[68] We may mention that the logicians have often endeavoured to introduce definitions of this sort into mathematics; but in mathematics, as in physics, their attempts have always been condemned. For instance, the definition of an inductive number, also given by Dr. Whitehead, was severely criticised by Poincaré (in “Science and Method”), who remarked that the object of a definition was to define, and that Whitehead’s definition being circular, defined nothing.
[69] In the case of Fresnel, his ignorance of non-Euclidean geometry might be claimed to lessen the force of the argument, but this contention would not hold for Lorentz or for the theoretical physicists of the latter part of the nineteenth century.
[70] It is important to understand the precise significance of the principle of entropy. Keeping in mind the fact that the principle refers to probabilities and not to absolute certainties, we may express its general formulation as follows: For all changes of a system the total entropy either is increased (irreversible changes) or else remains constant (reversible changes). At first sight it might appear as though the transformation of heat into work in the steam engine, or the withdrawal of heat from a colder body and its transfer to a warmer one, as in the case of refrigerating machines, would constitute flagrant violations of the principle. But, as a matter of fact, such is not the case. For, with the steam engine, the unnatural change exhibited by the transformation of heat into work is counterbalanced by a natural change illustrated by the fall of heat from generator to condenser. In fine, the total entropy thus remains constant (in a perfectly reversible cycle, for instance in a Carnot cycle), or else suffers an increase. Again, with the refrigerating machine, the unnatural passage of heat from the cold to the warmer body is compensated by a natural transformation; this is represented by the transformation into heat of the mechanical work performed on the apparatus. So once again the total entropy is increased, or, at best, remains constant. Still more striking cases, where the principle would appear to be at fault, are illustrated by the endothermic reactions in chemistry. Contrary to what we should expect, they absorb heat so that low-grade heat energy is transformed into higher-grade chemical energy; but here again compensating influences are at work. We cannot dwell further on these points. The reader who is interested in studying them must consult some standard work on thermodynamics.
[71] This decrease of the visual angle under the influence of relative motion would, however, apply only provided the observer were moving in a direction parallel to that of the pole, or at least yielding a component parallel to this direction.
[72] We do not wish to indulge in endless repetitions, but we must note once more that by absolute length of classical science we are referring solely to length once it has been defined by the norms of practical congruence. We do not imply absolute metaphysical spatial length, which is meaningless in science.
[73] A light year is an astronomical unit of length defined by the distance a wave of light covers in one year.
[74] We are considering only flat space-time far from matter.
[75] Recent experiments conducted by Kennedy have shown that Miller’s conclusions were erroneous, and that the null result of the original experiment was completely confirmed. (See Proc. Nat. Acad. of Sci., Nov., 1926.)
[76] It might be argued that a rotating disk is an accelerated body and that, in spite of this fact, no force acts upon it. This view would, however, be incorrect. Cohesive forces are acting between the various molecules of the disk, and were it not for the existence of these forces, the various molecules would fly off along tangents and would pursue Galilean motions; the disk would cease to exist. It is the presence of the cohesive forces which compels the molecules to describe circles, that is, to pursue accelerated or non-Galilean motions.
[77] This was in full accord with the principle of entropy.