CHAPTER VIII
SPACE, TIME, AND RELATIVITY
(_Paper read to Section A at the Manchester Meeting of the British Association, 1915, and later before the Aristotelian Society_)
FUNDAMENTAL Problems concerning space and time have been considered from the standpoints created by many different sciences. The object of this paper is the humble one of bringing some of these standpoints into relation with each other. This necessitates a very cursory treatment of each point of view.
Mathematical physicists have evolved their theory of relativity to explain the negative results of the Morley-Michelson experiment and of the Trouton experiment. Experimental psychologists have considered the evolution of spatial ideas from the crude sense-data of experience. Metaphysicians have considered the majestic uniformity of space and time, without beginning and without end, without boundaries, and without exception in the truths concerning them; all these qualities the more arresting to our attention from the confused accidental nature of the empirical universe which is conditioned by them. Mathematicians have studied the axioms of geometry, and can now deduce all that is believed to be universally true of space and of time by the strictest logic from a limited number of assumptions.
These various lines of thought have been evolved with surprisingly little interconnection. Perhaps it is as well. The results of science are never quite true. By a healthy independence of thought perhaps we sometimes avoid adding other people's errors to our own. But there can be no doubt that the normal method of cross-fertilising thought is by considering the same, or allied problems to our own, in the form which they assume in other sciences.
Here I do not propose to enter into a systematic study of these various chapters of science. I have neither the knowledge nor the time.
First, let us take the ultimate basis of any theory of relativity. All space measurement is from stuff in space to stuff in space. The geometrical entities of empty space never appear. The only geometrical properties of which we have any direct knowledge are properties of those shifting, changeable appearances which we call things in space. It is the sun which is distant, and the ball which is round, and the lamp-posts which are in linear order. Wherever mankind may have got its idea of an infinite unchangeable space from, it is safe to say that it is not an immediate deliverance of direct observation.
There are two antagonistic philosophical ways of recognising this conclusion.
One is to affirm that space and time are conditions for sensible experience, that without projection into space and time sensible experience would not exist. Thus, although it may be true to say that our knowledge of space and time is given in experience, it is not true to say that it is deduced from experience in the same sense that the Law of Gravitation is so deduced. It is not deduced, because in the act of experiencing we are necessarily made aware of space as an infinite given whole, and of time as an unending uniform succession. This philosophical position is expressed by saying that space and time are _a priori_ forms of sensibility.
The opposed philosophical method of dealing with the question is to affirm that our concepts of time and space are deductions from experience, in exactly the same way as the Law of Gravitation is such a deduction. If we form exact concepts of points, lines and surfaces, and of successive instants of time, and assume them to be related as expressed by the axioms of geometry and the axioms for time, then we find that we have framed a concept which, with all the exactness of which our observations are capable, expresses the facts of experience.
These two philosophic positions are each designed to explain a certain difficulty. The _a priori_ theory explains the absolute universality ascribed to the laws of space and time, a universality not ascribed to any deduction from experience. The experiential theory explains the derivation of the space-time concepts without introducing any other factors beyond those which are admittedly present in framing the other concepts of physical science.
But we have not yet done with the distinctions which in any discussion of space or time must essentially be kept in mind. Put aside the above question as to how these space-time concepts are related to experience--What are they when they are formed?
We may conceive of the points of space as self-subsistent entities which have the indefinable relation of being occupied by the ultimate stuff (matter, I will call it) which is there. Thus, to say that the sun is _there_ (wherever it is) is to affirm the relation of occupation between the set of positive and negative electrons which we call the sun and a certain set of points, the points having an existence essentially independent of the sun. This is the absolute theory of space. The absolute theory is not popular just now, but it has very respectable authority on its side--Newton, for one--so treat it tenderly.
The other theory is associated with Leibniz. Our space concepts are concepts of relations between things in space. Thus there is no such entity as a self-subsistent point. A point is merely the name for some peculiarity of the relations between the matter which is, in common language, said to be in space.
It follows from the relative theory that a point should be definable in terms of the relations between material things. So far as I am aware, this outcome of the theory has escaped the notice of mathematicians, who have invariably assumed the point as the ultimate starting ground of their reasoning. Many years ago I explained some types of ways in which we might achieve such a definition, and more recently have added some others. Similar explanations apply to time. Before the theories of space and time have been carried to a satisfactory conclusion on the relational basis, a long and careful scrutiny of the definitions of points of space and instants of time will have to be undertaken, and many ways of effecting these definitions will have to be tried and compared. This is an unwritten chapter of mathematics, in much the same state as was the theory of parallels in the eighteenth century.
In this connection I should like to draw attention to the analogy between time and space. In analysing our experience we distinguish events, and we also distinguish things whose changing relations form the events. If I had time it would be interesting to consider more closely these concepts of events and of things. It must suffice now to point out that things have certain relations to each other which we consider as relations between the space extensions of the things; for example, one space can contain the other, or exclude it, or overlap it. A point in space is nothing else than a certain set of relations between spatial extensions.
Analogously, there are certain relations between events which we express by saying that they are relations between the temporal durations of these events, that is, between the temporal extensions of the events. [The durations of two events A and B may one precede the other, or may partially overlap, or may one contain the other, giving in all six possibilities.] The properties of the extension of an event in time are largely analogous to the extension of an object in space. Spatial extensions are expressed by relations between objects, temporal extensions by relations between events.
The point in time is a set of relations between temporal extensions. It needs very little reflection to convince us that a point in time is no direct deliverance of experience. We live in durations, and not in points. But what community, beyond the mere name, is there between extension in time and extension in space? In view of the intimate connection between time and space revealed by the modern theory of relativity, this question has taken on a new importance.
I have not thought out an answer to this question. I suggest, however, that time and space embody those relations between objects on which depends our judgment of their externality to ourselves. Namely, location in space and location in time both embody and perhaps necessitate a judgment of externality. This suggestion is very vague, and I must leave it in this crude form.
_Diverse Euclidean Measure Systems_
Turning now to the mathematical investigations on the axioms of geometry, the outcome, which is most important for us to remember, is the great separation which it discloses between non-metrical projective geometry, and metrical geometry. Non-metrical projective geometry is by far the more fundamental. Starting with the concepts of points, straight lines, and planes (of which not all three need be taken as indefinable), and with certain very simple non-metrical properties of these entities--such as, for instance, that two points uniquely determine a straight line--nearly the whole of geometry can be constructed. Even quantitative coordinates can be introduced, to facilitate the reasoning. But no mention of distance, area, or volume, need have been introduced. Points will have an order on the line, but order does not imply any settled distance.
When we now inquire what measurements of distance are possible, we find that there are different systems of measurement all equally possible. There are three main types of system: any system of one type gives Euclidean geometry, any system of another type gives Hyperbolic (or Lobatchewskian) geometry, any system of the third type gives Elliptic geometry. Also different beings, or the same being if he chooses, may reckon in different systems of the same type, or in systems of different types. Consider the example which will interest us later. Two beings, A and B, agree to use the same three intersecting lines as axes of _x_, _y_, _z_. They both employ a system of measurement of the Euclidean type, and (what is not necessarily the case) agree as to the plane at infinity. That is, they agree as to the lines which are parallel. Then with the usual method of rectangular Cartesian axes, they agree that the coordinates of P are the lengths ON, NM, MP. So far all is harmony. A fixes on the segment OU_{1}, on O_x_, as being the unit length, and B on the segment OV_{1}, on O_x_. A calls his coordinates (_x_, _y_, _z_), and B calls them (X, Y, Z).
Then it is found [since both systems are Euclidean] that, whatever point P be taken,
X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.]
They proceed to adjust their differences, and first take the _x_-coordinates. Obviously they have taken different units of length along O_x_. The length OU_{1}, which A calls one unit, B calls β units. B changes his unit length to OU_{1}, from its original length OV_{1}, and obtains X = _x_. But now, as he must use the same unit for all his measurements, his other coordinates are altered in the same ratio. Thus we now have
X = _x_, Y = γ_y_/β, Z = δ_z_/β.
The fundamental divergence is now evident. A and B agree as to their units along O_x_. They settled that by taking along that axis a given segment OU_{1} as having the unit length. But they cannot agree as to what segment along O_y_ is equal to OU_{1}. A says it is OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ.
The result is that A's spheres
_x_^2 + _y_^2 + _z_^2 = _r_^2,
are B's ellipsoids,
X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2, _i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2.
Thus the measurement of angles by the two is hopelessly at variance.
If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular axes at O, namely that from which they started. If γ = δ, but β ≠ γ, then there are a singly infinite number of common rectangular axes found by rotating the axes round O_x_. This is, for us, the interesting case. The same phenomena are reproduced by transferring to any parallel axes.
The root of the difficulty is, that A's measuring rod, which for him is a rigid invariable body, appears to B as changing in length when turned in different directions. Similarly all measuring rods, satisfactory to A, violate B's immediate judgment of invariability, and change according to the same law. There is no way out of the difficulty. Two rods ρ and σ coincide whenever laid one on the other; ρ is held still, and both men agree that it does not change. But σ is turned round. A says it is invariable, B says it changes. To test the matter, ρ is turned round to measure it, and exactly fits it. But while A is satisfied, B declares that ρ has changed in exactly the same way as did σ. Meanwhile B has procured two material rods satisfactory to him as invariable, and A makes exactly the same objections.
We shall say that A and B employ diverse Euclidean metrical systems.
The most extraordinary fact of human life is that all beings seem to form their judgments of spatial quantity according to the same metrical system.
_Relativity in Modern Physics_
Owing to the fact that points of space are incapable of direct recognition, there is a difficulty--apart from any abstract question of the nature of space--in deciding on the motion to be ascribed to any body. Even if there be such a thing as absolute position, it is impossible in practice to decide directly whether a body's absolute position has changed. All spatial measurement is relative to matter.
Newton's laws of motion in their modern dress evade this difficulty by asserting that a framework of axes of coordinates can be defined by their relations to matter such that, assuming these axes to be at rest, and all velocities to be measured relatively to them, the laws hold. The same expedient has to be employed for time, namely, the laws hold when the measurement of the flow of time is made by the proper reference to periodic events. Thus the laws assert that the framework and the natural clock adapted for their use have been successfully found.
But, if one framework will do, an infinity of others serve equally well; namely, not only--as is of course the case--all those at rest relatively to the first framework, but also all those which move without relative rotation with uniform velocity relatively to the first. This whole set of frameworks is on a level in respect to Newton's laws. We will call them Dynamical frameworks.
Now, suppose there are two observers, A and B. They agree in their non-metrical projective geometry, _e.g._, what A calls a straight line so does B. They also both apply a Euclidean metrical system of measurement to this space. Their two metrical systems also agree in having the same plane at infinity, that is, lines which are parallel for A are also parallel for B. Furthermore, they have both successfully applied Newton's laws to the movement of matter, and agree in having the same sets of dynamical axes. But the framework (among these sets) which A chooses to regard as at rest is different from the frame (among the same sets) which B so regards.
Without alteration of their respective judgment of rest, they choose their co-ordinate axes so that the origins (O for A, and O′ for B) are in relative motion along OO′, which is the axis of _x_ for both.
Further, since OO′ is the line of symmetry of their diverse Euclidean systems, we assume that the two measure-systems agree for planes perpendicular to OO′, _i.e._, we assume a symmetry round OO′. Then if, for A at O, the distance OO′ be ξ, the relations at any instant between A's coordinates (_x_, _y_, _z_) and B's coordinates (_x′_, _y′_, _z′_) for the same point P are given by
_x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_.
Also, according to A's clock, O′ is moving forward with a uniform velocity _v_, and we measure A's time from the instant of the coincidence of O and O′.
Thus
ξ = _vt_,
and
_x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_.
We now consider B's clock, and ask for the most general supposition which is consistent with the fact that their judgments as to the fact of uniform motion are in agreement.
We do not assume that events in various parts of space which A considers to be simultaneous are so considered by B. But we assume that at any point P, with coordinates (_x_, _y_, _z_) for A, there is a determinate relation between B's time T and _x_, _y_, _z_, _t_.
Put
T = ƒ(_x_, _y_, _z_, _t_).
Write
P = δT/δ_x_, Q = δT/δ_y_, R = δT/δ_z_, S = δT/δ_t_.
Now suppose that the point P is moving, and that (_u_{1}, _u_{2}, _u_{3}) is its set of component velocities along the axes according to A's "space and clock" system, and (U_{1}, U_{2}, U_{3}) is its set of component velocities according to B's "space and clock" system. Then by mere differentiation it follows after a short mathematical deduction that
U_{1} = {(_d_β/_dt_)(_x_ - _vt_) + β(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, U_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, U_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}.
But we have assumed that, whatever the direction of the resultant velocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3}) and (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform.
Hence it is easily proved that β, P, Q, R, S are independent of the coordinates (_x_, _y_, _z_) and of the time _t_. In other words, they are constant.
Hence we obtain
U_{1} = β(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
and
T = P_x_ + Q_y_ + R_z_ + S_t_.
But we assumed that OO′, _i.e._, O_x_, is an axis of symmetry. It follows from this assumption that
Q = R = 0.
We thus obtain the simplified results
T = P_x_ + S_t_, } U_{1} = β(_u_{1} - _v_)/(P_u_{1} + S), } (I) U_{2} = _u_{2}/(P_u_{1} + S), } U_{3} = _u_{3}/(P_u_{1} + S). }
Here we remember that (_u_{1}, _u_{2}, _u_{3}) are the velocities of any particle according to A's "space and clock" system, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same point according to B's "space and clock" system. We have obtained the most general relations consistent with the facts that (1) they both employ Euclidean systems, related as described above, and (2) they agree in their judgments on the uniformity of velocity.
We now compare their judgments on the magnitudes of velocities.
Let the magnitude of the velocity of P be V according to A's judgment, and V′ according to B's’ judgment.
Then
V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2, V′^2 = U_{1}^2 + U_{2}^2 + U_{3}^2.
Also we can put
_u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V,
where (_l_, _m_, _n_) have nothing to do with the magnitude V, but simply depend on the direction of motion. In fact (_l_, _m_, _n_) are the "direction cosines" of the velocity according to A's judgment. By substituting in the above equation for V^2 we see that
_l_^2 + _m_^2 + _n_^2 = 1.
Now, substituting for (_u_{1}, _u_{2}, _u_{3}) in the equations (I) above, and squaring and adding, and eliminating _m_^2 + _n_^2 by the relation just found, we at once find
V′^2 = ((β^2 - 1)V^2_l_^2 - 2β^2V_vl_ + β^2_v_^2 + V^2)/(PV_l_ + S)^2.
It is thus seen that in general the relation of V′ to V depends on the direction cosine _l_. Now _l_ is the cosine of the angle which the direction of the velocity V makes with O_x_, according to A's judgment.
The meaning of this relation is, that if A discharges, from guns at the point P, shells with a given muzzle velocity V according to his judgment, B will consider that their muzzle velocities are different from each other, except in the case of pairs of guns equally inclined to the axis OO′. Instances of this type of diversity of judgment can be noted any day by any one who looks out of the window of a railway carriage, and forgets that he is travelling.
Now, suppose the velocity V′ bears a relation to the velocity V, which is independent of _l_. Then _l_ must disappear from the above formula. There are two conditions to be satisfied
One condition is
V^2 = β^2_v_^2/(β^2 - 1),
or in a more convenient form
β^2 = 1/(1 - _v_^2/V^2).
The meaning of this condition is, that there is one, and only one, muzzle velocity V (according to A's judgment), namely, the muzzle velocity given by the above formula, which can have the property that B will judge that all the guns are firing in their diverse directions with one common muzzle velocity.
Let us now suppose that V has this peculiar value: that is, if we look on this value V as known, we must suppose that β is given by the second of the above formulæ.
The other condition allows P and S to be put in the forms
P = -β_v_/λV^2, S = β/λ,
where
V′ = λV.
Thus we have the bundle of formulæ
β^2 = 1/(1 - _v_^2/V^2), T = β{_t_ - _vx_/V^2}/λ, V′ = λV.
The value which we give to λ is purely a matter for the adjustment of units. If we want A and B to agree in their judgments of the magnitude of this peculiar muzzle-velocity, we put λ = 1.
We then get the formulæ usually adopted, namely
β^2 = 1/(1 - _v_^2/V^2), } T = β{_t_ - _vx_/V^2}, } (II) V′ = V. }
But if we prefer that A and B should reckon (according to A's judgment) in the same units of time, we put λ = β, and obtain
β^2 = 1/(1 - _v_^2/V^2), T = (_t_ - _vx_)/V^2, V′ = βV.
But A and B are in any case in such hopeless difficulties over their comparisons of time-judgments that the detail of using the same units does not help them much. Accordingly the formulæ marked (II) are those used. Thus A and B agree in their judgments as to the magnitude of one special velocity V, whatever may be the direction in which the entity possessing it is moving.
In order to reach this measure of agreement, they have to disagree as to their space judgments and their time judgments. The root cause of their disagreement is their diverse judgment as to which axis system is to be taken at rest for the purpose of measuring velocities.
Before discussing the nature of the disagreement disclosed in formula (II), let us ask why we should bring these difficulties on our heads by supposing that two people in relative motion, who both (for the purpose of measuring velocities) assume that they are at rest, should agree in their judgments in respect to this special velocity V.
Such an agreement has no counterpart in any of our obvious judgments made from railway carriages. Surely we can wait till the contingency occurs before discussing the confusion which it creates.
But the contingency has occurred. It occurs when we consider the velocity of light. Perhaps I may venture to remind a philosophical society that light moves so very quickly that it is difficult to consider its velocity at all. So we need not be surprised that this peculiar fact concerning its velocity is not more obvious.
Now V being the velocity of light, unless _v_ is large, _v_/V (and still more _v_^2/V^2) will be quite inappreciable. The only velocity ready to hand which is big enough to give _v_/V an appreciable value is the velocity of the earth in its orbit.
Many diverse experiments have been made, and they all agree in concluding that a man who assumes the earth to be at rest will find by measurement that the velocity of light is the same in all directions. Furthermore, when the same man turns his attention to interstellar or interplanetary phenomena, and assumes the sun to be at rest, he will again find the velocity of light to be the same in all directions. These are well-attested experiments made at long intervals of time.
This is the exact contingency contemplated above.
Again the velocity of light _in vacuo_ has recently taken on a new dignity. It used to be one among other wave velocities such as the velocity of sound in air, or in water, or the velocity of surface waves in water. But Clerk Maxwell discovered that all electromagnetic influences are propagated with the velocity of light, and now modern physical science half suspects that electromagnetic influences are the only physical influences which relate the changes in the physical world. Accordingly the velocity of light becomes the fundamental natural velocity, and experiment shows that our judgment of its magnitude is not affected by our choice of the framework at rest, so long as we keep to a set of dynamical axes. These experiments on light have been confirmed by other electromagnetic experiments not involving light.
Thus we are driven to equations (II), where V is the velocity of light.
The first conclusion to be drawn from equations (II) is that two people who make different choices of bodies at rest will disagree as to their measuring rods in the way described above. There is no peculiar difficulty about that. The only wonder is that all people agree so well in their judgments as to metrical systems. A mathematical angel would naturally expect incarnate men to be in violent disagreement on this subject.
But the case of time is different. For simplicity of statement we speak of A as at O, and B as at O′. We remember that O′ is moving relatively to O with velocity _v_ in direction OO′. Suppose A and B are looking in this direction; and they both measure their time from the instant when they met, as O′ passed over O. Then we have
T = ((_t_ - _vx_)/V^2)/((1 - _v_^2)/V^2).
Now, suppose we consider all the events all over space which A considers to have happened simultaneously at the time _t_. The events of this set which occurred anywhere on a plane perpendicular to OO′ at a distance _x_ in front of O (according to A's reckoning), will have occurred according to B's reckoning at the time T as given above. Let us fix our attention on the fact that B does not consider all these events to be simultaneous. For let T_{1}, and T_{2} be B's times for such events on planes _x__{1}, and _x__{2}. Then
T_{1} - T_{2} = _v_(_x__{2} - _x__{1})/(V^2 - _v_^2).
Thus if _x__{2} be greater than _x__{1}, T_{2} is less than T_{1}. Thus B judges the more distant events in front of him to have happened earlier than the nearer events in front of him, and _vice versa_ for the events behind. This disturbance of the judgment of simultaneity is the fundamental fact. Obviously the measurement of time intervals is a detail compared to simultaneity. A may think a sermon long, and B may think it short, but at least they should both agree that it stopped when the clock hand pointed at the hour. The worst of the matter is that so far as any test can be applied there is no method of discriminating between the validities of their judgments.
Thus we are confronted with two distinct concepts of the common world, A's space-time concept, and B's space-time concept. Who is right? It is no use staying for an answer. We must follow the example of the wise old Roman, and pass on to other things.
Thus estimates of quantity in space and time, and, to some extent, even estimates of order, depend on the individual observer. But what are the crude deliverances of sensible experience, apart from that world of imaginative reconstruction which for each of us has the best claim to be called our real world? Here the experimental psychologist steps in. We cannot get away from him. I wish we could, for he is frightfully difficult to understand. Also, sometimes his knowledge of the principles of mathematics is rather weak, and I sometimes suspect---- No, I will not say what I sometimes think: probably he, with equal reason, is thinking the same sort of thing of us.
I will, however, venture to summarise conclusions, which are, I believe, in harmony with the experimental evidence, both physical and psychological, and which are certainly suggested by the materials for that unwritten chapter in mathematical logic which I have already commended to your notice. The concepts of space and time and of quantity are capable of analysis into bundles of simpler concepts. In any given sensible experience it is not necessary, or even usual, that the whole complete bundle of such concepts apply. For example, the concept of externality may apply without that of linear order, and the concept of linear order may apply without that of linear distance.
Again, the abstract mathematical concept of a space-relation may confuse together distinct concepts which apply to the given perceptions. For example, linear order in the sense of a linear projection from the observer is distinct from linear order in the sense of a row of objects stretching across the line of sight.
Mathematical physics assumes a given world of definitely related objects, and the various space-time systems are alternative ways of expressing those relations as concepts in a form which also applies to the immediate experience of observers.
Yet there must be one way of expressing the relations between objects in a common external world. Alternative methods can only arise as the result of alternative standpoints; that is to say, as the result of leaving something added by the observer sticking (as it were) in the universe.
But this way of conceiving the world of physical science, as composed of hypothetical objects, leaves it as a mere fairy tale. What is really actual are the immediate experiences. The task of deductive science is to consider the concepts which apply to these data of experience, and then to consider the concepts relating to these concepts, and so on to any necessary degree of refinement. As our concepts become more abstract, their logical relations become more general, and less liable to exception. By this logical construction we finally arrive at conceptions, (i) which have determinate exemplifications in the experience of the individuals, and (ii) whose logical relations have a peculiar smoothness. For example, conceptions of mathematical time, of mathematical space, are such smooth conceptions. No one lives in "an infinite given whole," but in a set of fragmentary experiences. The problem is to exhibit the concepts of mathematical space and time as the necessary outcome of these fragments by a process of logical building up. Similarly for the other physical concepts. This process builds a common world of conceptions out of fragmentary worlds of experience. The material pyramids of Egypt are a conception, what is actual are the fragmentary experiences of the races who have gazed on them.
So far as science seeks to rid itself of hypothesis, it cannot go beyond these general logical constructions. For science, as thus conceived, the divergent time orders considered above present no difficulty. The different time systems simply register the different relations of the mathematical construct to those individual experiences (actual or hypothetical) which could exist as the crude material from which the construct is elaborated.
But after all it should be possible so to elaborate the mathematical construct so as to eliminate specific reference to particular experiences. Whatever be the data of experience, there must be something which can be said of them as a whole, and that something is a statement of the general properties of the common world. It is hard to believe that with proper generalisation time and space will not be found among such properties.
_Commentary added on reading the Paper before the Aristotelian Society_
The first six pages of the paper consist of a summary of ideas which ought to be in our minds while considering problems of time and space. The ideas are mostly philosophical, and the summary has been made by an amateur in that science; so there is no reason to ascribe to it any importance except that of a modest reminder. There are only two points in this summary to which I would draw attention.
On pp. 192 and 193 there occurs--
"Wherever mankind ... unending uniform succession."
If I understand Kant rightly--which I admit to be very problematical--he holds that in the act of experience we are aware of space and time as ingredients necessary for the occurrence of experience. I would suggest--rather timidly--that this doctrine should be given a different twist, which in fact turns it in the opposite direction--namely, that in the act of experience we perceive a whole formed of related differentiated parts. The relations between these parts possess certain characteristics, and time and space are the expressions of some of the characteristics of these relations. Then the generality and uniformity which are ascribed to time and space express what may be termed the uniformity of the texture of experience.
The success of mankind--modest though it is--in deducing uniform laws of nature is, so far as it goes, a testimony that this uniformity of texture goes beyond those characteristics of the data of experience which are expressed as time and space. Time and space are necessary to experience in the sense that they are characteristics of our experience; and, of course, no one can have our experience without running into them. I cannot see that Kant's deduction amounts to much more than saying that "what is, is"--true enough, but not very helpful.
But I admit that what I have termed the "uniformity of the texture of experience" is a most curious and arresting fact. I am quite ready to believe that it is a mere illusion; and later on in the paper I suggest that this uniformity does not belong to the immediate relations of the crude data of experience, but is the result of substituting for them more refined logical entities, such as relations between relations, or classes of relations, or classes of classes of relations. By this means it can be demonstrated--I think--that the uniformity which must be ascribed to experience is of a much more abstract attenuated character than is usually allowed. This process of lifting the uniform time and space of the physical world into the status of logical abstractions has also the advantage of recognising another fact, namely, the extremely fragmentary nature of all direct individual experience.
My point in this respect is that fragmentary individual experiences are all that we know, and that all speculation must start from these _disjecta membra_ as its sole datum. It is not true that we are directly aware of a smooth running world, which in our speculations we are to conceive as given. In my view the creation of the world is the first unconscious act of speculative thought; and the first task of a self-conscious philosophy is to explain how it has been done.
There are roughly two rival explanations. One is to assert the world as a postulate. The other way is to obtain it as a deduction, not a deduction through a chain of reasoning, but a deduction through a chain of definitions which, in fact, lifts thought on to a more abstract level in which the logical ideas are more complex, and their relations are more universal. In this way the broken limited experiences sustain that connected infinite world in which in our thoughts we live. There are three more remarks while on this point I wish to make--
(i) The fact that immediate experience is capable of this deductive superstructive must mean that it itself has a certain uniformity of texture. So this great fact still remains.
(ii) I do not wish to deny the world as a postulate. Speaking without prejudice, I do not see how in our present elementary state of philosophical advance we can get on without middle axioms, which, in fact, we habitually assume.
My position is, that by careful scrutiny we should extrude such postulates from every part of our organised knowledge in which it is possible to do without them.
Now, physical science organises our knowledge of the relations between the deliverances of our various senses. I hold that in this department of knowledge such postulates, though not entirely to be extruded, can be reduced to a minimum in the way which I have described.
I have not the slightest knowledge of theories respecting our emotions, affections, and moral sentiments, and I can well believe that in dealing with them further postulates are required. And in practice I recognise that we all make such postulates, uncritically.
(iii) The next paragraphs on pp. 193 and 194 are as follows--
"The opposed philosophical method ... physical science."
It will be noted that, in the light of what has just been stated, the first of these paragraphs (which, I hope, faithfully expresses the experimental way of approaching the problem) really obscures the point which I have been endeavouring to make. The phrase, "If we form the exact concepts of points, etc.," is fatally ambiguous as between the method of postulating entities with assigned relations, and the method of forming logical constructions, and thus reaching points, etc., as the result of a chain of definitions.
Turning now to pp. 194-195, we come to the following paragraphs--
"The other theory ... eighteenth century."
We note again that the relational theory of space from another point of view brings us back to the idea of the fundamental space-entities as being logical constructs from the relations between things. The difference is, that this paragraph is written from a more developed point of view, as it implicitly assumes the things in space, and conceives space as an expression of certain of their relations. Combining this paragraph with what has gone before, we see that the suggested procedure is first to define "things" in terms of the data of experience, and then to define space in terms of the relations between things.
This procedure is explicitly assumed in the next short paragraph: "In this connection ... from the events."
The gist of the remaining paragraphs of this section is contained in the paragraph at the bottom of p. 196: "The point in time ... new importance."
The sentence, "We live in durations, and not in points," can be amplified by the addition, "We live in space-extensions and not in space-points."
It must be noted that "whole and part" as applied to extensions in space or time must be different from the "all and some" of logic, unless we admit points to be the fundamental entities. For "spatial whole and spatial part" can only mean "all and some" if they really mean "all the points and some of the points." But if extensions and their relations are more fundamental than points, this interpretation is precluded. I suggest that "spatial whole and spatial part" is intimately connected with the fundamental relation between things from which our space ideas spring.
The relation of space whole to space part has many formal properties which are identical with the properties of "all and some." Also when points have been defined, we can replace it by the conception of "all the points and some of the points." But the confusion between the two relations is fatal to sound views on the subject.
_Diverse Euclidean Measure Systems_
The next section deals with the measure systems applicable to space.
A measure system is a group of congruent transformations of space into itself. Consider a rigid body occupying all space. Let this body be moved in any way so that the particles of the body which occupied points P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2}, Q_{3}, etc. Then any point P_{1} in space is uniquely related to the corresponding point Q_{1} in space by a one-to-one transformation with certain characteristics. By the aid of these transformations we can achieve the definition of distance in a way which definitely determines the distance between any two points, provided that we can define what we mean by a congruent transformation without introducing the idea of distance. If we introduced the idea of distance, we should simply say that a congruent transformation is one which leaves all distances unchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1}, Q_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}.
But mathematicians have succeeded in defining congruent transformations without any reference to distance.
There are alternative groups of such congruent transformations, and each group gives a different measure system for space. The distance P_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system, and will not equal it for another measure system. All these different measure systems are on the same level, equally applicable. A being with a strong enough head could think of them all at once as applying to space. The result so far as it interests us in respect to the theory of relativity is explained on pp. 197-200, ending with "The most extraordinary fact ... same metrical system." This final sentence bears on Poincaré's assertion that the measure system adopted is purely "conventional." I presume that by "conventional" a certain arbitrariness of choice is meant; and in that case, I must express entire dissent. It is true that within the circle of geometrical ideas there is no means of giving any preference to any one measure system, and any one is as good as any other. But it is not true that if we look at a normal carriage wheel, and at an oval curve one foot broad and ten feet long, we experience any arbitrariness of judgment in deciding which has approximately the form of a circle. Accordingly to Poincaré the choice between them, as representing a circle, is entirely conventional.
Again, we equally form immediate judgments as to whether a body is approximately rigid. We know that a paving stone is rigid, and that a concertina is not rigid. This again necessitates a determinate measure system, selected from among the others.
Accordingly we conclude that (i) each being does, in fact, employ a determinate measure system, which remains the same, except possibly for very small variations, and (ii) the measure systems of different human beings agree, to within the limits of our observations. These conclusions are not the less extraordinary because no plain man has ever doubted them.
It is an interesting subject to investigate exactly what are the fundamental uniformities of experience which necessitate this conclusion. It is not so easy as it looks, since we have to divest ourselves of all aid of scientific hypothesis if our conclusions are to be demonstrative.
_Relativity in Modern Physics_
Pp. 201-202, "Owing to the fact ... which B so regards."
The fundamental formulæ for the theory of relativity are the relations between diverse co-ordinate systems given on p. 203, and formulæ II at the bottom of p. 207. The general explanation of one method in which these formulæ arise--namely, Einstein's method--is given on pp. 201-211. Namely, we seek the condition that for all dynamical axes the velocity of light should be the same, and the same in all directions. It should be noted that the experiments which, so far as they go, confirm these formulæ, can also be explained in another way which makes the theory of relativity unnecessary. We need only ascribe to the ether a certain property of contraction in the direction of motion, and the thing is done. So no one need be bludgeoned into accepting the rather bizarre doctrine of relativity, nor indeed any other scientific generalisation. The good old homely ether, which we all know, can in this case serve the purpose. Just as an author of genius, if he lives long enough, survives the inevitable accusation of immorality, so the ether by dint of persistence has outlived all reputation of extravagance. But if we detach ourselves from the glib phraseology concerning it, the scientific ether is uncommonly like the primitive explanation of the soul, as a little man inside us, which can sometimes be caught escaping in the form of a butterfly. As soon as the ether has to be patched up with special properties to explain special experiments, its scientific use is problematical, and its philosophic use is _nil_.
Philosophically the ether seems to me to be an ambitious attempt to give a complete explanation of the physical universe by making an elephant stand on a tortoise. Scientifically it has a perfectly adequate use by veiling the extremely abstract character of scientific generalisations under a myth, which enables our imaginations to work more freely. I am not advocating the extrusion of ether from our scientific phraseology, even though at special points we have to abandon it.
But the key to the reasons why it is worth while to consider seriously the doctrine of relativity is to be found on pp. 209, 210: "Again the velocity of light ... not involving light." Namely, we have begun to suspect that all physical influences require time for their propagation in space. This generalisation is a long way from being proved. Gravitation stands like a lion in the path. But if it be the case, then all idea of an immediate presentation to us of an aspect of the world as it in fact is, must necessarily be abandoned. What we perceive at any instant is already ancient history, with the dates of the various parts hopelessly mixed.
We must add to this the difficulty of determining what is at rest and what is in motion, and the further difficulty of determining a definite uniform flow of time. It is no use discussing this matter as though, but for the silly extravagant doctrine of relativity, everything would be plain sailing. It isn't. You may be quite sure that when, after prolonged study, you endeavour to give the simplest explanation of a grave difficulty, you will be accused of extravagance. I have no responsibility for the doctrine of relativity, and hold no brief for it, but it has some claim to be considered as a comparatively simple way out of a scientific maze.
In the first place, we use the Newtonian dynamical sets of axes, and the Newtonian clock to extricate ourselves partially from the difficulties of rest, motion, and time. These have proved capable of scientific determination within the limits of our experimental accuracy. Thus the only thing left over is the choice of the axes at rest, which is a completely indeterminate problem on Newtonian principles.
Again, so far as we can at present guess by adopting the theory that all metrical influence is electromagnetic, all influences are propagated with the velocity of light _in vacuo_. This electromagnetic hypothesis is by no means established, but it gives the simplest of all possible results in respect to the propagation of influence, which we therefore adopt.
But what dynamical axes are we taking as at rest? Now our practical choice gives a range of relative velocities small compared to that of light. So except for certain refined experiments it does not matter. There are two possibilities--
(i) We may assume that one set of axes are at rest, and that the others will show traces of motion in respect to the velocity of light; or--
(ii) That the velocity of light is the same in all directions whichever be the dynamical axes assumed.
The first supposition is negatived by experiment, and hence we are driven to the second supposition; which immediately lands us in the whole theory of relativity.
But if we will not have this theory we must reject the earlier supposition that the velocity of light _in vacuo_ is the same in all directions. This we do, in fact, by assuming an ether, and assuming a certain law for its modification. Then we, in fact, adopt the first supposition so far as to hold that there are dynamical axes specially at rest, namely, at rest relatively to the undisturbed ether. Then an assumed law for the modification of the ether so alters the velocity of light that we explain why no dynamical axes show traces of motion.
I wish now to go back to the point which I made a few minutes ago, that what we perceive at any instant is ancient history with its dates hopelessly mixed. In the earlier part of my comments I emphasised the point that our only data as to the physical world are our sensible perceptions. We must not slip into the fallacy of assuming that we are comparing a given world with given perceptions of it. The physical world is in some general sense of the term a deduced concept.
Our problem is, in fact, to fit the world to our perceptions, and not our perceptions to the world.
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