It was Minkowski who first formulated all this in a form susceptible of use in connection with the theory of relativity. His starting point lies in the distinction between the point and the event. Mr. Francis has brought this out rather well in his essay, being the only competitor to present the Euclidean geometry as a real predecessor of Newtonian science, rather than as a mere part of the Newtonian system. I think his point here is very well taken. As he says, Euclid looked into the world about him and saw it composed of points. Ignoring all dynamic considerations, he built up in his mind a static world of points, and constructed his geometry as a scientific machine for dealing with this world in which motion played no part. It could to be sure be introduced by the observer for his own purposes, but when so introduced it was specifically postulated to be a matter of no moment at all to the points or lines or figures that were moved. It was purely an observational device, intended for the observer’s convenience, and in the bargain a mental device, calling for no physical action and the play of no force. So far as Euclid in his daily life was obliged to take cognizance of the fact that in the world of work-a-day realities motion existed, he must, as a true Greek, have looked upon this as a most unfortunate deviation of the reality from his beautiful world of intellectual abstraction, and as something to be deplored and ignored. Even in their statuary the Greeks clung to this idea. A group of marvelous action, like the Laocoon, they held to be distinctly a second rate production, a prostitution of the noble art; their ideal was a figure like the majestic Zeus—not necessarily a mere bust, be it understood, but always a figure in repose without action. Their statuary stood for things, not for action, just as their geometry stood for points, not for events.
Galileo and Newton took a different viewpoint. They were interested in the world as it is, not as it ought to be; and if motion appears to be a fundamental part of that world, they were bound to include it in their scheme. This made it necessary for them to pay much more attention to the concept of time and its place in the world than did the Greeks. In the superposition process, and even when he allowed a curve to be generated by a moving point, the sole interest which Euclid had in the motion was the effect which was to be observed upon his static figures after its completion. In this effect the rate of the motion did not enter. So all questions of velocity and time are completely ignored, and we have in fact the curious spectacle of motion without time.
To Galileo and Newton, on the other hand, the time which it took a body to pass from one point of its path to another was of paramount importance. The motion itself was the object of their study, and they recognized the part played by velocity. But Galileo and Newton were still sufficiently under the influence of Euclid to fit the observed phenomena of motion, so far as they could, upon Euclid’s static world of points. This they effected by falling in with the age-old procedure of regarding time and space as something entirely disassociated and distinct. The motion of an object—in theory, of a point—was to be recorded by observing its successive positions. With each of these positions a time was to be associated, marking the instant at which the point attained that position. But in the face of this association, space and time were to be maintained as entirely separate entities.
The Four-Dimensional World of Events
This severe separation of time and space Minkowski has now questioned, with the statement that the elements of which the external world is composed, and which we observe, are not points at all, but are events. This calls for a revision of our whole habit of thought. It means that the perceptual world is four-dimensioned, not three-dimensioned as we have always supposed; and it means, at the very least, that the distinction between time and space is not so fundamental as we had supposed.
[This should not impress us as strange or incomprehensible. What do we mean when we say that a plane is two-dimensional? Simply that two coordinates, two numbers, must be given to specify the position of any point of the plane. Similarly for a point in the space of our accustomed concepts we must give three numbers to fix the position—as by giving the latitude and longitude of a point on the earth and its height above sea-level. So we say this space is three-dimensional. But a material body is not merely somewhere; it is somewhere now,][182] or was somewhere yesterday, or will be somewhere tomorrow. The statement of position for a material object is meaningless unless we at the same time specify the time at which it held that position. [If I am considering the life-history of an object on a moving train, I must give three space-coordinates and one time-coordinate to fix each of its positions.][182] And each of its positions, with the time pertaining to that position, constitutes an event. The dynamic, ever-changing world about us, that shows the same aspect at no two different moments, is a world of events; and since four measures or coordinates are required to fix an event, we say this world of events is four-dimensional. If we wish to test out the soundness of this viewpoint, we may well do so by asking whether the naming of values for the four coordinates fixes the event uniquely, as the naming of three under the old system fixes the point uniquely.
Suppose we take some particular event as the one from which to measure, and agree upon the directions to be taken by our space axes, and make any convention about our time-axis which subsequent investigation may show to be necessary. Certainly then the act of measuring so many miles north, and so many west, and so many down, and so many seconds backward, brings us to a definite time and place—which is to say, to a definite event. Perhaps nothing “happened” there, in the sense in which we usually employ the word; but that is no more serious than if we were to locate a point with reference to our familiar space coordinate system, and find it to lie in the empty void of interstellar space, with no material body occupying it. In this second case we still have a point, which requires, to insure its existence and location, three coordinates and nothing more; in the first case we still have an event, which requires for its existence and definition four coordinates and nothing more. It is not an event about which the headline writers are likely to get greatly excited; but what of that? It is there, ready and waiting to define any physical happening that falls upon it, just as the geometer’s point is ready and waiting to define any physical body that chances to fall upon it.