PLATONISM AFTER PLATO Aristotle's Account of Platonism.—Since Plato refused to write any formal exposition of his own metaphysic, our knowledge of its final shape has to be derived from the statements of Aristotle, which are confirmed by scanty remains of the earliest Platonists preserved in the Neo-Platonist commentaries on Aristo tle. These statements can, unfortunately, only be interpreted conjecturally. According to Aristotle (Metaphysics, A 987, b 18 25) Plato's doctrine of Forms was, in its general character, not very different from Pythagoreanism, the Forms being actually called Numbers. The two points on which Aristotle regards Plato as disagreeing with the Pythagoreans are, that (i) whereas the Pythagoreans said that numbers have as their constituents, the unlimited (67rEtpov) and the limit Plato taught that the forms have as constituents "the One" and the "great and small"; (2) the Pythagoreans had said that things are numbers, but Plato intercalated between his Forms (or Numbers) and sensible things an intermediate class of "mathematicals." It is curious, that in connection with the former difference Aristotle dwells mainly on the substitution of the "duality of the great-and-small" for the "unlimited," not on the much more significant point that the "One," which the Pythagoreans regarded as the simplest com plex of unlimited and limit, is treated by Plato as itself the element of limit. He further adds that the "great-and-small" is, in his own technical terminology, the "matter," the One, the formal constituent, in a Number.
If we could be sure how much of the polemic against Number Forms in Metaphysics M-N is aimed directly at Plato, we might add considerably to this bald statement of his doctrine, but un luckily it is certain that much of the polemic is concerned with the teaching of Speusippus and Xenocrates. It is not safe, there fore, to ascribe to Plato statements other than those with which Aristotle explicitly credits him. We have then to interpret, if we can, two main statements: (I) the statement that the Forms are Numbers; (2) the statement that the constituents of a Num ber are the "great-and-small" and "the One."
Light is thrown on the first statement if we recall the corpus cular physics of the Timaeus and the "mixture" of the Philebus. In the Timaezts, in particular, the behaviour of bodies is ex plained by the geometrical structure of their corpuscles, and the corpuscles themselves, are analysed into complexes built up out of two types of elementary triangle, which are the simplest "ele ments" of the narrative of Timaeus. Now a triangle, being deter mined in everything but "absolute magnitude" by the numbers which express the ratio of its sides, may be regarded as a triplet of numbers.' If we remember then, that the triangles determine the character of bodies, and are, themselves, determined by num bers, we may see why the ultimate Forms on which the character of Nature depends should be said to be Numbers, and also what is meant by the "mathematicals" intermediate between the Forms and sensible things. According to Aristotle, these "mathematicals" differ from Forms because they are many, whereas the Form is one, from sensible things in being unchanging. This is exactly how the geometer's figure differs at once from the type it em bodies and from a visible thing. There is, for example, only one type of triangle whose sides have the ratios 3:4:5, but there may be as many "pure" instances of the type as there are triplets of numbers exhibiting these ratios; and again, the geometrical tri angles which are such "pure" instances of the type, unlike sensible three-sided figures, embody the type exactly and un changingly. A mathematical physicist may thus readily be led to what seems to be Plato's view that the relations of numbers are the key to the whole mystery of nature, as is actually said in the Epinomis (99oe).
We can now, perhaps, see the motive for the further departure from Pythagoreanism. It is clear that the Pythagorean parallelism 'Thus the two fundamental triangles of Timaeus may be called the triplets (1,1g/2) (IA/3,2) respectively.