, Planck’s constant; and contrast them with the constants of mathematics, such as
or
. Whereas the constants of mathematics can be calculated to any degree of approximation we choose, the values we assign to the constants of physics can never be considered absolutely rigorous. It is not merely because physical observations are necessarily inaccurate or because conditions of observation may not be perfectly ideal owing to the presence of contingent influences; other reasons of a deeper order are involved. The fact is that we are by no means certain that nature is amenable to rigorous mathematical laws and that the so-called constants of physics represent other than average values. The triumphs of theoretical physicists suggest that a mathematisation of physics is permissible as a first approximation; but a closer microscopic survey might prove that this appearance of mathematical purity and simplicity in nature was due to our crude macroscopic survey of phenomena. For instance, it is a well-known fact that if conditions are sufficiently chaotic, the chaos will generate simplicity when we view things from a macroscopic standpoint; it is only when we wish to view things microscopically that the chaos appears and mathematical methods become impossible.
As we mentioned earlier in the chapter, the kinetic theory of gases furnishes us with an apt illustration. The molecules are rushing hither and thither, bounding off one another in the most capricious way.[139] It would be quite impossible to foresee the history of a given molecule; and yet, thanks to this very chaos, the macroscopic laws of perfect gases are exceedingly simple. We may infer, therefore, that the constants of physics which enter into our macroscopic laws represent mere average effects and differ essentially from those of pure mathematics. More generally, even under ideal conditions of observation the world of physics can never be assimilated to the world of pure mathematics.
The simplification which ensues from a macroscopic survey of nature allows us to understand the reason for those periodic swings which take place in theoretical science between the physics of the general principles and the atomistic viewpoint. If we view nature atomistically, it will be our desire to interpret phenomena in terms of attractions and repulsions between molecules, atoms or electrons. Following the success of Newton’s treatment of planetary motions, the scientists of the eighteenth and early nineteenth century endeavoured to pursue this method by reducing the entire world of physical science to attractions and repulsions between discrete particles. All sorts of laws were appealed to: inverse-cube laws and laws involving still higher powers. But nature was not so simple as scientists had hoped, and paid very little heed to the difficulties of mathematical analysis. So there arose an opposing school of scientists, who substituted general principles embodying wider concepts such as energy, entropy, action, for the atomistic theories of their predecessors.
The argument of this new school of thought ran somewhat along the following lines: “By concentrating more and more upon your world of atoms and molecules you are losing sight of those general principles which experiment has revealed. Would it not be better, therefore, to abandon your hopelessly complex speculations, and accept as a starting point these fundamental principles, which can be deduced from macroscopic investigation? We grant that by so doing you will be substituting a macroscopic for a microscopic view of nature; but by reason of the mathematical difficulties into which you have been led and the numerous hypotheses you have had to make in order to compensate for your ignorance of the microscopic mechanisms, it might be the safer course to content yourselves, for the time being, at any rate, with the macroscopic viewpoint.”
We may illustrate the new viewpoint somewhat as follows: A life-insurance company does not know and has no means of foretelling when one of its clients will die; yet, when instead of considering the case of an individual client (which would correspond to the microscopic or atomistic viewpoint) we consider men as a whole (the macroscopic point of view), certain general death statistics can be established. By attaching too much importance to one individual, we should lose sight of this general principle, and premiums could never be fixed. The analogy is far from perfect; but it is helpful in that it shows us that in certain cases, at least, the macroscopic point of view can yield us the information we require. A better illustration would be given in the following very free translation of a passage from Poincaré. As he tells us: