Hegel's Lectures on the History of Philosophy · Georg Wilhelm Friedrich Hegel
B. PYTHAGORAS AND THE PYTHAGOREANS.
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.
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.”
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.
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.
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.
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.
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.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.