Concerning the infinite multitude of Forms
Parmenides says to Socrates: ‘I think that the reason why you think that each Form is one is like this (Oimai se ek tou toioude hen hekaston eidos oiesthai einai): When many things appear to you to be large (hotan poll’ atta megala soi doxêi einai), there perhaps seems to be some Form, which is one and the same (mia tis isȏs dokei idea hê autê einai), as you look on them all (epi panta idonti); whence (hothen) you believe the large is one (hen to mega hêgêi einai)’. Socrates: ‘True (Alêthê legeis)’. Parmenides: ‘And what about the large itself (Ti d’ auto to mega) and all the other large things (kai t’alla ta megala), if in the same way (ean hȏsautȏs) you look in your mind at all of them (têi psuchêi epi panta idêis), will not again some large appear (ouchi hen ti au mega phaneitai) by which they all appear large (hȏi tauta panta megala phainesthai)?’ Socrates: ‘It seems so (Eoike)’. Parmenides: ‘So another Form of largeness (Allo ara eidos megethous) will come to view (anaphanêsetai), produced alongside the largeness itself (par’ auto te to megethos gegonos) and the things participating in it (kai ta metechonta autou); and over and above all these (kai epi toutois pasin), again (au), a different one (heteron), by which they all will be large (hȏi tauta panta megala estai). And so you won’t have one of each Form (kai ouketi dê hen hekaston soi tȏn eidȏn estai), but their multitude will be infinite (alla apeira to plêthos).’ (132a1-b2)
Aristotle says in Metaphysics A: ‘But as for those who posit the Ideas (hoi de tas ideas tithemenoi), firstly, in seeking to grasp the causes of the things around us (prȏton men zêtountes tȏnde tȏn ontȏn labein tas aitias), they introduced others equal in number to these (hetera toutois isa ton artithmon ekomisan), as if a man who wanted to count things (hȏsper ei tis arithmein boulomenos) thought he would not be able to do it while they were few (elattonȏn men ontȏn oioito mê dunêsesthai), but tried to count them when he had added to their number (pleiȏ de poiêsas arithmoiê) (990a34-b4) … of the more accurate arguments (hoi de akribesteroi tȏn logȏn), some lead to Ideas of relations (hoi men tȏn pros ti poiousi ideas), of which we say there is no independent class (hȏn ou phamen einai kath’ hauto genos), and others introduce the ‘third man’ (hoi de ton triton anthrȏpon legousi) (b15-17) … and if the Ideas and the particulars that share in them have the same form (kai ei men t’auto eidos tȏn ideȏn kai tȏn metechontȏn), there will be something common to these (estai ti koinon) … But if they have not the same form (ei de mê to auto eidos), they must have only the name in common (homȏnuma an eiê), and it is as if one were to call both Callias and a wooden image a ‘man’ (kai homoion hȏsper an ei tis kaloi anthrȏpon ton te Kallian kai to xulon), without observing any community between them (mêdemian koinȏnian epiblepsas autȏn)’ (991a2-8, tr. W. D. Ross).
A fragment from Aristotle’s De Ideis elucidates the ‘third man argument’: ‘There will be a third man (tritos anthrȏpos estai) both apart from each individual man (para te ton kath’ hekasta), such as Socrates and Plato (hoion Sȏkratê kai Platȏna), and apart from the Form (kai para tên idean), which is itself one in number (hêtis kai autê mia kat’ arithmon esti).’ (Alexander of Aphrodisias, in Metaph. 83, W. D. Ross, Aristotelis fragmenta selecta, Oxford University Press 1955 (1970), p. 125.)