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] e. From this the Pythagoreans proceed to the ten, another form of this tetrad. As the four is the perfect form of three, this fourfold, thus perfected and developed so that all its moments shall be accepted as real differences, is the number ten (_δεκάς_), the real tetrad. Sextus (adv. Math. IV. 3; VII. 94, 95) says: “Tetraktus means the number which, comprising within itself the four first numbers, forms the most perfect number, that is the number ten; for one and two and three and four make ten. When we come to ten, we again consider it as a unity and begin once more from the beginning. The tetraktus, it is said, has the source and root of eternal nature within itself, because it is the Logos of the universe, of the spiritual and of the corporeal.” It is an important work of thought to show the moments not merely to be four units, but complete numbers; but the reality in which the determinations are laid hold of, is here, however, only the external and superficial one of number; there is no Notion present although the tetraktus does not mean number so much as idea. One of the later philosophers, Proclus, (in Timæum, p. 269) says, in a Pythagorean hymn:— “The divine number goes on,”...“Till from the still unprofaned sanctuary of the Monad It reaches to the holy Tetrad, which creates the mother of all that is; Which received all within itself, or formed the ancient bounds of all, Incapable of turning or of wearying; men call it the holy Dekad.”
What we find about the progression of the other numbers is more indefinite and unsatisfying, and the Notion loses itself in them. Up to five there may certainly be a kind of thought in numbers, but from six onwards they are merely arbitrary determinations.
2. _Application of the System to the Universe_. This simple idea and the simple reality contained therein, must now, however, be further developed in order to come to reality as it is when put together and expanded. The question now meets us as to how, in this relation, the Pythagoreans passed from abstract logical determinations to forms which indicate the concrete use of numbers. In what pertains to space or music, determinations of objects formed by the Pythagoreans through numbers, still bear a somewhat closer relation to the thing, but when they enter the region of the concrete in nature and in mind, numbers become purely formal and empty.
a. To show how the Pythagoreans constructed out of numbers the system of the world, Sextus instances (adv. Math. X. 277-283), space relations, and undoubtedly we have in them to do with such ideal principles, for numbers are, in fact, perfect determinations of abstract space. That is to say, if we begin with the point, the first negation of vacuity, “the point corresponds to unity; it is indivisible and the principle of lines, as the unity is that of numbers. While the point exists as the monad or One, the line expresses the duad or Two, for both become comprehensible through transition; the line is the pure relationship of two points and is without breadth. Surface results from the threefold; but the solid figure or body belongs to the fourfold, and in it there are three dimensions present. Others say that body consists of one point” (_i.e._ its essence is one point), “for the flowing point makes the line, the flowing line, however, makes surface, and this surface makes body. They distinguish themselves from the first mentioned, in that the former make numbers primarily proceed from the monad and the undetermined duad, and then points and lines, plane surfaces and solid figures, from numbers, while they construct all from one point.” To the first, distinction is opposition or form set forth as duality; the others have form as activity. “Thus what is corporeal is formed under the directing influence of numbers, but from them also proceed the definite bodies, water, air, fire, and the whole universe generally, which they declare to be harmonious. This harmony is one which again consists of numeral relations only, which constitute the various concords of the absolute harmony.”
We must here remark that the progression from the point to actual space also has the signification of occupation of space, for “according to their fundamental tenets and teaching,” says Aristotle (Metaph. I. 8), “they speak of sensuously perceptible bodies in nowise differently from those which are mathematical.” Since lines and surfaces are only abstract moments in space, external construction likewise proceeds from here very well. On the other hand, the transition from the occupation of space generally to what is determined, to water, earth, &c., is quite another thing and is more difficult; or rather the Pythagoreans have not taken this step, for the universe itself has, with them, the speculative, simple form, which is found in the fact of being represented as a system of number-relations. But with all this, the physical is not yet determined.
b. Another application or exhibition of the essential nature of the determination of numbers is to be found in the relations of music, and it is more especially in their case that number constitutes the determining factor. The differences here show themselves as various relations of numbers, and this mode of determining what is musical is the only one. The relation borne by tones to one another is founded on quantitative differences whereby harmonies may be formed, in distinction to others by which discords are constituted. The Pythagoreans, according to Porphyry (De vita Pyth. 30), treated music as something soul-instructing and scholastic [Psychagogisches und Pädagogisches]. Pythagoras was the first to discern that musical relations, these audible differences, are mathematically determinable, that what we hear as consonance and dissonance is a mathematical arrangement. The subjective, and, in the case of hearing, simple feeling which, however, exists inherently in relation, Pythagoras has justified to the understanding, and he attained his object by means of fixed determinations. For to him the discovery of the fundamental tones of harmony are ascribed, and these rest on the most simple number-relations. Iamblichus (De vita Pyth. XXVI. 115) says that Pythagoras, in passing by the workshop of a smith, observed the strokes that gave forth a particular chord; he then took into consideration the weight of the hammer giving forth a certain harmony, and from that determined mathematically the tone as related thereto.[44] And finally he applied the same, and experimented in strings, by which means there were three different relations presented to him—Diapason, Diapente, and Diatessaron. It is known that the tone of a string, or, in the wind instrument, of its equivalent, the column of air in a reed, depends on three conditions; on its length, on its thickness, and on the amount of tension. Now if we have two strings of equal thickness and length, a difference in tension brings about a difference in sound. If we want to know what tone any string has, we have only to consider its tension, and this may be measured by the weight depending from the string, by means of which it is extended. Pythagoras here found that if one string were weighted with twelve pounds, and another with six (_λόγος διπλάσιος_, 1: 2) it would produce the musical chord of the octave (_διὰ πασῶν_); the proportion of 8: 12, or of 2: 3 (_λόγος ἡμιόλιος_) would give the chord of the fifth (_διὰ πέντε_); the proportion of 9 : 12, or 3: 4 (_λόγος ἐπίτριτος_), the fourth (_διὰ τεσσάρων_).[45] A different number of vibrations in like times determines the height and depth of the tone, and this number is likewise proportionate to the weight, if thickness and length are equal. In the first case, the more distended string makes as many vibrations again as the other; in the second case, it makes three vibrations for the other’s two, and so it goes on. Here number is the real factor which determines the difference, for tone, as the vibration of a body, is only a quantitatively determined quiver or movement, that is, a determination made through space and time. For there can be no determination for the difference excepting that of number or the amount of vibrations in one time; and hence a determination made through numbers is nowhere more in place than here. There certainly are also qualitative differences, such as those existing between the tones of metals and catgut strings, and between the human voice and wind instruments; but the peculiar musical relation borne by the tone of one instrument to another, in which harmony is to be found, is a relationship of numbers.
From this point the Pythagoreans enter into further applications of the theory of music, in which we cannot follow them. The _à priori_ law of progression, and the necessity of movement in number-relations, is a matter which is entirely dark; minds confused may wander about at will, for everywhere ideas are hinted at, and superficial harmonies present themselves and disappear again. But in all that treats of the further construction of the universe as a numerical system, we have the whole extent of the confusion and turbidity of thought belonging to the later Pythagoreans. We cannot say how much pains they took to express philosophic thought in a system of numbers, and also to understand the expressions given utterance to by others, and to put in them all the meaning possible. When they determined the physical and the moral universe by means of numbers, everything came into indefinite and insipid relationships in which the Notion disappeared. In this matter, however, so far as the older Pythagoreans are concerned, we are acquainted with the main principles only. Plato exemplifies to us the conception of the universe as a system of numbers, but Cicero and the ancients always call these numbers the Platonic, and it does not appear that they were ascribed to the Pythagoreans. It was thus later on that this came to be said; even in Cicero’s time they had become proverbially dark, and there is but little after all that is really old.
c. The Pythagoreans further constructed the heavenly bodies of the visible universe by means of numbers, and here we see at once the barrenness and abstraction present in the determination of numbers. Aristotle says (Met. I. 5), “Because they defined numbers to be the principles of all nature, they brought under numbers and their relationships all determinations and all sections, both of the heavens and of all nature; and where anything did not altogether conform, they sought to supply the deficiency in order to bring about a harmony. For instance, as the Ten or dekad appeared to them to be the perfect number, or that which embraces the whole essence of numbers, they said that the spheres moving in the heavens must be ten; but as only nine of these are visible, they made out a tenth, the Antichthone (_ἀντίχθονα_).” These nine are, first the milky way, or the fixed stars, and after that the seven stars which were then all held to be planets: Saturn, Jupiter, Mars, Venus, Mercury, the Sun, Moon, and in the last and ninth place, the Earth. The tenth is thus the Antichthone, and in regard to this it must remain uncertain whether the Pythagoreans considered it to be the side of the Earth which is turned away, or as quite another body.
Aristotle says, in reference to the specially physical character of these spheres (De cœlo II. 13 and 9), “Fire was by the Pythagoreans placed in the middle, but the Earth was made a star that moved around this central body in a circle.” This circle is, then, a sphere, which, as the most perfect of figures, corresponds to the dekad. We here find a certain similarity to our ideas of the solar system, but the Pythagoreans did not believe the fire to be the sun. “They thus,” says Aristotle, “rely, not on sensuous appearance, but on reasons,” just as we form conclusions in accordance with reasons as opposed to sensuous appearances; and indeed this comes to us still as the first example of things being in themselves different from what they appear. “This fire, that which is in the centre, they called Jupiter’s place of watch. Now these ten spheres make, like all that is in motion, a tone; but each makes a different one, according to the difference in its size and velocity. This is determined by means of the different distances, which bear an harmonious relationship to one another, in accordance with musical intervals; by this means an harmonious sound arises in the moving spheres”—a universal chorus.
We must acknowledge the grandeur of this idea of determining everything in the system of the heavenly spheres through number-relations which have a necessary connection amongst themselves, and have to be conceived of as thus necessarily related; it is a system of relations which must also form the basis and essence of what can be heard, or music. We have, comprehended here in thought, a system of the universe; the solar system is alone rational to us, for the other stars are devoid of interest. To say that there is music in the spheres, and that these movements are tones, may seem just as comprehensible to us as to say that the sun is still and the earth moves, although both are opposed to the dictates of sense. For, seeing that we do not see the movement, it may be that we do not hear the notes. And there is little difficulty in imagining a universal silence in these vast spheres, since we do not hear the chorus, but it is more difficult to give a reason for not hearing this music. The Pythagoreans say, according to the last quoted passage of Aristotle, that we do not hear it because we live in it, like the smith who gets accustomed to the blows of his hammer. Since it belongs to our substance and is identical with ourselves, nothing else, such as silence, by which we might know the other, comes into relationship with us, for we are conceived of as entirely within the movement. But the movement does not become a tone, in the first place, because pure space and time, the elements in movement, can only raise themselves into a proper voice, unstimulated from without, in an animate body, and movement first reaches this definite, characteristic individuality in the animal proper; and, in the next place, because the heavenly bodies are not related to one another as bodies whose sound requires for its production, contact, friction, or shock, in response to which, and as the negation of its particularity its own momentary individuality resounds in elasticity; for heavenly bodies are independent of one another, and have only a general, non-individual, free motion.
We may thus set aside sound; the music of the spheres is indeed a wonderful conception, but it is devoid of any real interest for us. If we retain the conception that motion, as measure, is a necessarily connected system of numbers, as the only rational part of the theory, we must maintain that nothing further has transpired to the present day. In a certain way, indeed, we have made an advance upon Pythagoras. We have learned from Kepler about laws, about eccentricity, and the relation of distances to the times of revolution, but no amount of mathematics has as yet been able to give us the laws of progression in the harmony through which the distances are determined. We know empirical numbers well enough, but everything has the semblance of accident and not of necessity. We are acquainted with an approximate rule of distances, and thus have correctly foretold the existence of planets where Ceres, Vesta, Pallas, &c., were afterwards discovered—that is, between Mars and Jupiter. But astronomy has not as yet found in it a consistent sequence in which there is rationality; on the other hand, it even looks with disdain on the appearance of regularity presented by this sequence, which is, however, on its own account, a most important matter, and one which should not be forgotten.
d. The Pythagoreans also applied their principle to the Soul, and thus determined what is spiritual as number. Aristotle (De anim. I. 2) goes on to tell that they thought that solar corpuscles are soul, others, that it is what moves them; they adopted this idea because the corpuscles are ever moving, even in perfect stillness, and hence they must have motion of their own. This does not signify much, but it is evident from it that the determination of self-movement was sought for in the soul. The Pythagoreans made a further application of number-conceptions to the soul after another form, which Aristotle describes in the same place as follows:—“Thought is the one, knowledge or science is the two, for it comes alone out of the one. The number of the plane is popular idea, opinion; the number of the corporeal is sensuous feeling. Everything is judged of either by thought, or science, or opinion, or feeling.” In these ideas, which we must, however, ascribe to later Pythagoreans, we may undoubtedly find some adequacy, for while thought is pure universality, knowledge deals with something “other,” since it gives itself a determination and a content; but feeling is the most developed in its determinateness. “Now because the soul moves itself, it is the self-moving number,” yet we never find it said that it is connected with the monad.
This is a simple relationship to number-determinations. Aristotle instances (De anim. I. 3) one more intricate from Timæus: “The soul moves itself, and hence also the body because it is bound up with body; it consists of elements and is divided according to harmonic numbers, and hence it has feeling and an immediately indwelling (_σύμφυτον_) harmony. In order that the whole may have an harmonious movement, Timæus has bent the straight line of harmony (_εὐθυωρίαν_) into a circle, and again divided off from the whole circle two circles, which are doubly connected; and the one of these circles is again divided into seven circles, so that the movements of the soul may resemble those of the heavens.” The more definite significance of these ideas Aristotle unfortunately has not given; they contain a profound knowledge of the harmony of the whole, but yet they are forms which themselves remain dark, because they are clumsy and unsuitable. There is always a forcible turning and twisting, a struggle with the material part of the representation, as there is in mythical and distorted forms: nothing has the pliability of thought but thought itself. It is remarkable that the Pythagoreans have grasped the soul as a system which is a counterpart of the system of the heavens. In Plato’s Timæus this same idea is more definitely brought forward. Plato also gives further number-relations, but not their significance as well; even to the present day no one has been able to make any particular sense out of them. An arrangement of numbers such as this is easy, but to give to it a real significance is difficult, and, when done, it always must be arbitrary.
There is still something worthy of attention in what is said by the Pythagoreans in reference to the soul, and this is their doctrine of the transmigration of souls. Cicero (Tusc. Quæst. I. 16) says: “Pherecydes, the teacher of Pythagoras, first said that the souls of men were immortal.” The doctrine of the transmigration of souls extends even to India, and, without doubt, Pythagoras took it from the Egyptians; indeed Herodotus (II. 123) expressly says so. After he speaks of the mythical ideas of the Egyptians as to the lower world, he continues: “The Egyptians were the first to say that the soul of man is immortal, and that, when the body disappears, it goes into another living being; and when it has gone through all the animals of land and sea, and likewise birds, it again takes the body of a man, the period being completed in 3000 years.” Diogenes Laertius says in this connection (VIII. 14) that the soul, according to Pythagoras, goes through a circle. “These ideas,” proceeds Herodotus, “are also found amongst the Greeks; there are some who, earlier or later, have made use of this particular doctrine, and have spoken of it as if it were their own; I know their names very well, but I will not mention them.” He undoubtedly meant Pythagoras and his followers. In the sequel, much that is given utterance to is fictitious: “Pythagoras himself is said to have stated that his former personality was known to him.
Hermes granted him a knowledge of his circumstances before his birth. He lived as the son of Hermes, Æthalides, and then in the Trojan war as Euphorbus, the son of Panthous, who killed Patroclus, and was killed by Menelaus; in the third place he was Hermotimus; fourthly, Pyrrhus, a fisherman of Delos; in all he lived 207 years. Euphorbus’ shield was offered up to Apollo by Menelaus, and Pythagoras went to the temple and, from the mouldering shield, showed the existence of signs, hitherto not known of, by which it was recognized.”[46] We shall not treat further of these very various and foolish stories.
As in the case of the brotherhood copied from the Egyptian priesthood, so must we here set aside this oriental and un-Greek idea of the transmigration of souls. Both were too far removed from the Greek spirit to have had a place and a development there. With the Greeks, the consciousness of a higher, freer individuality has become too strong to allow any permanence to the idea of metempsychosis, according to which, man, this independent and self-sufficing Being, takes the form of a beast. They have, indeed, the conception of men as becoming springs of water, trees, animals, &c., but the idea of degradation which comes as a consequence of sin, lies at its root. Aristotle (De anim. I. 3) shortly and in his own manner deals with and annihilates this idea of the Pythagoreans. “They do not say for what reason soul dwells in body, nor how the latter is related to it. For owing to their unity of nature when one acts the other suffers: one moves and the other is moved, but none of this happens in what is mutually contingent. According to the Pythagorean myths any soul takes to any body, which is much like making architects take to flutes. For crafts must necessarily have tools and soul body; but each tool must have its proper form and kind.” It is implied in the transmigration of souls that the organization of the body is something accidental to the human soul; this refutation by Aristotle is complete. The eternal idea of metempsychosis had philosophic interest only as the inner Notion permeating all these forms, the oriental unity which appears in everything; we have not got this signification here, or at best we have but a glimmering of it. If we say that the particular soul is, as a definite thing, to wander about throughout all, we find firstly, that the soul is not a thing such as Leibnitz’ Monad, which, like a bubble in the cup of coffee, is possibly a sentient, thinking soul; in the second place an empty identity of the soul-thing such as this has no interest in relation to immortality.
3. _Practical Philosophy_. As regards the practical philosophy of Pythagoras, which is closely connected with what has gone before, there is but little that is philosophic known to us. Aristotle (Magn. Moral. I. 1) says of him that “he first sought to speak of virtue, but not in the right way, for, because he deduced the virtues from numbers, he could not form of them any proper theory.” The Pythagoreans adopted ten virtues as well as ten heavenly spheres. Justice, amongst others, is described as the number which is like itself in like manner (_ἴσακις ἴσος_); it is an even number, which remains even when multiplied with itself. Justice is pre-eminently what remains like itself; but this is an altogether abstract determination, which applies to much that is, and which does not exhaust the concrete, thus remaining quite indeterminate.
Under the name of the “Golden words,” we have a collection of hexameters which are a succession of moral reflections, but which are rightly ascribed to later Pythagoreans. They are old, well-known, moral maxims, which are expressed in a simple and dignified way, but which do not contain anything remarkable. They begin with the direction “to honour the immortal gods as they are by law established,” and further, “Honour the oath and then the illustrious heroes;” elsewhere they go on to direct “honour to be paid to parents and to relatives,” &c.[47] Such matter does not deserve to be regarded as philosophy, although it is of importance in the process of development.