**PYTHAGORAS**

PYTHAGORAS , whose influence in ancient and modern times is my subject in this chapter , was intellectually one of the most important 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 Samps 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 . 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 . 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

1 The Greek cities of Sicily were in danger from the Carthaginians , but in Italy this danger was not felt to be imminent . Aristotle says , of him that he ” first worked at mathematics and arithmetic , and afterwards , at one time , condescended to the wonder working practiced by pherecydes . "

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 Pythagorcan 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 , cach 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 appli cable to the real world ; morcover 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 3³ + 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 min inches . Then m²n² = 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 4p² = 2² , there fore , n²2p² and therefore is even , contra hyp . Therefore no fraction m’n will measure the hypotenuse . The above proof is substantially that in Euclid , Book X.¹

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 + 6² . 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.

1 But not by Euclid . See Heath , Greek Mathematics . The above proof was probably known to Plato . 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.¹ 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 Ile is addicted to arithmetic . Rationalistic as opposed to apocalyptic religion has been , ever ” Self – evident ” was substituted by Franklin for Jefferson’s ” sacred and undeniable.

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 analyzed , 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 .