A History of Western Philosophy

by Bertrand Russell

  “Substance,” in a word, is a metaphysical mistake, due to transference to the world-structure of the structure of sentences composed of a subject and a predicate.

  I conclude that the Aristotelian doctrines with which we have been concerned in this chapter are wholly false, with the exception of the formal theory of the syllogism, which is unimportant. Any person in the present day who wishes to learn logic will be wasting his time if he reads Aristotle or any of his disciples. None the less, Aristotle’s logical writings show great ability, and would have been useful to mankind if they had appeared at a time when intellectual originality was still active. Unfortunately, they appeared at the very end of the creative period of Greek thought, and therefore came to be accepted as authoritative. By the time that logical orginality revived, a reign of two thousand years had made Aristotle very difficult to dethrone. Throughout modern times, practically every advance in science, in logic, or in philosophy has had to be made in the teeth of the opposition from Aristotle’s disciples.

  CHAPTER XXIII

  Aristotle’s Physics

  IN this chapter I propose to consider two of Aristotle’s books, the one called Physics and the one called On the Heavens. These two books are closely connected; the second takes up the argument at the point at which the first has left it. Both were extremely influential, and dominated science until the time of Galileo. Words such as “quintessence” and “sublunary” are derived from the theories expressed in these books. The historian of philosophy, accordingly, must study them, in spite of the fact that hardly a sentence in either can be accepted in the light of modern science.

  To understand the views of Aristotle, as of most Greeks, on physics, it is necessary to apprehend their imaginative background. Every philosopher, in addition to the formal system which he offers to the world, has another, much simpler, of which he may be quite unaware. If he is aware of it, he probably realizes that it won’t quite do; he therefore conceals it, and sets forth something more sophisticated, which he believes because it is like his crude system, but which he asks others to accept because he thinks he has made it such as cannot be disproved. The sophistication comes in by way of refutation of refutations, but this alone will never give a positive result: it shows, at best, that a theory may be true, not that it must be. The positive result, however little the philosopher may realize it, is due to his imaginative preconceptions, or to what Santayana calls “animal faith.”

  In relation to physics, Aristotle’s imaginative background was very different from that of a modern student. Now-a-days, a boy begins with mechanics, which, by its very name, suggests machines. He is accustomed to motor-cars and aeroplanes; he does not, even in the dimmest recesses of his subconscious imagination, think that a motor-car contains some sort of horse in its inside, or that an aeroplane flies because its wings are those of a bird possessing magical powers. Animals have lost their importance in our imaginative pictures of the world, in which man stands comparatively alone as master of a mainly lifeless and largely subservient material environment.

  To the Greek, attempting to give a scientific account of motion, the purely mechanical view hardly suggested itself, except in the case of a few men of genius such as Democritus and Archimedes. Two sets of phenomena seemed important: the movements of animals, and the movements of the heavenly bodies. To the modern man of science, the body of an animal is a very elaborate machine, with an enormously complex physico-chemical structure; every new discovery consists in diminishing the apparent gulf between animals and machines. To the Greek, it seemed more natural to assimilate apparently lifeless motions to those of animals. A child still distinguishes live animals from other things by the fact that they can move of themselves; to many Greeks, and especially to Aristotle, this peculiarity suggested itself as the basis of a general theory of physics.

  But how about the heavenly bodies? They differ from animals by the regularity of their movements, but this may be only due to their superior perfection. Every Greek philosopher, whatever he may have come to think in adult life, had been taught in childhood to regard the sun and moon as gods; Anaxagoras was prosecuted for impiety because he thought that they were not alive. It was natural that a philosopher who could no longer regard the heavenly bodies themselves as divine should think of them as moved by the will of a Divine Being who had a Hellenic love of order and geometrical simplicity. Thus the ultimate source of all movement is Will: on earth the capricious Will of human beings and animals, but in heaven the unchanging Will of the Supreme Artificer.

  I do not suggest that this applies to every detail of what Aristotle has to say. What I do suggest is that it gives his imaginative background, and represents the sort of thing which, in embarking on his investigations, he would expect to find true.

  After these preliminaries, let us examine what it is that he actually says.

  Physics, in Aristotle, is the science of what the Greeks called “phusis” (or “physis”), a word which is translated “nature,” but has not exactly the meaning which we attach to that word. We still speak of “natural science” and “natural history,” but “nature” by itself, though it is a very ambiguous word, seldom means just what “phusis” meant. “Phusis” had to do with growth; one might say it is the “nature” of an acorn to grow into an oak, and in that case one would be using the word in the Aristotelian sense. The “nature” of a thing, Aristotle says, is its end, that for the sake of which it exists. Thus the word has a teleological implication. Some things exist by nature, some from other causes. Animals, plants, and simple bodies (elements) exist by nature; they have an internal principle of motion. (The word translated “motion” or “movement” has a wider meaning than “locomotion”; in addition to locomotion it includes change of quality or of size.) Nature is a source of being moved or at rest. Things “have a nature” if they have an internal principle of this kind. The phrase “according to nature” applies to these things and their essential attributes. (It was through this point of view that “unnatural” came to express blame.) Nature is in form rather than in matter; what is potentially flesh or bone has not yet acquired its own nature, and a thing is more what it is when it has attained to fulfilment. This whole point of view seems to be suggested by biology: the acorn is “potentially” an oak.

  Nature belongs to the class of causes which operate for the sake of something This leads to a discussion of the view that nature works of necessity, without purpose, in connection with which Aristotle discusses the survival of the fittest, in the form taught by Empedocles. This cannot be right, he says, because things happen in fixed ways, and when a series has a completion, all preceding steps are for its sake. Those things are “natural” which “by a continuous movement originated from an internal principle, arrive at some completion” (199b).

  This whole conception of “nature,” though it might well seem admirably suited to explain the growth of animals and plants, became, in the event, a great obstacle to the progress of science, and a source of much that was bad in ethics. In the latter respect, it is still harmful.

  Motion, we are told, is the fulfilling of what exists potentially. This view, apart from other defects, is incompatible with the relativity of locomotion. When A moves relatively to B, B moves relatively to A, and there is no sense in saying that one of the two is in motion while the other is at rest. When a dog seizes a bone, it seems to common sense that the dog moves while the bone remains at rest (until seized), and that the motion has a purpose, namely to fulfil the dog’s “nature.” But it has turned out that this point of view cannot be applied to dead matter, and that, for the purposes of scientific physics, no conception of an “end” is useful, nor can any motion, in scientific strictness, be treated as other than relative.

  Aristotle rejects the void, as maintained by Leucippus and Democritus. He then passes on to a rather curious discussion of time. It might, he says, be maintained that time does not exist, since it is composed of past and future, of which one no longer exists while t
he other does not yet exist. This view, however, he rejects. Time, he says, is motion that admits of numeration. (It is not clear why he thinks numeration essential.) We may fairly ask, he continues, whether time could exist without the soul, since there cannot be anything to count unless there is some one to count, and time involves numeration. It seems that he thinks of time as so many hours or days or years. Some things, he adds, are eternal, in the sense of not being in time; presumably he is thinking of such things as numbers.

  There always has been motion, and there always will be; for there cannot be time without motion, and all are agreed that time is uncreated, except Plato. On this point, Christian followers of Aristotle were obliged to dissent from him, since the Bible tells us that the universe had a beginning.

  The Physics ends with the argument for an unmoved mover, which we considered in connection with the Metaphysics. There is one unmoved mover, which directly causes a circular motion. Circular motion is the primary kind, and the only kind which can be continuous and infinite. The first mover has no parts or magnitude and is at the circumference of the world.

  Having reached this conclusion, we pass on to the heavens.

  The treatise On the Heavens sets forth a pleasant and simple theory. Things below the moon are subject to generation and decay; from the moon upwards, everything is ungenerated and indestructible. The earth, which is spherical, is at the centre of the universe. In the sublunary sphere, everything is composed of the four elements, earth, water, air, and fire; but there is a fifth element, of which the heavenly bodies are composed. The natural movement of the terrestrial elements is rectilinear, but that of the fifth element is circular. The heavens are perfectly spherical, and the upper regions are more divine than the lower. The stars and planets are not composed of fire, but of the fifth element; their motion is due to that of spheres to which they are attached. (All this appears in poetical form in Dante’s Paradiso.)

  The four terrestrial elements are not eternal, but are generated out of each other—fire is absolutely light, in the sense that its natural motion is upward; earth is absolutely heavy. Air is relatively light, and water is relatively heavy.

  This theory provided many difficulties for later ages. Comets, which were recognized as destructible, had to be assigned to the sublunary sphere, but in the seventeenth century it was found that they describe orbits round the sun, and are very seldom as near as the moon. Since the natural motion of terrestrial bodies is rectilinear, it was held that a projectile fired horizontally will move horizontally for a time, and then suddenly begin to fall vertically. Galileo’s discovery that a projectile moves in a parabola shocked his Aristotelian colleagues. Copernicus, Kepler, and Galileo had to combat Aristotle as well as the Bible in establishing the view that the earth is not the centre of the universe, but rotates once a day and goes round the sun once a year.

  To come to a more general matter: Aristotelian physics is incompatible with Newton’s “First Law of Motion,” originally enunciated by Galileo. This law states that every body, left to itself, will, if already in motion, continue to move in a straight line with uniform velocity. Thus outside causes are required, not to account for motion, but to account for change of motion, either in velocity or in direction. Circular motion, which Aristotle thought “natural” for the heavenly bodies, involves a continual change in the direction of motion, and therefore requires a force directed towards the centre of the circle, as in Newton’s law of gravitation.

  Finally: The view that the heavenly bodies are eternal and incorruptible has had to be abandoned. The sun and stars have long lives, but do not live for ever. They are born from a nebula, and in the end they either explode or die of cold. Nothing in the visible world is exempt from change and decay; the Aristotelian belief to the contrary, though accepted by medieval Christians, is a product of the pagan worship of sun and moon and planets.

  CHAPTER XXIV

  Early Greek Mathematics and Astronomy

  I AM concerned in this chapter with mathematics, not on its own account, but as it was related to Greek philosophy—a relation which, especially in Plato, was very close. The preeminence of the Greeks appears more clearly in mathematics and astronomy than in anything else. What they did in art, in literature, and in philosophy, may be judged better or worse according to taste, but what they accomplished in geometry is wholly beyond question. They derived something from Egypt, and rather less from Babylonia; but what they obtained from these sources was, in mathematics, mainly rules of thumb, and in astronomy records of observations extended over very long periods. The art of mathematical demonstration was, almost wholly, Greek in origin.

  There are many pleasant stories, probably unhistorical, showing what practical problems stimulated mathematical investigations. The earliest and simplest relates to Thales, who, when in Egypt, was asked by the king to find out the height of a pyramid. He waited for the time of day when his shadow was as long as he was tall; he then measured the shadow of the pyramid, which was of course equal to its height. It is said that the laws of perspective were first studied by the geometer Agatharcus, in order to paint scenery for the plays of Aeschylus. The problem of finding the distance of a ship at sea, which was said to have been studied by Thales, was correctly solved at an early stage. One of the great problems that occupied Greek geometers, that of the duplication of the cube, originated, we are told, with the priests of a certain temple, who were informed by the oracle that the god wanted a statue twice as large as the one they had. At first they thought simply of doubling all the dimensions of the statue, but then they realized that the result would be eight times as large as the original, which would involve more expense than the god had demanded. So they sent a deputation to Plato to ask whether anybody in the Academy could solve their problem. The geometers took it up, and worked at it for centuries, producing, incidentally, much admirable work. The problem is, of course, that of determining the cube root of 2.

  The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the a’s and the b’s; each starts with 1. The next a, at each stage, is formed by adding the last a and b already obtained; the next b is formed by adding twice the previous a to the previous b. The first 6 pairs so obtained are (1,1), (2, 3), (5, 7), (12, 17), (29, 41), (70, 99). In each pair, 2a2-b2 is 1 or -1. Thus b/a is nearly the square root of 2, and at each fresh step it gets nearer. For instance, the reader may satisfy himself that the square of 99/70 is very nearly equal to 2.

  Pythagoras—always a rather misty figure—is described by Proclus as the first who made geometry a liberal education. Many authorities, including Sir Thomas Heath,* believe that he probably discovered the theorem that bears his name, to the effect that, in a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. In any case, this theorem , was known to the Pythagoreans at a very early date. They knew also that the sum of the angles of a triangle is two right angles.

  Irrationals other than the square root of two were studied, in particular cases, by Theodorus, a contemporary of Socrates, and in a more general way by Theaetetus, who was roughly contemporary with Plato, but somewhat older. Democritus wrote a treatise on irrationals, but very little is known as to its contents. Plato was profoundly interested in the subject; he mentions the work of Theodorus and Theaetetus in the dialogue called after the latter. In the Laws (819-820), he says that the general ignorance on this subject is disgraceful, and implies that he himself began to know about it rather late in life. It had of course an important bearing on the Pythagorean philosophy.

  One of the most important consequences of the discovery of irrationals was the invention of the geometrical theory of proportion by Eudoxus (ca. 408 - ca. 355 B.C.). Before him, there was only the arithmetical theory of proportion. According to this theory, the ratio of a
to b is equal to the ratio of c to d if a times d is equal to b times c. This definition, in the absence of an arithmetical theory of irrationals, is only applicable to rationals. Eudoxus, however, gave a new definition not subject to this restriction, framed in a manner which suggests the methods of modern analysis. The theory is developed in Euclid, and has great logical beauty.

  Eudoxus also either invented or perfected the “method of exhaustion,” which was subsequently used with great success by Archimedes. This method is an anticipation of the integral calculus. Take, for example, the question of the area of a circle. You can inscribe in a circle a regular hexagon, or a regular dodecagon, or a regular polygon of a thousand or a million sides. The area of such a polygon, however many sides it has, is proportional to the square on the diameter of the circle. The more sides the polygon has, the more nearly it becomes equal to the circle. You can prove that, if you give the polygon enough sides, its area can be got to differ from that of the circle by less than any previously assigned area, however small. For this purpose, the “axiom of Archimedes” is used. This states (when somewhat simplified) that if the greater of two quantities is halved, and then the half is halved, and so on, a quantity will be reached, at last, which is less than the smaller of the original two quantities. In other words, if a is greater than b, there is some whole number n such that 2n times b is greater than a.