[A distinguished English mathematician, philosopher, and educator, Whitehead collaborated with Bertrand Russell on the latter's Principia Mathematica (1910-13), a three-volume treatise on the relationship of logic to mathematics that would eventually inspire the Austrian philosopher Ludwig Wittgenstein in his ground-breaking studies. In the following essay, originally published in the Times Educational Supplement in 1920, Whitehead seeks to explain the major principles of Einstein's work.]

Einstein's work may be analysed into three factors—a principle, a procedure, and an explanation. This discovery of the principle and the procedure constitute an epoch in science. I venture, however, to think that the explanation is faulty, even although it formed the clue by which Einstein guided himself along the path from his principle to his procedure. It is no novelty to the history of science that factors of thought which guided genius to its goal should be subsequently discarded. The names of Kepler and Maupertuis at once occur in illustration.

What I call Einstein's principle is the connexion between time and space which emerges from his way of envisaging the general fact of relativity. This connexion is entirely new to scientific thought, and is in some respects very paradoxical. A slight sketch of the history of ideas of relative motion will be the shortest way of introducing the new principle. Newton thought that there was one definite space within which the material world adventured, and that the sequence of its adventures could be recorded in terms of one definite time. There would be, therefore, a meaning in asking whether the sun is at rest or is fixed in this space, even although the questioner might be ignorant of the existence of other bodies such as the planets and the stars. Furthermore, there was for Newton an absolute unique meaning to simultaneity, so that there can be no ambiguity in asking, without further specification of conditions, which of two events preceded the other or whether they were simultaneous. In other words, Newton held a theory of absolute space and of absolute time. He explained relative motion of one body with respect to another as being the difference of the absolute motions of the two bodies. The greatest enemy to his absolute theory of space was his own set of laws of motion. For it is a well-known result from these laws that it is impossible to detect absolute uniform motion. Accordingly, since we fail to observe variations in the velocities of the sun and stars, it follows that any one of them may with equal right be assumed to be either at rest or moving in any direction with any velocity which we like to suggest. Now, a character which never appears in the play does not require a living actor for its impersonation. Science is concerned with the relations between things perceived. If absolute motion is imperceptible, absolute position is a fairy tale, and absolute space cannot survive the surrender of absolute position.

So far our course is plain: we give up absolute space, and conceive all statements about space as being merely expositions of the internal relations of the physical universe. But we have to take account of two very remarkable difficulties which mar the simplicity of this theoretical position. In the first place there seems to be a certain absoluteness about rotation. The fact of this absoluteness is inherent in Newton's laws of motion, and the deducted consequences from these premises have received ample confirmation. For example, the effect of the rotation of the earth is manifested in phenomena which appear to have no connexion with extraneous astronomical bodies. There is the bulge of the earth at its equator, the invariable directions of rotation for cyclones and anti-cyclones, the rotation of the plane of oscillation of Foucault's pendulum, and the north-seeking property of the gyrocompass. The mass of evidence is decisive, and no theory which burkes it can stand as an adequate explanation of observed facts. It is not so obvious how to combine these facts of rotation with any principle of relativity.

Secondly, the ether contributes another perplexity just where it might have helped us. We might have regained the right quasi-absoluteness of motion by measuring velocity relatively to the ether. The facts of rotation could have thus received an explanation. But all attempts to measure velocity relatively to the ether have failed to detect it in circumstances when, granting the ordinary hypotheses, its effects should have been visible. Einstein showed that the whole series of perplexing facts concerning the ether could be explained by adopting new formulae connecting the spatial and temporal measurements made by observers in relative motion to each other. These formulae had been elaborated by Larmor and Lorentz, but it was Einstein who made them the foundation of a novel theory of time and space. He also discovered the remarkable fact that, according to these formulae, the velocity of light in vacuous space would be identical in magnitude for all these alternative assumptions as to rest or motion. This property of light became the clue by which his researches were guided. His theory of simultaneity is based on the transmission of light signals, and accordingly the whole structure of our concept of nature is essentially bound up with our perceptions of radiant energy.

In view of the magnificent results which Einstein has achieved it may seem rash to doubt the validity of a premiss so essential to his own line of thought. I do, however, disbelieve in this invariant property of the velocity of light, for reasons which have been partly furnished by Einstein's own later researches. The velocity of light appears in this connexion owing to the fact that it occurs in Maxwell's famous equations, which express the laws governing electromagnetic phenomena. But it is an outcome of Einstein's work that the electro-magnetic equations require modification to express the association of the gravitational and electro-magnetic fields. This is one of his greatest discoveries. The most natural deduction to make from these modified equations is that the velocity of light is modified by the gravitational properties of the field through which it passes, and that the absolute maximum velocity which occurs in the Maxwellian form of the equations has in fact a different origin which is independent of any special relation to light or electricity. I will return to this question later.

Before passing on to Einstein's later work a tribute should be paid to the genius of Minkowski. It was he who stated in its full generality the conception of a four-dimensional world embracing space and time, in which the ultimate elements, or points, are the infinitesimal occurrences in the life of each particle. He built on Einstein's foundations, and his work forms an essential factor in the evolution of relativistic theory.

Einstein's later work is comprised in what he calls the theory of general relativity. I will summarize what appear to me as the essential components of his thought, at the same time warning my readers of the danger of misrepresentation which lies in such summaries of novel ideas. It is safer to put it as my own way of envisaging the theory. What are time and space? They are the names for ways of conducting certain measurements. The four dimensions of nature as conceived by Minkowski express the fact that four measurements with a certain peculiar type of mutual independence are required to formulate the relations of any infinitesimal occurrence to the rest of the physical universe. These ways of measurement can be indefinitely varied by change of character, so that four independent measurements of one character will specify an occurrence just as well as four other measurements of some other character. A set of four measurements of a definite character which assigns a special type to each of the four measurements will be called a measure-system. Thus there are alternative measure-systems, and each measure-system embraces, for the specification of each infinitesimal occurrence, four assigned measurements of separate types, called the coordinates of that occurrence. The change from one measure-system to another appears in mathematics as the change from one set of variables (pl, p2, p3, p4) to another set of variables (q1, q2, q3, q4), the variables of the p-system being functions of the q-system, and vice versa. In this way all the quantitative laws of the physical universe can be expressed either in terms of the p-variables or in terms of the q-variables. If a suitable measure-system has been adopted, one of the measurements, say p4, will appear to us as a measurement of time, and the remaining measurements (p1, p2, p3) will be measurements of space, which are adequate to determine a point. But different measure-systems have this property of subdivision into spatial and temporal measurements according to the different circumstances of the observers. It follows that what one observer means by space and time is not necessarily the same as what another observer may mean. It is to be observed that not every change of measure-system involves a change in the meanings of space and time. For example, let (p1, p2, p3, p4) and (q1, q2, q3, q4) be the measurements in two systems which determine the same event-particle, as I will name an infinitesimal occurrence. The two measurements of time, p4 and q4, may be identical or may differ only by a constant; and the spatial set of the p-system, namely (pl, p2, p3), may be functions of the spatial set of the q-system, namely (q1, q2, q3) with q4 excluded and vice versa. In this case the two systems subdivide into the same space and the same time. I will call such two systems "consentient." A measure-system which has the property for a suitable observer of thus subdividing itself I will call "spatio-temporal." I am unaware whether Einstein would accept these distinctions and definitions. If he would not I have failed to understand his theory. At the same time I would maintain them as necessary to relate the mathematical theory with the facts of physical experience.

What can we mean by space as an enduring fact, within which the varying phenomena of the universe are set at successive times? I will call space as thus conceived "timeless space." All the measure-systems of a consentient spatio-temporal set will agree in specifying the same timeless space; but two spatio-temporal systems which are not consentient specify distinct timeless spaces. A point of a timeless space must be something which for all time is designated by a definite set of values for the three spatial coordinates of an associated measure-system. Let (pl, p2, p3, p4) be such a measure-system, then a point of the timeless p-space is to be designated by a definite specification of values for the coordinates in the set (p1, p2, p3), giving the same entity for all values of p4. Furthermore, according to Minkowski's conception, the life of the physical universe can be specified in terms of the intrinsic properties and mutual relations of event-particles and of aggregates of event-particles. Our problem then is narrowed down to this: how can we define the points of the timeless p-space in terms of event-particles and aggregates of event-particles? Evidently there is but one solution. The point (pl, p2, p3) of the timeless p-space must be the set of event-particles indicated by giving p4 every possible value in (pl, p2, p3, p4), while pl, p2, p3 are kept fixed to the assigned coordinates of the point. Two consequences follow from this definition of a point. In the first place, a point of timeless space is not an entity of any peculiar ultimate simplicity; it is a collection of event-particles.

Years ago, in a communication to the Royal Society in 1906 ["Mathematical Concepts of the Material World"], I pointed out that the simplicity of points was inconsistent with the relational theory of space. At that time, so far as I am aware, the two inconsistent ideas were contentedly adopted by the whole of the scientific and philosophic worlds. To say that the event-particle (pl, p2, p3, p4) occupies, or happens at, the point (pl, p2, p3) merely means that the event-particle is one of the set of event-particles which is the point. The second consequence of the definition is that if the p-system and the q-system are spatio-temporal systems which are not consentient, the p-points and the q-points are radically distinct entities, so that no p-point is the same as any q-point. A complete explanation is thus achieved of the paradoxes in spatial measurement involved in the comparison of measurements of spatial distances between event-particles as effected in a p-space and a q-space. The ordinary formulae which we find in the early chapters of text-books on dynamics only look so obvious because this radical distinction between the different spaces has been ignored.

We can now make a further step and distinguish between an instantaneous p-space and the one timeless p-space. Suppose that p4 has a fixed value, then evidently every p-point is occupied by one and only one event-particle for which p4 has this value. This event-particle has the p1, p2, p3 belonging to its p-point and also the assigned value of p4 as its four coordinate measurements which specify it. It is evident, therefore, that the set of event-particles which all occur at the assigned p-time p4 but have among them all possible spatial coordinates together reproduce in their mutual spatial relations all the peculiarities of the relations between the points of the timeless p-space. Such a set of event-particles form the instantaneous p-space occurring at the p-time p4. They are the instantaneous points of the instantaneous space. Also, all the instantaneous p-spaces, for different values of p4, are correlated to each other in pointwise fashion by means of the timeless points which intersect each instantaneous space in one event-particle. An instantaneous space of some appropriate measure-system is the ideal limit of our outlook on the world when we contract our observation to be as nearly instantaneous as possible. We may conclude this part of our discussion by noting that there are three distinct meanings which may be in our mind when we talk of space, and it is mere erroneous confusion if we do not keep them apart. We may mean by space either (i.) the unique four-dimensional manifold of event-particles or (ii.) an assigned instantaneous space of some definite spatio-temporal measure-system, or (iii.) the timeless space of some definite spatio-temporal measure-system.

We now turn to the consideration of time. So long as we keep to one spatio-temporal measure-system no difficulty arises; the sets of event-particles, which are the sets of instantaneous points of successive instantaneous p-spaces ("p" being the name of the measure-system), occur in the ordered succession indicated by the successive values of p4 (the p-time). The paradox arises when we compare the p-time p4 with the q-time q4 of the spatio-temporal q-system of measurement, which is not consentient with the p-system. For now if (pl, p2, p3, p4) and (q1, q2, q3, q4) indicate the same event-particle q4 can be expressed in terms of (pl, p2, p3, p4) where p4 and at least one of the spatial set (pl, p2, p3) must occur as effective arguments to the function which expresses the value of q4. Thus when we keep p4 fixed, and vary (pl, p2, p3) so as to run over all the event-particles of a definite instantaneous p-space, the value of q4 alters from event-particle to event-particle. Thus two event-particles which are contemporaneous in p-time are not necessarily contemporaneous in q-time. In relation to a given event-particle E all other event-particles fall into three classes—(l) there is the class of event-particles which precede E according to the time-reckonings of all spatio-temporal measure-systems; (2) there is the class of event-particles which are contemporaneous with E in some spatio-temporal measure-system or other; (3) there is the class of event-particles which succeed E according to the time-reckonings of all spatio-temporal measure-systems. The first-class is the past and the third class is the future. The second class will be called the class co-present with E. The whole class of event-particles co-present with E is not contemporaneous with E according to the time-reckoning of any one definite measure system. Furthermore, no velocity can exist in nature, in whatever spatio-temporal measure-system it be reckoned, which could carry a material particle from one to the other of two mutually co-present event-particles. If El and E2 be a pair of mutually co-present event-particles, then El precedes E2 in some time-systems and E2 precedes El in other time-systems and El and E2 are contemporaneous in the remaining time-systems. The properties of co-present event-particles are undeniably paradoxical. We have, however, to remember that these paradoxes occur in connexion with the ultimate baffling mystery of nature—its advance from the past to the future through the medium of the present.

For any assigned observer there is yet a fourth class of event-particles—namely, that class of event-particles which comprises all nature lying within his immediate present. It must be remembered that perception is not instantaneous. Accordingly such a class is a slab of nature comprised between two instantaneous spaces belonging to the spatio-temporal measure-system which accords with the circumstances of his observation. I have elsewhere [in "Inquiry Concerning the Principles of Natural Knowledge"] called such a class a "duration."

The physical properties of nature arise from the fact that events are not merely colourless things which happen and are gone. Each event has a character of its own. This character is analysable in two components:—(1) There are the objects situated in that event; and (2) there is the field of activity of the event which regulates the transference of the objects situated in it to situations in subsequent events. It is essential to grasp the distinction between an object and an event. An object is some entity which we can recognize, and meet again; an event passes and is gone. There are objects of radically different types, but we may confine our attention to material physical objects and to scientific objects such as electrons. Space and time have their origin in the relations between events. What we observe in nature are the situations of objects in events. Physical science analyses the fields of activity of events which determine the conditions governing the transference of objects. The whole complex of events viewed in connexion with their characters of activity takes the place of the material ether of the science of the last century. We may call it the ether of events.

Now the spatial and temporal relations of event-particles to each other are expressed by the existence in space (in whatever sense that term is used) of points, straight lines, and planes. The qualitative properties and relation of these spatial elements furnish the set conditions which are a necessary pre-requisite of measurement. For it must be remembered that measurement is essentially the comparison of operations which are performed under the same set assigned conditions. If there is no possibility of assigned conditions applicable to different circumstances, there can be no measurement. We cannot, therefore, begin to measure in space until we have determined a non-metrical geometry and have utilized it to assign the conditions of congruence agreeing with our sensible experience. Practical measurement merely requires practical conformity to definite conditions. The theoretical analysis of the practice requires the theoretical geometrical basis. For this reason I doubt the possibility of measurement in space which is heterogeneous as to its properties in different parts. I do not understand how the fixed conditions for measurement are to be obtained. In others words, I do not see how there can be definite rules of congruence applicable under all circumstances. This objection does not touch the possibility of physical spaces of any uniform type, non-Euclidean or Euclidean. But Einstein's interpretation of his procedure postulates measurement in heterogeneous physical space, and I am very sceptical as to whether any real meaning can be attached to such a concept. I think that it must be a certain feeling for the force of this objection which has led certain men of science to explain Einstein's theory by postulating uniform space of five dimensions in which the universe is set. I cannot see how such a space, which has never entered into experience, can get over the difficulty.

There is, however, another way which obtains results identical with Einstein's to an approximation which includes all that is observable by our present methods. The only difference arises in the case of the predicted shifting of lines towards the red end of the spectrum. Here my theory makes no certain prediction. A particle vibrating in the atmosphere of the sun under an assigned harmonic force would experience an increase of apparent inertia in the ratio of 1 to 3/5ga/c² , if vibrating radially, and in the ratio of I to 2/5ga/c² , if vibrating transversely to the sun's radius, where a is the sun's radius, g is the acceleration due to gravity, and c is the critical velocity which we may roughly call the velocity of light. If we assume that the internal vibration of a molecule can be crudely represented in this fashion, and if we may assume that the internal forces of the molecule are not themselves affected in a compensatory manner by the gravitational field, then we may expect a shifting of lines towards the red end of the spectrum somewhere between three-fifths and two-fifths of Einstein's predicted amount—namely, a shift and a broadening. But both these assumptions are evidently very ill-founded. The theory does not require that any space should be other than Euclidean, and starts from the general theory of time and space which is explained in my work already cited.

I start from Einstein's great discovery that the physical field in the neighbourhood of an event-particle should be defined in terms often elements, which we may call by the typical name Jao where a and o are each written for any one of the four suffixes 1, 2, 3, 4. According to Einstein such elements merely define the properties of space and time in the neighbourhood. I interpret them as defining in Euclidean space a definite physical property of the field which I call the "impetus." I also follow Einstein in utilizing general methods of transformation from one measure-system to another, and in particular from one spatio-temporal system to another. But the essence of the divergence of the two methods lies in the fact that my law of gravitation is not expressed as the vanishing of an invariant expression, but in the more familiar way by the expression of the ten elements Jao- in terms of two functions of which one is the ordinary gravitational potential and the other is what I call the "associate potential," which is obtained by substituting the direct distance for the inverse distance in the integral definition of the gravitational potential. The details of the methods and other results are more suitable for technical exposition.