This is the whole of the Pythagorean philosophy taken generally. We now have to come to closer quarters, and to consider the determinations, or universal significance. In the Pythagorean system numbers seem partly to be themselves allied to categories—that is, to be at once the thought-determinations of unity, of opposition and of the unity of these two moments. In part, the Pythagoreans from the very first gave forth universal ideal determinations of numbers as principles, and recognized, as Aristotle remarks (Metaph. I. 5), as the absolute principles of things, not so much immediate numbers in their arithmetic differences, as the principles of number, i.e. their rational differences. The first determination is unity generally, the next duality or opposition. It is most important to trace back the infinitely manifold nature of the forms and determinations of finality to their universal thoughts as the most simple principles of all determination. These are not differences of one thing from another, but universal and essential differences within themselves. Empirical objects distinguish themselves by outward form; this piece of paper can be distinguished from another, shades are different in colour, men are separated by differences of temperament and individuality. But these determinations are not essential differences; they are certainly essential for the definite particularity of the things, but the whole particularity defined is not an existence which is in and for itself essential, for it is the universal alone which is the self-contained and the substantial. Pythagoras began to seek these first determinations of unity, multiplicity, opposition, &c. With him they are for the most part numbers; but the Pythagoreans did not remain content with this, for they gave them the more concrete determinations, which really belong to their successors. Necessary progression and proof are not to be sought for here; comprehension, the development of duality out of unity are wanting. Universal determinations are only found and established in a quite dogmatic form, and hence the determinations are dry, destitute of process or dialectic, and stationary.
a. The Pythagoreans say that the first simple Notion is unity (μονάς); not the discrete, multifarious, arithmetic one, but identity as continuity and positivity, the entirely universal essence. They further say, according to Sextus (adv. Math. X. 260, 261): “All numbers come under the Notion of the one; for duality is one duality and triplicity is equally a ‘one,’ but the number ten is the one chief number. This moved Pythagoras to assert unity to be the principle of things, because, through partaking of it, each is called one.” That is to say, the pure contemplation of the implicit being of a thing is the one, the being like self; to all else it is not implicit, but a relationship to what is other. Things, however, are much more determined than being merely this dry “one.” The Pythagoreans have expressed this remarkable relationship of the entirely abstract one to the concrete existence of things through “simulation” (μίμησις). The same difficulty which they here encounter is also found in Plato’s Ideas; since they stand over against the concrete as species, the relation of concrete to universal is naturally an important point. Aristotle (Metaph. I. 6) ascribes the expression “participation” (μέθεξις) to Plato, who took it in place of the Pythagorean expression “simulation.” Simulation is a figurative, childish way of putting the relationship; participation is undoubtedly more definite. But Aristotle says, with justice, that both are insufficient; that Plato has not here arrived at any further development, but has only substituted another name. “To say that ideas are prototypes and that other things participate in them is empty talk and a poetic metaphor; for what is the active principle that looks upon the ideas?” (Metaph. I. 9). Simulation and participation are nothing more than other names for relation; to give names is easy, but it is another thing to comprehend.
b. What comes next is the opposition, the duality (δυάς), the distinction, the particular; such determinations have value even now in Philosophy; Pythagoras merely brought them first to consciousness. Now, as this unity relates to multiplicity, or this being-like-self to being another, different applications are possible, and the Pythagoreans have expressed themselves variously as to the forms which this first opposition takes.
(α) They said, according to Aristotle (Metaph. I. 5): “The elements of number are the even and the odd; the latter is the finite” (or principle of limitation) “and the former is the infinite; thus the unity proceeds from both and out of this again comes number.” The elements of immediate number are not yet themselves numbers: the opposition of these elements first appears in arithmetical form rather than as thought. But the one is as yet no number, because as yet it is not quantity; unity and quantity belong to number. Theon of Smyrna[42] says: “Aristotle gives, in his writings on the Pythagoreans, the reason why, in their view, the one partakes of the nature of even and odd; that is, one, posited as even, makes odd; as odd, it makes even. This is what it could not do unless it partook of both natures, for which reason they also called the one, even-odd” (ἀρτιοπέριττον).
(β) If we follow the absolute Idea in this first mode, the opposition will also be called the undetermined duality (ἀόριστος δυάς). Sextus speaks more definitely (adv. Math. X. 261, 262) as follows: “Unity, thought of in its identity with itself (κατ̓ αὐτότητα ἑαυτῆς), is unity; if this adds itself to itself as something different (καθ̓ ἑτερότητα), undetermined duality results, because no one of the determined or otherwise limited numbers is this duality, but all are known through their participation in it, as has been said of unity. There are, according to this, two principles in things; the first unity, through participation in which all number-units are units, and also undetermined duality through participation in which all determined dualities are dualities.” Duality is just as essential a moment in the Notion as is unity. Comparing them with one another, we may either consider the unity to be form and duality matter, or the other way; and both appear in different modes. (αα) Unity, as the being-like-self, is the formless; but in duality, as the unlike, there comes division or form. (ββ) If, on the other hand, we take form as the simple activity of absolute form, the one is what determines; and duality as the potentiality of multiplicity, or as multiplicity not posited, is matter. Aristotle (Met. I. 6) says that it is characteristic of Plato that “he makes out of matter many, but with him the form originates only once; whereas out of one matter only one table proceeds, whoever brings form to matter, in spite of its unity, makes many tables.” He also ascribes this to Plato, that “instead of showing the undetermined to be simple (ἀντὶ τοῦ ἀπείρου ὡς ἑνός), he made of it a duality—the great and small.”
(γ) Further consideration of this opposition, in which Pythagoreans differ from one another, shows us the imperfect beginning of a table of categories which were then brought forward by them, as later on by Aristotle. Hence the latter was reproached for having borrowed these thought-determinations from them; and it certainly was the case that the Pythagoreans first made the opposite to be an essential moment in the absolute. They further determined the abstract and simple Notions, although it was in an inadequate way, since their table presents a mixture of antitheses in the ordinary idea and the Notion, without following these up more fully. Aristotle (Met. I. 5) ascribes these determinations either to Pythagoras himself, or else to Alcmæon “who flourished in the time of Pythagoras’ old age,” so that “either Alcmæon took them from the Pythagoreans or the latter took them from him.” Of these antitheses or co-ordinates to which all things are traced, ten are given, for, according to the Pythagoreans, ten is a number of great significance:—
1. The finite and the infinite.
2. The odd and the even.
3. The one and the many.
4. The right and the left.
5. The male and the female.
6. The quiescent and the moving.
7. The straight and the crooked.
8. Light and darkness.
9. Good and evil.
10. The square and the parallelogram.
This is certainly an attempt towards a development of the Idea of speculative philosophy in itself, i.e. in Notions; but the attempt does not seem to have gone further than this simple enumeration. It is very important that at first only a collection of general thought-determinations should be made, as was done by Aristotle; but what we here see with the Pythagoreans is only a rude beginning of the further determination of antitheses, without order and sense, and very similar to the Indian enumeration of principles and substances.
(δ) We find the further progress of these determinations in Sextus (adv. Math. X. 262-277), when he speaks about an exposition of the later Pythagoreans. It is a very good and well considered account of the Pythagorean theories, which has some thought in it. The exposition follows these lines: “The fact that these two principles are the principles of the whole, is shown by the Pythagoreans in manifold ways.”
א. “There are three methods of thinking things; firstly, in accordance with diversity, secondly, with opposition, and thirdly, according to relation. (αα) What is considered in its mere diversity, is considered for itself; this is the case with those subjects in which each relates only to itself, such as horse, plant, earth, air, water and fire. Such matters are thought of as detached and not in relation to others.” This is the determination of identity with self or of independence. (ββ) “In reference to opposition, the one is determined as evidently contrasting with the other; we have examples of this in good and evil, right and wrong, sacred and profane, rest and movement, &c. (γγ) According to relation (πρός τι), we have the object which is determined in accordance with its relationship to others, such as right and left, over and under, double and half. One is only comprehensible from the other; for I cannot tell which is my left excepting by my right.” Each of these relations in its opposition, is likewise set up for itself in a position of independence. “The difference between relationship and opposition is that in opposition the coming into existence of the ‘one’ is at the expense of the ‘other,’ and conversely. If motion is taken away, rest commences; if motion begins, rest ceases; if health is taken away, sickness begins, and conversely. In a condition of relationship, on the contrary, both take their rise, and both similarly cease together; if the right is removed, so also is the left; the double goes and the half is destroyed.” What is here taken away is taken not only as regards its opposition, but also in its existence. “A second difference is that what is in opposition has no middle; for example, between sickness and health, life and death, rest and motion, there is no third. Relativity, on the contrary, has a middle, for between larger and smaller there is the like; and between too large and too small the right size is the medium.” Pure opposition passes through nullity to opposition; immediate extremes, on the other hand, subsist in a third or middle state, but in such a case no longer as opposed. This exposition shows a certain regard for universal, logical determinations, which now and always have the greatest possible importance, and are moments in all conceptions and in everything that is. The nature of these opposites is, indeed, not considered here, but it is of importance that they should be brought to consciousness.