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.
It is now comprehensible that Christians sought and found the Trinity in this threefold nature. It has often been made a superficial reason for objecting to them; sometimes the idea of the Trinity as it was present to the ancients, was considered as above reason, as a secret, and hence, too high; sometimes it was deemed too absurd. But from the one cause or from the other, they did not wish to bring it into closer relation to reason. If there is a meaning in this Trinity, we must try to understand it. It would be an anomalous thing if there were nothing in what has for two thousand years been the holiest Christian idea; if it were too holy to be brought down to the level of reason, or were something now quite obsolete, so that it would be contrary to good taste and sense to try to find a meaning in it. It is the Notion of the Trinity alone of which we can speak, and not of the idea of Father and Son, for we am not dealing with these natural relationships.
d. The Four (τετράς) is the triad but more developed, and hence with the Pythagoreans it held a high position. That the tetrad should be considered to be thus complete, reminds one of the four elements, the physical and the chemical, the four continents, &c. In nature four is found to be present everywhere, and hence this number is even now equally esteemed in natural philosophy. As the square of two, the fourfold is the perfection of the two-fold in as far as it—only having itself as determination, i.e. being multiplied with itself—returns into identity with itself. But in the triad the tetrad is in so far contained, as that the former is the unity, the other-being, and the union of both these moments, and thus, since the difference, as posited, is a double, if we count it, four moments result. To make this clearer, the tetrad is comprehended as the τετρακτύς, the efficient, active four (from τέτταρα and ἄγω); and afterwards this is by the Pythagoreans made the most notable number. In the fragments of a poem of Empedocles, who originally was a Pythagorean, it is shown in what high regard this tetraktus, as represented by Pythagoras, was held:
“If thou dost this,
It will lead thee in the path of holy piety. I swear it
By the one who to our spirit has given the Tetraktus,
Which has in it eternal nature’s source and root.”[43]