, taken from totally different regions of mathematics, from geometry and from logarithms, are suddenly found to manifest extraordinary relationships.
But this introduction of imaginary numbers into mathematical analysis was soon to lead to still more wonderful discoveries. Cauchy undertook a general study of functions in which the variable would be a complex number, that is to say, a real number plus an imaginary one. He uncovered thereby a new universe of functions, called functions of a complex variable, possessing the strangest properties; and it was seen that our older functions were but particular instances of these more general types. But for these new functions to possess the essential attributes of our older ones, a certain mathematical restriction had to be placed upon them. The restrictions turned out to be expressed by the same mathematical relationship which defines the potential distribution in Newton’s law of gravitation, namely, Laplace’s equation.[137] This equation crops up again in the theory of conformal representation, in the theory of heat, in the theory of elasticity, etc. Unsuspected relationships thus appear to be springing up on all sides. Incidentally we may note that Riemann’s great discoveries in the theory of functions take their start from this analogy existing between the functions of a complex variable and the laws of the potential distribution around matter.
The theory of the functions of a complex variable constitutes one of the most extensive domains of pure mathematics, with which the names of some of the greatest mathematicians are associated, such as Cauchy, Riemann, Weierstrass, Jacobi, Abel, Hermite and Poincaré. Numerous problems pertaining to the ordinary functions receive a very rapid solution when we consider these functions as restricted cases of the more general functions of a complex variable. As time went on, a number of these functions manifesting strange yet simple properties were brought to light, always by the same process of successive generalisation. First came the elliptical functions discovered by Abel and Jacobi, then the modular functions of Hermite, and finally the automorphous functions of Poincaré. Once again the discovery of these new functions revealed unexpected relationships between domains of mathematics which had appeared to be totally estranged; and Poincaré established surprising relationships between the automorphous functions, certain problems of the theory of numbers, and non-Euclidean geometry.
Incidentally, the popular belief that mathematicians, by appealing to imaginary magnitudes and the like, are abandoning the world of reality for one of shadows, is belied by the fact that one of the most elementary mechanical problems, namely, the oscillatory motion of an ordinary pendulum, can be solved mathematically only by an appeal to elliptical functions, in spite of the fact that imaginary quantities are part and parcel of their make-up.[138] It is the same with a number of electrical problems of widespread industrial importance; likewise, with Einstein’s law of gravitation, the orbits of the planets cannot be calculated unless we appeal to elliptical functions. Similar considerations would apply to Poincaré’s automorphous functions. However, this point being granted, it is of course obvious that the mathematician is not primarily interested in knowing whether his abstract speculations have any counterpart in the real objective world, any more than the poet wishes to know whether his dreams will come true. The sole object of his investigations is to explore all rational possibilities and to co-ordinate into one consistent whole various mathematical edifices which at first blush might appear to be totally estranged from one another.
We have insisted particularly on this generalising tendency of mathematics, as it is the one which will play a prominent part when we consider the methodology of mathematical physics as applied to the world of reality. Here we must recall that however mysterious it may seem, nature appears to be amenable to mathematical investigation and to be governed by rigid mathematical laws, at least to a first approximation. So far as scientists are concerned, this belief is not the outcome of religious or philosophical presuppositions. Rather is it a belief which is forced upon our minds by the triumphs of theoretical physics, the first grand example of which was afforded by Newton’s celestial mechanics. As soon as this susceptibility to law was recognised in nature, the avowed aim of science was to discover the unknown laws and in this manner allow us to foresee and to foretell, hence also to forestall, and to cease living in a world of unexpected miracles. In those realms where laws were finally established, science displaced superstition; and wherever, for one reason or another, laws could not be found, superstition continued to reign supreme. A case in point is afforded by meteorology, which in spite of recent progress remains an extremely backward science. As a result an astronomer, who would never think of praying for a solar eclipse, might still pray for a cloudless day favourable for his observations.
Now, in order to render nature amenable to mathematical treatment, it is necessary that we should succeed in reducing the various natural phenomena to common terms. This is done by seeking differences of quantity beneath differences of quality. When and only when this quantitative reduction has been accomplished can science proceed with its investigations, its deductions and inductions. In the case of the objective universe of physics, this process of reducing quality to quantity leads us to conceive of the objective universe as one of electromagnetic vibrations and of molecules, atoms and electrons acting on one another according to fixed laws, rushing hither and thither in space or vibrating round fixed points. The objective universe of science is thus noiseless, lightless, odourless. All the qualities and values are conceived of as arising from the interactions existing between our sense organs and the motions of the outside world, just as the impact of a body would reveal itself as sound to a gong gifted with consciousness.
This quantitative reduction is, however, but a first step; and even after it has been accomplished, nature might still defy mathematical investigation. The fact is that mathematics consists in compounding the similar with the similar; wide heterogeneities even among quantities would render an appeal to human mathematics useless. Thus, the game of chess, where the various pieces can move in widely different ways, is not subject to mathematical treatment. The fact that theoretical science is possible proves that similarity and unity can be found in nature.
Then again, nature must be simple, or at least simple to a first approximation. Theoretically, simplicity cannot exist in nature, since the whole influences the part and the part influences the whole. But in certain cases it has been found permissible to neglect a number of influences owing to their minuteness, and science has thus been able to progress. It is because in meteorology this restriction of active influences to a minimum appears impossible that long-range weather prediction is any man’s guess.
But there is yet another condition which must be found in nature, and that is continuity. Mathematics is capable of attacking problems where discontinuities are present, but the technical difficulties are very great; and this is the chief reason why the theory of numbers presents such obstacles. On the other hand, where we are dealing with problems of continuity the mathematician feels more at ease; and that superb mathematical instrument known as the differential equation becomes applicable. Incidentally we can understand why it was that Planck’s discovery of quantum phenomena, or discontinuous jumps in the processes of nature, was such unwelcome news to theoretical physicists; the differential equation had lost its power.
Even now we are not at the end of our difficulties. Assuming that nature manifests unity, simplicity and continuity, we can only investigate her secrets provided we are faced with mathematical problems that we can solve. Even such relatively simple problems as that of the motion of four bodies attracting one another under Newton’s law, have thus far defied all attempts. Again, our differential equations are of a very simple species. They suffice only for the simplest type of problems, when, for instance, the future position of a body moving under the action of a given force depends solely on its circumstances of motion in its initial position. But what if the future position of the body were to depend on its entire past history? Phenomena of this type, subjected as they are to hereditary influences, are well known in physics, being illustrated by the hysteresis, or fatigue, of metals; and of course in biology they are the rule. Such problems can be attacked to-day, at any rate in the most elementary cases, following Volterra’s discoveries in integro-differential equations. This opens up a new domain of science, that known as “hereditary mathematics.”