Mathematical deductions are mind-born; they pertain to reason and are not dependent on experience. When, therefore, we assume that our mathematical deductions and operations will be successful in portraying the workings of nature, we are assuming that nature also is rational, and that therefore a definite parallelism or correspondence exists between the two worlds, the mathematical and the physical. A priori, there appears to be no logical necessity why any such parallelism should exist. Here, however, we are faced with a situation over which it is useless to philosophise. Success has attended the efforts of mathematical physicists in so large a number of cases that, however marvellous it may appear, we can scarcely escape the conclusion that nature must be rational and susceptible to mathematical law. In [Pg xiii] fact, were this not the case, prevision would be impossible and science non-existent. It may be that nature is only approximately rational; it may be that her appearance of rationality is due to the very crudeness of our observations and that more refined experiments would yield a very different picture. Eddington goes so far as to suggest that the difficulties which confront us in the study of quantum phenomena may indeed be due to the fact that nature is found to be irrational when we seek to examine her processes in a microscopic way. This is a possibility which we cannot afford to reject. But at any rate, as long as our theories appear to be verified by experiment, we must proceed as though nature were rational, and hope for the best. Now it must not be thought that the introduction of the mathematical instrument into our study of nature creates any essential departure from the commonplace method of ordinary deductive and inductive reasoning. There is no particular mystery about mathematical analysis; its only distinguishing feature is that it is more trustworthy, more precise, and permits us to proceed farther and along safer lines. Consider, for example, the well-known change of colour from red to white displayed by the light radiated through an aperture made in a heated enclosure, as the temperature increases. From this elementary fact of observation Planck, thanks to mathematical analysis, was able to deduce the existence of light quanta and thence the possibility that all processes of change were discontinuous, and that a body could only rotate with definite speeds. Obviously, commonplace reasoning unaided by mathematics would never have led us even to suspect these extraordinary results.