Translated with introduction and notes by A.E. Taylor

Teleology and the Formative Principle← TOC→The Peculiarities of Plato


Mathematicians, Pythagoreans, and Eleatics

At the same time, and even earlier, the so-called Pythagoreans attached themselves to the mathematics and were the first to advance that science76 by their education, in which they were led to suppose that the principles of mathematics are the principles of all things. So as numbers are logically first among these principles, and as they fancied they could perceive in numbers many analogues of what is and what comes into being, much more readily than in fire and earth and water (such and such a property of number being justice, such and such another soul or mind, another opportunity, and so on, speaking generally, with all the other individual cases), and since they further observed that the properties and determining ratios of harmonies depend on numbers--since, in fact, everything else manifestly appeared to be modelled in its entire character on numbers, and numbers to be the ultimate77 (986 a) things in the whole Universe, they became convinced that the elements of numbers are the elements of everything, and that the whole "Heaven"78 is harmony and number. So, all the admitted analogies they could show between numbers and harmonies and the properties or parts of the "Heaven" and the whole order of the universe, they collected and accommodated to the facts; if any gaps were left in the analogy, they eagerly caught at some additional notion, so as to introduce connection into their system as a Whole. I mean, e. g., that since the number 10 is thought to be perfect, and to embrace the whole essential nature of the numerical system, they declare also that the number of revolving heavenly bodies is ten, and as there are only nine79 visible, they invent the Antichthon as a tenth. But I have discussed this subject more in detail elsewhere.80 I only enter on it here for the purpose of discovering from these philosophers as well as from the others what principles they assume, and how those principles fit into our previous classification of causes. Well, they, too, manifestly regard number as a principle, both in the sense that it is the material of things, and in the sense that it constitutes their properties and states. The elements of number are, they think, the Even and the Odd, the former being unlimited, the latter limited. Unity is composed of both factors, for, they say, it is both even and odd. Number is derived from unity, and numbers, as I have said, constitute the whole "Heaven."

Other members of the same school say that the principles are ten, which they arrange in a series of corresponding pairs:81

Limit—The Unlimited

Alcmaeon of Crotona appears to have followed the same line of thought, and must either have borrowed the doctrine from them or they from him, since Alcmaeon was contemporary with the old age of Pythagoras. His views were very similar to theirs. He says, in fact, that most things human form pairs, meaning pairs of opposites. He does not, however, like the Pythagoreans, give a precise list of these, but mentions at random any that occur to him, e. g., White-Black, Sweet-Bitter, Good-Bad, Great-Small. Thus in other cases he merely threw out indefinite suggestions, but the Pythagoreans further undertook ­(986 b)  to explain how many and what the opposites are. From both, then, we can learn this much: that the opposites are the principles of things, but only from the latter how many, and what these are. They have not clearly explained in detail how these opposites are to be reduced to our previous classification of causes, but they appear to treat their elements as the material of things; for they say that Being82 is composed and fashioned out of them as inherent constituent factors. The meaning, then, of those ancients who asserted that the elements of the universe are a plurality can be sufficiently perceived from the foregoing exposition. But there are some83 who expressed the view that the all is one single entity, though they differed among themselves both in respect of the merits of their doctrine, and in respect of its logical character. Now, a discussion of their views is not strictly relevant to our present inquiry into causation, for, unlike some of the physicists84 who postulate the unity of Being, and yet treat of its derivation from the one substance as its material cause, they maintain the doctrine in a different sense. Those physicists assume, also of course, the existence of motion, since they treat of the derivation of the All, but this school declares that the All is motionless. Still, one observation at least is relevant to our present inquiry. Parmenides appears to conceive of the One in a formal sense, Melissus in a material. Hence the former calls it limited, the latter unlimited. Xenophanes, who was the first of them to teach the doctrine of unity (for they say that Parmenides had been his disciple), did not make any definite pronouncement, and seems to have formed the notion of neither of these entities, but gazing up at the whole Heaven85, declared that the One is God. As I said, then, for the purposes of the present investigation this school may be disregarded. Two of them we may disregard altogether as a little too naïve,86 viz., Xenophanes and Melissus, but Parmenides appears, perhaps, to speak with greater insight. For, since he claims that Non-being, as contrasted with Being, is nothing, he is forced to hold that Being is one, and that nothing else exists--a doctrine on which we have spoken more fully and clearly in our course on Physics.87 But, as he is obliged to adapt his views to sensible appearances, he assumes that things are one from the point of view of reason, but many from that of sensation, and thus reintroduces a duality of principles and causes, the Hot and Cold, by which he means, e.g., fire and earth. Of these he co-ordinates the Hot with Being, its counterpart (987 a) with Non-being.

Now, from the account we have just given, and by a comparison of the thinkers who have previously concerned themselves with the subject, we have arrived at the following result. From the earliest philosophers we have learned of a bodily principle (for water, fire, and the like, are bodies), which some of them88 regard as a single principle, others89 as a plurality, though both schools treat these principles as bodily. From others we have learned, in addition to this principle, of a source of motion, and this also is regarded by some 90of them as one, but by others91 as twofold. They all, down to the Italian92 school and exclusive of them, treated the subject in a rather ordinary93 way. As I have said, they only employed two kinds of cause, and the second of these, the source of motion, some of them regarded as one, others as two. The Pythagoreans likewise maintained a duality of principles, but they added, and this is peculiar to them, the notion that the limited, the unlimited, the one are not predicates of some other entity, such as fire, or earth, or something else of that kind, but that the Unlimited and the One themselves are the substance of the things of which they are predicated. This is why, according to them, number is the substance of everything.

This was the doctrine they proclaimed on these points. They also began to discuss the what of things, and to give definitions of it, but their method of procedure was extraordinarily crude. Their definitions were superficial, and they regarded anything to which a term under examination first applied as the essential nature of the object in question, as if one were to think that "double of" and "the number 2" are the same thing, on the ground that 2 is the first number which is double of another. But, methinks, it is not the same thing to be double of something as it is to be the number 2. If it were, then one thing would be many,94 a consequence which actually followed in their system.95 So much, then, is what may be learned from the earlier thinkers and their successors.

Taylor's Notes

(Taylor's footnotes have been converted to endnotes)


[76]. On the nature and extent of the Pythagorean mathematics, see particularly Cantor, Geschichte der Mathematik, I., pp. 137-169. Milhaud, Les Philosophes-Géomètres de la Grèce, bk. 1, ch. 2. Note that Aristotle never professes to know anything of the philosophical or scientific views of Pythagoras himself. On the sources of his knowledge of the "so-called Pythagoreans, " consult Burnet, op. cit. p. 321.

[77]. πρῶτοι; literally, "first," as above.

[78]. οὐρανός; literally, "heaven" meant to the early Greek physicists the whole collection of bodies comprised within the apparent vault of the sky. We must not translate by "universe," since it was commonly held that much which exists is outside the οὐρανός. An equivalent term of later date, probably of Pythagorean origin, is κόσμος.

[79]. viz., Earth, Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn, circle of Fixed Stars.

[80]. In a now lost work, "On the Pythagoreans."

[81]. On the meaning of this numerical cosmology of the Pythagoreans the student will find most enlightenment in the works of Burnet and Milhaud, previously referred to, and the section on Pythagoreanism in Bäumker, Das Problem der Materie in der Griechischen Philosophie. He should note, also, the fundamental initial error which vitiated all ancient arithmetical theory, viz., the view that 1, and not 0, is the first of the series of integers. This view is connected partly with the defects of Greek arithmetical notation, partly with an erroneous assumption, tacitly made by all Greek logicians, as to the "existential import" of predication.

[82]. τὴν οὐσίαν

[83]. viz., the Eleatics.

[84]. i. e., the Ionian Monists, from Thales to Heraclitus.

[85]. Or, perhaps, "contemplating the Universe as a whole."

[86] "Naïveté" (ἀγροικία) is a technical term with Aristotle, for want of acquaintance with formal logic. On the particular logical fallacy to which he objected in Melissus, see Burnet, op. cit. p. 341, and on his misapprehension of the second part of the poem of Parmenides, p. 195 of the same work. Plato, it should be said, held the same view as to the relative merits of these two philosophers. See Theaetetus, 183e.

[87]. Physics, I., 3, 186a 3, ff.

[88] The Milesians, Heraclitus, Diogenes.

[89]  Empedocles, Anaxagoras, the Atomists.

[90]  i. e., Anaxagoras.

[91]  i. e., Empedocles.

[92]  i. e., the Pythagoreans of Magna Graecia.

[93]  μετριώτερον. If the text is correct, this must mean "unsatisfactorily," though the word will hardly bear that sense. There is a rival MS. reading μαλακώτερον, "rather feebly," and Alexander of Aphrodisias appears to be explaining a reading μοναχώτερον, "rather one‑sidedly."

[94]  For, if every number which is double of another is the number 2 the single number 2 must be identical with an infinity of other even numbers, 4, 6, 8 . . . .

[95]  The way in which this occurred was that the same number was identified, on the strength of different fanciful analogies, with a variety of different objects. Thus 1 was "the point," but it was also "the soul."

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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