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,
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contra hyp. Therefore no fraction m/n will measure the hypotenuse. The above proof is
substantially that in Euclid, Book X. *
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 nineteenthcentury 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 selfevident,» it is modelling itself on Euclid. The eighteenth-century doctrine of natural rights is a search for Euclidean axioms in politics. †The form of Newton Principia, in spite of its admittedly empirical
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* But not by Euclid. See Heath, Greek Mathematics. The above proof was probably known to Plato.
€ «Self-evident» was substituted by Franklin for Jefferson’s «sacred and undeniable.»
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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, Saint Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant 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 world; 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.
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CHAPTER IV Heraclitus
TWO opposite attitudes towards the Greeks are common at the present day. One, which was practically universal from the Renaissance until very recent times, views the Greeks with almost superstitious reverence, as the inventors of all that is best, and as men of superhuman genius whom the moderns cannot hope to equal. The other attitude, inspired by the triumphs of science and by an optimistic belief in progress, considers the authority of the ancients an incubus, and maintains that most of their contributions to thought are now best forgotten. I cannot myself take either of these extreme views; each, I should say, is partly right and partly wrong. Before entering upon any detail, I shall try to say what sort of wisdom we can still derive from the study of Greek thought.
As to the nature and structure of the world, various hypotheses are possible. Progress in metaphysics, so far as it has existed, has consisted in a gradual refinement of all these hypotheses, a development of their implications, and a reformulation of each to meet the objections urged by adherents of rival hypotheses. To learn to conceive the universe according to each of these systems is an imaginative delight and an antidote to dogmatism. Moreover, even if no one of the hypotheses can be demonstrated, there is genuine knowledge in the discovery of what is involved in making each of them consistent with itself and with known facts. Now almost all the hypotheses that have dominated modern philosophy were first thought of by the Greeks; their imaginative inventiveness in abstract matters can hardly be too highly praised. What I shall have to say about the Greeks will be said mainly from this point of view; I shall regard them as giving birth to theories which have had an independent life and growth, and which, though at first somewhat infantile, have proved capable of surviving and developing throughout more than two thousand years.
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The Greeks contributed, it is true, something else which proved of more permanent value to abstract thought: they discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible. But in connection with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from what appeared self-evident, not inductively from what had been observed. Its amazing successes in the employment of this method misled not only the ancient world, but the greater part of the modern world also. It has only been very slowly that scientific method, which seeks to reach principles inductively from observation of particular facts, has replaced the Hellenic belief in deduction from luminous axioms derived from the mind of the philosopher. For this reason, apart from others, it is a mistake to treat the Greeks with superstitious reverence. Scientific method, though some few among them were the first men who had an inkling of it, is, on the whole, alien to their temper of mind, and the attempt to glorify them by belittling the intellectual progress of the last four centuries has a cramping effect upon modern thought.
There is, however, a more general argument against reverence, whether for the Greeks or for anyone else. In studying a philosopher, the right attitude is neither reverence nor contempt, but first a kind of hypothetical sympathy, until it is possible to know what
