What is most important to us is the Pythagorean philosophy—not the philosophy of Pythagoras so much as that of the Pythagoreans, as Aristotle and Sextus express it. The two must certainly be distinguished, and from comparing what is given out as Pythagorean doctrine, many anomalies and discrepancies become evident, as we shall see. Plato bears the blame of having destroyed Pythagorean philosophy through absorbing what is Pythagorean in it into his own. But the Pythagorean philosophy itself developed to a point which left it quite other than what at first it was. We hear of many followers of Pythagoras in history who have arrived at this or that conclusion, such as Alcmæon and Philolaus; and we see in many cases the simple undeveloped form contrasted with the further stages of development in which thought comes forth in definiteness and power. We need, however, go no further into the historical side of the distinction, for we can only consider the Pythagorean philosophy generally; similarly we must separate what is known to belong to the Neo-Platonists and Neo-Pythagoreans, and for this end we have sources to draw from which are earlier than this period, namely the express statements found in Aristotle and Sextus.
The Pythagorean philosophy forms the transition from realistic to intellectual philosophy. The Ionic school said that essence or principle is a definite material. The next conclusion is (_α_) that the absolute is not grasped in natural form, but as a thought determination. (_β_) Then it follows that determinations must be posited while the beginning was altogether undetermined. The Pythagorean philosophy has done both.
1. _The System of Numbers_. Thus the original and simple proposition of the Pythagorean philosophy is, according to Aristotle (Metaph. I. 5), “that number is the reality of things, and the constitution of the whole universe in its determinations is an harmonious system of numbers and of their relations.” In what sense is this statement to be taken? The fundamental determination of number is its being a measure; if we say that everything is quantitatively or qualitatively determined, the size and measure is only one aspect or characteristic which is present in everything, but the meaning here is that number itself is the essence and the substance of things, and not alone their form. What first strikes us as surprising is the boldness of such language, which at once sets aside everything which to the ordinary idea is real and true, doing away with sensuous existence and making it to be the creation of thought. Existence is expressed as something which is not sensuous, and thus what to the senses and to old ideas is altogether foreign, is raised into and expressed as substance and as true Being. But at the same time the necessity is shown for making number to be likewise Notion, to manifest it as the activity of its unity with Being, for to us number does not seem to be in immediate unity with the Notion.
Now although this principle appears to us to be fanciful and wild, we find in it that number is not merely something sensuous, therefore it brings determination with it, universal distinctions and antitheses. The ancients had a very good knowledge of these. Aristotle (Metaph. I. 6) says of Plato: “He maintained that the mathematical elements in things are found outside of what is merely sensuous, and of ideas, being between both; it differs from what is sensuous in that it is eternal and unchangeable, and from ideas, in that it possesses multiplicity, and hence each can resemble and be similar to another, while each idea is for itself one alone.” That is, number can be repeated; thus it is not sensuous, and still not yet thought. In the life of Pythagoras, this is further said by Malchus (46, 47): “Pythagoras propounded philosophy in this wise in order to loose thought from its fetters. Without thought nothing true can be discerned or known; thought hears and sees everything in itself, the rest is lame and blind. To obtain his end, Pythagoras makes use of mathematics, since this stands midway between what is sensuous and thought, as a kind of preliminary to what is in and for itself.” Malchus quotes further (48, 53) a passage from an early writer, Moderatus: “Because the Pythagoreans could not clearly express the absolute and the first principles through thought, they made use of numbers, of mathematics, because in this form determinations could be easily expressed.” For instance, similarity could be expressed as one, dissimilarity as two. “This mode of teaching through the use of numbers, whilst it was the first philosophy, is superseded on account of its mysterious nature. Plato, Speusippus, Aristotle, &c., have stolen the fruits of their work from the Pythagoreans by making a simple use of their principle.” In this passage a perfect knowledge of numbers is evident.
The enigmatic character of the determination through number is what most engages our attention. The numbers of arithmetic answers to thought-determinations, for number has the “one” as element and principle; the one, however, is a category of being-for-self, and thus of identity with self, in that it excludes all else and is indifferent to what is “other.” The further determinations of number are only further combinations and repetitions of the one, which all through remains fixed and external; number, thus, is the most utterly dead, notionless continuity possible; it is an entirely external and mechanical process, which is without necessity. Hence number is not immediate Notion, but only a beginning of thought, and a beginning in the worst possible way; it is the Notion in its extremest externality, in quantitative form, and in that of indifferent distinction. In so far, the one has within itself both the principle of thought and that of materiality, or the determination of the sensuous. In order that anything should have the form of Notion, it must immediately in itself, as determined, relate itself to its opposite, just as positive is related to negative; and in this simple movement of the Notion we find the ideality of differences and negation of independence to be the chief determination. On the other hand, in the number three, for instance, there are always three units, of which each is independent; and this is what constitutes both their defect and their enigmatic character. For since the essence of the Notion is innate, numbers are the most worthless instruments for expressing Notion-determinations.
Now the Pythagoreans did not accept numbers in this indifferent way, but as Notion. “At least they say that phenomena must be composed of simple elements, and it would be contrary to the nature of things if the principle of the universe pertained to sensuous phenomena. The elements and principles are thus not only intangible and invisible, but altogether incorporeal.”[41] But how they have come to make numbers the original principle or the absolute Notion, is better shown from what Aristotle says in his Metaphysics (I. 5), although he is shorter than he would have been, because he alleges that elsewhere (infra., p. 214) he has spoken of it. “In numbers they thought that they perceived much greater similitude to what is and what takes place than in fire, water, or earth; since a certain property of numbers (_τοιονδὶ πάθος_) is justice, so is it with (_τοιονδὶ_) the soul and understanding; another property is opportunity, and so on. Since they further saw the conditions and relations of what is harmonious present in numbers, and since numbers are at the basis of all natural things, they considered numbers to be the elements of everything, and the whole heavens to be a harmony and number.” In the Pythagoreans we see the necessity for one enduring universal idea as a thought-determination. Aristotle (Met. XII. 4), speaking of ideas, says: “According to Heraclitus, everything sensuous flows on, and thus there cannot be a science of the sensuous; from this conviction the doctrine of ideas sprang. Socrates is the first to define the universal through inductive methods; the Pythagoreans formerly concerned themselves merely with a few matters of which they derived the notions from numbers—as, for example, with what opportuneness, or right, or marriage are.” It is impossible to discern what interest this in itself can have; the only thing which is necessary for us as regards the Pythagoreans, is to recognize any indications of the Idea, in which there may be a progressive principle.
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.
ב. “Now since these three represent three different genera, the subjects and the two-fold opposite, there must be a higher genus over each of them which takes the first place, since the genus comes before its subordinate kinds. If the universal is taken away, so is the kind; on the other hand, if the kind, not the genus, for the former depends on the latter, but not the contrary way.” (_αα_) “The Pythagoreans have declared the one to be the highest genus of what is considered as in and for itself” (of subjects in their diversity); this is, properly speaking, nothing more than translating the determinations of the Notion into numbers. (_ββ_) “What is in opposition has, they say, as its genus the like and the unlike; rest is the like, for it is capable of nothing more and nothing less; but movement is the unlike. Thus what is according to nature is like itself; it is a point which is not capable of being intensified (_ἀνεπίτατος_); what is opposed to it is unlike. Health is like, sickness is unlike. (_γγ_) The genus of that which is in an indifferent relationship is excess and want, the more and the less;” in this we have the quantitative relation just as we formerly had the qualitative.
ג. We now come for the first time to the two opposites: “These three genera of what is for itself, in opposition and in relationship, must now come under”—yet simpler, higher—“genera,” _i.e._ thought-determinations. “Similarity reduces itself to the determination of unity.” The genus of the subjects is the very being on its own account. “Dissimilarity, however, consists of excess and want, but both of these come under undetermined duality;” they are the undetermined opposition, opposition generally. “Thus from all these relationships the first unity and the undetermined duality proceed;” the Pythagoreans said that such are found to be the universal modes of things. “From these, there first comes the ‘one’ of numbers and the ‘two’ of numbers; from the first unity, the one; from the unity and the undetermined duality the two; for twice the one is two. The other numbers take their origin in a similar way, for the unity over moves forward, and the undetermined duality generates the two.” This transition of qualitative into quantitative opposition is not clear. “Hence underlying these principles, unity is the active principle” or form, “but the two is the passive matter; and just as they make numbers arise from them, so do they make the system of the world and that which is contained in it.” The nature of these determinations is to be found in transition and in movement. The deeper significance of this reflection rests in the connection of universal thought-determinations with arithmetic numbers—in subordinating these and making the universal genus first.
Before I say anything of the further sequence of these numbers, it must be remarked that they, as we see them represented here, are pure Notions. (_α_) The absolute, simple essence divides itself into unity and multiplicity, of which the one sublates the other, and at the same time it has its existence in the opposition. (_β_) The opposition has at the same time subsistence, and in this is found the manifold nature of equivalent things. (_γ_) The return of absolute essence into itself is the negative unity of the individual subject and of the universal or positive. This is, in fact, the pure speculative Idea of absolute existence; it is this movement: with Plato the Idea is nothing else. The speculative makes its appearance here as speculative; whoever does not know the speculative, does not believe that in indicating simple Notions such as these, absolute essence is expressed. One, many, like, unlike, more or less, are trivial, empty, dry moments; that there should be contained in them absolute essence, the riches and the organization of the natural, as of the spiritual world, does not seem possible to him who, accustomed to ordinary ideas, has not gone back from sensuous existence into thought. It does not seem to such a one that God is, in a speculative sense, expressed thereby—that what is most sublime can be put in those common words, what is deepest, in what is so well known, self-evident and open, and what is richest, in the poverty of these abstractions.
It is at first in opposition to common reality that this idea of reality as the manifold of simple essence, has in itself its opposition and the subsistence of the same; this essential, simple Notion of reality is elevation into thought, but it is not flight from what is real, but the expression of the real itself in its essence. We here find the Reason which expresses its essence; and absolute reality is unity immediately in itself. Thus it is pre-eminently in relation to this reality that the difficulties of those who do not think speculatively have become so intense. What is its relation to common reality? What has taken place is just what happens with the Platonic Ideas, which approximate very closely to these numbers, or rather to pure Notions. That is to say, the first question is, “Numbers, where are they? Dispersed through space, dwelling in independence in the heaven of ideas? They are not things immediately in themselves, for a thing, a substance, is something quite other than a number: a body bears no similarity to it.” To this we may answer that the Pythagoreans did not signify anything like that which we understand by prototypes—as if ideas, as the laws and relations of things, were present in a creative consciousness as thoughts in the divine understanding, separated from things as are the thoughts of an artist from his work. Still less did they mean only subjective thoughts in our consciousness, for we use the absolute antithesis as the explanation of the existence of qualities in things, but what determines is the real substance of what exists, so that each thing is essentially just its having in it unity, duality, as also their antithesis and connection. Aristotle (Met. I. 5, 6) puts it clearly thus: “It is characteristic of the Pythagoreans that they did not maintain the finite and the infinite and the One, to be, like fire, earth, &c., different natures or to have another reality than things; for the Infinite and the abstract One are to them, the substance of the things of which they are predicated. Hence too, they said, Number is the essence of all things. Thus they do not separate numbers from things, but consider them to be things themselves. Number to them is the principle and matter of things, as also their qualities and forces;” hence it is thought as substance, or the thing as it is in the reality of thought.
These abstract determinations then became more concretely determined, especially by the later philosophers, in their speculations regarding God. We may instance Iamblichus, for example, in the work _θεολογούμενα ἀριθμητικῆς_, ascribed to him by Porphyry and Nicomachus. Those philosophers sought to raise the character of popular religion, for they inserted such thought-determinations as these into religious conceptions. By Monas they understood nothing other than God; they also call it Mind, the Hermaphrodite (which contains both determinations, odd as well as even), and likewise substance, reason, chaos (because it is undetermined), Tartarus, Jupiter, and Form. They called the duad by similar names, such as matter, and then the principle of the unlike, strife, that which begets, Isis, &c.
c. The triad (_τριάς_) has now become a most important number, seeing that in it the monad has reached reality and perfection. The monad proceeds through the duad, and again brought into unity with this undetermined manifold, it is the triad. Unity and multiplicity are present in the triad in the worst possible way—as an external combination; but however abstractly this is understood, the triad is still a profound form. The triad then is held to be the first perfect form in the universal. Aristotle (De Cœlo I. 1) puts this very clearly: “The corporeal has no dimension outside of the Three; hence the Pythagoreans also say that the all and everything is determined through triplicity,” that is, it has absolute form. “For the number of the whole has end, middle, and beginning; and this is the triad.” Nevertheless there is something superficial in the wish to bring everything under it, as is done in the systematization of the more modern natural philosophy. “Therefore we, too, taking this determination from nature, make use of it in the worship of the gods, so that we believe them to have been properly apostrophized only when we have called upon them three times in prayer. Two we call both, but not all; we speak first of three as all. What is determined through three is the first totality (_πᾶν_); what is in triple form is perfectly divided. Some is merely in one, other is only in two, but this is All.” What is perfect, or has reality, is its identity, opposition and unity, like number generally; but in triplicity this is actual, because it has beginning, middle, and end. Each thing is simple as beginning; it is other or manifold as middle, and its end is the return of its other nature into unity or mind; if we take this triplicity from a thing, we negate it and make of it an abstract construction of thought.