The organisation of thought, educational and scientific · Alfred North Whitehead
Q = R = 0.
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.
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.
"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.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.