Pythagoras
From
History of Western Philosophy by Bertrand Russell (Chapter 3, page 38)
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, 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 has been
closely linked in commerce with Miletus. Croton was famous for medicine; a
certain Demoedes 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 thy 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 souls[3]
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, wa
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 his 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 h was immeasurably
influential, and the two were not so separate as they seem to a modern mind.
Most
scientists, 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 beings with Pythagoras.
Pythagoras,
as every 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 4p2 = 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 od 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 were 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.
[1]
The Greek cities of Sicily were in danger from the Carthaginians, but in Italy
this danger was not felt to be imminent.
[2] 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.’
[3] Clown: What is the opinion of Pythagoras
concerning wildfowl?
Malvolio: That
the soul of our grandma 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)
[4]
Quoted from Burnet’s Early Greek
Philosophy.
[5]
Cornford, op. cit., p. 201
[6] Early Greek Philosophy, p. 108
[7]
But not by Euclid. See Heath, Greek
Mathematics. The above proof was probably known to Plato.
[8]
‘Self-Evident’ was substituted by Franklin for Jefferson’s ‘sacred and
undeniable.’
