Pythagoreanism

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

"[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."

"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."

"[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..."

"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."

"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 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.""

"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."

"[T]he numbers were intended to be mathematical... though... not separated from... things of sense. ...[T]hey were not mere predicates of something else, but had an independent reality... "They did not hold that the limited and the unlimited and the one were... substances, such as fire, water... [etc.,] but... the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything.""

"Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things. According to the Pythagoreans, things are made of numbers in the same sense as they were made of fire, air, or water in the theories of their predecessors."

"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."

"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..."

"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... 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."

"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."

"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."

"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 criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each... a number. But in the very same regions of the heavens were... things having magnitude which were also numbers. How can this be, since Justice has no magnitude? This means... the Pythagoreans... failed to give... clear account of the relation between these... fanciful analogies and their quasi-geometrical construction of the universe."

"[W]hat distinguished the Pythagoreanism of this period from its earlier form was that it sought to adapt... to the new theory of "elements." ...[T]his ...makes it necessary ...to take up ...consideration of the system ...in connexion with the pluralists."

"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."

"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."

"[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."

"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."

"The most interesting point in the theory is... the use... of the ... identified... with the "sphere of the universe," or... in the Philolaic fragment, with the "hull of the sphere." ...[I]t must be taken in close connexion with the word "" applied to the central fire. The structure of the world was compared to the building of a ship..."

"In the Phaedo we read that the "true earth,"... looked at from above, is "many-coloured like the balls that are made of twelve pieces of leather." In the Timaeus... "Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it." ...[T]he approaches more nearly to the than any other of the regular solids. The twelve pieces of leather used to make a ball would... be s; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. This points to the Pythagoreans having had at least the rudiments of the "" formulated later by Eudoxos."

"They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids. That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of straight and curved in the "table of opposites" under the head of Limit and Unlimited points in the same direction."

"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."

"[T]he Pythagoreans adopted the pentagram or pentalpha as their symbol. The use... in later magic is well known; and Paracelsus... employed it as a symbol of health, which is... what the Pythagoreans called it."

"The view that the soul is a "harmony," or... attunement, is intimately connected with the theory of the four elements. It cannot have belonged to the earliest... Pythagoreanism; for... in Plato’s Phaedo, it is... inconsistent with the idea that the soul can exist independently of the body. It is... opposite of the belief that "any soul can enter any body." ...[F]rom the Phaedo... it was accepted by Simmias and Kebes, who had heard Philolaos at Thebes, and by Echekrates of Phleious, who was the disciple of Philolaos and Eurytos."

"The account of the doctrine given by Plato is... in accordance with the view that it was of medical origin. Simmias says: "Our body being... strung and held together by the warm and the cold, the dry and the moist... [etc.,] our soul is a sort of temperament and attunement of these, when... mingled... well and in due proportion. If, then, our soul is an attunement,... when the body has been relaxed or strung up out of measure by diseases and other ills, the soul must... perish at once." This is... an application of the theory of Alkmaion, and is in accordance with... the Sicilian school of medicine. It completes the evidence that the Pythagoreanism of the end of the fifth century was an adaptation of the old doctrine to the new principles introduced by Empedokles."

"The planetary system which Aristotle attributes to "the Pythagoreans" and Aetios to Philolaos is... remarkable. The earth is no longer in the middle of the world; its place... taken by a central fire, which is not... the sun. Round this fire revolve ten bodies. First comes the Antichthon or , and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the five planets, and the heaven of the fixed stars. We do not see the central fire and the antichthon because... [our] side of the earth... is always turned away from them.., explained by the analogy of the moon. ...[M]en living on the other side of it would never see the earth. ...[A]ll these bodies rotate on their axes in the same time as they revolve round the central fire."

"Plato gives a description of the earth and its position... entirely opposed to... [antichthon theory], but is accepted... by Simmias the disciple of Philolaos. It is undoubtedly... Pythagorean... and marks... advance on the Ionian views then current at Athens. ...Plato states it as ...a novelty that the earth does not require ...support ...to keep it in its place. ...Anaxagoras had not been able to shake himself free of that idea, and Demokritos still held it."

"The... inference from the Phaedo would... be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaos... If so, the doctrine of the central fire would belong to a somewhat later generation of the school, and Plato may have learnt it from Archytas and his friends after he had written the Phaedo."

"[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."

"It seems probable that the... earth’s revolution round the central fire... originated in the account... by Empedokles of the sun's light. The two... are brought into... connexion by Aetios, who says... Empedokles believed in two suns, while Philolaos believed in two or... three. The theory of Empedokles... gives two inconsistent explanations of night."

"The central fire received a number of mythological names. ...[W]e are dealing with a real scientific hypothesis. It was a great thing... that the phenomena could best be "saved" by a central luminary, and that the earth must... be a revolving sphere like the planets. [W]e are almost tempted to say that the identification of the central fire with the sun... suggested for the first time in the Academy, is a mere detail in comparison. The great thing was that the earth should... take its place among the planets... once... done.., we can... search for the true "hearth" of the planetary system... It is probable... that... this theory... made it possible for Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis, and it was... Aristotle’s reversion to the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that the Pythagorean theory put him on the right track."

"The existence of the antichthon was... a hypothesis intended to account for... eclipses. ...Aristotle says that the Pythagoreans invented it... to bring the number of revolving bodies up to ten; but that is a... sally... Aristotle... knew better. In his work on the Pythagoreans... he said... eclipses of the moon were caused sometimes by.... the earth and sometimes by... the antichthon... the same statement was made by Philip of Opous..."

"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."

"Anaximenes... assumed... existence of dark planets to account for the frequency of s, and Anaxagoras... revived that view. Certain Pythagoreans had placed these dark planets between the earth and the central fire... to account for their invisibility, and the next stage was to reduce them to a single body. ...[A]gain ...the Pythagoreans tried to simplify the hypotheses of ...predecessors."

"We must not assume ...Pythagoreans made the sun, moon, and planets, including the earth, revolve in the opposite direction to the heaven of the fixed stars. ...Alkmaion is said to have agreed with "some of the mathematicians" in holding this view, but it is never ascribed to Pythagoras or even to Philolaos."

"The old theory was that all... heavenly bodies revolved... from east to west, but that the planets revolved more slowly the further they were removed from the heavens, so... those... nearest the earth are "overtaken" by those that are further away. This view was... maintained by Demokritos, and that it was... Pythagorean... follow[s] from... the "harmony of the spheres." [W]e cannot attribute this theory in... later form to the Pythagoreans of the fifth century, but we have... testimony of Aristotle... that those Pythagoreans whose doctrine he knew believed... heavenly bodies produced s in their courses. ...[V]elocities of these bodies depended on the distances between... [which] corresponded to the intervals of the . He... implies that the heaven of the fixed stars takes part in the concert; for... "the sun, the moon, and the stars, so great in magnitude and in number as they are..." ...[T]he slower bodies give out a deep note and the swifter a high note."

"[P]revailing tradition gives the high note of the octave to the heaven of the fixed stars... [I]t follows that all the heavenly bodies revolve in the same direction, and... their velocity increases in proportion to their distance from the centre."