Hegel’s History of Philosophy: Greek Philosophy

B. Pythagoras and the Pythagoreans.

The later Neo-Pythagoreans have written many extensive biographies of Pythagoras, and are especially diffuse as regards the Pythagorean brotherhood. But it must be taken into consideration that these often distorted statements must not be regarded as historical. The life of Pythagoras thus first comes to us in history through the medium of the ideas belonging to the first centuries after Christ, and more or less in the style in which the life of Christ is written, on the ground of ordinary actuality, and not in a poetic atmosphere; it appears to be the intermingling of many marvellous and extravagant tales, and to take its origin in part from eastern ideas and in part from western. In acknowledging the remarkable nature of his life and genius and of the life which he inculcated on his followers, it was added that his dealings were not with right things, and that he was a magician and one who had intercourse with higher beings. All the ideas of magic, that medley of unnatural And natural, the mysteries which pervade a clouded, miserable imagination, and the wild ideas of distorted brains, have attached themselves to him.

However corrupt the history of his life, his philosophy is as much so. Everything engendered by Christian melancholy and love of allegory has been identified with it. The treatment of Plato in Christian times has quite a different character. Numbers have been much used as the expression of ideas, and this on the one hand has a semblance of profundity. For the fact that another significance than that immediately presented is implied in them, is evident at once; but how much there is within them is neither known by him who speaks nor by him who seeks to understand; it is like the witches’ rhyme (one time one) in Goethe’s “Faust.” The less clear the thoughts, the deeper they appear; what is most essential, but most difficult, the expression of oneself in definite conceptions, is omitted. Thus Pythagoras’ philosophy, since much has been added to it by those who wrote of it, may similarly appear as the mysterious product of minds as shallow and empty as they are dark. Fortunately, however, we have a special knowledge of the theoretic, speculative side of it, and that, indeed, from Aristotle and Sextus Empiricus, who have taken considerable trouble with it. Although later Pythagoreans disparage Aristotle on account of his exposition, he has a place above any such disparagement, and therefore to them no attention must be given.

In later times a quantity of writings wore disseminated and foisted upon Pythagoras. Diogenes Laërtius (VIII. 6, 7) mentions many which were by him, and others which were set down to him in order to obtain authority for them. But in the first place we have no writings by Pythagoras, and secondly it is doubtful whether any ever did exist. We have quotations from. these in unsatisfactory fragments, not from Pythagoras, but from Pythagoreans. It cannot be decisively determined which developments and interpretations belonged to the ancients and which to the moderns; yet with Pythagoras and the ancient Pythagoreans the determinations were not worked out in so concrete away as later.

As to the life of Pythagoras, we hear from Diogenes Laërtius (VIII. 1-3, 45) that he flourished about the 60th Olympiad (540 B.C.). His birth is usually placed in the 49th or 50th Olympiad (584 B.C.) ; by Larcher in Tennemann (Vol. I., pp. 413, 414), much earlier — in the 43rd Olympiad (43, 1, i.e. 608 B.C.). He was thus contemporaneous with Thales and Anaximander. If Thales’ birth were in the 38th Olympiad and that of Pythagoras in the 43rd, Pythagoras was only twenty-one years younger than he; he either only differed by a couple of years from Anaximander (01. 42, 3) in age, or the latter was twenty-six years older. Anaximenes was from twenty to twenty-five years younger than Pythagoras. His birthplace was the Island of Samos, and hence he belonged to the Greeks of Asia Minor, which place we have hitherto found to be the seat of philosophy. Pythagoras is said by Herodotus (IV., 93 to 96) to have been the son of Mnesarchus, with whom Zalmoxis served as slave in Samos; Zalmoxis obtained freedom and riches, became ruler of the Getæ, and asserted that he and his people would not die. He built a subterranean habitation and there withdrew himself from his subjects; after four years he re-appeared; hence the Getans believed in immortality. Herodotus thinks, however, that Zalmoxis was undoubtedly much older than Pythagoras.

His youth was spent at the court of Polyerates, under whose rule Samos was brought, not only to wealth, but also to the possession of culture and art. In this prosperous period, according to Herodotus (III., 39), it possessed a fleet of a hundred ships. His father was an artist or engraver, but reports vary as to this, as also as to his country, some saying that his family was of Tyrrhenian origin and did riot go to Samos till after Pythagoras’ birth. That may be as it will, for his youth was spent in Samos and he must hence have been naturalized there, and to it he belongs. He soon journeyed to the main land of Asia Minor and is said there to have become acquainted with Thales. From thence he travelled to Phoenicia and Egypt, as Iamblichus (III., 13,14) says in his biography of Pythagoras. With both countries Asia Minor had many links, commercial and political, and it is related that he was recommended by Polyerates to King Amasis, who, according to Herodotus (II. 154), attracted many Greeks to the country, and had Greek troops and colonies. The narratives of further journeys into the interior of Asia, to the Persian magicians and Indians, seem to be altogether fabulous, although travelling, then as now, was considered to be a means of culture. As Pythagoras travelled with a scientific purpose, it is said that he had himself initiated into nearly all the mysteries of Greeks and of Barbarians, and thus he obtained admission into the order or caste of the Egyptian priesthood.

These mysteries that we meet with amongst the Greeks, and which are hold to be the sources of much wisdom, appear in their religion to have stood in the relationship of doctrine to worship. This last existed in offerings and solemn festivals only, but to ordinary conceptions, to a consciousness of these conceptions, there is no transition visible unless they were preserved in poems as traditions. The doctrines themselves, or the act of bringing the actual home to the conception, seems to have been confined to the mysteries; we find it to be the case, however, that it is not only the ideas as in our teaching, but also the body that is laid claim to — that there was brought home to mail by sending him to wander amongst his fellow-men, both the abandonment of his sensuous consciousness and the purification and sanctification of the body. Of philosophic matter, however, there is as little openly declared as possible, and just. as we know the system of freemasonry, there is no secret in those mysteries.

His alliance with the Egyptian priesthood had a most important influence upon Pythagoras, not through the derivation of profound speculative wisdom therefrom, but by the idea obtained through it of the realization of the moral consciousness of man; the individual, he learned, must attend to himself, if inwardly and to the outer world he is to be meritorious and to bring himself, morally formed and fashioned, into actuality. This is a conception which he subsequently carried out, and it is as interesting a matter as his speculative philosophy. Just as the priests constituted a particular rank and were educated for it, they also had a special rule, which was binding throughout the whole moral life. From Egypt Pythagoras thus without doubt brought the idea of his Order, which was a regular community brought together for purposes of scientific and moral culture, which endured during the whole of life. Egypt at that time was regarded as a highly cultured country, and it was so when compared with Greece; this is shown even in the differences of caste which assumes a division amongst the great branches of life and work, such as the industrial, scientific and religious. But beyond this, we need not seek great scientific knowledge amongst the Egyptians, nor think that Pythagoras got his science there. Aristotle (Metaph. I.) only says that “in Egypt mathematical sciences first commenced, for there the nation of priests had leisure.”

Pythagoras stayed a long time in Egypt, and returned from thence to Samos; but he found the internal affairs of his own country in confusion, and left it soon after. According to Herodotus’ account (III. 45-47), Polyerates bad — not as tyrant — banished many citizens from Samos, who sought and found support amongst the Lacedæmonians, and a civil war had broken out. The Spartans had, at an earlier period, given assistance to the others, for, as Thucydides says (1. 18), to them thanks were generally ascribed for having abolished the rule of the few, and caused a reversion to the system of giving public power to the people ; later on they did the opposite, abolishing democracy and introducing aristocracy. Pythagoras’ family was necessarily involved in these unpleasant relations, and a condition of internal strife was not congenial to Pythagoras, seeing that he no longer took an interest in political life, and that he saw in it an unsuitable soil for carrying. out his plans. He traversed Greece, and betook himself from thence to Italy, in the lower parts of which Greek colonies from various states and for various motives had settled, and there flourished as important trading towns, rich in people and possessions.

In Crotona he settled down, and lived in independence, neither as a statesman, warrior, nor political lawgiver to the people, so far as external life was concerned., but as a public teacher, with the provision that his teaching should not be taken up with mere conviction, but should also regulate the whole moral life of the individual. Diogenes Laërtius says that he first gave himself the name filosofoς, instead of sofoς; and men called this modesty, as if he thereby expressed, not the possession of wisdom, but only the struggle towards it, as towards an end which cannot be attained. But sofos at the same time means a wise man, who is also practical, and that not in his own interest only, for that requires no wisdom, seeing that every sincere and moral man does what is best from his own point of view. Thus filosofoς signifies more particularly the opposite to participation in practical matters, that is in public affairs. Philosophy is thus not the love of wisdom, as of something which one sets oneself to acquire; it is no unfulfilled desire. Filosofoς means a man whose relation to wisdom is that of making it his object; this relationship is contemplation, and not mere Being; but it must be consciously that men apply themselves to this. The man who likes wine (filoinoς) is certainly to be distinguished from the man who is full of wine, or a drunkard. Then does filoinoς signify only a futile aspiration for wine?

What Pythagoras contrived and effected in Italy is told us by later eulogists, rather than by historians. In the history of Pythagoras by Malchus (this was the Syrian name of Porphyry) many strange things are related, and with the Neo-Platonists the contrast between their deep insight and their belief in the miraculous is surprising. For instance, seeing that the later biographers of Pythagoras had already related a quantity of marvels, they now proceeded to add yet more to these with reference to his appearance in Italy. It appears that they were exerting themselves to place him, as they afterwards did with Apollonius of Tyana, in opposition to Christ. For the wonders which they tell of him seem partly to be an amplification of those in the New Testament, and in part they are altogether absurd. For instance, they make Pythagoras begin his career in Italy with a miracle. When he landed in the Bay of Tarentum, at Crotona, be encountered fishermen on the way to the town who had caught nothing. He called upon them to draw their nets once more, and foretold the number of fishes that would be found in them. The fishermen, marvelling at this prophecy, promised him that if it came true they would do whatever he desired. It came to pass as he said, and Pythagoras then desired them to throw the fishes alive back into the sea, for the Pythagoreans ate no flesh. And it is further related as a miracle which then took place, that none of the fishes whilst they were out of the water died during the counting. This is the kind of miracle that is recorded, and the stories with which his biographers fill his life are of the same silly nature. They then make him effect such a general impression upon the mind of Italy, that all the towns reformed upon their luxurious and depraved customs, and the tyrants partly gave up their powers voluntarily, and partly they were driven out. They thereby, however, commit such historical errors as to make Charondas and Zaleucus, who lived long before Pythagoras, his disciples; and similarly to ascribe the expulsion and death of the tyrant Phalaris to him, and to his action.

Apart from these fables, there remains as an historic fact, the great work which he accomplished, and this he did chiefly by establishing a school, and by the great influence of his order upon the principal part of the Greco-Italian states, or rather by means of the rule which was exercised in these states through this order, which lasted for a very long period of time. It is related of him that he was a very handsome man, and of a majestic appearance, which captivated as much as it commanded respect. With this natural dignity, nobility of manners, and the calm propriety of his demeanour, he united external peculiarities, through which he seemed a remarkable and mysterious being. He wore a white linen garment, and refrained from partaking of certain foods. Particular personality, as also the externalities of dress and the like, are no longer of importance; men let themselves be guided by general custom and fashion. since it is a matter outside of and indifferent to them not to have their own will here; for we hand over the contingent to the contingent, and only follow the external rationality that consists in identity and universality. To this outward personality there was added great eloquence and profound perception ; not only did lie undertake to impart this to his individual friends, but he proceeded to bring a general influence to bear on public culture, both in regard to understanding and to the whole manner of life and morals. He not merely instructed his friends, but associated them in a particular life in order to constitute them into persons and make them skilful in business and eminent in morals. The Institute of Pythagoras grew into a league which included all men and all life in its embrace.. for it was an elaborately fashioned piece of work, and excellently plastic in design.

Of the regulations of Pythagoras’ league, we have descriptions from his successors, more especially from the Neo-Platonists, who are particularly diffuse as regards its laws. The league had, on the whole, the character of a voluntary priesthood, or a monastic order of modern times. Whoever wished to be received was proved in respect of his education and obedience, and information was collected about his conduct, inclinations, and occupations. The members were subject to a special training, in which a difference was made amongst those received, in that some were exoteric and some esoteric. These last were initiated into the highest branches of science, and since political operations were not excluded from the order, they were also engaged in active politics ; the former had to go through a novitiate of five years. Each member must have surrendered his means to the order, but he received them again on retiring, and in the probationary period silence was enjoined (ecemuqia).

This obligation to cease from idle talk may be called an essential condition for all culture and learning; with it men must begin if they wish to comprehend the thoughts of others and relinquish their own ideas. We are in the habit of saying that the understanding is cultivated through questioning, objecting and replying, &c., but in fact, it is not thus formed, but made from without.

What is inward in man is by culture got at and developed; hence though he remains silent, he is none the poorer in thought or denser of mind. He rather acquires thereby the power of apprehension, and comes to know that his ideas and objections are valueless ; and as he learns that such ideas are valueless, he ceases to have them. Now the fact that in Pythagoras there is a separation between those in the course of preparation and those initiated, as also that silence is particularly enjoined, seems most certainly to indicate that in his brotherhood both were formal elements and not merely as present in the nature of things, as might occur spontaneously in the individual without any special law or the application of any particular consideration. But here it is important to remark that Pythagoras may be regarded as the first instructor in Greece who introduced the teachings of science; neither Thales, who was earlier than he, nor his contemporary Anaximander taught scientifically, but only imparted their ideas to their friends. There were, generally speaking no sciences at that time ; there was neither a science of philosophy, mathematics, jurisprudence or anything else, but merely isolated propositions and facts respecting these subjects. What was taught was the use of arms, theorems, music, the singing of Homer’s or Hesiod’s songs, tripod chants, &c., or other arts. This teaching is accomplished in quite another way. Now if we said that Pythagoras had introduced the teaching of science amongst a people who, though like the Greeks, untaught therein, were not stupid but most lively, cultured and loquacious, the external conditions of such teaching might in so far be given as follows: — (a) He would distinguish amongst those who as yet had no idea of the process of learning a science, so that those who first began should be excluded from that which was to be imparted to those further on; and (b) he would make them leave the unscientific mode of speaking of such matters, or their idle prattle, alone, and for the first time study science. But the fact that this action both appeared to be formal and likewise required to be made such, was, on account of its unwonted character, a necessary one, just because the followers of Pythagoras were not only numerous, necessitating a definite form and order, but also, generally speaking, they lived continually together. Thus a particular form was natural to Pythagoras, because it was the very first time that a teacher in Greece arrived at a totality, or a new principle, through the cultivation of the intelligence, mind and will. This common life had not only the educational side and that founded on the exercise of physical ingenuity or skill, but included also that of the moral culture of practical men. But even. now everything relating to morality appears and is or becomes altogether formal, or rather this is so in as far as it is consciously thought of as in this relation, for to be formal is to be universal, that which is opposed to the individual. It appears so particularly to him who compares the universal and the individual and consciously reflects over both, but this difference disappears for those living therein, to whom it is ordinary habit.

Finally, we have sufficient and full accounts of the outward forms observed by the Pythagoreans in their common life and also of their discipline. For much of this, however, we are indebted to the impressions of later writers. In the league, a life regulated in all respects was advocated. First of all, it is told us, that the members made themselves known by a similar dress — the white linen of Pythagoras. They had a very strict order for, each day, of which each hour had its work. The morning, directly after rising, was set aside for recalling to memory the history of the previous day, because what is to be done in the day depends chiefly on the previous day; similarly the most constant self-examination was made the duty of the evening in order to find whether the deeds done in the day were right or wrong. True culture is not the vanity of directing so much attention to oneself and occupying oneself with oneself as an individual, but the self-oblivion that absorbs oneself in the matter in hand and in the universal; it is this consideration of the thing in hand that is alone essential, while that dangerous, useless, anxious state does away with freedom. They had also to learn by heart from Homer and from Hesiod; and all through the day they occupied themselves much with music — one of the principal parts of Greek education and culture. Gymnastic exercises in wrestling, racing, throwing, and so on, were with them also enforced by rule. They dined together, and here, too, they had peculiar customs, but of these the accounts are different. Honey and bread were made their principal food, and water the principal, and indeed only, drink ; they must thus have entirely refrained from eating meat as being associated with metempsychosis. A distinction was also made regarding vegetables — beans, for example, being forbidden. On account of this respect for beans, they were much derided, yet in the subsequent destruction of the political league, several Pythagoreans, being pursued, preferred to die than to damage a field of beans.

The order, the moral discipline which characterized them, the common intercourse of men, did not, however, endure long; for even in Pythagoras life-time the affairs of his league must have become involved, since he found enemies who forcibly overthrew him. He drew down upon him, it is said, the envy of others, and was accused of thinking differently from what he seemed to indicate, and thus of having an arrière pensée. The real fact of the case was that the individual belonged, not entirely to his town, but also to another. In this catastrophe, Pythagoras himself, according to Tennemann (Vol. I. p. 414), met his death in the 69th Olympiad (504, B.C.) in a rising of the people against these aristocrats; but it is uncertain whether it happened in Crotona or in Metapontum, or in a war between the Syracusans and the Agrigentines. There is also much difference of opinion about the age of Pythagoras, for it is given sometimes as 80, and sometimes as 104.1 For the rest, the unity of the Pythagorean school, the friendship of the members. and the connecting bond of culture have even in later times remained, but not in the formal character of a league, because what is external must pass away. The history of Magna Graecia is in general little known, but even in Plato’s time we find Pythagoreans appearing at the head of states or as a political power.

The Pythagorean brotherhood had no relation with Greek public and religious life, and therefore could not endure for long: in Egypt and in Asia exclusiveness and priestly influence have their home, but Greece, in its freedom, could not lot the Eastern separation of caste exist. Freedom here is the principle of civic life, but still it is not yet determined as principle in the relations of public and private law. With us the individual is free since all are alike before the law; diversity in customs, in political relations and opinions may thus exist, and must indeed so do in organic states. In democratic Greece, on the contrary, manners, the external mode of life, necessarily preserved a certain similarity, and the stamp of similarity remained impressed on these wider spheres ; for the exceptional condition of the Pythagoreans, who could not take their part as free citizens, but were dependent on the plans and ends of a combination and led an exclusive religious life, there was no place in Greece. The preservation of the mysteries certainly belonged to the Eumolpidæ, and other special forms of worship to other particular families, but they were not regarded in a political sense as of fixed and definite castes, but as priests usually are, politicians, citizens, men like their fellows; nor, as with the Christians, was the separation of religious persons driven to the extreme of monastic rule. In ordinary civic life in Greece, no one could prosper or maintain his position who held peculiar principles, or even secrets, and differed in outward modes of life and clothing; for what evidently united and distinguished them was their community of principles and life — whether anything was good for the commonwealth or not, was by them publicly and openly discussed. The Greek s are above having particular clothing, maintaining special customs of washing, rising, practising music, and distinguishing between pure and impure foods. This, they say, is partly the affair of the particular individual and of his personal freedom, and has no common end in view, and partly it is a general custom and usage for everybody alike.

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 (a) that the absolute is not grasped in natural form, but as a thought determination. (b) Then it follows that determinations must be posited while the beginning was altogether undetermined. The Pythagorean philosophy has done both.

1. Thus the original and simple proposition of the Pythagorean philosophy is, according to Aristotle (Metaph. 1. 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. 1. 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 farther (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 moist 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.” 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 (toiondi paqoς) is justice, so is it with (toiondi) 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. 1. 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 (monas) ; 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 11 one.” The Pythagoreans have expressed this remarkable relationship of the entirely abstract one to the concrete existence of things through “simulation” (mimhsiς). 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. 1. 6) ascribes the expression “participation” (meqexiς) 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. g). 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 (duaς), 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 from the very first gave forth universal ideal determinations of numbers as principles, and recognized, as Aristotle remarks (Metaph. 1. 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 (monaς) ; 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 that 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” (mimhsiς). 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. 1. 6) ascribes the expression “participation” (meqexiς) 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.

(a) They said, according to Aristotle (Metaph. 1. 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 Smyrna1 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” (artioperitton).

(b) If we follow the absolute Idea in this first mode, the opposition will also be called the undetermined duality (aoristos duaς). Sextus speaks more definitely (adv. Math. X. 261, 262) as follows: “Unity, thought of in its identity with itself (kat autothta eauthς), is unity; if this adds itself to itself as something different (kaq eterothta), 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. (aa) Unity, as the being-like-self, is the formless ; but in duality, as the unlike, there comes division or form. (bb) 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. 1. 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 (anti tou apeirou ws enoς), he made of it a duality — the great and small.”

(g) Further consideration of this opposition, in which Pythagoreans differ from ono 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. 1. 5) ascribes these determinations either to Pythagoras himself, or else to Alcmæon “who flourished in the time of Pythagoras’ old acre,” so that 11 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, ton is a number of great significance:

1. The finite and the infinite.
2. The odd and the oven.
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.

(d) 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. (aa) 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. (bb) “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. (gg) According to relation (pros ti), 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.” (aa) “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. (bb) “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 (anepitatoς) ; what is opposed to it is unlike. Health is like, sickness is unlike. (gg) 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 ono is two. The other numbers take their origin in a similar way, for the unity ever 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. (a) 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. (b) The opposition has at the same time subsistence, and in this is found the manifold nature of equivalent things. (g) 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 these 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. 1. 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 Qeologoumena ariqmhtikhs, 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 (trias) 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 it has absolute form. “For the number of the whole bas 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 (pan) ; 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 bas 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 wore 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 are not dealing with these natural relationships.

d. The Four (tetraς) 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 tetraktus, the efficient, active four (from tettara and agw) ; and afterwards this is by the Pythagoreans made the most notable number. In the fragments of a poem or 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.”

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 (dekaς), 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. 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 mid 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. 1. 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 j ustified 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. 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 (logos diplasioς, 1:2) it would produce the musical chord of the octave (dia paswn) ; the proportion of 8:12, or of 2:3 (logos hmiolioς) would give the chord of the fifth (dia pente); the proportion of 9:12, or 3:4 (logos epitritoς), the fourth (dia tessarwn). 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 a 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 wore 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.

e. 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 (anticqona).” 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 coelo II. 13 and g), “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 beard, 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. 1. 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. 1. 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 (sumfuton) harmony. In order that the whole may have an harmonious movement, Timaeus has bent the straight line of harmony (euquwrian) 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 (11. 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 Laërtius 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 1 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, Ythalides, and then in the Trojan war as Euphorbus, the son of Pauthous, 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.” 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. 1. 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. 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. 1. 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 (isakis isoς) ; 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 m 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. Such matter does not deserve to be regarded as philosophy, although it is of importance in the process of development.

The transition from the form of outward morals to morality as existent, is more important. As in Thales’ time, law-givers and administrators of states were pre-eminent in possessing a physical philosophy, so we see that with Pythagoras practical philosophy is advocated as the means of constituting a moral life. There we have the speculative Idea, the absolute essence, in its reality, and in a definite, sensuous existence ; and similarly the moral life is submerged in actuality as the universal spirit of a people, and as their laws and rule. In Pythagoras, on the contrary we have the reality of absolute essence raised, in speculation, out of sensuous reality, and expressed, though still imperfectly, as the essence of thought. Morality is likewise partly raised out of actuality as ordinarily known ; it is certainly a moral disposition of all actuality, but as a brotherhood, and not as the life of a people. The Pythagorean League is an arbitrary existence and not a part of the constitution recognized by public sanction; and in his person Pythagoras isolated himself as teacher, as he also did his followers. The universal consciousness, the spirit of a people, is the substance of which the accident is the individual consciousness; the speculative is thus the fact that pure, universal law is absolute, individual consciousness, so that this last, because it draws therefrom its growth and nourishment, becomes universal self-consciousness. These two sides do not, however, come to us in the form of the opposition; it is first of all in morality that there is properly this Notion of the absolute individuality of consciousness which does everything on its own account. But we see that it was really present to the mind of Pythagoras that the substance of morality is the universal, from an example in Diogenes Laërtius (VIII. 16). “A Pythagorean answered to the question of a father who inquired as to the best education he could give his son, that it should be that which would make him the citizen of a well-regulated State.” This answer is great and true; to the great principle of living in the spirit of one’s people, all other circumstances are subordinate. Now-a-days men try to keep education free from the spirit of the times, but they cannot withdraw themselves from this supreme power, the State, for even if they try to separate themselves, they unconsciously remain beneath this universal. The speculative meaning of the practical philosophy of Pythagoras thus is, that in this signification, the individual consciousness shall obtain a moral reality in the brotherhood. But as number is a middle thing between the sensuous and Notion, the Pythagorean brotherhood is a middle between universal, actual morality and maintaining that in true morality the individual, as an individual, is responsible for his own behaviour; this morality ceases to be universal spirit. If we wish to see practical philosophy reappear, we shall find it; but, on the whole, we shall not see it become really speculative until very recent times.

We may satisfy ourselves with this as giving us an idea of the Pythagorean system. I will, however, shortly give the principal points of the criticism which Aristotle (Met. 1. 8) makes upon the Pythagorean number-form. He says justly, in the first place: “If only the limited and the unlimited, the even and odd are made fundamental ideas, the Pythagoreans do not explain how movement arises, and how. without movement and change there can be coming into being and passing away, or the conditions and activities of heavenly objects.” This defect is significant ; arithmetical numbers are dry forms and barren principles in which life and movement are deficient. Aristotle says secondly, “From number no other corporeal determinations, such as weight and lightness, are conceivable;” or number thus cannot pass into what is concrete. “They say that there is no number outside of those in the heavenly spheres.” For instance, a heavenly sphere and a virtue, or a natural manifestation in the earth, are determined as one and the same number. Each of the first numbers may be exhibited in each thing or quality; but in so far as number is made to express a further determination, this quite abstract, quantitative difference becomes altogether formal; it is as if the plant were five because it has five stamens. This is just as superficial as are determination through elements or through particular portions of the globe; it is a method as formal as that by which men now try to apply the categories of electricity, magnetism, galvanism, compression and expansion, of manly and of womanly, to everything. It is a purely empty system of determination where reality should be dealt with.

To Pythagoras and his disciples there are, moreover, many scientific conclusions and discoveries ascribed, which, however, do not concern us at all. Thus, according to Diogenes Laërtius (VIII. 14, 27), he is said to have known that the morning and evening star is the same, and that the moon derives her light from the sun. We have already mentioned what he says of music. But what is best known is the Pythagorean Theorem; it really is the main proposition in geometry, and cannot be regarded like any other theorem. According to Diogenes, (VIII. 12), Pythagoras, on discovering the theorem, sacrificed a hecatomb, so important did he think it; and it may indeed seem remarkable that his joy should have gone so far us to ordain a great feast to which rich men and all the people were invited. It was worth the trouble; it was a rejoicing, a feast of spiritual cognition — at the cost of the oxen.

Other ideas which are brought forward by the Pythagoreans casually and without any connection, have no philosophic interest, and need only be mentioned. Aristotle, for instance, says (Phys. IV. 6) that “the Pythagoreans believed in an empty space which the heavens inspire, and an empty space which separates natural things and brings about the distinction between continuous and discrete; it first exists in numbers and makes them to be different.” Diogenes Laërtius (VIII. 26-28) says much more, all of which is dull; this is like the later writers, who, generally speaking, take up what is external and devoid of any intellectual meaning. “The air which encircles the earth is immovable” (aseiston, at least through itself) “and diseased, and all that is in it is mortal; but what is highest is in continual movement, pure and healthy, and in it everything is immortal — divine. Sun, moon and the other stars are gods, for in them warmth has predominance and is the cause of life. Man is related to the gods because he participates in warmth, and hence God cares for us. A ray penetrates from the sun through the thick and cold ether and gives life to everything; they call air, cold ether, the sea and moisture, thick ether. The soul is a detached portion of ether.”

Translated by E.S. Haldane and Frances H. Simson, published by K. Paul Trench, Trübner in 1894.

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