Pythagoras and the Pythagoreans – A Brief History Charles H. Kahn

Pythagoras and the Pythagoreans – A Brief History

Charles H. Kahn

Hackett Publishing Company 2001

Pg 2

  • Eduard Zeller – Pythagorean community essentially a religious sect. Doubtful over Pythagorean’s his scientific achievements

Pg 3

  • Burkert traces P cosmology and number-philosophy reported by Aristotle to Philolaus in the middle of the late 5th century.
  • B – Pythagoras as a shamanistic figure, charismatic spiritual leader and organizer – but contributed nothing to mathematics or philosophy.
  • The whole heavens as ‘harmonia and number’

Pg 4

  • Soul as immortal and hence potentially divine (Greek tradition of deathless is an attribution of the Gods)
  • P’s view of soul developed most in Plato’s Phaedo. (plus Republic and Phaedrus)
  • Mathematical and musical concept – developed in Plato’s Timaeus where the world soul is structured by musical ratios and the world body is organized by elementary triangles
  • The cunning geometry in Timaeus  and elaborate arithmetic = ‘certainly plato’s own invention’ plus recollection and immortality in terms of cognitive grasp of eternal Forms.

Pg 5 – Pythagaros and the Pythagorean way of life

  • 3 lives of Pythagoras in late antiquity: Diogenes Laertius, Porphyry and Iamblichus (each one more marvellous than predecessor) Zeller – further the account from P’s own time the fuller the account of P becomes)

Pg 6

  • For P’s – 2 rational beings: humans, gods, and people like Pythagaros


  • Pythagoreans were bound together by cult practices and specific burial rights.
  • Those who come together to listen – ‘akousma’
  • ‘Friends have all things in common’ (do not break bread for bread brings friends together)

Pg 10

  • Few akousmata point in the direction of mathematics or natural philosophy but a communal way of life
  • This way of life cont. down to Plato’s times. The only reference to Pythag in Plato’s work ; referred to someone  ‘who was beloved for his instruction as a leader of culture ans education, whose followers down to present are renowned for the way of life called ‘Pythagorean’

Pg 12

  • Greece – became distinguished for his practice of sacrificial ritual in the sanctuaries

Pg 13

  • The picture of Pythagoras changes radically with Plato and his school where he becomes also a the creator of mathematical philosophy (not just religious leader)
  • Plato: the mathematical sciences and specifically astronomy and harmonics are said to be ‘sister sciences, as the Pythagoreans assert and we agree’
  • Philebus –  Socrates offers a description of dialectic based upon the principle that all things are derived ‘from one and from many, having Limited and Unlimited built into their nature

Pg 14

  • Limited and Unlimited are, as we shall see, fundamental principles in the Pythagorean cosmology of Philolaus
  • Kahn – is Pythagoras as mathematical philosopher essentially a fabrication of Plato’s school
  • For Burkert; Pythagoras is indeed at the origin of the conception of the soul as immortal and as reborn in different animal forms but he is not in any significant sense a mathematical thinker and author of the view of the universe based upon number and proportion.

Pg 15

  • Carl Huffman developed Burkert’s position; it is Philolaus, a century after Pythagoras, who became first Pythagorean to enter tradition of Pre-Socratic cosmology and he do so as an innovator with no debt to Pythagoras.
  • We must accept that P is not likely to be a creation of Plato and his disciples. Ina passage traced back to Aristotle, Iamblichus reports the existence of two rival schools of Pythagoreans both of whom claim to be the true followers of Pythagoras.
  • Aksousmata – claim that mathematical school derives from Hippasus

(lived early 5th cent)

pg 17

  • Empedocles – two generations after Pythagoras – documentation shows how unmistakably how the same individual can figure as a religious prophet and natural philosophy (who appears to be inspired but idiosyncratic follower of Pythagoras) – abstained from meat and blood sacrifices

Pg 18

  • This dual role of Empedocles shows that it is historically possible (in the mentality of archaic Greece, what strikes us as a religious extreminism is entirely compatible with important work in the new cosmology and philosophy)

Pg 20

  • Connection to Orpheus – no good evidence for Orphic poetry before Pythagorean time – so should we in fact take into account the claims that Orphic poetry was composed by Pythagoras.  A similar biew put forward by Herodotus that cults called Orphic are Pythagorean.
  • Diff – Orphic based on written texts. And no space for mythologies in P or Ionian philosophy

Pg 22

  • A distinct tradition – the lyre of Orpheus becomes the symbol for Pythagorean cosmic music

Pg 23

Pythagorean Philosophy before Plato

  • No documents on Pythag – therefore we can only begin with the first known Pythagorean book – This is the work of Philolaus in the last half of the 5th century. Burkert has confirmed the authenticity of a substantial amount of quotes and extended by Carl Huffman analysis of the development of cosmological theories in the 5th cent
  • Kahn refers to this as the system of Philolaus

Pg 24

  • Diogenes Laertius tells us that Philolaus’ book begins with ‘On the Nature oF Things’: ‘Nature is the world order was fitted together harmoniously from unlimited things and also from limiting ones, both the world order as a whole and all things within it’
  • This is familiar from Pre-Socratic texts. Unlimited – starting point from which the world develops in Milesian cosmology. The contrasting notion of the limit is emphasized by Parmenides as mark of perfection on being.  = Combination of Ionian philosophy with Eleatic ontology and are joined together by means of harmony or consonance. Harmonia is known as the principle cosmic union from the slightly earlier system of Empedocles and earlier still from the fragments of Haraclitus. But Philolaus work receives a Pythagorean development of numerical ratios and musical scales.
  • Philolaus offers a logical argument in support of his opening thesis; the world cannot be derived from either unlimiting origins or limiting principles alone, since it contains examples of both.
  • Therefore, all we can know if the reality of his 2 fundamental principles is that their existence is a necessary condition for the coming into being of everything else.

Pg 25

  • Harmony thus serves here the same function as for Empedocles and also Heraclitus: to produce unity out of multiplicity by bringing diverse and discordant elements in to agreement with one another.
  • For Philoaus – Harmonia  is in numerical form ‘all things which are known have number; for nothing can be understood without number’ (arranged according to ratios that correspond to the three basic musical consonances)
  • If we add the integers together, their sum is the number that according to Aristotle in Met the Pythagoreans regard as perfect (1+2+3+4=10)
  • Continuing the cosmic sketch of Philolaus –  since 10 is perfect there must be 10 heavenly bodies. Fire first most precious thing and since the centre and circumference are the places of honor, there must be fire in the centre of the cosmos (the hearth of the universe) and also in the outer most sphere of the stars
  • Added a counter earth – invisible as is the fire = whole universe is harmony and number as Aristotle reports. Not stated, but clearly assumed that the periodic motion of these bodies around the central Hearth somehow instantiate the ratios of musical concord so that their’ revolutions produce the cosmic music of the spheres’
  • Copernicus – ‘Philolaus the Pythagorean’ Originally names his system as ‘astonomia Pythagorica or Philolaica’
  • Not geocentric = led some to believe genius of early Pythagoreans while others denied such an advanced cosmic world picture. Kahn – it does show ‘remarkable freedom of speculative imagination departing the usual assumption that the earth must occupy the centre of the heavens’

Pg 27

  • Aristotle reports that for the Pythagoreans – ‘all is number’ or ‘imitate number’
  • Sextus Empiricus quotes a verse ‘all things resemble number’
  • In a literal quotations from Philolaus – it is by means of numbers and proportion that the cosmos becomes organized and knowable for us’ Hence the process by which the cosmos came into existence seems to have been conceived as analogous to a generation of numbers’ ‘The first thing harmoniously fitted together, the one in the centre of the sphere , is called Hestia, the hearth’
  • Kahns account follows Aristotle assuming that the Pythagoreans generate the heavens by the same process that generates the natural number, so that for Philolaus the ‘one in the centre of the sphere’ is  both the central fire,or Hearth, and also the first integer (harmony – corresponding to the ratio (2:1) Thus Aristotle reports that Pythagoreans  construct the whole heavens out of numbers, but not out of monads, fort they assume monads have magnitude’ (Metaphysics)
  • This was challenged by Huffman –  claims it was not Philolaus that confused things with numbers but Aristotle who then attributed this confusion to the Pythagoreans.
  • That the central fire points to the number one but is not identical to it –  it is, Huffman claims, ‘impossible to imagine that he (philo) confused the arithmetic unit to the central fire. For if he did, his arithmetical unit is more than a bare monad with position; it is also fiery and orbited by ten bodies.

Pg 28

  • Is Aristotle mistaken when he reports that the Pythagoreans generate the physical universe out of numbers? The texts are too few and fragmentary for us to be sure.
  • Risky to rely on A when we cannot see original texts but Kahn inclined to accept his account of numerical cosmology, despite Huffman’s doubts.
  • For P – numbers are both universals and particulars – there are many ones in the cosmos and the universe is one itself. (1st one is central fire)
  • Mistake to think of Ps as offering a theory of what numbers are or how they are related to things, such as physical bodies.  P tells us only how the first numbers came into being or more precisely, how the primordial instance of each of the first ten integers was constructed as a fundamental part of the cosmic order.
  • The One is prior to numbers proper which divided into even (unlimited) and odd (limiting) the one itself however is both even-odd
  • Cosmogony begins as the numbers are generate when the unlimited is drawn in (or breathed in) by the limiting principle. This the cosmos arises from the One by breathing, like a new born animal) The heavens take shape as breath and void are drawn from the unlimited
  • The One is not thought as an abstract entity but as a fiery unit with a definite position in the centre of the sphere thus no distinction is here made of the generation of numbers, the emergence of geometric points and the production of sensible magnitudes. Such refinements will be the work of Plato.
  • Burkert – ‘mythology in scientific clothing’ (agree)
  • Huffman – however, describes Philolaus system as ‘ the most impressive example of presocratic  speculative astronomy and that he may have even been ahead of his contemporaries

Pg 30

  • Kahn – praise excessive for a system so strange – but muse remember that they were speculating in an age of breathtaking intellectual exploration
  • Timaeus – Plato abandons central fire and the counter earth but keeps world soul according to musical numbers and the workd body being made from elementary triangles

Pg 31


  • ‘ the source and root of ever flowing nature’ (same number represented by all three sides of an equilateral triangle)

Pg 32

  • the T is a complete symbol for the musical-numerical order of the cosmos
  • NO evidence P introduced P’s theorem to Greece

Pg 33

  • Only Philaus can be reasonably traced back to Pythagoras


  • Unlimited as a cosmic starting point – Xenophanes denied breathing for his cosmic god – plausible to see this as a polemic response to Pythagoras. If so, it could only be the theory of P himself. (this is not proof)
  • More significant in proving the theories is the cosmic interpretation of harmonia in the thought of herclitus for this def implies a musical interpretation of the world order as represented in the tuning of the lyre

Pg 38

  • In the absence of reliable documentation this can only be an inference that such theories were his own. More plausible that some nameless/Philolaus Pythagorean invented the theories

Plato and Archytus

  • Plato’s maths hevily indebted to Pythagoras. Cites them by name.

Pg 42

  • In music theory most relative is most def Philoaus (Archytus would have taken it from there)
  • Archytus – did work on physical acoustics

Pg 45

  • Plato to Republic –  previous thinkers understood that no sound could come from no blog ‘but many sounds are not recognizable  by our nature…some because of their magnitude that exceeds our hearing’ (trying to claim why we cannot hear the musical attunement)

Pg 52

  •  The Pythagorean way of following God – understood in socaratic terms as the pursuit of the unity of virtue in wisdom

Pg 53

  • Plato’s conception of Philosophy merges with Orphic and Pythagorean streams
  • P’s mathematics is complexified  in Plato’s works

Pg 55

  • The study of Pythagorean mathematics is prep for Platonic dialectic, which has an entirely new object: incorporeal realm of the Form.

Pg pg 56

  • Timaeus – rich in Pythagorean number and cosmic geometry. The world soul is, from which the human soul is eventually derived from, made up from a series of odd and even integers beg with 2 and 3 and proceeding through their squares. 1 is not the first as for Greeks number implied plurality. Finally then, the world represents a dictatonic scale but its basic structure is given by mathematical and physical considerations. (T = a genuine Pythagorean blend)

Pg 58

  • The one and many are the fundamental principlies that underlie all rational thought and discourse corresponding to the principles of the Limited and Unlimited

Pg 39 Pythagorean in the early academy


  • Burkert – Pythagoreans studied by Aristotle have no notion of incorporeal principles; the heavens are as a whole are numbers, and there are no numbers ‘distinct from sensible things’. In A’s view – it was Plato who made the mistake of separating the incorporeal, and it was Plato who replaced Pythagorean Unlimited by the Indeterminate Dyad

Pg 61

  • P’s Forms in relation to Pythagoras. Sources are inconsistent here – A often speaks as if Forms were identified with numbers. But also a text from Thoephrastus that locates numbers above the Forms.

Mathematics, music and astonomy

Pg 153

  • Ptolemy – preserves the essential features of the Pythagorean harmonic theory; a concern with the physics of sound


Pg 158

  • Pico – Pythagorean joins forces with the Cabalistic lore where words and numbers of holy scripture are deciphered according to a mystical system
  • Reuchlin’s idea to cover the name with Pythagoras – complete the works of Ficino and Pico who had made Plato known in Italy – by the re-birth of Pythagoras ‘ but this will require knowledge of Hebrew Cabala ‘since the philosophy of Pythagoras was drawn from the teachings of chaldean science’

(eastern wisdom)

  • The humanists restored the image of P as omniscient sage familiar from late antiquity.

Pg 163

  • Kepler  – Pythagoreanism is not a vague theosophy but instead a philosophical approach to natural science.
  • Pg 170 = we can understand how Kepeler himself could seehis work, as the realization of the Pythagorean dream; the natural oreder of the universe finally stands revealed as a tremendous exemplification of musical numbers, and these numbers are now confirmed by precise observations
  • Pg 171 = In a sense Kepler was the last Pythagorean but a true one. His success in uncovering mathematical regularities in the labyrinth of precise observations collected by Tycho Bahue

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