Pythagoras, whose influence in ancient and modern times is my subject in this chapter, was intellectually one of the most important men that ever lived, both when he was wise and when he was unwise. Mathematics, in the sense of demonstrative deductive argument, begins with him, and in him is intimately connected with a peculiar form of mysticism. The influence of mathematics on philosophy, partly owing to him, has, ever since his time, been both profound and unfortunate.

Let us begin with what little is known of his life. He was a native of the island of Samos, and flourished about 532 B.C. Some say he was the son of a substantial citizen named Mnesarchos, others that he was the son of the god Apollo; I leave the reader to take his choice between these alternatives. In his time Samos was ruled by the tyrant Polycrates, an old ruffian who became immensely rich, and had a vast navy.

Samos was a commercial rival of Miletus; its traders went as far afield as Tartessus in Spain, which was famous for its mines. Polycrates became tyrant of Samos about 535 B.C., and reigned until 515 B.C. He was not much troubled by moral scruples; he got rid of his two brothers, who were at first associated with him in the tyranny, and he used his navy largely for piracy. He profited by the fact that Miletus had recently submitted to Persia. In order to obstruct any further westward expansion of the Persians, he allied himself with Amasis, king of Egypt. But when Cambyses, king of Persia, devoted his full energies to the conquest of Egypt, Polycrates realized that he was likely to win, and changed sides. He sent a fleet, composed of his political enemies, to attack Egypt; but the crews mutinied and returned to Samos to attack him. He got the better of them, however, but fell at last by a treacherous appeal to his avarice. The Persian satrap at Sardes represented that he intended to rebel against the Great King, and would pay vast sums for the help of Polycrates, who went to the mainland for an interview, was captured and crucified.

Polycrates was a patron of the arts, and beautified Samos with remarkable public works. Anacreon was his court poet. Pythagoras, however, disliked his government, and therefore left Samos. It is said, and is not improbable, that Pythagoras visited Egypt, and learnt much of his wisdom there; however that may be, it is certain that he ultimately established himself at Croton, in southern Italy.

The Greek cities of southern Italy, like Samos and Miletus, were rich and prosperous; moreover they were not exposed to danger from the Persians.1 The two greatest were Sybaris and Croton. Sybaris has remained proverbial for luxury; its population, in its greatest days, is said by Diodorus to have amounted to 300,000, though this is no doubt an exaggeration. Croton was about equal in size to Sybaris. Both cities lived by importing Ionian wares into Italy, partly for consumption in that country, partly for re-export from the western coast to Gaul and Spain. The various Greek cities of Italy fought each other fiercely; when Pythagoras arrived in Croton, it had just been defeated by Locri. Soon after his arrival, however, Croton was completely victorious in a war against Sybaris, which was utterly destroyed (510 B.C.). Sybaris had been closely linked in commerce with Miletus. Croton was famous for medicine; a certain Democedes of Croton became physician to Polycrates and then to Darius.

At Croton Pythagoras founded a society of disciples, which for a time was influential in that city. But in the end the citizens turned against him, and he moved to Metapontion (also in southern Italy), where he died. He soon became a mythical figure, credited with miracles and magic powers, but he was also the founder of a school of mathematicians.2 Thus two opposing traditions disputed his memory, and the truth is hard to disentangle.

Pythagoras is one of the most interesting and puzzling men in history. Not only are the traditions concerning him an almost inextricable mixture of truth and falsehood, but even in their barest and least disputable form they present us with a very curious psychology. He may be described, briefly, as a combination of Einstein and Mrs Eddy. He founded a religion, of which the

main tenets were the transmigration of souls3 and the sinfulness of eating beans. His religion was embodied in a religious order, which, here and there, acquired control of the State and established a rule of the saints. But the unregenerate hankered after beans, and sooner or later rebelled.

Some of the rules of the Pythagorean order were:

1. To abstain from beans.

2. Not to pick up what was fallen.

3. Not to touch a white cock.

4. Not to break bread.

5. Not to step over a crossbar.

6. Not to stir the fire with iron.

7. Not to eat from a whole loaf.

8. Not to pluck a garland.

9. Not to sit on a quart measure.

10. Not to eat the heart.

11. Not to walk on highways.

12. Not to let swallows share one's roof.

13. When the pot is taken off the fire, not to leave the mark of it in the ashes, but to stir them together.

14. Do not look in a mirror beside a light.

15. When you rise from the bedclothes, roll them together and smooth out the impress of the body.4

All these precepts belong to primitive tabu-conceptions.

Cornford (From Religion to Philosophy) says that, in his opinion, 'The School of Pythagoras represents the main current of that mystical tradition which we have set in contrast with the scientific tendency.' He regards Parmenides, whom he calls 'the discoverer of logic', as 'an offshoot of Pythagoreanism, and Plato himself as finding in the Italian philosophy the chief source of his inspiration.' Pythagoreanism, he says, was a movement of reform in Orphism, and Orphism was a movement of reform in the worship of Dionysus. The opposition of the rational and the mystical, which runs all through history, first appears, among the Greeks, as an opposition between the Olympic gods and those other less civilized gods who had more affinity

with the primitive beliefs dealt with by anthropologists. In this division, Pythagoras was on the side of mysticism, though his mysticism was of a peculiarly intellectual sort. He attributed to himself a semi-divine character, and appears to have said: 'There are men and gods, and beings like Pythagoras.' All the systems that he inspired, Cornford says, 'tend to be otherworldly, putting all value in the unseen unity of God, and condemning the visible world as false and illusive, a turbid medium in which the rays of heavenly light are broken and obscured in mist and darkness.'

Dikaiarchos says that Pythagoras taught 'first, that the soul is an immortal thing, and that it is transformed into other kinds of living things; further, that whatever comes into existence is born again in the revolutions of a certain cycle, nothing being absolutely new; and that all things that are born with life in them ought to be treated as kindred.'5 It is said that Pythagoras, like St Francis, preached to animals.

In the society that he founded, men and women were admitted on equal terms; property was held in common, and there was a common way of life. Even scientific and mathematical discoveries were deemed collective, and in a mystical sense due to Pythagoras even after his death. Hippasos of Metapontion, who violated this rule, was shipwrecked as a result of divine wrath at his impiety.

But what has all this to do with mathematics? It is connected by means of an ethic which praised the contemplative life. Burnet sums up this ethic as follows:

We are strangers in this world, and the body is the tomb of the soul, and yet we must not seek to escape by self-murder; for we are the chattels of God who is our herdsman, and without His command we have no right to make our escape. In this life, there are three kinds of men, just as there are three sorts of people who come to the Olympic Games. The lowest class is made up of those who come to buy and sell, the next above them are those who compete. Best of all, however, are those who come simply to look on. The greatest purification of all is, therefore, disinterested science, and it is the man who devotes himself to that, the true philosopher, who has most effectually released himself from the 'wheel of birth.'6

The changes in the meanings of words are often very instructive. I spoke above about the word 'orgy'; now I want to speak about the word 'theory'. This was originally an Orphic word, which Cornford interprets as 'passionate sympathetic contemplation'. In this state, he says, 'The spectator is identified

with the suffering God, dies in his death, and rises again in his new birth.' For Pythagoras, the 'passionate sympathetic contemplation' was intellectual, and issued in mathematical knowledge. In this way, through Pythagoreanism, 'theory' gradually acquired its modern meaning; but for all who were inspired by Pythagoras it retained an element of ecstatic revelation. To those who have reluctantly learnt a little mathematics in school this may seem strange; but to those who have experienced the intoxicating delight of sudden understanding that mathematics gives, from time to time, to those who love it, the Pythagorean view will seem completely natural even if untrue. It might seem that the empirical philosopher is the slave of his material, but that the pure mathematician, like the musician, is a free creator of his world of ordered beauty.

It is interesting to observe, in Burnet's account of the Pythagorean ethic, the opposition to modern values. In connection with a football match, modern-minded men think the players grander than the mere spectators. Similarly as regards the State: they admire more the politicians who are the contestants in the game than those who are only onlookers. This change of values is connected with a change in the social system—the warrior, the gentleman, the plutocrat, and the dictator, each has his own standard of the good and the true. The gentleman has had a long innings in philosophical theory, because he is associated with the Greek genius, because the virtue of contemplation acquired theological endorsement, and because the ideal of disinterested truth dignified the academic life. The gentleman is to be defined as one of a society of equals who live on slave labour, or at any rate upon the labour of men whose inferiority is unquestioned. It should be observed that this definition includes the saint and the sage, insofar as these men's lives are contemplative rather than active

Modern definitions of truth, such as those of pragmatism and instrumentalism, which are practical rather than contemplative, are inspired by industrialism as opposed to aristocracy.

Whatever may be thought of a social system which tolerates slavery, it is to gentlemen in the above sense that we owe pure mathematics. The contemplative ideal, since it led to the creation of pure mathematics, was the source of a useful activity; this increased its prestige, and gave it a success in theology, in ethics, and in philosophy, which it might not otherwise have enjoyed.

So much by way of explanation of the two aspects of Pythagoras: as religious prophet and as pure mathematician. In both respects he was immeasurably influential, and the two were not so separate as they seem to a modern mind.

Most sciences, at their inception, have been connected with some form of false belief, which gave them a fictitious value. Astronomy was connected with astrology, chemistry with alchemy. Mathematics was associated with a more refined type of error. Mathematical knowledge appeared to be certain, exact, and applicable to the real world; moreover it was obtained by mere thinking, without the need of observation. Consequently, it was thought to supply an ideal, from which every-day empirical knowledge fell short. It was supposed, on the basis of mathematics, that thought is superior to sense, intuition to observation. If the world of sense does not fit mathematics, so much the worse for the world of sense. In various ways, methods of approaching nearer to the mathematician's ideal were sought, and the resulting suggestions were the source of much that was mistaken in metaphysics and theory of knowledge. This form of philosophy begins with Pythagoras.

Pythagoras, as everyone knows, said that 'all things are numbers'. This statement, interpreted in a modern way, is logically nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music, and the connection which he established between music and arithmetic survives in the mathematical terms 'harmonic mean' and 'harmonic progression'. He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares and cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or, as we should more naturally say, shot) required to make the shapes in question. He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.

The greatest discovery of Pythagoras, or of his immediate disciples, was the proposition about right-angled triangles, that the sum of the squares on the sides adjoining the right angle is equal to the square on the remaining side, the hypotenuse. The Egyptians had known that a triangle whose sides are 3, 4, 5 has a right angle, but apparently the Greeks were the first to observe that 32 + 42 = 52, and, acting on this suggestion, to discover a proof of the general proposition.

Unfortunately for Pythagoras, his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side. Let us suppose each side an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m2/n2 = 2. If m and n have a common factor, divide it out, then either m or n must be odd. Now m2 = 2n2, therefore m2 is even, therefore m is even, therefore n is odd. Suppose m = 2p. Then 4 p2 = 2n2, therefore, n2 = 2p2 and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. The above proof is substantially that in Euclid, Book X.7

This argument proved that, whatever unit of length we may adopt, there are lengths which bear no exact numerical relation to the unit, in the sense that there are no two integers m, n, such that m times the length in question is n times the unit. This convinced the Greek mathematicians that geometry must be established independently of arithmetic. There are passages in Plato's dialogues which prove that the independent treatment of geometry was well under way in his day; it is perfected in Euclid. Euclid, in Book II, proves geometrically many things which we should naturally prove by algebra, such as (a + b)2 = a2 + 2ab + b2. It was because of the difficulty about incommensurables that he considered this course necessary. The same applies to his treatment of proportion in Books V and VI. The whole system is logically delightful, and anticipates the rigour of nineteenth-century mathematicians. So long as no adequate arithmetical theory of incommensurables existed, the method of Euclid was the best that was possible in geometry. When Descartes introduced co-ordinate geometry, thereby again making arithmetic supreme, he assumed the possibility of a solution of the problem of incommensurables, though in his day no such solution had been found.

The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems that are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers. When the Declaration of Independence says 'we hold these truths to be self-evident', it is modelling itself on Euclid. The eighteenth-century doctrine of natural rights is a search for Euclidean axioms in politics.8 The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source. Personal religion is derived from ecstasy, theology from mathematics; and both are to be found in Pythagoras.

Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also

reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

The combination of mathematics and theology, which began with Pythagoras, characterized religious philosophy in Greece, in the Middle Ages, and in modern times down to Kant. Orphism before Pythagoras was analogous to Asiatic mystery religions. But in Plato, St Augustine, Thomas Aquinas, Descartes, Spinoza, and Leibniz there is an intimate blending of religion and reasoning, of moral aspiration with logical admiration of what is timeless, which comes from Pythagoras, and distinguishes the intellectualized theology of Europe from the more straightforward mysticism of Asia. It is only in quite recent times that it has been possible to say clearly where Pythagoras was wrong. I do not know of any other man who has been as influential as he was in the sphere of thought. I say this because what appears as Platonism is, when analysed, found to be in essence Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the senses, is derived from him. But for him, Christians would not have thought of Christ as the Word; but for him, theologians would not have sought logical proofs of God and immortality. But in him all this is still implicit. How it became explicit will appear as we proceed.

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