History Of Science

"While most s emphasized the reality of change — in particular, the Atomists, followers of and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the '). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical (2b = a + c), geometrical (b^2 = ac), and a harmonical (\frac{2}{b} = \frac{1}{a} + \frac{1}{c}) proportion that they interpreted philosophically and socially."

"Hippasos made Fire the first principle, and... from the Timaeus... it would be possible to represent air and water as forms of fire. The other element is... earth, not air, as... it was in early Pythagoreanism. That would be a... result of the discovery of atmospheric air by Empedokles and of his general theory of the elements. It would... explain the... fact... that Aristotle identifies the two "forms" spoken of by Parmenides with Fire and Earth."

"[T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics. The most grandiose ambition they conceived was to explain all the properties of Nature in arithmetical terms alone. This was the aim of the Pythagoreans... [T]hey... knew that the phenomena of the Heavens recurred in a cyclical manner; and... discovered ...that the sound of a vibrating string ...is simply related to the length ...and its 'harmonics' always go with simple fractional lengths. ...[S]ince the Pythagoreans were a religious brotherhood... they thought that this search would lead to more than explanations alone. If one discovered the mathematical harmonies in things, one should... discover how to put oneself in harmony with Nature. ...[T]hey had ...positive grounds for thinking that both astronomy and acoustics were at the bottom arithmetical; and the study of simple fractions was called 'music' right down until the late Middle Ages."

"It has been no easy task to revise this volume in such a way as to make it more worthy of the favour with which it has been received. Most of it has had to be rewritten in the light of certain discoveries made since the publication of the first edition, above all, that of the extracts from Menon’s Iατρικά, which have furnished, as I believe, a clue to the history of Pythagoreanism."

"It is usually maintained that the Platonic or Socratic philosophy, like the rest of Greek speculation, was original, indigenous, owing very little to any outside influence. But the quest and life and faith of Socrates were as un-Greek as anything could possibly be: that was one of the reasons why the Greeks killed him: the essence of his life belonged to a world unknown to them, and therefore dangerous in their eyes […] There is only one “philosopher” whose doctrines, both practical and theoretical, appear to have resembled Plato’s in spirit and aim as well as in substance; and that one is Pythagoras. It is noteworthy that Pythagoras is the only great thinker of Greece whom Plato never criticises, but of whom he speaks with the greatest deference and respect […] instancing him as the great example of a teacher whose teaching had in it living truth enough to inspire a band of devoted disciples, and to transform their lives as well as their beliefs. And every one of those doctrines, which we know formed the “gospel” of Pythagoras and of the Pythagorean brotherhood at Crotona, was an almost exact reproduction of the cardinal doctrines of the Indian Vidya and the Indian Yoga—so much so that Indian Vedantists today do not hesitate to claim Pythagoras as one of themselves, one of their great expounders, whose very name was only the Greek form of the Indian title Pitta Guru, or Father-Teacher.’"

"The tradition confirms... the importance of the in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing its construction and claiming the discovery as his οwn."

"Aristotle is... decided as to Pythagorean points having magnitude. "They construct the whole world out of numbers... but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.""

"Aristotle is... decided in his opinion that Pythagoreanism was intended to be a cosmological system like the others. "Though the Pythagoreans... made use of less obvious s and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does." They apply their first principles entirely to these things, "agreeing... with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens," though "the first principles and causes of which they made use were... adequate to explain realities of a higher order than the sensible.""

"Simplicius, with the poem of Parmenides before him, corrects Aristotle by substituting Light and Darkness for Fire and Earth... Parmenides... calls one "form" Light, Flame, and Fire, and the other Night, and we... consider whether these can be identified with the Pythagorean Limit and Unlimited. We have... reason to believe that... the world breathing belonged to the earliest form of Pythagoreanism, and... identifying this "boundless breath" with Darkness, which stands... for the Unlimited. "Air" or mist was always regarded as the dark element. And that which gives definiteness to the vague darkness is... light or fire, and this may account for the prominence given to that element by Hippasos. We may probably conclude... that the Pythagorean distinction between the Limit and the Unlimited... made its first appearance in this crude form. If... we identify darkness with the Limit, and light with the Unlimited, as most critics do, we get into insuperable difficulties."

"The Pythagorean Order was simply, in its origin, a religious fraternity... and not, as has sometimes been maintained, a political league. Nor had it anything to do with the "Dorian aristocratic ideal." Pythagoras was an Ionian, and the Order was originally confined to Achaian states. Nor is there the slightest evidence that the Pythagoreans favoured the aristocratic rather than the democratic party. The main purpose... was to secure for... members a more adequate satisfaction of the religious instinct than... the State religion. It was... an institution for the cultivation of holiness. ...[I]t resembled an Orphic society, though it seems that Apollo, rather than Dionysos, was the chief Pythagorean god. That is doubtless why the Krotoniates identified Pythagoras with Apollo Hyperboreios. ...[H]owever, an independent society within a Greek state was apt to be brought into conflict with the larger body. The only way in which it could then assert its right to exist was... by securing the control of the sovereign power. The history of the Pythagorean Order... is, accordingly, the history of an attempt to supersede the State..."

"The Phaedo is dedicated... to Echekrates and the Pythagorean society at Phleious, and it is evident that Plato in his youth was impressed by the religious side of Pythagoreanism, though the influence of Pythagorean science is not clearly marked till a later period."

"There is no reason to believe that the detailed statements which have been handed down with regard to the organisation of the Pythagorean Order rest upon any historical basis... The distinction of grades within the Order, variously called Mathematicians and Akousmatics, Esoterics and Exoterics, Pythagoreans and Pythagorists, is an invention designed to explain how there came to be two widely different sets of people, each calling themselves disciples of Pythagoras, in the fourth century B.C. So, too, the statement that the Pythagoreans were bound to inviolable secrecy, which goes back to Aristoxenos, is intended to explain why there is no trace of the Pythagorean philosophy proper before Philolaos."

"[A] good many fragments of... Aristoxenos and Dikaiarchos are embedded in the mass. These writers were both disciples of Aristotle; they were natives of Southern Italy, and contemporary with the last generation of the Pythagorean school. Both wrote accounts of Pythagoras; and Aristoxenos, who was personally intimate with the last representatives of scientific Pythagoreanism, also made a collection of the sayings of his friends."

"[I]t is... incredible that the heaven of the fixed stars should have been regarded as stationary. That would have been the most startling paradox that any scientific man had yet propounded, and we should have expected the comic poets and popular literature generally to raise the cry of atheism... [W]e should have expected Aristotle to say something... He made the circular motion of the heavens the... keystone of his system, and would have regarded... a stationary heaven as blasphemous. ...[H]e argues against those who, like the Pythagoreans and Plato, regarded the earth as in motion; but he does not attribute the view that the heavens are stationary to any one. There is no necessary connexion between the two ideas. All the heavenly bodies may be moving as rapidly as we please, provided that their relative motions are such as to account for the phenomena."

"Platonic elements which have crept into later accounts... are of two kinds. First... genuine Academic formulae... as... identification of the Limit and the Unlimited with the One and the Indeterminate Dyad; ...secondly ...the Neoplatonic doctrine which represents it as an opposition between God and Matter. ...[N]o one will any longer attribute these doctrines to the Pythagoreans of the fifth century."

"Aristotle shows... how the theory originated... that some thought there might be a considerable number of bodies revolving round the centre, though invisible because of the intervention of the earth, and... they accounted... for there being more eclipses of the moon than of the sun. ...Aristotle regarded the two hypotheses as of the same nature."

"When discussing the Pythagorean system, Aristotle always refers it to "the Pythagoreans," not to Pythagoras himself. ...[T]his was intentional ...Pythagoras himself is only thrice mentioned in the whole Aristotelian corpus, and in only one... is any philosophical doctrine ascribed to him. ...Aristotle ...is quite clear that what he knew as the Pythagorean system belonged in the main to the days of Empedokles, Anaxagoras, and Leukippos; for ...he goes on to describe the Pythagoreans as "contemporary with and earlier than them.""

"The Pythagoreans held, [Aristotle] tells us that there was "boundless breath" outside the heavens, and that it was inhaled by the world. In substance, this is the doctrine of Anaximenes, and... it was that of Pythagoras... Xenophanes denied it. ...[F]urther development of the idea is ...due to Pythagoras ...We are told that, after the first unit had been formed ...the nearest part of the Boundless was first drawn in and limited; and... the Boundless thus inhaled... keeps the units separate from each other. It represents the interval between them. This is a... primitive way of describing... discrete quantity."

"In... Aristotle... the Boundless is also... the void or empty. This identification of air and the void is a confusion... in Anaximenes... too. We find also... the other confusion... air and vapour. ...Pythagoras identified the Limit with fire, and the Boundless with darkness. We are told by Aristotle that Hippasos made Fire the first principle... Parmenides... attributes... two primary "forms," Fire and Night. ...Light and Darkness appear in the Pythagorean table of opposites under the heads of the Limit and the Unlimited respectively."

"[T]hink, then, of a "field" of darkness or breath marked out by luminous units ...which the starry heavens would naturally suggest."

"The identification of breath with darkness ...is a strong proof of the primitive character of the doctrine; for in the sixth century darkness was supposed ...a sort of vapour, while in the fifth, its true nature was ...known. Plato... makes the Pythagorean Timaios describe mist and darkness as condensed air."

"In the Timaeus... the slowest of the heavenly bodies appear the fastest and vice versa; and, as this... is... a Pythagorean [speaking], we might suppose the theory of a composite movement to have been anticipated by some... [in] that school."

"[T]he problem... is still extremely difficult."

"According to Aristotle, the Pythagoreans said Things are numbers, though that does not appear to be the doctrine of the fragments of "Philolaos." According to them, things have number, which make them knowable, while their real essence is... unknowable. ...[B]ut ...things are numbers seems meaningless. We have seen reason for believing that it is due to Pythagoras..., though we did not feel able to say... clearly what he meant..."

"[W]e have to... interpret what Aristotle tells us in the spirit of Plato, and... consider how the doctrine... is related to the systems which had preceded it. ...[This] delicate operation... has been made... safer by recent discoveries in the early history of mathematics and medicine."

"We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself."

"At the same time, Pythagoreanism had taken root in Hellas. Lysis... remained at Thebes, where Simmias and Kebes had heard Philolaos, and there was an important community of Pythagoreans at Phleious. Aristoxenos was personally acquainted with the last generation of the school, and mentioned by name Xenophilos the Chalkidian from , with Phanton, Echekrates, Diokles, and Polymnestos of Phleious. They were all, he said, disciples of Philolaos and Eurytos. Plato was on friendly terms with these men, and dedicated the Phaedo to them. Xenophilos was the teacher of Aristoxenos..."

"We know... Philolaos wrote on "numbers"; for Speusippos followed him in the account he gave of the Pythagorean theories on that subject. It is probable... he busied himself... with arithmetic, and... his geometry was... primitive... Eurytos was his disciple, and... his views were... crude."

"It seems natural to suppose... the Pythagorean elements of Plato’s Phaedo and Gorgias come mainly from Philolaos. Plato makes Sokrates express surprise that Simmias and Kebes had not learnt from him why it is unlawful for a man to take his [own] life, and it seems to be implied that the Pythagoreans at Thebes used the word "philosopher" in the... sense of... seeking to find... release from the burden of this life? It is... probable that Philolaos spoke of the body... as the tomb... of the soul. ...[H]e taught the old Pythagorean religious doctrine in some form, and... likely... laid stress upon knowledge as a means of release. ...Plato ...is by far the best authority ...on the subject."

"Sokrates does not introduce the theory as a novelty. The reality of the "ideas" is the... reality "we are always talking about," and they are explained in a peculiar vocabulary... of a school."

"Ionian philosophers... had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire as the basic element. The Pythagoreans had taken a more abstract direction, postulating that number... was the basic stuff behind phenomena; this numerical atomism... had come under attack by the followers of Parmenides of Elea... The fundamental tenet of the was the unity and permanence of being... contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic... who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility."

"Plato was intimate with these men and was deeply impressed by their religious teaching, though... he did not adopt it... He was still more attracted by the scientific side of Pythagoreanism, and... this exercised a great influence on him. His own system in its final form had many points of contact with it, as he is careful to mark in the ' But... he is apt to develop Pythagoreanism on lines of his own, which may or may not have commended themselves to Archytas, but are no guide to the views of Philolaos and Eurytos. He is not careful... to claim the authorship of his own improvements in the system. He did not believe that cosmology could be an exact science, and he... therefore... credit[s] Timaios the Lokrian, or "ancient sages"... with theories which... had their birth in the Academy."

"We... exceeded the limits... by tracing the history of Pythagoreanism... to... where it becomes practically indistinguishable from the earliest form of ; but it was necessary... to put the statements of our authorities in their true light."

"Aristoxenos is not likely... mistaken with regard to the opinions of the men he had known personally, and Aristotle’s statements must have had some foundation."

"Philolaos... is a sufficiently remarkable figure... and has... been spoken of as a "precursor of Copernicus.""

"The School of Pythagoras, in our opinion, represents the main current of that mystical tradition which we have set in contrast with the scientific tendency. The terms 'mystical' and 'scientific,' ...are ...not to be understood as if ...all the philosophers we class as mystic were unscientific. The fact that we regard Parmenides, the discoverer of Logic, as an offshoot of Pythagoreanism, and Plato... as finding in the Italian philosophy the chief source of his inspiration, will be enough to refute such a misunderstanding. Moreover, the Pythagorean School... developed a scientific doctrine closely resembling the Milesian Atomism; and Empedocles, again, attempted to combine the two types of philosophy."

"[T]hroughout the mystical systems inspired by Orphism, we... find the fundamental contrast between... principles of Light and Darkness, identified with Good and Evil. This cosmic dualism is the counterpart of the dualism in the... soul; for... physis and soul... are... identical in substance. The soul in its pure state consists of fire, like the divine stars from which it falls; in its impure state, throughout... reincarnation, it... is infected with the baser elements, and weighed down... In the cosmologies... the manifold world of sense will be viewed as a degradation from the purity of real being. Such systems will tend to be other-worldly, putting all value in the unseen unity of God, and condemning the visible world as false and illusive, a turbid medium... obscured in mist and darkness. These characteristics are common to all the systems which came out of the Pythagorean movement—Pythagoreanism proper, and the philosophies of Parmenides, Empedocles, and Plato."

"The doctrines of mysticism are secret, because they are not cold, abstract beliefs, or articles in a creed, which can be taught and explained by intellectual processes... The 'truth' which mysticism guards is... only... learnt by being experienced (παθεῖν μαθεῖν); it is... not an intellectual, but an emotional experience—that invasive, flooding sense of oneness, of reunion and communion with... the life of the world... Being an emotional, non-rational state, it is indescribable, and incommunicable save by suggestion. To induce that state, by the stimulus of collective excitement and all the pageantry of dramatic ceremonial, is the aim of mystic ritual. The 'truth' can only come to those who submit themselves to these... because it is... to be immediately felt, not conveyed by dogmatic instruction. For that reason only... 'mysteries' are reserved to the initiate, who have undergone 'purification,' ...a state of mind which fits them for the consummate experience. Pythagoreanism presents... an attempt to intellectualise... Orphism, while preserving its social form, and... spirit... Orphism ceases to be a cult, and becomes a Way of life. As a revival, Pythagoreanism means a return to an earlier simplicity... simple enough to adapt itself to a new movement of the spirit. Pythagoreanism is... a complex phenomenon, containing the germs of several tendencies... philosophies that emerged from the school... separating towards divergent issues, or intertwined in ingenious reconciliations. Our analysis must take account of three strata, superimposed... Dionysus, Orpheus, Pythagoras. From Dionysus come the unity of all life, in the cycle of death and rebirth, and the conception of the or collective soul, immanent in the group as a whole, and yet something more than any or all... To Orpheus is due the shift of focus from earth to heaven, the substitution for the vivid, emotional experience of the renewal of life in nature, of the worship of a distant and passionless perfection in the region of light, from which the soul, now immortal, is fallen into the body of this death, and which it aspires to regain by the formal observances of asceticism. But the Orphic still clung to the emotional... reunion and... ritual that induced it, and... to the passionate spectacle (theoria) of the suffering God. Pythagoras gave a new meaning to theoria... as the passionless contemplation of rational, unchanging truth... a 'pursuit of wisdom' (philosophia). The way of life is still also a way of death; but now... death to the emotions and lusts... and a release of the intellect to soar into the untroubled of theory... by which the soul can 'follow God' (ἕπεσθαι θεῷ)... beyond the stars. Orgiastic ritual... drives a... nail into the coffin of the soul, and binds it... to its earthly prison-house. ...[O]only certain ascetic prescriptions of the Orphic askesis are retained, to symbolise a turning away from lower desires, that might enthral... reason."

"It is certain that the Theory of Numbers originated in the school of Pythagoras."

"The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason."

"Those who dwelt in the common auditorium adopted this oath: "I swear by the discoverer of the Tetraktys, which is the spring of all our wisdom; The perennial fount and root of Nature.""

"There is no such doubt... [in] his school. Aristotle says they used the formula in a cosmological sense. The world... was made of numbers in the same sense as others had said it was made of "four roots" or "innumerable seeds." It will not do to dismiss this as mysticism."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"They thought they found in numbers, more than in fire, earth, or water, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number."

"None of Pythagoras' own work has survived, but the ideas fathered on him by his followers would be the most potent in modern history. Pure knowledge, the Pythagoreans argued, was the purification (catharsis) of the soul... rising above the data of the human senses. The pure essential reality... was found only in the realm of numbers. The simple, wonderful proportion if numbers would explain the harmonies of music... [T]hey introduced the musical terminology of the octave, the fifth, the fourth, expressed as 2:1, 3:1, and 4:3. ..."

"They [the Pythagoreans] say the things themselves are Numbers and do not place the objects of mathematics between forms and sensible things. ...Since again, they saw that the modifications and the ratios of the musical scales were expressible in numbers—since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number... and the whole arrangement of the heavens they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent."

"These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states."

"In Copernicus' time Pythagoreans still believed that the only way to truth was by mathematics."

"The Neo-Pythagoreans treated all the divisions of philosophy. In Metaphysics they held that the Unit and the (indeterminate) Two are the basis of all things. the Unit being the form, and the Two the matter. ...The Unit being the prior principle may be identified with Deity, and, as such, was thought of either as the former [creator] of indefinite matter into individual things, or, as in Neo-Platonism, as the transcendent origin of the derivative Unit and Two. Another mode of conception was to identify the numbers with the Platonic Ideas and then to think of the Unit as comprehending them in the same manner as the mind comprehends its thoughts and gives them form. In Logic the Neo-Pythagoreans were for the most part imitators of Aristotle. Their Physics was Aristotelian and Stoic. Their Anthropology was Platonic. In Ethics and Politics they merely reechoed the Academy and the Lyceum with Stoic additions. In all this Neo-Pythagoreanism has little originality."

"Whatever we may think of Pythagoras, the Pythagoreans of the fifth century were scientific men, and they must have meant something... definite. ...[T]hey used the words Things are numbers in a ...non-natural sense, but there is no difficulty in such a supposition."