does not ascend to a superior abode, or sink to an inferior place, it must float about in the air until it enters another body." (Zeller 1881: 485, fn2)

Unlike the Pythagoreans, the Orphics committed their teachings to writing from the beginning, as is shown, aside from the evidence cited in n. 29 above, by the vase paintings showing a scribe standing before Orpheus' singing head, Linforth 122ff. (Incidentally, their use of writing gives us a historical terminus post quem.) (Burkert 1972: 130, fn 55)

Regrettable that the reverse wasn't the case.

If one believes, with Nietzsche, in a primal opposition of "Apollonian" and "Dionysian," then Pythagoras and Orphism must stand in the same polar relationship; and if, under the influence of later evidence, one regards the philosophy of number and the foundation of exact science as the essential ingredient of Pythagoreanism, the antithesis of Apollonian rationality and Dyonysian mysticism fits in very nicely. We must bear in mind, however, that as the Greeks thought of them, Apollo and Dionysus were brothers; the supposed clear differentiation of Pythagoreanism from Orphism is simply not attested in the oldest sources. (Burkert 1972: 132)

Neat analogy. Didn't know that they were brothers.

Was there present at its beginning the significant semantic innovation whereby the "soul," as distinguished from the body and independently of it, is regarded as the "complete coalescence of life-soul and consciousness" - a world away from the Homeric conception - or is "soul" primarily a mysterious, meta-empirical Self, independent of consciousness, as some important witnesses seem to indicate? (Burkert 1972: 134)

Footnote regarding meta-empirical selfhood: "Only exceptional persons like Pythagoras remember their previous incarnations" (ibid, 134, fn 79).

It is often seen as matter for regret that miraculous tales have attached themselves to the figure of Pythagoras and make it difficult for the scholar to disentangle the thread of historicity from the web of legend and fiction. In the circumstances it is very tempting to use expressions like "neo-Pythagorean" or "late antiquity" to classify these tales. One feels confident, in any case, that they represent a [|] secondary growth, layers that must be stripped off until what the scientific historian recognizes as "facts" can be seen. Only in a few cases has it been recognizd that these miraculous stories do not conceal but reveal reality, that they give us a clue to the impression made on contemporaries by an actual person, and that they may even contain facts of a special character. The Pythagoras legend is the oldest available layer of the tradition on Pythagoras; it is attested earlier than any of the "historical" details of his life in Aristoxenus and Dicaerchus, and is presupposed by the Platonizing reinterpretation of Pythagoras in the Old Academy. (Burkert 1972: 136-137)

It is tempting to peel away the layers of mysticism and supernaturalism, yet these are at the very core of the subject.

Rostagni mentions the report that Pythagoreans avoided pronouncing Pythagoras' name (Verbo 231; Iam. VP 88, probably from Aristotle; also Iam. VP 150, 255). (Burkert 1972: 137, fn 96)

This makes me think that the right thing to do when writing about Pythagoras would be to avoid mentioning him by name as much as possible.

Heraclides has Pythagoras tell this tale: he had once been Aethalides, the son of Hermes, and received from the latter the gift of remembering everything, both in life and in death. Thus he knew that, as Euphorbus, he had been slain by Menelaus in the Trojan War, and that he had subsequently been Hermotimus, then Pyrrhus, a fisherman of Delos, and finally Pythagoras. Here we are on the shaky ground of Academic and Peripatetic controversy. Dicaerchus and Clearchus give a very different list of Pythagoras' previous incarnations: Euphorbus, Prandrus, Aethalides, a beautiful prostitute named Alco, and Pythagoras. It looks as though each one treated the Pythagoras tradition as his whim or fantasy dictated. (Burkert 1972: 138)

These make it possible to write a biography of Pythagoras without mentioning his most famous reincarnation. Though, as information about earlier incarnations is scarce, some "whim or fantasy" would be necessary.

Aethalides is a son of Hermes. He has the privilege that his soul may dwell part of the time on earth and part of the time in Hades; this is reminiscent of the Dioscuri, not of metempsychosis. Aethalides belongs to Lemnos; this may have something to do with the supposed Tyrrhenian origin of Pythagoras (ch. II 2, n. 12). (Burkert 1972: 138, fn 102)

"The Dioscuri were regarded as helpers of humankind and held to be patrons of travellers and of sailors in particular, who invoked them to seek favourable winds." (Wiki) - Certainly fits with Iamblichus' story of Pythagoras' travel from Phoinicia to Egypt, i.e. the sailors intending to sell him into slavery, but perceive something holy in his presence, which conjures prosperous winds for the sails.

Heraclides (fr. 89) has Hermotimus find the shield, in the sanctuary of Apollo at Didyma, corrupted in Tert. An. 28 to "Delphi." This has the appearance of being secondary, and if so the shield story is at least older than Heraclides. (Burkert 1972: 140, fn 109)

Nowadays named Didim on the Aegean coast of western Turkey, featuring a still impressive picture of the ruins of said temple.

Aristotle records the following items:

1. Pythagoras was called "Hyperbolean Apollo" by the Crotoniates.
2. At the same hour on the same day he was seen both in Croton and in Metapontum. [|]
3. When Pythagoras stood up among the spectators at Olympia, people saw that one of his thighs was of gold.
4. He reminded Myllias of Croton that he had been King Midas.
5. He stroked a white eagle in Croton.
6. As Pythagoras was crossing the Casas River, the river hailed him in an audible voice, "Greetings, Pythagoras!"
7. As a ship was entering the harbor of Metapontum, he predicted that a dead man would be found in it.
8. In Caulonia, he correctly predicted the appearance of a white bear. [|]
9. In Etruria, he bit a poisonous snake to death.
10. After predicting to the Pythagoreans the outbreak of civil strife (οτάοις) he disappeared to Metapontum without anyone's seeing him go. [...]
11. Pythagoras took from Abaris, the priest of Apollo from the country of the Hyperboreans, the arrow with which he traveled, and thus established himself as the Hyperborean Apollo.
12. "They say of the man who bought Pythagoras' house and tore it down, that he did not dare tell anyone what he saw, but that as a rest of this crime he was convicted of sacrilege by the Crotonians and executed. For he was convicted of having stolen the golden chin which had fallen from the god's statue." Pythagoras' house is inviolate, like a sanctuary of the mysteries; the transgressor dies the death of a scapegoat.

These "miracles" are portents without interpretations, revelation and occultation at once. At a certain moment there is a glimpse of the divine - the gleam of the golden thigh, the greeting of the river god, the arrival of the Hyperborean. Superhuman powers are evident in Pythagoras' prophecies, in his mastery of the animals, and in his control of space and time, as well as in the numinous dread that [|] attaches to his house. There is always something enigmatic aout the meaning of these miracles, which is apparently revealed to the insider but not explained to the uninitiated. (Burkert 1972: 141-144)

A useful rundown of the fabulous stories connected with Pythagoras. Some of these are news to me.

An epic poem entitled Arimaspeia, by Aristeas of Proconnesus, [|] was in circulation in the early sixth century B.C. Aristeas told how, possessed by Apollo (φοιβόλαμπτος γενόμενος) he had traveled to the country of the Issedones in the far north, and learned from them about the Arimaspi, the griffins, and the Hyperboreans who lived still further north. Herodotus adds a local legend from Proconnesus, to the effect that Aristeas ded, and soon after was seen traveling abroad, while his body was found to have vanished. After seven years he appeared in town again, bringing his Arimaspeia, and then disappeared again. Herodotus also was told, in Metapontum, that Aristeas had appeared there and bidden the natives to build an altar to Apollo. "For (he said) Apollo had visited them alone among the Italians, and he himself had accompanied him in th form of a raven [...] And even now there stands in the agora, near the statue of Apollo, a statue inscribed with the name of Aristeas, and there are laurel bushes round about." Herodotus calculates that the appearance of ARisteas in Metapontum occurred 240 years after his disappearance in Proconnesus. (Burkert 1972: 147-148)

Apollo could possess people?

Later reports are unequivocal: the soul leaves the body and hovers about in the air "in the form of a bird." Perhaps the contradiction itself, the failure to smooth over difficulties, and the lack of a clear separation of body and soul, are signs of an archaic way of thinking. (Burkert 1972: 149)

Linnutee.

According to Herodotus, without ever eating, Abaris carried Apollo's arrow all over the world, but as early as Heraclides it was said that he flew on this arrow, and this version is regarded as the original one. It may be, though, that just as in the case of Aristeas, the tradition was self-contradictory from the beginning. There were alternative ways to report the activities of the miracle-worker; Abaris could not perhaps "actually" fly, but he could claim the ability, and even, in ecstatic ritual, act it out, as it were, as a shaman. (Burkert 1972: 150)

"The Hyperborean, in return, presented the Samian, as though he equalled Apollo himself in wisdom, with the sacred arrow, on which the Greeks have fabulously related that he sat astride, and flew upon it, through the air, over rivers and lakes, forests and mountains" (EB I, 1797: 5).

A journey of the soul, in pure form, was attributed to Hermotimus of Clazomenae: His soul left his body and wandered about, while the body lay as though dead, until one day his enemies burned the body while the soul was absent. Hermotimus could predict future events; the Clazomenians built a sanctuary in his honor. (Burkert 1972: 152)

He "was a philosopher who first proposed, before Anaxagoras (according to Aristotle) the idea of mind being fundamental in the cause of change." (Wiki) - Possibly an early instance of this separation of the soul from the body, i.e. travelling into the "spirit realm".

The oldest form of transmission of the teachings of Pythagoras is represented by the acusmata, which are also called symbola, orally transmitted maxims or sayings. Our first tangible evidence about them goes back to about 400 B.C. Anaximander of Miletus (the younger), whom Xenophon names as one of those who could find the hidden meanings in Homer, also wrote an Explanation of Pythagorean Symbola. This shows that the tradition was pre-Platonic; and if these symbola already needed explanation, or allegorical interpretation, like the text of Homer, they must be much older, must go back in fact to pre-classical or archaic times. (Burkert 1972: 166)

For the time-traveller to cop.

This shows that the entire section in Iamblichus is full of Aristotelian material. It may be that the division of maxims into three categories is likewise Aristotle's work. He was interested in logical distinctions, and in particular in the early stages of conceptual definition. From the lines of parallel transmission emerges quite an extensive body of material. (Burkert 1972: 170)

I.e. "the acusmata are divided into three groups and given as answers to the quetions τί ἕστι; τί μάλιστα; and τί πρακτέον" (ibid, 167).

What are the Isles of the Blest? Sun and moon. What is the Oracle of Delphi? τετρακτύς ὅπερ ἐστὶν ἡ ἁρμονία ἐν ᾗ αἱ Σειρῆνες. Pythagoras is the Hyperborean Apollo. An earthquake is a mass meeting of the dead. The purpose of thunder is to threaten those in Tartarus, so that they will be afraid. The rainbow is the reflected splendor of the sun. [|] The sea is the tears of Cronus. The Great Bear and the Little Bear are the hands of Rhea. The Pleiades are the lyre of the Muses, and the planets are Persephone's dogs. The ring of bronze when it is struck is the voice of a daemon entrapped in it; and the ringing that people often hear in their ears is the voice of the κρείττονες. Old age is decrease, and youth is increase; health is retention of form, disease its destruction. Friendship is harmonious equality. The most just thing is to sacrifice, the wisest is number, the most beautiful, harmony, the strongest, insight, the best (in the sense of the most desired), happiness, the truest, that men are wicked, the holiest, a mallow leaf, the most beautiful shapes, circle and sphere. One ought to beget children, for it is our duty to leave behind, for the gods, people to worship them. One should put on the right shoe [|] first, not travel by the main roads (λεωφόροι) (no. 41), not dip one's hand into holy water (no. 44), not use the public baths (no. 45), and not help a person to unload but only to load up. One should not have children by a woman who wears gold jewelry (no. 21), not speak in the dark (no. 51). One should pour libations over the handle of the cup, refrain from wearing rings with depictions of gods (no. 9), and not "pursue" one's own wife, since the husband, in receiving her as a suppliant at the altar, has taken her under his protection. One should not sacrifice a white cock, because they are suppliants and sacred to the god Men. One should never give advice except with the best intent (for advice is sacred) (no. 63), nor make a detour on the way to the temple (no. 59). One should sacrifice and enter the temple barefoot (no. 3), and in battle hold one's place, so as to fall with wounds in the breast. Eat only the flesh of animals that may be sacrificed; abstain from beans; do not pick up food that falls from the table, for it belongs to the Heroes. Abstain from fish that are sacred (including τρίγλη, ἀκαλήφη, ἐρυθρῖνος, μελάνονρος); do not break bread; put salt on the table as a symbol of righteousness. (Burkert 1972: 170-172)

A rundown of the pythagorean symbols.

On the other hand, the word σύμβολον carries with it the suggestion of a "symbolic" interpretation. The word is not a late addition, however, but carries another implication as well, prior to any "symbolic" exegesis. In the realm of mystery religion, σύμβολα are "passwords" - specified formulas, sayings, ἐπῳδαί, which are given the initiate and which provide him assurance that by his fellows, and especially by the gods, his new, special status will be recognized. (Burkert 1972: 176)

These sayings enable pythagoreans to recognize each other.

Some taboos are attested from an earlier date, and a good many are widely spread folk tradition. Above all, the form of such authoritatively prescribed commandments and prohibitions, which are not supposed to be understood but merely obeyed, is primeval; it is entwined in the very roots of religious ritual. In what one does and does not do is manifested the identity of the group, the membership of the members and the exclusion of outsiders. The more selective the society, the more careful are the "taboos." Fasting, abstention from particular foods, and rules of sexual behavior play an important role. It is of first importance that the "wise man" - the priest, the hierophant, the shaman - who claims a special position in the social organization, gain and maintain, through a special ascetic regimen, the special powers that belong to him. (Burkert 1972: 178)

Also somewhat "phatic".

The τετρακτύς, a "tetrad" made up of unequal members, is a cryptic formula, only comprehensible to the initiated. The word inevitably reminds of τρικτύς, the "triad" of different sacrificial animals. Is the sacrificial art of the seer, involving the shedding of blood, superseded by a "higher," bloodless secret? The acusmata provide a hint toward an explanation: "What is the oracle of Delphi?" "The tetractys; that is, the harmony in which the Sirens sing" (Iam. VP 85). The later tradition is more explicit: The "tetrad" of the numbers 1, 2, 3, and 4, which add up to 10 (the "perfect triangle"), contains within itself at the same time the harmonic ratios of fourth, fifth, and octave. The Sirens produce the music of the spheres, the whole universe is harmony and number, ἀριθμῷ δέ πάντ' ἐπέοικεν. The tetractys has within it the secret of the world; and in this manner we can also understand the connection with Delphi, the seat of the highest and most secret wisdom. Perhaps Pythagorean speculation touched upon that focal point, or embodiment, of Delphic wisdom, the brone tripod of Apollo. Later sources speak of its mysterious ringing, which must have been "daemonic" for Pythagoreans. (Burkert 1972: 187)

Curious stuff all around. Burkert proceeds here from his interpretation of golden verses 46-47.

The mathematici have a different explanation:

Pythagoras, they say, came from Ionia and Samos at the time of the tyranny of Polycrates, when the civilization of Italy was flourishing, and the first men in the cities became his trsted associates. The older of these he addressed in simple style, since they had little leisure, being occupied with political affairs, and he saw that it was difficult to speak to them in terms of μαθήματα and proofs. He thought they would be better off for knowing how to act, even without knowing the reasons, just as persons under medical care get well even though they are not told the reason for every detail of their treatment. The younger men, however, who had time to put in the effort of learning, he addressed with proofs and μαθήματα. They themselves, then, the mathematici, are the successors of the latter goup, and the acusmatici of the former.

Hippasus, they said, only published, for his own aggrandizement, things that Pythagoras had taught long before. (Burkert 1972: 195)

A possible explanation for the schism based on Pythagoras' habit of addressing men, women, and children separately.

When we start looking for acusmatici, we think first of the Πυθαγορισταί who appear in leading or secondary roles in the Middle [|] Comedy. It is repeated again and again that they eat οὐδὲν ἔμψυχον, not even the meat of sacrificial animals, only a lot of wretched vegetables. Sometimes they scarcely eat anything at all; they drink plain water, attract attention with their silence and their στυγνότης, they wear a ragged τρίβων, go about barefoot, and are stiff with dirt since it is against their principles to bathe. Shabbiness turns into arrogance; they are typical ἀλαζόνες. (Burkert 1972: 198-199)

Their silence is conspicuous. Burkert has colourful adjectives for this group: "mendicant Pythagoreans", "dirty ragamuffins" (ibid, 199), and "low-class tatterdemalions" (ibid, 200).

Just as the Pythagorean feels himself a stranger on the earth, so the Cynic tries to free himself from all ties. Threads lead from here to the Stoa as well; both Zeno and Chrysippus were interested in aspects of Pythagoreanism. (Burkert 1972: 203)

I'm just visiting.

After Lycon, who must have been approximately contemporary with Aristotle, there are no more Pythagoreans of this type to be found. An echo of such activity can still be heard in Onesicritus, who named Pythagoras as one of the Greeks who, before Socrates and Diogenes, had taught doctrines like those of the Indian Gymnosophists. But Onesicritus is regarded as a Cynic. The tendency in Pythagoreanism represented by Diodorus of Aspendus was absorbed by Cynicism, which took shape as the form of the self-sufficient, world-despising βίος which suited the demands of the age. Meanwhile the spiritual power of Pythagoreanism found, through the interpretation of the Platonists, a new vehicle adaptable to changing times. (Burkert 1972: 204)

"Onesicritus (Greek: Ὀνησίκριτος; c. 360 BC – c. 290 BC), a Greek historical writer and Cynic philosopher, who accompanied Alexander the Great on his campaigns in Asia." (Wiki)

The situation is scarcely different with recollection (ἀνάμνησις) and related ideas. Here, too, there is a discernible Pythagorean background: Empedocles (fr. 129) says that Pythagoras, when he put all his spiritual power to work, could survey ten and twenty generations. Surely this assumes that he recalled his earlier incarnations. The later tradition tells of a system of memory training among the Pythagoreans. They triin the morning, or in the evening, to recall all the events of the past day, and even of the day before. (Burkert 1972: 213)

A familiar theme from Iamblichus, though I cannot find the exact quote. "Never let slumber approach thy wearied eye-lids, Ere thrice you reviewed what this day you did;" (Golden Verses, 40-41).

The other tradition sets in at just about the time of Aristotle. Aristippus dealt with Pythagoras in a book entitled Περὶ φυσιολόγων. Hecateus of Abdera and Anticlides, who wrote about Alexander the Great, represent Pythagoras as introducing geometry from Egypt. By this time a conception of Pythagoras had become dominat which was obviously quite unknown to the pre-Platonic writers. (Burkert 1972: 216)

Mankind's conception of Pythagoras is constantly changing.

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3. Philolaus

It has long been an established principle in the study of the pre-Socratic philosophers that only their original words, so far as they have been preserved, can provide an adequate foundation for interpretation and reconstruction. In any paraphrase lurk potential errors, for alteration of the form is bound to affect the content, whether because of adaptation to modern ways of thinking, or of a polemical bent which results in a reluctance to see the true sense, or of both. The special difficulty in the study of Pythagoreanism comes from the fact that this principle cannot be applied; whether an item of the tradition may be regarded as an authentic pronouncement of a Pythagorean must in each case be decided first on the basis of the indirect testimony. For the mass of writing that was fored and attributed to Pythagoras and his pupils was so vast that, contrary to ordinary methodological principles, in the case of any text purportedly composed by an early Pythagorean, the burden of proof lies with anyone who wishes to maintain its authenticity. (Burkert 1972: 218)

The exact reason I don't read translations from English (into Estonian) if I can at all avoid it. It is notable that some of those "forgeries" have been translated into English only in recent decades.

The opinion was widespread that no such book had been preserved, or even that Pythagoras avoided the written word on principle and recorded his teachings only in the minds of his [|] disciples. There is a Platonic coloration here, and one may suspect that this emphatic declaration served as a pretext for discarding certain Pythagorean writings that were felt to be an embarrassment because of their old-fashioned character. Delatte attempted to refer the various lines of Pythagorean verse, one of which was cited as early as Chrysippus, and most of which eventually found their way into the late compilation called Carmen aureum, to a 'Ιερὸς λόγος of the fifth century. (Burkert 1972: 218-219)

The first endonymic philosopher might have been so archaic that his writings were destroyed because they didn't stand up to the grandiose mythology that had grown around him.

While Anaximander and Parmenides, as well as Empedocles, survived almost exclusively because of the works that bore their names, in the case of Pythagoras what stands at the origin of the tradition is the picture of a "sage," unclear in many details but powerful for that very reason - an outline to be filled in by later generations. (Burkert 1972: 220)

Pythagoras is iconic exactly because we don't have his books to read ourselves, but an empty vessel to fill with our own imagination.

"Forgeries" are usually pieces of quasi-historical reconstruction; the Pythagorean pseudepigrapha, for example, show what people wanted to be regarded as Pythagorean. We can detect in them a certain tendency of interpretation, a general purpose which is also discernible in the distortions of the doxographical tradition. A genuine fragment must show itself so by standing aloof from this tendency and not being deducible from it. In this way one can establish the presumption that something is probably genuine. (Burkert 1972: 223)

The genuine article probably does not conform to expectations nor go to much against the grain.

"Pythagoras wrote three books: On Education, On Statesmanship, and On Nature." This so-called tripartitum was available to Heraclides Lembus, who also used Saturys, and also to the author who put together the account which is at the base of the lives of Pythagoras in Diogenes laertius and Hesychius. (Burkert 1972: 225)

Cool, cool, cool.

We have from Hermippus another story, older than that of the "tripartitum": "He (Philolaus) wrote one book, which according to Hermippus some writer said Plato the philosopher [...] bought from the relatives of Philolaus for forty Alexandrine minas of silver, and from which he copied his Timaeus" (D.L. 8.85). (Burkert 1972: 225)

Now I have a name to attach to this accusation.

The charge of plagiarism is like a philological discovery, a clever inspiration that is all the more effective if the similarity is not apparent on the surface. The keen-witted Kritiker detects the "theft" which completely deceives the man in the street. Therefore we must suppose that the book of Philolaus, too, upon which the Timaeus was said to be based, really did exist, but not thatit showed so thorough an agreement as does the "Timaeus Locrus" book. (Burkert 1972: 227)

This tickles me.

Thus Simplicius, in explication of the Aristotelian account of the Pythagorean system of the world, writes as follows (and quite similar words may be found in Asclepius and in an anonymous scholium to Aristotle):

This is the way he understood the Pythagoreans' theory himself. But those of them with more genuine knowledge understood by "central fire" the creative force which, from its mid position, produces life over the whole earth, and keeps warm the parts of it that tend to cool off. [...] And they used to call the earth a "star" because it too, like the stars, is a creator of time; for it is the cause of days and nights. The part of it which is shone upon by the sun makes day, and that which is in the cone produced by its shadow makes night. The Pythagoreans called the moon "counter-earth," as though to call it "the ethereal earth," and because it intercepts the sun's light, which belongs characteristically to the earth.

The expression about "more genuine" Pythagoreans here as been eagerly seized upon, because it seemed reasonable that there should have been a geocentic system as a precursor to the more complicated system described by Aristotle. (Burkert 1972: 232)

This makes a lot more sense than earlier explications of pythagorean cosmology. The "central fire" is the Earth's core, and the Moon is a "counter-earth" because it is in Earth's orbit.

It was no longer a day of lonely prophets but one of far-reaching debate. A statesman like Melissus of Samos might take part in it, or a poet like Ion of Chios; physicians, too, were beginning to formulate in written terms the scientific basis of their art. A book by a Pythagorean, in this period, cannot have been so much like an erratic boulder as a link in a long chain of tradition. (Burkert 1972: 239)

Which one is E. R. Clay? An erratic boulder or a link in a long chain?

This means - though the fact has scarcely ever been clearly recognized - that, if Pythagorean doctrine was not committed to writing before philolaus, then there did not exist, before Philolaus, any Pythagorean philosophy, in the Greek sense of the word, but only a different kind of thing: a lore or "wisdom" consisting of disconnected teachings about the world, gods, and human beings, having its foundation in a specific way of lifeand transmitted in individual maxims. This "wisdom" and way of life were variable in detail and lacked logical foundation or systematic and conceptual coherence; in fact they consisted in our familiar acusmata, the doctrine of transmigration, and the βίος Πυθαγόρειος in which they were rooted. (Burkert 1972: 240)

Hence the title of the book.

Further, Nicomachus ascribes to Philolaus a number system which represents a development of being in numerical stages: 1, point; 2, line; 3, plane surface; 4, solid; 5, ποιότης καὶ χρῶσις; 6, animation; 7, νοῶς καὶ ὑγίεια καὶ τὸ ὑπ' αὑτοῦ λεγόμενον φῶς; 8, ἔρως, φιλία, μῆτις, ἐπίνοια. This scheme contradicts the report of Aristotle, according to which the Pythagoreans called the plane surface χροιά, that is, they were not able to distinguish, even in terminology, between "surface" and "color." In general, this gradation of being, and in particular the order of geoetrical forms, is Platonic and not Pythagorean. In fact, Plutarch cites the very system here attributed to Philolaus, as Platonic. Nicomachus and Proclus made abundant use of this "Pythagorean" scheme, which may also have been included in a ἱερὸς λόγος attributed to Pythagoras. (Burkert 1972: 247)

Somewhat sad, this. The gradation of being is my favorite theme in pythagoreanism.

The φύσις in the cosmos has been put together harmoniously from unlimited and limiting (constituents), both the whole cosmos and all the things in it (fr. 1). Existing things must be, all of them, either limiting, or unlimited, or both limiting and unlimited; but they would not be unlimited only <or limiting only(?)>. Since, however, they are clearly neither made of limiting (constituents) only nor of unlimited only, it is therefore obvious that from both limiting and unlimited (constituents) the cosmos and the things in it were harmoniously put together [...] (fr. 2). This is the situation about φύσις and harmony: the being (ἐστώ) of things, which is eternal, and φύσις itself admit of [|] divine but not of human knowledge, except that it was impossible for any of the things that exist and are recognized by us to come to be if there were not the being (ἐστώ) of the things of which the cosmos was composed, both the limiting and the unlimited. And since these beginnings (ἀρχαί) were not alike or of the same kind, it would have been impossible for them to be put together harmoniously if harmony had not supervened - however it was that it came to be. It is not things that are alike and of the same kind that need harmony, but things unlike and different and of unequal speed; such things must be bonded together by harmony, if they are to be held together in a cosmos [...] (fr. 6).

What is striking here is the really heavy-handed insistence with which the leading ideas - "limiting and unlimited," "cosmos," "harmony" - are repeated. But far from giving cause for suspicion, this cautious procedure makes the impression of a genuine effort of thought, anxious to keep hold of the important points, and not seeking to make a display with carefully learned or borrowed fustian. This is the typical style of the pre-Socratics; "the amount of tedious repetition, even in Anaxagoras, is incredible." (Burkert 1972: 251-252)

define:fustian - pompous or pretentious speech or writing. Philolaus' writing is thick.

"Limit" and "unlimitedness" are not isolated as entities in themselves, congealed into an abstract substantive or hypostasized as intangible substance, but they are thought of as scattered or deployed, so to speak, in individual things, περαίίνοντα or πειραἄ. This is a basic difference between the thought of the pre-Socratics and that of the Platonic and Aristotelian schools, a difference brought out especially by Cherniss. By a word like θερμόν or ψυχρόν, for example, a pre-Socratic thinker does not mean an abstract entity, or an οὐσία, but quite concretely the sum of particular things characterized by the word. Only in Plato's dialectic was the foundation laid for separating qualities and quantities from objects and regarding them αὐτὰ καθ' αὑτά. For later ages these distinctions came to seem self-evident. (Burkert 1972: 254)

The ideal and material remain undifferentiated.

It is very easy to understand "limit" as a formative principle, and "unlimitedness" as a material principle. But in considering a question of authenticity we must differentiate between philosophical interpretation, which seeks to understand the author better than he understood himself, and philological interpretation, whose first duty is to understand and place historically what the author put down. The thought that god gives form to formless matter, by imposing limits," περατοῦν τὴν ὕλην ἄπειρον οὖσαν, is characteristic of the Platonists; that it does not go back to the Pythagoreans, and that it was not an accomplishment of theirs, as is often thought, to conceive for the first time, in explicit terms, of a formal principle, follows from the exposition of Aristotle, according to which the form-matter dichotomy is not applicable to their number theory. If the situation were different in the Philolaus fragments, this would be a serious cause for suspicion; but any such interpretation is impossible. (Burkert 1972: 255)

Admittedly, casting away the form-matter dichotomy is a difficult thing to do.

Aristotle remarked that the atomists, too, in a certain way made things out to be numbers. The "void" of the atomists is not a single, endless space, but the plurality of interstices which make divisibility and plurality possible. The cosmos grows by taking in material from outside. It is obviously the Pythagoreans who inspired the atomists to see in the motes in a sunbeam an indication of the nature of the soul-atoms. We have the testimony of a contemporary that Democritus studied with a Pythagorean; thus Philolaus and Leucippus are thrown close together. To be sure, Philolaus maintains a distance from the atomists; he does not speculate further about being, but looks for relationships in our given, familiar world and finds them in the ordering function of number. (Burkert 1972: 259)

"Democritus was acquainted with Philolaus, that he spoke with admiration of Pythagoras in a treatise called after him, and, in general, had made industrious use of the Pythagorean doctrines" (Zeller 1881: 363, fn3).

The Hippocratic writings illustrate how some thought of the numerical and musical ratios as bearing on the life of man. For example Regimen defines its basic question as the determination of the right relation between nourishment and activity:

If indeed [...] it were possible to discover for the constitution of each individual a due proporton of food to exercise, with no inaccuracy either of excess or of defects, this would mean, precisely, the key to health for human beings.

Health, in "precise" terms, is a numerical ratio; whoever knows the numbers has found all he needs. (Burkert 1972: 262)

The harmony of calories in and calories out.

Isocrates represents the mythical king Busiris as establishing the castes of priest, artisan, and warrior in Egypt: [.|.] Busiris took "all numbers," that is all the classes, in their state as numbered and ordered groups that would be useful in the government of society. (Burkert 1972: 265-266)

The triad of society.

Democritus uses metaphors from botany in speaking of the importance of the navel; the expression σπέρματος καταβολά occurs in the passage about Philolaus in Menon's history of medicine (A27); both Empedocles and Diogenes of Apollonia know the hierarchy of plant-animal-man; and the distinction between man and beast by the [|] criteion of νοῦς may be derived from Alcmaeon. (Burkert 1972: 270-271)

Cool, cool, cool. Further: "The hierarchy plant-animal-man-god is known in the east from Sumerian times (vase from Uruk, ANEP no. 502) to Iranian (J. Duchesne-Guillemin, East and West 13 [1962] 200)." (ibid, 270, fn 157)

A medical milieu is also the source of the doctrine that the ψυχή, "soul" or rather "life," is in fact a "harmony" of the bodily functions. It is ascribed by Macrobius to "Pythagoras and Philolaus"; this can come from reliable doxographical tradition, but also may be an inference from the Phaedo, where this doctrine is discussed. Scholars saw long ago that this is the only point where Echecrates interrupts the account of Socrates' last conversation: θανμαστῶς γάρ μου ὁ λόγος αὗτος ἀντιλαμβάνεται καὶ νῦν καὶ ἀεί, τὸ ἁρμονίαν τινὰ ἡμῶν εἶναι τὴν ψυχήν [...] This shows that, as Plato represents the matter, the soul-harmony doctrine was important for this Pythagorean from Phlius, who was a pupil of Philolau. If the order of the universe is ἁρμονία, then so is that of organic life, and the passage we have cited from the Hippocratic Regimen (n.114 above) shows how one could express the idea of "life" in musical terms, too. (Burkert 1972: 272)

Hing on keha harmoonia.

The content is simple, sometimes even trivial. Only one sentence, bearing on the theory of knowledge, is obscure, though not incomprehensible. Number "in the soul, in harmony with sense perception, makes everything knowable and mutually agreeable, working like a carpenter's square, fixing and loosing the proportions of things, each for itself separately, those that are unlimited and those that are limiting." (Burkert 1972: 273)

Relevant for the etymological connection between "harmony" and carpentry.

Here the proof of authenticity drawn, in an indirect way, from the doxographical tradition, is directly confirmed; there have been preserved for us remains of a book composed by Phiolaus in the pre-Platonic period, including both word-for-word fragments and doxographical reports, which advocates that philosophy of Limit and Unlimited, of number and harmony, to which [|] Plato alludes in the Philebus and which Aristotle ascribes to the Pythagoreans. This may well be the only written exposition of Pythagorean number theory before Plato. (Burkert 1972: 276-277)

Bumping Philebus upwards.

The author's own intentions and the borrowed philosophical terminology do not always fit harmoniously, so that much seems tedious or awkward, and much unsystematic or "eclectic." Only in the science of Archytas (who was the teacher of Eudoxus) and in the philosophical reinterpretation of Plato did Pythagoreanism attain to a form in which its real influence could develop. (Burkert 1972: 277)

Archytas more systematic than Philolaus.

If there is no direct evidence, can indirect testimony be found, for example reflections of Pythagorean teachings in the works of other philosophers? They may have taken over Pythagorean material, or entered into polemic against it. Since the day of Tannery scholars have been treading this path, with growing confidence. They attempt to discover doctrines of Pythagoras from their ifluence, as an astronomer sometimes infers the existence of a hithero unknown star from irregularities in the course of known planets. In this way a tempting chapter of the history of philosophy may be built; erratic boulders and unidentifiable gravel coalesce into a comprehensive structure. The suspected interaction of the Eleatics and Pythagoreans, in particular, becomes a living dialogue. Parmenides, the apostate Pythagorean, sets up his own system in opposition to that of the school; in response, the Pythagoreans revise their theories, only to be subjected to new attacks, by Zeno; this forces them to undertake further revision... (Burkert 1972: 278)

Very well put. I love it when imagery comes together like this.

But this is a very difficult thing to prove, even in the case of Plato, and seems to lead to nothing but further controversy; for the pre-Socratics, preserved only in sorry fragments, it is practically hopeless. (Burkert 1972: 279)

Need väikesed kivikesed, mida tuleb kätte võtta, sõrmitseda ja peos edasi kanda, et igat kühmu ja lohku põhjalikult tundma õppida.

Unknown quantities keep multiplying, for the nature and characteristics of Pythagoreanism, whose influence and diffusion one is trying to determine, are far from being clearly understood. In order to get any kind of start, one has to take something or other as presupposed, "given." Scholars have frequently regarded it as almost self-evident that the Pythagorean doctrine of Limit, Unlimited, and number must have existed from the day of Pythagoras in some form or other, which in any case was abstract and philosophical; in this way, the only question is to decide what aspects of it Parmenides and Zeno presuppose. (Burkert 1972: 269)

I wonder if half a century of study has changed this situation.

An epoch in which a unique development in the history of thought took place, like the period of something more than a hundred years between Pythagoras and Plato, surely saw inner transformations even in apparently stable traditions. (Burkert 1972: 280)

Timeframe.

Most important of the relationships between Pythagoreans and other groups are those with the Eleatics. Geographically these were close to the south Italian centers of the Pythagoreans, and the abstract, immaterial character of their philosophy naturally results in coincidences with a philosophy of number. In addition, the ancient traditon makes Parmenides and Zeno Pythagoreans, or at least pupils of Pythagoreans. (Burkert 1972: 280)

Muy importante.

Xenophanes is reacting against the naive, anthropomorphic conception of the gods. The principal criteria of a living being, along with the ability to see, hear, and apprehend psychologically (νοεῖν), are breath and motion. The god, "as a whole," exercises the former functions, but not the latter. There is no more reason to suppose the denial that he breathes is directed polemically against contemporary philosophers than the denial that he moves about. (Burkert 1972: 281)

Noted only because today I caught myself musing over whether Jakobson's functions could be actually be rearranged according to my vision of his scheme, and specifically if his "physical channel and psychological connection" couldn't be divvied up between the opposites of contact (physical channel) and context (psychological connection). His phrasing of "common context" might give way to an broader interpretation of his referential function.

Parmenides has been exploited much more as a source for Pythagorean philosophy, though in a different way. Tannery maintained that the doxa section of his poem, with the specific statement that its teachings are "deceptive" (fr. 8.52), was a doxography, from a hostile point of view, of Pythagorean cosmology. Later writers claimed that [|] in Parmenides' particular manner of developing his argument there could be seen a reflection of Pythagorean mathematics. Then, finally, Raven sought to explain the Parmenidean predicates of Being as a polemical expression against the Pythagorean doctrine of opposites. (Burkert 1972: 281-282)

Oo-wee! Something to take account of when going over Parmenides' poem.

According to the view of Raven, Parmenides arrived at his pronouncements on the ἐόν as an alternative to the Pythagorean doctrine of opposites. The ἐόν is limited, single, indivisible, and cannot come into being or perish, while in Pythagoreanism Limit stands over against the Unlimited, and in a cosmogonic process the One develops into a Many. (Burkert 1972: 283)

Really giving off that "the unity of plurality is totality" vibe.

Parmenides puts his knowledge into competition with older wisdom. Light and Night represent the realms of life and death. The goddess "sends the souls, now from the visible into the invisible, now back again." This means that the existence of the soul is antecedent to the cycle of life and death, and implies a kind of transmigration. (Burkert 1972: 284)

Highly symbolic.

κλαῦτά τε καὶ κώκυσα ἰδὼν ἀσυνήθεα χῶρον

- this is Empedocles' reaction, as expressed in the Katharmoi, to entrance into human life; and a Pythagorean acusma calls our birth a punishment. Such a "puritanical" attitude to life, which sees our existence mainly as a burden and a punishment, can scarcely be called anything but Pythagorean, especially in southern Italy. (Burkert 1972: 284)

"For as we came into the present life for the purpose of punishment, it is necessary that we should be punished." (Iamblichus 1818: 45)

There is no law of reciprocal interaction in the field of thougt, which could make it possible for us, by inference, to fill adequately the gaps in our tradition, as the law of gravitation enables astronomers to calculate the position or movement of an unknown star. In fact, the suspicion persists that the lacuna in the tradition about early Pythagoreanism is not an accident. If we cannot get a clear idea of the philosophy and science of Pythagoras, it is because Plato and Aristotle did not consider him a philosopher. If we cannot find a clue to the philosophy of Limit and Unlimited and their harmony achieved through number, before the day of Philolaus, it is because this doctrine, in this bastract form, was first created as Philolaus worked to formulate anew, with the help of fifth-century φυσιολογία, a view of the world that came to him, somehow, from Pythagoras. (Burkert 1972: 298)

Peirce, likewise, considered Pythagoras a mere law-giver.

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4. Astronomy and Pythagoreanism

The earliest connected discussions of astronomical matters are found in the earliest works of Plato, and it is not merely a coincidence that almost all the important astronomers of later times were Platonists. The Greek idea of a general structure of the world is set forth here in all its essential features: the earth is spherical and rests, free of support, at the center of the sphere of the fixed stars; the planets ae stationed in concentric paths at varying distances; and their apparent irregularities are explained by mathematical principles. (Burkert 1972: 300)

Astronomy is not the first thing that comes to mind when thinking of Plato.

Eudemus must have found Democritus' arrangement of the planets incorrect, too, because of the special position of Venus. The "correct" one was, however, included in the system of Philolaus; for the sequence of the ten "divine bodies" was, in the unanimous testimony of Aristotle and the doxographers, central fire, counter-earth, earth, moon, sun, five planets, heaven of the fixe stars. If we consider only the portion between earth and heaven, this is the order accepted by Eudoxus, Plato, and Aristotle. In addition, this is the system which Aristotle ascribes simply to "the Pythagoreans;" so nothing seems in the way of the assumption that Eudemus meant the same Pythagoreans, and his report that "the Pythagoreans" had established the order of the planets referred precisely to the system of Philolaus. (Burkert 1972: 313)

The "central fire" is still a question mark.

Scholars have seized upon the assertion of the "more genuine" Pythagoreans, that the central fire is a force in the interior of the earth, named 'Εστία. In fact, this epithet is applied to the earth a few times in the fifth century, and Empedocles spoke of fires beneath the earth. Here, it is thought, we have traces of a geocentric system belonging to the early Pythagoreans, which we should postulate anyway and which displays a suitable mixture of myth and science. Nevertheless, the basis of the reconstruction, the report of the "genuine Pythagoreans," is an artificial reinterpretation of the reports of Aristotle, and without independent value as a source. The only other point is the name of 'Εστία, but this is comprehensible as an expression of the central location of the earth, without the idea of a central fire. (Burkert 1972: 317)

Hestia.

Eudoxus went on from there; but the admiration of the perfect circle takes us back to a much earlier period. Alcmaeon spoke of the imperfect circle, of the failure to join beginning and end which is the cause of a man's death (fr. ,), [|] as well as of the eternal circular movement of the divine stars (A12). A Pythagorean acusma says that circle and sphere are the most beautiful shapes. But even before Pythagoras, in Anaximander, the marvelous properties of the circle keep the earth in equilibrium; the cycles of day and year are even older ideas; and Homer himself speaks of the "sacred circle." Thus in the postulate of uniform circular movement, which formed Eudoxus' point of departur for the solution of his problem, there is a reminiscence of more ancient speculations. (Burkert 1972: 331-332)

What's the symbolism behind joining beginning and end? Metempsychosis?

The oldest Babylonian text yet known that refers to the "signs" of the zodiac, not to constellations, is a horoscope from the year 410 B.C. Van der Waerden takes expressions like "at the end of Pisces," which occur some decades earlier, as evidence for the introduction of the twelve signs before that date, in place of the ancient names of constellations. (Burkert 1972: 334)

Surprisingly late.

The only way open to progress in astronomy was to abandon physical explanations based on the necessary laws of movement and to adopt purely mathematical description. The result was the Greek mathematical theory of planetary motion, a tremendous achievement. It was not possible, however, to find one's way back from its complexities to simple physical laws; so that, from Aristotle's time on, the two-world theory was dominant, regarding the realm of the heavens as wholly different from and foreign to ours. Only with Galileo and Newton did astronomy once more, from the heliocentric standpoint, align itself with physics. Plato thought it was an inescapable conclusion that the orderly movement of the stars is due to beings with souls; it is a voluntary, chosen order. Here sophisticated Greek science harks back to the pre-scientific way of thinking and comes to rest in it. (Burkert 1972: 335)

If it (the motion of planets) cannot be explained, it must be because they're gods.

The doctrine in question, according to the consistent testimony of Aristotle and the doxographers, was that our earth is "one of the [|] stars" and along with the moon, the sun, five planets, and an invisible "counter-earth" revolved about a "central fire." The earth a planet! This seems to anticipate Copernicus' momentous discovery, nad ne involuntarily regards the Philolaic system as an attempt to explain, in as clear a way as possible, certain specific astronomical observations. (Burkert 1972: 337-338)

Crazy!

In other words, this Pythagorean system, which expressly denies a parallax as the result of the earth's movement, cannot provide any theory of planetary movement and has no intention of doing so. In this respect it is to be classified with pre-Eudoxan astronomy, in which the capricious prancings of the planets were simply taken to be inexplicable. (Burkert 1972: 339)

A lovely phrase.

Philolaus taught that the moon "is inhabited all around, as the earth is in our zone, by creatures and plants that are larger and more beautiful, for living creatures no the moon are fifteen times as strong, and eliminate no excrement. Their day is proportionately longer." Herodotus of Heraclea, in the fifth century, wrote that "women on the moon lay eggs, and their offspring are fifteen times as large as we are." Herodorus presupposes the story that Helen, who was born from an egg, had fallen from the moon; a similar story was told of the Nemean Lion. As support for his theory that the moon was an inhabited "earth," Anaxagoras cited not only the observation of a fallen meteorite, but the story of the Nemean Lion. The Pythagorean acusma that the sun and moon are the "Isles ofthe Blest" belongs in this context. (Burkert 1972: 346)

Insectoid aliens?

Proclus and, after him, Damascius report that Philolaus "dedicated" certain geometrical figures to particular gods - the angle of the triangle to Cronus, Hades, Ares, and Dionysus, the angle of the square to Rhea, Demeter, and Hestia, and the angle of the dodecagon to Zeus. Damasciu adds that the semicircle was sacred to the Dioscuri. One would quickly reject this late testimony, if it were not corroborated by a very ancient piece of evidence. Eudoxus mentions that in the Pythagorean doctrine the angle of a triangle belongs to Hades, Dionysus, and Ares, and that of the square to Rhea, Aphrodite, Demeter, Hestia, and Hera, that of the dodecagon to Zeus, and that of the 56-angled figure (the hekkapentekontagonion) to the baneful Typhon. This remarkable doctrine is thus attested for pre-Platonic Pythagoreans by a contemporary of Plato. Scholars from Boeckh to Zeller scarcely knew what to make of it, till Tannery, Newbold, Boll, and Olivieri pointed out the connection with astrology. According to an astrological procedure often repeated, with certain variations, triangles and squares are inscribed in the zodiac and are then associated with elements and planets. There are four τρίγωνα and three τετράγωνα; a triangle spans four signs, a square three. This seems to explain the striking connection of three goddesses with the square, and of four gods with the triangle, in Philolaus (even though this precise correspondence is not attested in the Eudoxus passage). The dodecagon, which corresponds to Zeus, is the whole zodiac with its twelve signs. (Burkert 1972: 349)

Interesting stuff, though I have difficulties visualizing how this would work out (on the zodiac wheel?).

When we look beyond the facade of analysis and explication of the harmony of the spheres, what we find is neither empirical nor mathematical science, but eschatology. In the religion of Zarathustra, the paradise to which the soul ascends is called garo demana, "House of Psalms." It was related of Pythagoras that in his dying hour he aske that the monochord be played: "Souls cannot ascend without music." (Burkert 1972: 357)

I don't think I've met this tidbit before.

There remains to consider the acusma which asks the question, "What are the Isles of the Blest?" and answers "The sun and moon." This places the Beyond in the orderly cosmos; it represents the same desirfor stability that forges a theology of the soul out of myths about the soul and puts ritual taboos together into a "way of life" (βίος). (Burkert 1972: 363)

A pretty harsh estimation.

This separation of the Isles of the Blest from the realm of the dead is doubtless ancient, and presumably the earliest form of the school's doctrine, and therefore the total picture, along with the acusmata, makes consistent sense. Of course it is wholly pre-astronomical, as is shown even by the simple association of "sun and moon," as though they were islands in the same sea. There is n hint of the multi-storied universe which is standard in the later tradition. Hence, once mor, we find that the acusmata represent a strand of tradition independent of the later tradition, and also of Empedocles and Plato, and evince a Pythagoreanism still innocent of the scientific view of the world. (Burkert 1972: 364)

Pythagoreanism, once again, is exceedingly archaic and primitive.

The souls making their way along the heavenly path naturally find themselves in the company of the stars, and many have the impression that the stars enter and leave the sky by two doors. An indication of Pythagorean influence on Plato is the identification of the road "up" with that "to the right." (Burkert 1972: 365)

Indeed, on the various "The Broad and Narrow Way" paintings, the path to "Heaven" is on the right (cf. Matthew 7:13).

Then comes the Timaeus: the Demiurge creates as many souls as there are stars, for each star a soul, puts each into its star "as in a wagon," and thus shows it "the nature of the universe." Then it must leave its star, to be incarnated on earth; then after a period of trial on earth or another planet it may claim the promised return to its σύννομος ἀστήρ. (Burkert 1972: 366)

Timaeus is indeed wild. I wonder if this motif has anything to do with the expression "heaven gained a star/received another angel".

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5. Pythagorean Musical Theory

It is a striking paradox that music, which is the most spontaneous expression of physical activity, at the same time admits, or rather even challenges, the most rigorous mathematical analysis. (Burkert 1972: 369)

Very poetic.

In the one passage where he explicitly names οἱ Πυθαγόρειοι, Plato credits the Pythagoreans with a mathematical theory of music; this is one of the few fixed points in the reconstruction of Pythagoreanism before Plato. As early as Xenocrates, the crucial discovery was attributed to Pythagoas himself; and though this testimony is treated with great reserve, still it is generally regarded as established that the first natural law to be formulated mathematically - the relation between pitch and the length of a vibrating string - was a discovery of the Pythagorean school. (Burkert 1972: 371)

Valuable data-points.

What Plato desiderates is not an analysis of audible music but pure number theory, above and beyond experience. (Burkert 1972: 372)

Desires? Back-forming esiderata into a verb.

In the Timaeus, Plato carried out this program, at least by way of suggestion, using a series of numbers derived from the ultimate principles, which arrayed themselves in a scale without audible sound, the numerically harmonic structural pattern of the world, the "world soul." Succeeding ages [|] regarded the construction of the world soul in the Timaeus as one of the most illustrious examples of Plato's "Pythagorean" wisdom; but his own words, in the Republic, show that he went beyond the teachings of the Pythagoreans in an independent way. (Burkert 1972: 372-373)

Indeed, I've seen plenty of instances where the world-soul doctrine is attributed to Pythagoras (and the pythagoreans), whereas later commentators expressly say what Burkert is saying here - that it is a purely Platonic formulation.

But even in Archytas, in his view, considerations of number theory, rather [|] than empirically exact measurements, were most important; and the monochord was probably not invented until after Archytas' time. (Burkert 1972: 373-374)

Harsh.

Actually, the Aristotelian problema, in describing a similar experiment only speaks of ἠχώ, echo; and the resonance of hollow vessels was used in the Greek theater. (Burkert 1972: 378)

nxw

Lasus of Hermione, who became prominent in the time of the Peisisatidae (Htd. 7.6), was a close contemporary of Pythagoras. He is never called a Pythagorean, but was doubtless among the earliest Greek musicologists. What distinguishes the Pythagoreans was apparently not a special knowledge, inaccessile to others. Rather, something which may well have lost its interest for professional musicians came to be prized among them as a fundamental insight into the nature of reality. The wondrous potency of music, which moves the world and compels the spirit, captures in the net of number - this was a cardinal element of the secret of the universe revealed to the wise Pythagoras. (Burkert 1972: 378)

Makes sense. It's not that the pythagoreans had some "very secret" knowledge, but that they framed already available knowledge in very special light (or, more properly, sound - harmony).

The attraction and the significance of this theory lie not in the theory itself but in the orderly, rational pattern that it reveals. Order and pattern, however, which the human spirit craves, are to be found not only in the form of conceptual rigor and neatly logical structure, but, at an earlier level, in richness of mutual allusivenes and interconnection, where things fit together "symbolically." Thus the interrelation of number and music can be conceived, earlier than any mathematically oriented natural science and quite apart from it, as an aspect of the universal orderliness of the cosmos. (Burkert 1972: 399)

Phraseology for why I like triadism.

Van der Waerden draws attention to the tetractys, which has its roots ni the ancient stratum of the acusmata tradition. The "Fourness" which is the "harmony" which the Sirens sing, suggests the numbers 1, 2, 3, 4, which group themselves into the fundamental concords 2 : 1, 3 : 2, and 4 : 3, and thus comprehend the orderliness not only of music but of the universe; and the sum of these four numbers is 10, the "perfect" number. The tradition of the acusmta is independent of Philolaus, and leads back, past him, to the oldest stratum of Pythagoreanism; and the idea of the music of the cosmos is also of great antiquity. According to a report of Eudemus, the Pythagoreans emphasized that the fourth, the fifth, and the octave are comprised in the number 9, because 2 + 3 + 4 = 9; and here, too, it is clearly number as such, not proportion, that is the significant thing. Not only Hippasus but Archytas as well classified the intervals by use of individual numbers. The earliest Pythagorean musical theory is not founded on mathematics or on experimental physics, but on "reverence" for certain numbers in their roles in music and cosmology; and this situation is never completely abandoned. (Burkert 1972: 400)

Firstness, Secondness, Thirdness, and Fourness.

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6. Pythagorean Number Theory and Greek Mathematics

What is the origin of the firmly rooted conviction that Pythagoreanism was the source of Greek mathematics? This question is easy to answer: it came from the educational tradition. Everyone comes upon the name of Pythagoras for the first time in school mathematics; and this has been true from the earliest stages of the Western cultural tradition. None of the ancient textbooks which formed the basis of the medieval curriculum forgets Pythagoras. He is the companion of Arithmetica in Martianus Capella; and according to Isidore he was the first, among the Greeks, to sketch out the doctrine of number, which was then set forth in detail by Nicomachus. This takes us back to the origin of this tradition; Nicomachus, who is himself called a Pythagorean, begins his Arithmetic, which was much used as a schoolbook, with praise of the Master. Boethius' Arithmetic, drawn largely from Nicomachus, also names Pythagoras in its first line. (Burkert 1972: 406)

Didn't know that it was so from the very first mathematics textbooks.

The Ars geometriae bearing the name of Boethius, though obviously not composed before the High Middle Ages, even presents an early version of the Arabic numerals as an invention of the "Pythagorici," and describes the method of calculating with these apices on an abacus, called mensa Pythagorea - perhaps the most striking of the anarchronisms in which the Pythagorean tradition is so rich. (Burkert 1972: 406)

Pythagoras invented numbers, don't yer know?

In Iamblichus we read, ἐκαλεῖτο δἐ ἡ γεωμετρία πρὸς Πυθαγόρου ἱστορία, and Tannery translated this, "geometry was called 'the tradition according to Pythagoras.'" In consideration of the context in Iamblichus, he interpreted this to mean that before Hippocrates of Chios there was publishd a treatise on geometry with the title The Tradition according to Pythagoras. This Pythagorean textbook was the cornerstone of Tannery's reconstruction, and it has continued to play a part right down to the present day. It owes its existence, however, to an obvious mistake in translation; it is impossible to take πρὸς Πυθαγόρου ἱστορία together, and the meaning must be "geometry was called ἱστορία by Pythagoras." (Burkert 1972: 408)

An entry for the list of non-existent books.

The attempt at purely logical argumentation, a systematic progression from one thought to another, and the advancement of proofs and conclusions in conscious contradiction to the evidence of the senses make their first appearance in Parmenides. Kurt von Fritz has shown how νοεῖν, which previously meant an intuitive comprehension, first becam logical "thinking" in Parmenides. (Burkert 1972: 424)

Curious. Mõistmine → mõtlemine.

On the other hand, there is one remarkable type of arithmetic that appears exclusively in the Pythagorean tradition, in which numbers are represented by figures made with counters or pebbles, ψῆφοι. Aristotle knows of "triangular numbers"; and the "perfect" number 10, in its deployment inthe form of the "tetractys," was certainly presented as a triangular number long before Aristotle. And what at first seems merely a game does lead to arithmetical combinations that are by no means trivial. For example, if the odd numbers, when added successively in a pebble figure, make a square each time, this means discovery of the rule for the series of square numbers; and, if in the ἑτερομήκεις ἀριθμοι, constructed in similar fashion from the even numbers, one recognizes the triangular numbers, doubled, then he has the formula for the sum of triangular numbers, a special case in the arithmetic series. (Burkert 1972: 427)

Hence, perhaps, why the odd number is considered better than the even.

The suspicion remains that the theorem had more than a mathematical significance in Pythagoras' school, and that the numbers involved seemed in a cryptic way meaningful. The formula of "Pythagoras" points in this direction, as it belongs to the context of the pebble figures, like the form of the tradtion that only mentions the triangle with the sides 3, 4, and 5. In fact, this fits especially well with the kind of number speculation we learn of from Aristotle, where 3 is male, 4 is female, and 5, which mysteriously unites them in the Pythagorean triangle, is "marriage." Plato's "nuptial number" obviously presupposes this interpretation. (Burkert 1972: 429)

So that's what "marriage" and/or "health" means!

The graphic procedure with ψῆφοι makes it possible to formulate impressively generalizations about numbers; but it also "reveals" each fact without deducing one from the other in an bstract chain of reasoninrg. It is the element of the unforeseeable which gives number games their appearance of something profound and secret. The "occult" charm of mathematics comes from the fact that the human mind forgets its own way of proceeding and loses sight of its own preconceptions; for alert mathematical analysis, that which fills the naive mind with amazement is seen as tautologous and therefore self-evident. What we find among the Pythagoreans is amazement and "reverence" for certain numbers and their properties and interrelations. "Even" and "odd" are united in "marriage"; and to them this means that cosmic forces are at work. A scheme of proof could hardly be anything but annoying because it would show the result as the logical consequence of the preconceptions, and reduce it to banality. (Burkert 1972: 433)

Or, in other words, the mathematical secrets of numbers are transcendental.

By analysis of Euclid, Oskar Becker has reconstructed a set of theorems which has been widely heralded as proof of the existence of a deductive Pythagorean arithmetic, namely the "doctrine of odd and even," as developed in Euclid 9.21-34. These propositions stand isolated in Euclid, a trivial appendage to the sophisticated number theory of books 7-9, which culminates in the proof tat there are infinitely many prime numbers (9.20). The propositions about the even and the odd are only once applied by Euclid, in the proposition about perfect numbers (9.36), and in a proof of irrationality, given as an appendix, which Aristotle already knew. (Burkert 1972: 434)

Prop. 9.21: Sum of Even Numbers is Even

Of course the Pythagoreans knew that odd plus odd makes even, and that odd plus even gives odd - they demonstrated this with their pebble diagrams - but they did not deduce one proposition from the other. They saw, directly, that the "male" odd number showed itself dominant in association with the "female" even number. (Burkert 1972: 435)

Huh. Even numbers can only make more even numbers (when added) but even numbers can make both odd and even numbers.

Becker imagines that they represented even numbers by equal numbers of white and black pebbles; but an arrangement in two rows is even more striking (cf. Pl. Euthyphro 12d, where even number is defined as ἰσοσκελής and odd as σκαληνός): [...] (But see also above, ch. I 2, n. 31. The "power" of theodd number depends on the "one" that it has in its "middle.") (Burkert 1972: 435, fn 49)

Indeed, quite striking. I've made the "one"/"middle" pebble red for emphasis.

There remains the division of numbers into odd and even, generally, in which one might, in spite of all, see the beginning of number theory; it is confidently attributed to Pythagoras himself. But this very point can be refuted by hilological means. In all Pythagorean speculation the odd number is more highly valued; it is what "sets the limit," is the male element, and stands in the "column of the good." In all this is preserved, as shown both by anthropological parallels from folklore and linguistic observation, an ancient and widespread piece of number lore. (Burkert 1972: 437)

Now I'm imagining them setting pebbles into these double rows, and whichever column (the one on the right?) receives the last, odd pebble becomes the good/male column that literally "sets the limit" - it goes one further than the last even pair in both columns.

The term λόγος, in its mathematical sense of "relation, ratio, proportion," has been attributed by von Fritz to the Pythagoreans, and, conjecturalyl, to Pythagoras himself. [.|.] The answer is simple: they wee terms used in the calculation of interest. Whoever lends money expects to get his principal back and a specified fraction of it in addition. This could be ἐπίτριτον (4/3, or 33⅓%), ἐπίπεμπτον (6/5, or 20%), more usually ἔφεκτον (7/6, or 16⅔%), sometimes ἐπόγδοον (9/8, or 12½%), at the lowest ἐπιδέκατον τόκον (interest of 1/10; this was what the gods received). It is certain that the practice of loaning money at interest went back before the time of Solon; and, though there is no direct evidence, it can hardly be doubted that the expressions mentioned were in use that early - long before the day of Pythagoras. [...] But the calculation of interest is in fact called λογίζεσθαι: λογίσωμαι [|] τοὺς τόκους, says Strepsiades. The officials who calculate the interest on temple loans are called λογισταί, and the calculation itself is λόγος. (Burkert 1972: 438-440)

It sounds awfully lot like the word "logos" springs from the tithes given to temples.

The connection of proportion and music, resulting in the equation of διάστημα and λόγος, remains to the credit of the Pythagoreans; and in one aspect of the theory of proportion, the doctrine of "means," Pythagorean influence is a possibility. The three μεσότητες, the arithmetic, geometric, and harmonic means, are generally regarded in the tradition as a discovery of Pythagoras. The fact that all three means have a role in the Timaeus could rouse suspicion about the tradition involving Pythagoras himself. But Theatetus already knew the system of the three means, and used it, in a rather forced mann, as the point of departure for his classification of irrational lines; thus the [|] means are presumably older, and they are closely related to Pythagorean music theory. The name of the "harmonic mean" is to be explained directly from the latter; the Mese is the harmonic mean of the octave Nete-Hyptae. (Burkert 1972: 440)

A common theme in recent chapters: something may be definitely known to the pythagoreans, but not originate from them.

Above all, Archytas presupposes a whole series of arithmetical propositions and expressly cites an auxiliary theorem. Tannery, who called attention to the proof of Archytas transmitted by Boethius, concluded from this that Archytas must have had a kind of Elements of Arithmetic, and van der Waerden undertook to reconstruct, systematically, this number theory presupposed by Archytas, coming to the conclusion that in all essentials the material of Euclid's seventh and eighth books must already have been in existence; book 7, he thought, "existed in written form before 400 B.C." and "had been taken over by Euclid without significant alteration." Book 8 was the work of Archytas himself. If this were correct, we should have an imposing edifice of Pythagorean arithmetic of an entirely different kind from what Speusippus, Aristotle, Theo, and Nicomachus lead us to expect. (Burkert 1972: 443)

"Book 7 deals with elementary number theory: divisibility, prime numbers and their relation to composite numbers, Euclid's algorithm for finding the greatest common divisor, finding the least common multiple. Book 8 deals with the construction and existence of geometric sequences of integers." (Wiki)

In any case, athetesis is no suitable way to get rid of the problem. (Burkert 1972: 446)

define:athetesis - The act or fact of setting aside as spurious; rejection as invalid. // How have I not met this word before?

The name "number theory," ἀριθμῶν θεωρία, appears to come from Xenocrates, following Plato in his demand for a "pure," logical and deductive treatment of numbers, above and beyond the realm of sense impression - θέα τῆς τῶν ἀριθμῶν φύσεως. Thus the number theory stemming from Archytas - like, [|] in a sense, Pythagorean philosophy in general - achieved its final form in Platonism. Before Archytas there were number games accompanied by the "interpretation" of, and "reverence" for, number. (Burkert 1972: 446-447)

Harsh.

In fact, the discovery of the irrational is also ascribed to Pythagoreans, or even to Pythagoras himself; but the actual situation is extremely hard to grasp because of the profusion of ancient legend and allegory, and the modern conjectures they have inspired. The ancients speak of this situation in terms of "secrecy" and "treason," the moderns of the "Grundlagenkrisis der griechischen Mathematik." (Burkert 1972: 454)

Typical.

An important prop for this theory of a Grundlagenkrisis, and also a useful chronological point of refernece, was the interpretation of the polemics of Zeno and Elea as relevant to the history of mathematics. This was inaugurated by Helmut Hasse and Heinrich Scholz, who argued that his critique was directed specifically against some "unclear" (unsauber) mathematics of infintesimals, by means of which the Pythagoreans supposedly attempted to escape the consequences of irrationality. If this is correct, the discovery of the irrational must have taken place before 460 B.C., which would fit in well with the conjectural dating of Hippasus. (Burkert 1972: 456)

I.e. "the Eleatic system seems to presuppose Pythagoreanism" (Zeller 1881: 512).

Plutarch, who is our oldest witness for this, speaks of the secrecy, and the prohibition of putting doctrines down in writing, in the Pythagorean group: "And when their treatment of the abstruse and mysterious processes of geometry had been divulged to a certain unworthy person, they said the gods threatened to punish such lawlessness and impiety with some signal and widespread calamity." We cannot equate this episode with that of Hippasus; the latter was a Pythagorean, so that his initiation into the "difficult and secret procedures" was therefore not any kind of "divulgement," and his death was not a κοινὸν κακόν. (Burkert 1972: 457)

Plutarch is damnably late (1st century AD).

The scholia to Euclid have preserved an abbreviated version of the Greek text, and Iamblichus, too, knows the tradition Pappus is following. But he also has another, according to which the traitor was only symbolically killed - a tomb was erected with his name. Iamblicus sets the three versions side by side - the symbolic "death" of the betrayer of irrationality, the drowning of the man who revealed the dodecahedron, and the drowning of the one who divulged the fact of irrationality. In addition, Iamblichus has a different story about how Pythagorean geometry became known, without any quarrel or catastrophe: permission wsa given an impoverhed Pythagorean to earn a living by giving lessons in geometry. It is conjectured that this version was originally related to Hippocrates of Chios. (Burkert 1972: 458)

Context for the claim that the Pythagorean community practised this "symbolic death" for every member who was not accepted during the initial years of silent study. This may have been invented wholesale from this passage from Iamblichus mentioned here.

Numerous dodecahedra made of bronze have been found in Gaul and thereabouts; and one made of stone has been found in northern Italy, dating back to prehistoric times. Their significance and use is unclear; the best conjecture seems to be that they were a kind of dice, used for oracular or mantic purposes. In Plato's Timaeus the dodecahedron appears unexpectedly as the image of the whole (55c); it is widely supposed that the Pythagorean tradition was in his mind, the one that presupposed the Hippasus story and was not without relation to the Italo-Gallic region. The dodecahedron may well have been important as a σύμβολον in the Pythagorean school, like the pentagram; Hippasus' offense was in analyzing the sacred object, publicly, by mathematical means. (Burkert 1972: 460)

This is referenced in the Futurama movie Bender's Game, i.e. the Anti-Backwards Crystal, with it's own quasi-Pythagorean symbolism (3 - Growth; 7 - Banish Foes; 12 - Mirror Mania).

One is νοῦς and οὐσία; two is δόξα; three is the number of the whole - beginning, middle, and end; four is justice - equal times equal - but it is also, in the form of the tetractys, the "whole nature of numbers"; five is marriage, as the first combination of odd and even, male and female; seven is opportunity (καιρός) and also Athena, as the "virginal" prime number; ten is the perfect number, which [|] comprehends the whole nature of number and determines the structure of the cosmos, and with it ends the symbolic interpretation of numbers. (Burkert 1972: 467-468)

Eight and nine once again missing.

Greece, too, has its primeval, ritually significant symbolic numbers. Even in the Homeric epics one notices the preference for certain numbers, especially 3 and 9, and also 5 and 7 - the odd numbers. The number 8 hardly occurs at all, and also apparently played no part in the ancient Pythagorean tradition. Certain numbers belong to certain gods; the cult of Aollo and that of Dionysus were dominated by the numbers 7 and 9. (Burkert 1972: 474)

The universally neglected 8.

Mathematical astronomy, after some stirring in the fifth century, was only brought to full development by Eudoxus; music theory was at first more a number game than a science; and the "philosophiae naturalis principia mathematica," in the sense of Newton, were never attainable to the Greeks, even though in the Timaeus it seems that Plato dreamed of something of this kind. Modern perspectives distort our view of the ancient "wisdom" of Pythagoras, as in fact it had soon become distorted in antiquity. (Burkert 1972: 479)

Another general evaluation.

The tradition of Pythagoras as a philosopher and scientist is, from the historical point of view, a mistake. But the fascination that surrounded, and still surrounds, the name of Pythagoras does not come, basically, from specific scientific connotations, or from the rational method of mathematics, and certainly not from the success of mathematical physics. More important is the feeling that there is a kind of knowing which penetrates to the very ore of the universe, which offers truth as something at once beatific and comforting, and presents the human being as cradled in a universal harmony. (Burkert 1972: 482)

Hence why Pythagoras is still embraced by all sorts of mysticism, conspiracy theories, etc. That is, it is imagined that he took part in some "universal truths" (perhaps bestowed by the gods), which, as Burkert effectively points out, are very primitive and psychologically appealing.