1. To abstain from beans.
2. Not to pick up what has 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.
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* Aristotle says of him that he «first worked at mathematics and arithmetic, and afterwards, at one time, condescended to the wonder-working practised by Pherecydes.»
â Clown: What is the opinion of Pythagoras concerning wildfowl?
€ Malvolio: That the soul of our grandam might haply inhabit a bird.
Clown: What thinkest thou of his opinion?
Malvolio: I think nobly of the soul, and no way approve his opinion.
Clown: Fare thee well; remain thou still in darkness: thou shalt hold the opinion of Pythagoras ere I will allow of thy wits.
(Twelfth Night)
15. When you rise from the bedclothes, roll them together and smooth out the impress of the
body. *
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.» †It is said that Pythagoras, like Saint 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
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* Quoted from Burnet Early Greek Philosophy.
Cornford, op. cit., p. 201.
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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.'» *
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
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* Early Greek Philosophy, p. 108.
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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, intui-
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tion 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
