To this objection there are two answers, both of some importance.

(a) We have been considering, in the above account, the question of the _verifiability_ of physics. Now verifiability is by no means the same thing as truth; it is, in fact, something far more subjective and psychological. For a proposition to be verifiable, it is not enough that it should be true, but it must also be such as we can _discover_ to be true. Thus verifiability depends upon our capacity for acquiring knowledge, and not only upon the objective truth. In physics, as ordinarily set forth, there is much that is unverifiable: there are hypotheses as to (a) how things would appear to a spectator in a place where, as it happens, there is no spectator; () how things would appear at times when, in fact, they are not appearing to anyone; (?) things which never appear at all. All these are introduced to simplify the statement of the causal laws, but none of them form an integral part of what is _known_ to be true in physics. This brings us to our second answer.

(b) If physics is to consist wholly of propositions known to be true, or at least capable of being proved or disproved, the three kinds of hypothetical ent.i.ties we have just enumerated must all be capable of being exhibited as logical functions of sense-data. In order to show how this might possibly be done, let us recall the hypothetical Leibnizian universe of Lecture III. In that universe, we had a number of perspectives, two of which never had any ent.i.ty in common, but often contained ent.i.ties which could be sufficiently correlated to be regarded as belonging to the same thing. We will call one of these an "actual"

private world when there is an actual spectator to which it appears, and "ideal" when it is merely constructed on principles of continuity. A physical thing consists, at each instant, of the whole set of its aspects at that instant, in all the different worlds; thus a momentary state of a thing is a whole set of aspects. An "ideal" appearance will be an aspect merely calculated, but not actually perceived by any spectator. An "ideal" state of a thing will be a state at a moment when all its appearances are ideal. An ideal thing will be one whose states at all times are ideal. Ideal appearances, states, and things, since they are calculated, must be functions of actual appearances, states, and things; in fact, ultimately, they must be functions of actual appearances. Thus it is unnecessary, for the enunciation of the laws of physics, to a.s.sign any reality to ideal elements: it is enough to accept them as logical constructions, provided we have means of knowing how to determine when they become actual. This, in fact, we have with some degree of approximation; the starry heaven, for instance, becomes actual whenever we choose to look at it. It is open to us to believe that the ideal elements exist, and there can be no reason for _dis_believing this; but unless in virtue of some _a priori_ law we cannot _know_ it, for empirical knowledge is confined to what we actually observe.

(2) The three main conceptions of physics are s.p.a.ce, time, and matter.

Some of the problems raised by the conception of matter have been indicated in the above discussion of "things." But s.p.a.ce and time also raise difficult problems of much the same kind, namely, difficulties in reducing the haphazard untidy world of immediate sensation to the smooth orderly world of geometry and kinematics. Let us begin with the consideration of s.p.a.ce.

People who have never read any psychology seldom realise how much mental labour has gone into the construction of the one all-embracing s.p.a.ce into which all sensible objects are supposed to fit. Kant, who was unusually ignorant of psychology, described s.p.a.ce as "an infinite given whole," whereas a moment's psychological reflection shows that a s.p.a.ce which is infinite is not given, while a s.p.a.ce which can be called given is not infinite. What the nature of "given" s.p.a.ce really is, is a difficult question, upon which psychologists are by no means agreed. But some general remarks may be made, which will suffice to show the problems, without taking sides on any psychological issue still in debate.

The first thing to notice is that different senses have different s.p.a.ces. The s.p.a.ce of sight is quite different from the s.p.a.ce of touch: it is only by experience in infancy that we learn to correlate them. In later life, when we see an object within reach, we know how to touch it, and more or less what it will feel like; if we touch an object with our eyes shut, we know where we should have to look for it, and more or less what it would look like. But this knowledge is derived from early experience of the correlation of certain kinds of touch-sensations with certain kinds of sight-sensations. The one s.p.a.ce into which both kinds of sensations fit is an intellectual construction, not a datum. And besides touch and sight, there are other kinds of sensation which give other, though less important s.p.a.ces: these also have to be fitted into the one s.p.a.ce by means of experienced correlations. And as in the case of things, so here: the one all-embracing s.p.a.ce, though convenient as a way of speaking, need not be supposed really to exist. All that experience makes certain is the several s.p.a.ces of the several senses, correlated by empirically discovered laws. The one s.p.a.ce may turn out to be valid as a logical construction, compounded of the several s.p.a.ces, but there is no good reason to a.s.sume its independent metaphysical reality.

Another respect in which the s.p.a.ces of immediate experience differ from the s.p.a.ce of geometry and physics is in regard to _points_. The s.p.a.ce of geometry and physics consists of an infinite number of points, but no one has ever seen or touched a point. If there are points in a sensible s.p.a.ce, they must be an inference. It is not easy to see any way in which, as independent ent.i.ties, they could be validly inferred from the data; thus here again, we shall have, if possible, to find some logical construction, some complex a.s.semblage of immediately given objects, which will have the geometrical properties required of points. It is customary to think of points as simple and infinitely small, but geometry in no way demands that we should think of them in this way. All that is necessary for geometry is that they should have mutual relations possessing certain enumerated abstract properties, and it may be that an a.s.semblage of data of sensation will serve this purpose. Exactly how this is to be done, I do not yet know, but it seems fairly certain that it can be done.

The following ill.u.s.trative method, simplified so as to be easily manipulated, has been invented by Dr Whitehead for the purpose of showing how points might be manufactured from sense-data. We have first of all to observe that there are no infinitesimal sense-data: any surface we can see, for example, must be of some finite extent. But what at first appears as one undivided whole is often found, under the influence of attention, to split up into parts contained within the whole. Thus one spatial object may be contained within another, and entirely enclosed by the other. This relation of enclosure, by the help of some very natural hypotheses, will enable us to define a "point" as a certain cla.s.s of spatial objects, namely all those (as it will turn out in the end) which would naturally be said to contain the point. In order to obtain a definition of a "point" in this way, we proceed as follows:

Given any set of volumes or surfaces, they will not in general converge into one point. But if they get smaller and smaller, while of any two of the set there is always one that encloses the other, then we begin to have the kind of conditions which would enable us to treat them as having a point for their limit. The hypotheses required for the relation of enclosure are that (1) it must be transitive; (2) of two _different_ spatial objects, it is impossible for each to enclose the other, but a single spatial object always encloses itself; (3) any set of spatial objects such that there is at least one spatial object enclosed by them all has a lower limit or minimum, _i.e._ an object enclosed by all of them and enclosing all objects which are enclosed by all of them; (4) to prevent trivial exceptions, we must add that there are to be instances of enclosure, _i.e._ there are really to be objects of which one encloses the other. When an enclosure-relation has these properties, we will call it a "point-producer." Given any relation of enclosure, we will call a set of objects an "enclosure-series" if, of any two of them, one is contained in the other. We require a condition which shall secure that an enclosure-series converges to a point, and this is obtained as follows: Let our enclosure-series be such that, given any other enclosure-series of which there are members enclosed in any arbitrarily chosen member of our first series, then there are members of our first series enclosed in any arbitrarily chosen member of our second series.

In this case, our first enclosure-series may be called a "punctual enclosure-series." Then a "point" is all the objects which enclose members of a given punctual enclosure-series. In order to ensure infinite divisibility, we require one further property to be added to those defining point-producers, namely that any object which encloses itself also encloses an object other than itself. The "points" generated by point-producers with this property will be found to be such as geometry requires.

(3) The question of time, so long as we confine ourselves to one private world, is rather less complicated than that of s.p.a.ce, and we can see pretty clearly how it might be dealt with by such methods as we have been considering. Events of which we are conscious do not last merely for a mathematical instant, but always for some finite time, however short. Even if there be a physical world such as the mathematical theory of motion supposes, impressions on our sense-organs produce sensations which are not merely and strictly instantaneous, and therefore the objects of sense of which we are immediately conscious are not strictly instantaneous. Instants, therefore, are not among the data of experience, and, if legitimate, must be either inferred or constructed.

It is difficult to see how they can be validly inferred; thus we are left with the alternative that they must be constructed. How is this to be done?

Immediate experience provides us with two time-relations among events: they may be simultaneous, or one may be earlier and the other later.

These two are both part of the crude data; it is not the case that only the events are given, and their time-order is added by our subjective activity. The time-order, within certain limits, is as much given as the events. In any story of adventure you will find such pa.s.sages as the following: "With a cynical smile he pointed the revolver at the breast of the dauntless youth. 'At the word _three_ I shall fire,' he said. The words one and two had already been spoken with a cool and deliberate distinctness. The word _three_ was forming on his lips. At this moment a blinding flash of lightning rent the air." Here we have simultaneity--not due, as Kant would have us believe, to the subjective mental apparatus of the dauntless youth, but given as objectively as the revolver and the lightning. And it is equally given in immediate experience that the words _one_ and _two_ come earlier than the flash.

These time-relations hold between events which are not strictly instantaneous. Thus one event may begin sooner than another, and therefore be before it, but may continue after the other has begun, and therefore be also simultaneous with it. If it persists after the other is over, it will also be later than the other. Earlier, simultaneous, and later, are not inconsistent with each other when we are concerned with events which last for a finite time, however short; they only become inconsistent when we are dealing with something instantaneous.

It is to be observed that we cannot give what may be called _absolute_ dates, but only dates determined by events. We cannot point to a time itself, but only to some event occurring at that time. There is therefore no reason in experience to suppose that there are times as opposed to events: the events, ordered by the relations of simultaneity and succession, are all that experience provides. Hence, unless we are to introduce superfluous metaphysical ent.i.ties, we must, in defining what mathematical physics can regard as an instant, proceed by means of some construction which a.s.sumes nothing beyond events and their temporal relations.

If we wish to a.s.sign a date exactly by means of events, how shall we proceed? If we take any one event, we cannot a.s.sign our date exactly, because the event is not instantaneous, that is to say, it may be simultaneous with two events which are not simultaneous with each other.

In order to a.s.sign a date exactly, we must be able, theoretically, to determine whether any given event is before, at, or after this date, and we must know that any other date is either before or after this date, but not simultaneous with it. Suppose, now, instead of taking one event A, we take two events A and B, and suppose A and B partly overlap, but B ends before A ends. Then an event which is simultaneous with both A and B must exist during the time when A and B overlap; thus we have come rather nearer to a precise date than when we considered A and B alone.

Let C be an event which is simultaneous with both A and B, but which ends before either A or B has ended. Then an event which is simultaneous with A and B and C must exist during the time when all three overlap, which is a still shorter time. Proceeding in this way, by taking more and more events, a new event which is dated as simultaneous with all of them becomes gradually more and more accurately dated. This suggests a way by which a completely accurate date can be defined.

A____________________

B____________________

C________

Let us take a group of events of which any two overlap, so that there is some time, however short, when they all exist. If there is any other event which is simultaneous with all of these, let us add it to the group; let us go on until we have constructed a group such that no event outside the group is simultaneous with all of them, but all the events inside the group are simultaneous with each other. Let us define this whole group as an instant of time. It remains to show that it has the properties we expect of an instant.

What are the properties we expect of instants? First, they must form a series: of any two, one must be before the other, and the other must be not before the one; if one is before another, and the other before a third, the first must be before the third. Secondly, every event must be at a certain number of instants; two events are simultaneous if they are at the same instant, and one is before the other if there is an instant, at which the one is, which is earlier than some instant at which the other is. Thirdly, if we a.s.sume that there is always some change going on somewhere during the time when any given event persists, the series of instants ought to be compact, _i.e._ given any two instants, there ought to be other instants between them. Do instants, as we have defined them, have these properties?

We shall say that an event is "at" an instant when it is a member of the group by which the instant is const.i.tuted; and we shall say that one instant is before another if the group which is the one instant contains an event which is earlier than, but not simultaneous with, some event in the group which is the other instant. When one event is earlier than, but not simultaneous with another, we shall say that it "wholly precedes" the other. Now we know that of two events which are not simultaneous, there must be one which wholly precedes the other, and in that case the other cannot also wholly precede the one; we also know that, if one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third. From these facts it is easy to deduce that the instants as we have defined them form a series.

We have next to show that every event is "at" at least one instant, _i.e._ that, given any event, there is at least one cla.s.s, such as we used in defining instants, of which it is a member. For this purpose, consider all the events which are simultaneous with a given event, and do not begin later, _i.e._ are not wholly after anything simultaneous with it. We will call these the "initial contemporaries" of the given event. It will be found that this cla.s.s of events is the first instant at which the given event exists, provided every event wholly after some contemporary of the given event is wholly after some _initial_ contemporary of it.

Finally, the series of instants will be compact if, given any two events of which one wholly precedes the other, there are events wholly after the one and simultaneous with something wholly before the other. Whether this is the case or not, is an empirical question; but if it is not, there is no reason to expect the time-series to be compact.[17]

[17] The a.s.sumptions made concerning time-relations in the above are as follows:--

I. In order to secure that instants form a series, we a.s.sume:

(a) No event wholly precedes itself. (An "event" is defined as whatever is simultaneous with something or other.)

(b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third.

(c) If one event wholly precedes another, it is not simultaneous with it.

(d) Of two events which are not simultaneous, one must wholly precede the other.

II. In order to secure that the initial contemporaries of a given event should form an instant, we a.s.sume:

(e) An event wholly after some contemporary of a given event is wholly after some _initial_ contemporary of the given event.

III. In order to secure that the series of instants shall be compact, we a.s.sume:

(f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.

This a.s.sumption entails the consequence that if one event covers the whole of a stretch of time immediately preceding another event, then it must have at least one instant in common with the other event; _i.e._ it is impossible for one event to cease just before another begins. I do not know whether this should be regarded as inadmissible.

For a mathematico-logical treatment of the above topics, _cf._ N.

Wilner, "A Contribution to the Theory of Relative Position," _Proc.

Camb. Phil. Soc._, xvii. 5, pp. 441-449.

Thus our definition of instants secures all that mathematics requires, without having to a.s.sume the existence of any disputable metaphysical ent.i.ties.

Instants may also be defined by means of the enclosure-relation, exactly as was done in the case of points. One object will be temporally enclosed by another when it is simultaneous with the other, but not before or after it. Whatever encloses temporally or is enclosed temporally we shall call an "event." In order that the relation of temporal enclosure may be a "point-producer," we require (1) that it should be transitive, _i.e._ that if one event encloses another, and the other a third, then the first encloses the third; (2) that every event encloses itself, but if one event encloses another different event, then the other does not enclose the one; (3) that given any set of events such that there is at least one event enclosed by all of them, then there is an event enclosing all that they all enclose, and itself enclosed by all of them; (4) that there is at least one event. To ensure infinite divisibility, we require also that every event should enclose events other than itself. a.s.suming these characteristics, temporal enclosure is an infinitely divisible point-producer. We can now form an "enclosure-series" of events, by choosing a group of events such that of any two there is one which encloses the other; this will be a "punctual enclosure-series" if, given any other enclosure-series such that every member of our first series encloses some member of our second, then every member of our second series encloses some member of our first.

Then an "instant" is the cla.s.s of all events which enclose members of a given punctual enclosure-series.

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 cla.s.ses 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 ent.i.ties.

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 a.s.sume 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, s.p.a.ce, 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, s.p.a.ce, 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 _a.n.a.lysis 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 cla.s.s 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 repet.i.tion 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 cla.s.s 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 cla.s.s.

Since there certainly is the cla.s.s, while any other common property may be illusory, it is prudent, in order to avoid needless a.s.sumptions, to subst.i.tute the cla.s.s for the common property which would be ordinarily a.s.sumed. 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 a.s.sumes, since we do not deny them, but merely abstain from a.s.serting 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 fict.i.tious metaphysical ent.i.ties.