Translated with introduction and notes by A.E. Taylor

Mathematicians, Pythagoreans, and Eleatics← TOC→The Four Conceptions of Cause United in Aristotle


The Peculiarities of Plato

The said philosophies were succeeded by the system of Plato, which was for the most part in harmony with them, but had also some distinctive peculiarities by which it was discriminated from the philosophy of the Italians.96 In his youth Plato had been familiar with Cratylus and with the Heraclitean doctrines, according to which all things perceived by the senses are in incessant flux, and there is no such thing as scientific knowledge of them, and to this part of the doctrine he remained true through life. (987 b) Socrates, however, though confining his examination to questions of moral conduct, and giving no study to the nature of the universe as a whole, sought within the moral sphere for the universal, and was the first to concentrate his attention on definitions. Hence Plato, who succeeded him, conceived for the reason immediately to be mentioned that the objects thus defined cannot be any sensible things, but are of some different kind, since it is impossible that there should be a general definition of a sensible thing, as such things are incessantly changing. Hence he called this kind of things "Ideas," and held that all sensible things exist by the side of them and are named after them; for the multiplicity of things called by the same names as the Ideas exist, he holds, in consequence of their "participation" in them.97

In this theory of participation the only innovation lay in the name, for the Pythagoreans say that things exist by "imitation" of the numbers, and Plato by "participation" [a mere change of a word]. But what this "participation in" or "imitation of" the Ideas may be, they left for their successors to inquire.

Further, he teaches that the objects of mathematics exist as an intermediate class beside the Ideas and sensible things. They differ from sensible things in being eternal and immutable, and from the Ideas in this, that there is a multiplicity of similar mathematical objects, but each Idea is a single, self-subsisting entity.98

And since the Ideas are the causes of everything else, he thought that their constituent elements are the elements of everything. Their material principle, then, is the "Great and Small," but their formal principle the One. For the numbers [which are the Ideas]99 are derived from the former principle by participation in the One. In regarding the One as a substance, and not as a predicate of some other entity, his doctrine resembles Pythagoreanism, and also in holding that the numbers are the causes of Being in everything else. But it is peculiar to him to set up a duality instead of the single Unlimited, and to make the Unlimited consist of the Great and Small.100 It is a peculiarity, also, that he regards the Numbers as distinct from sensible things, whereas the Pythagoreans say that things themselves are number, and do not assert the existence of an intermediate class of mathematical objects. This treatment of the One and the Numbers as distinct from things, in which he differed from the Pythagoreans, and also the introduction of the "Ideas," were due to his logical101 studies (for his predecessors knew nothing of Dialectic); his conception of the second principle as a Duality, to the ease with which numbers other than primes can be generated from such a Duality as a matrix.102 (988 a)

Yet, the actual process is the reverse of this, and his suggested derivation has no logical foundation. According to his followers, the existence of a multiplicity of things is a consequence of matter, whereas each Form is only productive once for all. Yet, it is notorious that only one table103 can be fashioned from one and the same piece of timber, whereas he who impresses the form on it, though but a single workman, can make many tables. So with the relation of the male to the female; the latter is impregnated by a single coition, but one male can impregnate many females. And yet these relations are "copies" of those principles!

This, then, is the account which Plato gave of the questions we are now investigating. From our statement it is clear that he only employed two kinds of cause, the principle of the what and the material cause. (The Ideas, in fact, are the cause of the what in everything else, and the One in the Ideas themselves.) He also tells us what is the material substratum of which the Ideas are predicated in the case of sensible things, the One in the case of the Ideas, viz., that it is the duality of the "Great and Small." He further identified these two elements with the causes of good and evil, respectively, a line of research which, as we have said, had already been followed by some of his philosophical predecessors, e.g., Empedocles and Anaxagoras.

Taylor's Notes

(Taylor's footnotes have been converted to endnotes)


[96]. i. e., the Pythagoreans.

[97]. Cf. the fuller parallel passage, Metaphysics, M, 1078a 9 ff: “The theory of Ideas arose in the minds of its originators from their persuasion of the truth of the Heraclitean doctrine, that all sensible things are always in flux. Hence, they inferred, if there is to be scientific knowledge and rational comprehension of anything, there must be other entities distinct from those of sense, and they must be permanent. Now, Socrates confined his studies to the moral virtues, and was the first to attempt universal definition in connection with them. Among the physicists, Democritus had indeed just touched the fringe of the problem, and had given a sort of definition of heat and cold, and the Pythagoreans even earlier had discussed the definition of a few concepts, connecting them with their theory of numbers. They asked, e.g., what is opportunity, or justice, or marriage? But Socrates had a good reason for inquiring into the what of things. He was attempting to construct syllogisms, and the ‘what is it' is the starting-point of the syllogism . . . . There are, in fact, two things which must in justice be assigned to Socrates, inductive arguments and universal definition. For both of these have to do with the foundation of science. Socrates, however, did not regard his universals, or definitions, as separable from things; his successors made the separation, and called this class of objects `Ideas.’"

[98]. From the polemic against Plato, which occupies books M and N of the Metaphysics, particularly from M 2, 1076b, it appears that Aristotle understood Plato to distinguish between three kinds of entity, each of which is in its ultimate constitution a number, or ratio of numbers: (1) The sensible object, e.g., a visible round disc; (2) the "mathematical object," e.g., our visual imagination of a perfectly circular disc; (3) the Idea, e.g., the circle in the sense in which it is studied by the analytical geometer, and defined by its equation. (2) differs from (3) as "circles" from "the circle."

[99]. The text is τὰ εἴδη τοὺς ἀριθμούς, where either τὰ εἴδη or τοὺς ἀριθμούς is pretty clearly a gloss. I follow Zeller's reading. Christ has τὰ εἴδη [τοὺς ἀριθμούς], "the Ideas [which are the numbers]."

[100]. This "Great and Small," or principle of indefinite, variability, is regularly spoken of by Aristotle in the sequel as "the indeterminate Dyad" or "Duality." It corresponds exactly to the notion of "the variable" in modern Logic and Mathematics. The nearest equivalent phrase in the writings of Plato himself occurs at Philebus, 24e, where the ἄπειρον or indeterminate is characterized as "all things which appear to us to exist in a greater and a less degree, and admit the qualifications ‘intensely,' ‘gently,' ‘excessively' and the like." According to the ancient commentators, the foregoing account of the composition of the Ideas, which is not to be found explicitly in any of the Platonic writings, was given orally by Plato in lectures which were posthumously edited by Aristotle and others of his disciples.

[101]τὴν ἐν τοῖς λόγοις σκέψιν, "his inquiries in the domain of concepts," i. e., his study of the nature of logical definition and division.

[102]. A matrix, ἐκμαγεῖον, properly, a mass of material prepared to receive a mould or stamp, a Platonic term borrowed by Aristotle from Theaetetus, 191c; Timaeus, 50c. The clause "other than primes" is difficult to interpret, and has been treated as a mistaken gloss. I think, however, that it alludes to Parmenides, 143-4, where Plato deduces from the existence of 1, that of 2, and from these two that of the whole series of all the other integers which can be resolved into factors, whether odd or even; i. e., all except the primes. If this explanation is correct, and it appears to have been held by Bonitz (see his edition of the Metaphysics, Commentary, p. 94-5), this is one of several passages which refute the current assertion that the dialogue Parmenides is never cited by Aristotle. Another is N, 1091a 11, which unmistakably refers to the same passage of the Parmenides. It should, however, be observed that the two factors from which numbers are derived in that dialogue are not the number 1 and the Variable or "Indeterminate Duality," but the number 1, and the number 2, "the Ideal Duality." This conscious or unconscious perversion of Plato's theory of numbers recurs throughout the whole of the sustained polemic of Books M, N.

[103]. The illustration of the table is an echo of Republic, X., 596a, Aristotle is punning on the literal meaning of the word ὕλη, timber, which he employs as a technical term for the "material" from which a thing is produced.

Created for Peithô's Web from Aristotle on his predecessors; being the first book of his Metaphysics; tr. from the text edition of W. Christ, with introd. and notes by A. E. Taylor. Chicago, Open Court, 1907.Taylor's footnotes have been converted to endnotes. Greek unicode text entered with Peithô's Younicoder.
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