Religious Philosophy

"The science of geometry had... made considerable advances, but the old view of quantity as a sum of units had not been revised... so... [ such a] doctrine... was inevitable."

"Philolaos and his followers cannot have regarded the Unlimited in the old Pythagorean way as Air; for... they adopted the theory of Empedokles as to that "element," and accounted for it otherwise. ...[T]hey can hardly have regarded it as an absolute void; for that conception was introduced by the Atomists. ...[T]hey meant by the Unlimited the ', without analysing that... further."

"As the Unlimited is spatial, the Limit must be spatial too, and we should... expect... the point... line, and... surface were regarded as... forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is... that the point was not... a limit, but... the first product of the Limit and the Unlimited, and was identified with the arithmetical unit. According[ly]... the point has one dimension, the line two, the surface three, and the solid four... [i.e.,] Pythagorean points have magnitude... lines breadth, and... surfaces thickness. The whole theory... turns on the definition of the point as a unit “having position." ...[O]ut of such elements ...it seemed possible to construct a world."

"[T]his way of regarding the point... line, and... surface is closely bound... with... representing numbers by dots... in symmetrical patterns... attribut[ed]... to the Pythagoreans."

"When the Pythagoreans returned to Southern Italy, they must have found views... there which... demanded a partial reconstruction of their own system. ...Empedokles founded a philosophical society, but ...influence[d] ...the medical school of these regions; and ...Philolaos played a part in the history of medicine."

"Simplicius... preserved an explanation, in all probability Alexander’s... that they called the even number unlimited "because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on '. But, when the odd is added, it limits it; for it prevents its division into equal parts.""

"[W]e must not impute to the Pythagoreans... that even numbers can be halved indefinitely. They had... studied the properties of the decad, and... must have known that... 6 and 10 do not admit of this."

"In this way, then, the Odd and the Even were identified with the Limit and the Unlimited, and it is possible... Pythagoras... had taken this step... by... Unlimited he meant something spatially extended, and... identified... with air, night, or the void, so we are prepared to find... his followers also thought of the Unlimited as extended."

"Aristotle speaks of certain "elements"... of numbers, which were also the elements of things. ...Primarily, the "elements of number" are the Odd and the Even... identified in a somewhat violent way with the Limit and the Unlimited... the original principles of the Pythagorean cosmology. Aristotle tells us... the Even... gives things their unlimited character when... contained in them and limited by the Odd... [C]ommentators... understand... this to mean... the Even is... the cause of infinite divisibility. They get into great difficulties, however..."

"Aristotle... argues... if the Unlimited is... a reality, and not merely the predicate of some other reality, then every part of it must be unlimited... just as every part of air is air. The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens. Further than this, it is hardly safe to go."

"The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory... in the more developed form... attained in Plato’s Timaeus. If they were to retain their position as... leaders of medical study... they were bound to account for the elements."

"Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things."

"[T]he Pythagorean One, or Monad, splits into two principles, male and female, the Even and the Odd, which are the elements of all numbers and so of the universe. ...One is not simply a numerical unit, which gives rise to other numbers by ...addition. That conception belongs to the later atomistic number-doctrine ...In the earlier Pythagoreanism, we must think of the One (which is not itself a number at all) as analogous to Anaximander’s ἄπειρον. It is the primary, undifferentiated group-soul, or physis, of the universe, and numbers must arise from it by a process of differentiation or 'separating out' (ἀπόκρισις). Similarly, each of these numbers is not a collection of units, built up by addition, but itself a sort of minor group-soul—a distinct 'nature,' with various mystical properties. In the same way, it is by dividing up the whole interval of the octave that the harmonic proportions are determined."

"[T]he Pythagorean construction of the elements was... that... in Plato’s Timaeus. ...[T]here is good reason for believing they only knew three of the regular solids, the , the pyramid (), and the . Plato starts from fire and earth, and... the construction οf the elements proceeds... such... that the and the can easily be transformed into pyramids, while the cube and the dodecahedron cannot. ...[I]t follows that, while air and water pass readily into fire, earth cannot... and the dodecaedron is reserved for another purpose... This would... suit the Pythagorean system; for it would leave room for a dualism... outlined in the Second Part of the poem of Parmenides."

"Not one of the philosophical ideas in Part I of the commentary is peculiarly Neoplatonic. The doctrine of the Threeness of things... is found in Aristotle and goes back to the early Pythagoreans or to Homer even; paragraph 8 is mathematical in content rather than philosophical... although there is an allusion in it to the Monad as the principle of finitudes, again a very early Pythagorean doctrine; and these two paragraphs are the source of [Heinrich] Sitter's suggestion of the authorship of Proclus. As a matter of fact, the philosophical notions in Part I have been borrowed for the most part directly from Plato, with two or three exceptions that are Aristotelian... Plato's Theaetetus, Parmenides, and the Laws, are specifically mentioned. The Timaeus forms the background of much of the thought. And the Platonism of a mathematician of the turn of the third century A. D. need not surprise us, if we but recall Aristotle's accusation that the Academy tended to turn philosophy into mathematics."

"§1. The aim of Book X of Euclid's treatise on the Elements is to investigate the commensurable and incommensurable, the rational and irrational continuous quantities. This science (or knowledge) had its origin in the sect (or school) of Pythagoras, but underwent an important development at the hands of the Athenian, Theaetetus, who had a natural aptitude for this as for other branches of mathematics most worthy of admiration."

"§2. Since this treatise (i. e. Book X of Euclid.) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of nonidentity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account."

"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..."

"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."

"Pythagorean science... will inevitably reproduce the later and inconsistent conception of the atomic, indestructible, individual soul. This... was... present in Orphic religion, fallen from its first Dionysiac faith in the one continuous life in all things, towards the Olympian conception of athanasia. The later Pythagoreans of the fifth century 'construct the whole world out of numbers, but they suppose the units to have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.' ...at a loss, because they could not realise that this physical doctrine was ...a reflection of the belief in a plurality of immortal souls, which contradicted their older faith that Soul was a Harmony—a bond linking all things in one. This Soul had formerly been the One God manifest in the logos; now it is broken up into a multitude of individual atoms, each claiming an immortal and separate persistence. And the material world suffers a corresponding change. In place of the doctrine of procession from the Monad, bodies are built up out of numbers, now conceived as collections of ultimate units, having position and magnitude. Thus, Pythagoreanism is led... from a temporal monism to a spatial pluralism—a doctrine of number-atoms hardly distinguishable from the atoms of Leukippus and Democritus, who, as Aristotle says, like these Pythagoreans, 'in a sense make all things to be numbers and to consist of numbers.' But the development of this number-atomism was predestined by religious representations of the nature of soul older than Pythagoreanism itself, and already contained in the blend of Dionysiac and Olympian conceptions inherited by Pythagoras from Orphism."

"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."

"The tendency which impelled Pythagorean science towards a materialistic atomism is only the recoil of that same tendency which exalted Pythagoras, from his position as the indwelling daemon of his church, to the distant heaven of the immortals. It is the tendency to dualism. When God ceases to be the immanent Soul of the world, living and dying in its ceaseless round of change, and ascends to the region of immutable perfection, it is because man has acquired a soul of his own, a little indestructible atom of immortality, a self-subsistent individual. 'Nature' likewise loses her unity, continuity, and indwelling life, and is remodelled as an aggregate of little indestructible atoms of matter. But note the consequence: she, too, is now self-subsistent. The world of matter becomes the undisputed dominion of Destiny, or Chance, or Necessity—of Moira, ', . There is no place in it for the God who has vanished beyond the stars."

"The Pythagoreans had... a great veneration for the... words of the Master... but... veneration is often accompanied by a singular licence of interpretation."

"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.""

"[W]e cannot safely take Plato as our guide to the original meaning of the Pythagorean theory, though... from him alone... we can learn to regard it sympathetically."

"Aristotle... was... out of sympathy with Pythagorean ways of thinking, but took... great... pains to understand them. This was... because they played so great a part in the philosophy of Plato and his successors, and he had to make the relation of the two doctrines as clear as he could to... his disciples."

"[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."

"This sufficiently justifies... regarding the "fragments of Philolaos" with... more than suspicion."

"We know nothing of Timaios except what Plato tells us... and he may... be a fictitious character like the Eleatic Stranger."

"We are told that the other book which passed under the name of Pythagoras was really by Lysis."

"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."

"[W]e have... testimony that the five "Platonic figures,"... were discovered in the Academy. In the to Euclid... Pythagoreans only knew the , the pyramid (), and the , while the and the were discovered by Theaitetos."

"Philolaos wrote on medicine, and... while... influenced by the... Sicilian school, he opposed them from the Pythagorean standpoint. ...[H]e said... our bodies were composed only of the warm... [O]nly after birth... the cold was introduced by respiration. The connexion... with the old Pythagorean theory is obvious. Just as the Fire in the macrocosm draws in and limits the cold dark breath which surrounds the world... so do our bodies inhale cold breath... Philolaos made , blood, and the causes of disease..."

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

"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 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."

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

"The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and... therefore things are numbers. He is equally positive that these "things" are sensible things, and... are... the bodies of which the world is constructed. This construction... out of numbers was a real process in time, which the Pythagoreans described in detail."

"Plato had many enemies and detractors, and this literary device enabled them to bring against him the charge of plagiarism. Aristoxenos... made the extraordinary statement that most of the Republic was... found in a work by Protagoras. ...He seems also... the... source of the story that Plato bought "three Pythagorean books" from Philolaos and copied the Timaeus out of them. ...[A]ccounts... imply... Plato bought... either a book by Pythagoras, or... notes of his teaching..."

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

"It is... probable that we should ascribe to Pythagoras the Milesian view of a plurality of worlds, though... not... infinite ...Petron, one of the early Pythagoreans, said there were ...a hundred and eighty-three worlds arranged in a triangle; and Plato makes Timaios admit, when laying down ...only one world, that something might be urged in favour of ...five, as there are five regular solids."

"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 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."

"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 the fourth century, the chief seat of the school is at Taras, and we find the Pythagoreans heading the opposition to Dionysios of Syracuse. ...[In] this period... Archytas... the friend of Plato... almost realised, if he did not suggest, the ideal of the . ...He was also the inventor of mathematical mechanics."

"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."

"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."

"[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."

"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."