Now in all these theories of mathematical physics, the same type of procedure is invariably followed. Experimenters establish certain definite facts and detect precise numerical relationships between magnitudes, for example, between the intensity of an electric current flowing along a wire and the intensity and orientation of the magnetic field surrounding the wire. The mathematical physicist then enters upon the scene, assigns certain letters of the alphabet to the [Pg xii] physical entities involved (in the present case electric current designated by and magnetic intensity designated by ) and by this means translates the numerical relationships discovered by the experimenter into mathematical form. He thus obtains a mathematical relationship or equation which is assumed to constitute the mathematical image of the concrete physical phenomenon . His task will now be to extract from his mathematical equation or equations all their necessary mathematical consequences. In this way, provided his technique does not fail him, he may be led to new equations . These new equations , when translated back from the mathematical to the physical, will express new physical relationships . The mathematician assumes that just as his equations were the necessary mathematical consequences of his original equations , so also must the physical translation of constitute a physical phenomenon , which follows as a necessary consequence of the existence of the physical phenomenon . If occurs, must ensue. We thus understand the significance of a theory of mathematical physics. Its utility is to allow us to foresee and to foretell physical phenomena. In this way it suggests precise experiments which might never have been thought of, and permits us to anticipate new relationships and new laws and to discover new facts. From a philosophical point of view, by establishing a rational connection between seemingly unconnected phenomena, it enables us to detect the harmony and unity of nature which lie concealed under an outward appearance of chaos. Of course the experimenter in the first place must be very careful to give accurate information to the mathematician; for if by any chance his information should be only approximately correct, the mathematical translation would likewise be lacking in accuracy, and the mathematical consequences of a might be still further at variance with the world of physical reality. It is as though, when firing at a distant target, we were to point the rifle a wee bit too far to one side; the greater the range, the wider would be the divergence. Dangers of this sort are of course inevitable, for human observations are necessarily imperfect. In any case, therefore, the mathematician’s physical anticipations will always require careful checking up by subsequent experiment. Obviously, however, something much deeper is at stake than mere accuracy of observation.