Part 10

The correlation of the times of different private worlds so as to produce the one all-embracing time of physics is a more difficult matter. We saw, in Lecture III., that different private worlds often contain correlated appearances, such as common sense would regard as appearances of the same "thing." When two appearances in different worlds are so correlated as to belong to one momentary "state" of a thing, it would be natural to regard them as simultaneous, and as thus affording a simple means of correlating different private times. But this can only be regarded as a first approximation. What we call one sound will be heard sooner by people near the source of the sound than by people further from it, and the same applies, though in a less degree, to light. Thus two correlated appearances in different worlds are not necessarily to be regarded as occurring at the same date in physical time, though they will be parts of one momentary state of a thing. The correlation of different private times is regulated by the desire to secure the simplest possible statement of the laws of physics, and thus raises rather complicated technical problems; but from the point of view of philosophical theory, there is no very serious difficulty of principle involved.

The above brief outline must not be regarded as more than tentative and suggestive. It is intended merely to show the kind of way in which, given a world with the kind of properties that psychologists find in the world of sense, it may be possible, by means of purely logical constructions, to make it amenable to mathematical treatment by defining series or classes of sense-data which can be called respectively particles, points, and instants. If such constructions are possible, then mathematical physics is applicable to the real world, in spite of the fact that its particles, points, and instants are not to be found among actually existing entities.

The problem which the above considerations are intended to elucidate is one whose importance and even existence has been concealed by the unfortunate separation of different studies which prevails throughout the civilised world. Physicists, ignorant and contemptuous of philosophy, have been content to assume their particles, points, and instants in practice, while conceding, with ironical politeness, that their concepts laid no claim to metaphysical validity. Metaphysicians, obsessed by the idealistic opinion that only mind is real, and the Parmenidean belief that the real is unchanging, repeated one after another the supposed contradictions in the notions of matter, space, and time, and therefore naturally made no endeavour to invent a tenable theory of particles, points, and instants. Psychologists, who have done invaluable work in bringing to light the chaotic nature of the crude materials supplied by unmanipulated sensation, have been ignorant of mathematics and modern logic, and have therefore been content to say that matter, space, and time are "intellectual constructions," without making any attempt to show in detail either how the intellect can construct them, or what secures the practical validity which physics shows them to possess. Philosophers, it is to be hoped, will come to recognise that they cannot achieve any solid success in such problems without some slight knowledge of logic, mathematics, and physics; meanwhile, for want of students with the necessary equipment, this vital problem remains unattempted and unknown.

There are, it is true, two authors, both physicists, who have done something, though not much, to bring about a recognition of the problem as one demanding study. These two authors are Poincare and Mach, Poincare especially in his _Science and Hypothesis_, Mach especially in his _Analysis of Sensations_. Both of them, however, admirable as their work is, seem to me to suffer from a general philosophical bias. Poincare is Kantian, while Mach is ultra-empiricist; with Poincare almost all the mathematical part of physics is merely conventional, while with Mach the sensation as a mental event is identified with its object as a part of the physical world. Nevertheless, both these authors, and especially Mach, deserve mention as having made serious contributions to the consideration of our problem.

When a point or an instant is defined as a class of sensible qualities, the first impression produced is likely to be one of wild and wilful paradox. Certain considerations apply here, however, which will again be relevant when we come to the definition of numbers. There is a whole type of problems which can be solved by such definitions, and almost always there will be at first an effect of paradox. Given a set of objects any two of which have a relation of the sort called "symmetrical and transitive," it is almost certain that we shall come to regard them as all having some common quality, or as all having the same relation to some one object outside the set. This kind of case is important, and I shall therefore try to make it clear even at the cost of some repetition of previous definitions.

A relation is said to be "symmetrical" when, if one term has this relation to another, then the other also has it to the one. Thus "brother or sister" is a "symmetrical" relation: if one person is a brother or a sister of another, then the other is a brother or sister of the one. Simultaneity, again, is a symmetrical relation; so is equality in size. A relation is said to be "transitive" when, if one term has this relation to another, and the other to a third, then the one has it to the third. The symmetrical relations mentioned just now are also transitive--provided, in the case of "brother or sister," we allow a person to be counted as his or her own brother or sister, and provided, in the case of simultaneity, we mean complete simultaneity, _i.e._ beginning and ending together.

But many relations are transitive without being symmetrical--for instance, such relations as "greater," "earlier," "to the right of," "ancestor of," in fact all such relations as give rise to series. Other relations are symmetrical without being transitive--for example, difference in any respect. If A is of a different age from B, and B of a different age from C, it does not follow that A is of a different age from C. Simultaneity, again, in the case of events which last for a finite time, will not necessarily be transitive if it only means that the times of the two events overlap. If A ends just after B has begun, and B ends just after C has begun, A and B will be simultaneous in this sense, and so will B and C, but A and C may well not be simultaneous.

All the relations which can naturally be represented as equality in any respect, or as possession of a common property, are transitive and symmetrical--this applies, for example, to such relations as being of the same height or weight or colour. Owing to the fact that possession of a common property gives rise to a transitive symmetrical relation, we come to imagine that wherever such a relation occurs it must be due to a common property. "Being equally numerous" is a transitive symmetrical relation of two collections; hence we imagine that both have a common property, called their number. "Existing at a given instant" (in the sense in which we defined an instant) is a transitive symmetrical relation; hence we come to think that there really is an instant which confers a common property on all the things existing at that instant. "Being states of a given thing" is a transitive symmetrical relation; hence we come to imagine that there really is a thing, other than the series of states, which accounts for the transitive symmetrical relation. In all such cases, the class of terms that have the given transitive symmetrical relation to a given term will fulfil all the formal requisites of a common property of all the members of the class. Since there certainly is the class, while any other common property may be illusory, it is prudent, in order to avoid needless assumptions, to substitute the class for the common property which would be ordinarily assumed. This is the reason for the definitions we have adopted, and this is the source of the apparent paradoxes. No harm is done if there are such common properties as language assumes, since we do not deny them, but merely abstain from asserting them. But if there are not such common properties in any given case, then our method has secured us against error. In the absence of special knowledge, therefore, the method we have adopted is the only one which is safe, and which avoids the risk of introducing fictitious metaphysical entities.

LECTURE V

THE THEORY OF CONTINUITY

The theory of continuity, with which we shall be occupied in the present lecture, is, in most of its refinements and developments, a purely mathematical subject--very beautiful, very important, and very delightful, but not, strictly speaking, a part of philosophy. The logical basis of the theory alone belongs to philosophy, and alone will occupy us to-night. The way the problem of continuity enters into philosophy is, broadly speaking, the following: Space and time are treated by mathematicians as consisting of points and instants, but they also have a property, easier to feel than to define, which is called continuity, and is thought by many philosophers to be destroyed when they are resolved into points and instants. Zeno, as we shall see, proved that analysis into points and instants was impossible if we adhered to the view that the number of points or instants in a finite space or time must be finite. Later philosophers, believing infinite number to be self-contradictory, have found here an antinomy: Spaces and times could not consist of a _finite_ number of points and instants, for such reasons as Zeno's; they could not consist of an _infinite_ number of points and instants, because infinite numbers were supposed to be self-contradictory. Therefore spaces and times, if real at all, must not be regarded as composed of points and instants.

But even when points and instants, as independent entities, are discarded, as they were by the theory advocated in our last lecture, the problems of continuity, as I shall try to show presently, remain, in a practically unchanged form. Let us therefore, to begin with, admit points and instants, and consider the problems in connection with this simpler or at least more familiar hypothesis.

The argument against continuity, in so far as it rests upon the supposed difficulties of infinite numbers, has been disposed of by the positive theory of the infinite, which will be considered in Lecture VII. But there remains a feeling--of the kind that led Zeno to the contention that the arrow in its flight is at rest--which suggests that points and instants, even if they are infinitely numerous, can only give a jerky motion, a succession of different immobilities, not the smooth transitions with which the senses have made us familiar. This feeling is due, I believe, to a failure to realise imaginatively, as well as abstractly, the nature of continuous series as they appear in mathematics. When a theory has been apprehended logically, there is often a long and serious labour still required in order to _feel_ it: it is necessary to dwell upon it, to thrust out from the mind, one by one, the misleading suggestions of false but more familiar theories, to acquire the kind of intimacy which, in the case of a foreign language, would enable us to think and dream in it, not merely to construct laborious sentences by the help of grammar and dictionary. It is, I believe, the absence of this kind of intimacy which makes many philosophers regard the mathematical doctrine of continuity as an inadequate explanation of the continuity which we experience in the world of sense.

In the present lecture, I shall first try to explain in outline what the mathematical theory of continuity is in its philosophically important essentials. The application to actual space and time will not be in question to begin with. I do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities, but I do see reason to suppose that the continuity of actual space and time may be more or less analogous to mathematical continuity. The theory of mathematical continuity is an abstract logical theory, not dependent for its validity upon any properties of actual space and time. What is claimed for it is that, when it is understood, certain characteristics of space and time, previously very hard to analyse, are found not to present any logical difficulty. What we know empirically about space and time is insufficient to enable us to decide between various mathematically possible alternatives, but these alternatives are all fully intelligible and fully adequate to the observed facts. For the present, however, it will be well to forget space and time and the continuity of sensible change, in order to return to these topics equipped with the weapons provided by the abstract theory of continuity.

Continuity, in mathematics, is a property only possible to a _series_ of terms, _i.e._ to terms arranged in an order, so that we can say of any two that one comes _before_ the other. Numbers in order of magnitude, the points on a line from left to right, the moments of time from earlier to later, are instances of series. The notion of order, which is here introduced, is one which is not required in the theory of cardinal number. It is possible to know that two classes have the same number of terms without knowing any order in which they are to be taken. We have an instance of this in such a case as English husbands and English wives: we can see that there must be the same number of husbands as of wives, without having to arrange them in a series. But continuity, which we are now to consider, is essentially a property of an order: it does not belong to a set of terms in themselves, but only to a set in a certain order. A set of terms which can be arranged in one order can always also be arranged in other orders, and a set of terms which can be arranged in a continuous order can always also be arranged in orders which are not continuous. Thus the essence of continuity must not be sought in the nature of the set of terms, but in the nature of their arrangement in a series.

Mathematicians have distinguished different degrees of continuity, and have confined the word "continuous," for technical purposes, to series having a certain high degree of continuity. But for philosophical purposes, all that is important in continuity is introduced by the lowest degree of continuity, which is called "compactness." A series is called "compact" when no two terms are consecutive, but between any two there are others. One of the simplest examples of a compact series is the series of fractions in order of magnitude. Given any two fractions, however near together, there are other fractions greater than the one and smaller than the other, and therefore no two fractions are consecutive. There is no fraction, for example, which is next after 1/2: if we choose some fraction which is very little greater than 1/2, say 51/100 we can find others, such as 101/200, which are nearer to 1/2. Thus between any two fractions, however little they differ, there are an infinite number of other fractions. Mathematical space and time also have this property of compactness, though whether actual space and time have it is a further question, dependent upon empirical evidence, and probably incapable of being answered with certainty.

In the case of abstract objects such as fractions, it is perhaps not very difficult to realise the logical possibility of their forming a compact series. The difficulties that might be felt are those of infinity, for in a compact series the number of terms between any two given terms must be infinite. But when these difficulties have been solved, the mere compactness in itself offers no great obstacle to the imagination. In more concrete cases, however, such as motion, compactness becomes much more repugnant to our habits of thought. It will therefore be desirable to consider explicitly the mathematical account of motion, with a view to making its logical possibility felt. The mathematical account of motion is perhaps artificially simplified when regarded as describing what actually occurs in the physical world; but what actually occurs must be capable, by a certain amount of logical manipulation, of being brought within the scope of the mathematical account, and must, in its analysis, raise just such problems as are raised in their simplest form by this account. Neglecting, therefore, for the present, the question of its physical adequacy, let us devote ourselves merely to considering its possibility as a formal statement of the nature of motion.

In order to simplify our problem as much as possible, let us imagine a tiny speck of light moving along a scale. What do we mean by saying that the motion is continuous? It is not necessary for our purposes to consider the whole of what the mathematician means by this statement: only part of what he means is philosophically important. One part of what he means is that, if we consider any two positions of the speck occupied at any two instants, there will be other intermediate positions occupied at intermediate instants. However near together we take the two positions, the speck will not jump suddenly from the one to the other, but will pass through an infinite number of other positions on the way. Every distance, however small, is traversed by passing through all the infinite series of positions between the two ends of the distance.

But at this point imagination suggests that we may describe the continuity of motion by saying that the speck always passes from one position at one instant to _the next_ position at _the next_ instant. As soon as we say this or imagine it, we fall into error, because there is no _next_ point or _next_ instant. If there were, we should find Zeno's paradoxes, in some form, unavoidable, as will appear in our next lecture. One simple paradox may serve as an illustration. If our speck is in motion along the scale throughout the whole of a certain time, it cannot be at the same point at two consecutive instants. But it cannot, from one instant to the next, travel further than from one point to the next, for if it did, there would be no instant at which it was in the positions intermediate between that at the first instant and that at the next, and we agreed that the continuity of motion excludes the possibility of such sudden jumps. It follows that our speck must, so long as it moves, pass from one point at one instant to the next point at the next instant. Thus there will be just one perfectly definite velocity with which all motions must take place: no motion can be faster than this, and no motion can be slower. Since this conclusion is false, we must reject the hypothesis upon which it is based, namely that there are consecutive points and instants.[18] Hence the continuity of motion must not be supposed to consist in a body's occupying consecutive positions at consecutive times.

[18] The above paradox is essentially the same as Zeno's argument of the stadium which will be considered in our next lecture.

The difficulty to imagination lies chiefly, I think, in keeping out the suggestion of _infinitesimal_ distances and times. Suppose we halve a given distance, and then halve the half, and so on, we can continue the process as long as we please, and the longer we continue it, the smaller the resulting distance becomes. This infinite divisibility seems, at first sight, to imply that there are infinitesimal distances, _i.e._ distances so small that any finite fraction of an inch would be greater. This, however, is an error. The continued bisection of our distance, though it gives us continually smaller distances, gives us always _finite_ distances. If our original distance was an inch, we reach successively half an inch, a quarter of an inch, an eighth, a sixteenth, and so on; but every one of this infinite series of diminishing distances is finite. "But," it may be said, "_in the end_ the distance will grow infinitesimal." No, because there is no end. The process of bisection is one which can, theoretically, be carried on for ever, without any last term being attained. Thus infinite divisibility of distances, which must be admitted, does not imply that there are distances so small that any finite distance would be larger.

It is easy, in this kind of question, to fall into an elementary logical blunder. Given any finite distance, we can find a smaller distance; this may be expressed in the ambiguous form "there is a distance smaller than any finite distance." But if this is then interpreted as meaning "there is a distance such that, whatever finite distance may be chosen, the distance in question is smaller," then the statement is false. Common language is ill adapted to expressing matters of this kind, and philosophers who have been dependent on it have frequently been misled by it.

In a continuous motion, then, we shall say that at any given instant the moving body occupies a certain position, and at other instants it occupies other positions; the interval between any two instants and between any two positions is always finite, but the continuity of the motion is shown in the fact that, however near together we take the two positions and the two instants, there are an infinite number of positions still nearer together, which are occupied at instants that are also still nearer together. The moving body never jumps from one position to another, but always passes by a gradual transition through an infinite number of intermediaries. At a given instant, it is where it is, like Zeno's arrow;[19] but we cannot say that it is at rest at the instant, since the instant does not last for a finite time, and there is not a beginning and end of the instant with an interval between them. Rest consists in being in the same position at all the instants throughout a certain finite period, however short; it does not consist simply in a body's being where it is at a given instant. This whole theory, as is obvious, depends upon the nature of compact series, and demands, for its full comprehension, that compact series should have become familiar and easy to the imagination as well as to deliberate thought.

[19] See next lecture.