But when we examine the procedure of the lay philosopher who discusses scientific matters, we see that his procedure is entirely different. Taking examples at random, we find a contemporary philosopher telling us that the spatio-temporal system has to be regarded as limited, since “all existent things, as distinguished from pure forms or orders, are finite.”[131] But the average scientist would retort that with the same degree of plausibility we might say that the spatio-temporal system must be infinite, since the concept of infinity exists. At all events, it is instructive to contrast these loose arguments with Einstein’s painstaking labours which led him to the finite universe.
Another example is afforded by Bergson’s comments on the invariant velocity of light. He explains that inasmuch as the corpuscular theory of light has been rejected, light is a propagation and not a transference of matter. It is, then, only natural (according to Bergson) that its speed should be invariant. In his own words: “Why should it be affected by a certain too human way of perceiving and conceiving of things?” Bergson apparently forgets that it is only light in vacuo that moves with an invariant speed, and, even so, only when far from matter. A ray of light passing through water is also a propagation, and yet its speed is not invariant, as Fizeau’s experiment proved seventy years ago. Also, we might with equal plausibility apply Bergson’s arguments to sound waves. They also constitute propagations, and yet, once again, their speed is not invariant. In short, it would appear that the philosopher, even when he interprets the facts correctly, appears to be ignorant of so large a number of other facts that his philosophical conclusions rest on no solid foundation.
So much for the facts; but the facts are not all, since the theoretical scientist’s chief aim is to co-ordinate these facts. But, here again, the philosopher’s criticisms would suggest that, being unable to follow the series of mathematical deductions which lead the theoretical scientist to his conclusions, he fails to understand the necessity for these conclusions, and accordingly considers them to be wild guesses over which he will invariably suggest his own as an improvement.
Possibly a few definite illustrations will bring out with greater clarity some of the points we are endeavouring to stress. Let us revert, for example, to space-time, which proves itself so offensive to Bergson. His position, as already stated, is that space-time is a pure mathematical fiction in no wise demanded by the theory of relativity, which he accepts as sound physically, provided its interpretation is abandoned to lay philosophers. His arguments proceed somewhat as follows: You physicists have performed certain electromagnetic experiments; you mathematicians have deduced therefrom the Lorentz-Einstein transformations on purely formal grounds. Then you have plastered these transformations together and discovered a certain mathematical invariant. So far, well and good; now you must stop. It is for us philosophers to interpret these results. We will begin by ruling out your faulty inferences as to the existence of a four-dimensional continuum, space-time, of which space and time are mere abstractions. Follow our arguments carefully and we shall convince you. Possibly you mathematicians have not noticed that every observer is perfectly aware of a separate space and time wherever he goes, just as he is aware of a separate space and temperature. Were space-time one sole continuum, as you maintain, we could never be aware of space and of time, but only of space-time. This is contrary to the facts. Hence, space-time is a hoax, and your mathematics has led you astray.—Q. E. D.
But it is obvious that arguments of this sort can never have any merit, for the reason that they are based on a total misconception of the significance of dimensionality and manifolds. These are highly technical concepts which it requires more than a crude elementary knowledge of mathematics to grasp. However, even without going into details, we can easily convince ourselves of the fallacy of Bergson’s argument by applying it to the three-dimensional space of classical science. Thus, we know that wherever we are situated in three-dimensional space, we can always split it up into height, length and breadth; yet this does not preclude the fact that length, breadth and height are but abstractions from one single entity, namely, three-dimensional space. And, in the same way, as has been demonstrated by Minkowski, length, breadth, height and duration are but abstractions from four-dimensional space-time.
Another philosopher, Professor Broad, in his book, “Scientific Thought,” informs us: “(a) No matter what frame we choose, we shall need four independent pieces of information to place and date any instantaneous point-event. This fact is expressed by saying that Nature is a four-dimensional manifold; and nothing further is expressed thereby. (b) In whatever frame we choose we shall find that our four pieces of information divide into two groups; three of them are spatial and one is temporal. Thus we must be careful not to talk, or listen to, nonsense about ‘Time being a fourth dimension of Space.’”
Now it is agreed that whoever speaks of time as a fourth dimension of space is expressing himself very loosely. What we should say is: “Time is a fourth dimension of the space-time world.” But Broad’s argument does not suggest that it is this looseness of phraseology that offends him. His words would imply that it is the reference to time as a fourth dimension that must be branded as “nonsense.” His error in this respect is thus exactly the same as Bergson’s. As for his contention that the four-dimensional structure which relativity ascribes to nature means no more than that events need four co-ordinates to be placed and dated, the statement is scarcely correct. Were this trivial piece of information all that Minkowski was conveying when he referred to the world as four-dimensional, his discovery would have excited neither admiration nor criticism. It would have passed unnoticed by scientists, as expressing a mere platitude that could have been no news even to a child. The point that philosophers so persistently fail to understand is the difference between an amorphous continuum, such as a manifold of sounds or colours, and a metrical continuum, i.e., one with which a definite geometry is associated. What Minkowski did was to prove that, contrary to the belief of classical science, the world was a four-dimensional “metrical” continuum, i.e., one with which a four-dimensional space and time geometry was associated. It was this novel aspect of the world that implied the revolutionary conception of the fusion of space and time. And it was this aspect that entitled Minkowski to speak of time as a fourth dimension in a profound sense as well as in the trivial sense which, though never disputed by classical science, was never stressed on account of its artificial nature. Indeed, had it not been for this interpretation placed upon his words by Minkowski himself and by all scientists, the four-dimensional world would have appeared to be as artificial as a four-dimensional space-temperature continuum. It is granted that the various meanings attributed to the word “dimension” by mathematicians may cause some trouble to beginners, but, on the other hand, unless the four-dimensional aspect of nature in the relativistic sense is understood, it is quite useless to philosophise on the more advanced aspects of the theory. Before we try to run, we should at least learn to stand on our feet.
To take another illustration at random, it is the same when Professor Broad discusses the law of gravitation. Notwithstanding his condemnation of the conception of space-time, as one sole four-dimensional continuum, he sees no inconsistency in discussing the geometry of this four-dimensional continuum. At any rate, after telling the reader that a space may be flat or curved like an egg or like a sphere, he decides (p. 224) that the law
denotes a spherical curvature. Then, on a later page (p. 485), we are informed that the curvature is homaloidal. This taxes the imagination of the reader, for as “homaloidal” means flat, a space cannot be both homaloidal and spherical at one and the same time. However, it is to be presumed that Broad meant “homogeneous,” not “homaloidal,” and we shall interchange the two words accordingly. But even when amended in this way, his premises are totally incorrect, since the law