Aristotle’s testimony concerning the Pythagoreans in Metaphysics A should be viewed against the background of Plato’s Parmenides. A theory of Forms which young Socrates presents in the dialogue appears to be nothing new to Parmenides and Zeno; if they knew it, they must have known it as a Pythagorean theory.
Aristotle says in the 1st book of Metaphysics that the Pythagoreans ‘extend their vision to all things that exist, and of the existing things suppose some to be perceptible and others not perceptible’ (989b24-26); ‘they got their principles from non-sensible things’ (989b31, tr. Ross). In the Parmenides Socrates asks Zeno whether he agrees that there are two sorts of things, those we can see with our eyes and those we can’t, which he called Forms (eidê, 128e6-130a2). Although Socrates presented his question concerning the Forms as a challenge to both Zeno and Parmenides, the two listened to him in admiration (agamenous ton Sȏkratê, 130b7). After exposing the notion of Forms to criticism that Socrates could not answer, Parmenides told him: ‘I admired you for saying to Zeno that you would not allow inquiry to wander among the visible things and consider them, but rather concern those things which one would most especially grasp by rational account and consider to be Forms.’ (135d8-e4) As I have mentioned in an earlier entry (‘Plato’s Parmenides and Parmenides’ poem On nature’), the ancients viewed Parmenides as an associate of the Pythagoreans (DK I. Fr. A 4, pp. 218-9; A 12, p. 220; A 40a p. 225; A44 p. 225). Although Parmenides had to overcome the Pythagorean plurality, he appears to have appreciated Pythagoreans for getting ‘their principles from non-sensible things’.
According to Aristotle the Pythagoreans viewed numbers as ‘principles of all things’ (tȏn ontȏn archas ȏiêthêsan einai pantȏn, 985b25-6) – ‘such and such a modification of numbers being justice (to men toiondi tȏn arithmȏn pathos dikaiosunê), another being soul and reason (to de toiondi psuchê kai nous), another being an opportunity (heteron de kairos) and similarly all the other things, so to speak’ (kai tȏn allȏn hȏs eipein hekaston homiȏs, 985b29-31) – ‘for all other things seemed in their whole nature to be modelled on numbers’ (ta men alla tois arithmois ephaineto tên phusin aphomoiousthai pasan, 985b32-3) In the Parmenides Socrates suggested that the Forms are paradigms (paradeigmata) in relation to which all other things are modelled (ta de alla toutois eoikenai kai einai homoiȏmata, 132d2-3).
Aristotle says that Plato’s philosophy in most respects followed the Pythagoreans (ta men polla toutois akolouthousa, 987a30); ‘the Pythagoreans say that things exist by imitation of numbers (mimêsei ta onta phasi einai tȏn arithmȏn), and Plato says they exist by participation (Platȏn de methexei), changing the name (t’ounoma metabalȏn). But what the participation or the imitation of the Forms could be (tên mentoi ge methexin ê tên mimêsin hêtis an eiê tȏn eidȏn) they left an open question’ (apheisan en koinȏi zêtein, 987b11-14, tr. Ross). In Plato’s dialogue Parmenides dismissed the theory of ‘imitation’ with the words: ‘So the other things do not get a share of the Forms by likeness (ouk ara homoiotêti t’alla tȏn eidȏn metalambanei), but one must look for something else by which they get a share’ (alla ti allo dei zêtein hȏi metalambanei, 133a5-6); what that ‘something else’ might be, he does not say.
Aristotle says that Pythagoreans arranged their principles into two columns of opposites, among which we can find ‘one and plurality’, ‘resting and moving’, ‘good and bad’. Socrates in the Parmenides contemplates Forms as opposite to each other, ‘such as likeness and unlikeness, multitude and the one, rest and motion’ (128e5-129e1).
Aristotle says that the Pythagoreans viewed ‘the infinity itself (auto to apeiron) and the one itself (kai auto to hen) as the substance of things of which they are predicated (ousian einai toutȏn hȏn katêgorountai) … they began to discuss essence and define it (peri tou ti estin êrxanto legein kai horizesthai), but they did so too superficially (lian d’ haplȏs epragmateuthêsan); the first subject of which a given definition was predicable was the subject of the thing defined … thus the one will be many (polla to hen estai), which in fact happened to them (ho k’akeinois sunebainen, Met. A 987a18-27). The one that Parmenides discusses in Plato’s dialogue, the one that becomes ‘many’, is not the one of his poem, but the Pythagorean one.