[133] It is scarcely necessary to add that our awareness of sensations does not presuppose any knowledge of space or of our human body as an object situated in space. When, for instance, an infant who is beginning to emerge into consciousness feels a pain in its leg, then one in its arm, it is not supposed that it succeeds in localising the two sensations in the two respective limbs. Only much later will it succeed in localising its sensations. For the present we are assuming that the infant knows nothing of space or of its body; it is merely registering sensations, and the pain in its leg will appear to it to differ in some obscure qualitative way from the pain in its arm.
[134] Recently considerable progress has been made by Schrödinger in the interpretation of quantum phenomena within the atom, by means of a wave theory of matter, known as wave mechanics.
[135] By realism we mean “common-sense realism,” and not that monstrous distortion known as “neo-realism.”
[136] Of recent years certain philosophers known as logicians, Bertrand Russell in England, Couturat in France, among others, have stressed the logical aspect of mathematics. The question is whether they have not over-stressed it. That mathematical reasoning implies clear thinking and complies with the rules of logic has never been denied; nevertheless the assertion that all mathematical reasonings are of a purely deductive nature, and are reducible to the rules of logic, is an opinion which is by no means unanimous. Some of the greatest among the modern mathematicians, notably Poincaré and Klein, have protested vigorously against this view and have pointed out numerous cases of circularity in the arguments and lack of rigour in the definitions presented by the logicians. Over and above this aspect of the matter, they have maintained that the rules of a game are not everything in its make-up. To say that mathematics and logic are one and the same would be equivalent to maintaining that poetry was nothing but grammar, syntax and rules of versification, or that music was nothing but counterpoint and harmony. It is conceivable that we might acquire as thorough a knowledge of counterpoint and harmony as Beethoven may have possessed and yet be unable to compose a work rivalling any of his symphonies. We should have no hesitancy in granting that a Beethoven must obviously have been gifted with some mysterious faculty which had been denied us; and this faculty, whatever its essence, would relate to music, would be a part of music, since were all men lacking in it, there would be no great music. Under the circumstances, in spite of what might be called our logical knowledge of music, could we truthfully claim to have as thorough a knowledge of it as a Beethoven?
And it is exactly the same with pure mathematics. We know from experience that many persons, though possessing highly logical minds, are yet refractory to advanced mathematics. Were mathematics nothing but logic, this situation would seem extraordinary. Logistics, from a failure to see in mathematics anything but a series of rules and regulations with no creative faculty behind it, has been christened “thoughtless thinking” by its adversaries. But without wishing to take sides in a controversy for which the majority of persons evince but little interest, there is a point which the unprejudiced onlooker must perceive. With the sole exception of Hilbert, who, though opposing Russell’s views, defends opinions of a somewhat similar nature, none of the logicians have contributed to the constructive side of mathematics. This again appears somewhat strange when we recall that one of the earliest boasts of this school of thinkers was that logistics would give them wings. One cannot help but suspect that the logicians are lacking in some creative faculty of which they may not be conscious, and that as a result they perhaps occupy in mathematics a position analogous to that of the professor of counterpoint and harmony in music. Under the circumstances, it is questionable whether they possess a sufficient understanding of this difficult science to contribute any information of value.
Of course a charge of this sort cannot apply to Hilbert, whose great work in the creative regions of mathematics has proved him to be gifted with the creative faculty in addition to the purely formal dissecting faculty which all mathematicians, regardless of their tendencies, must necessarily possess. Owing to Hilbert’s attitude, the problem is generally regarded as controversial.
[137] In this case, however, Laplace’s equation is of the two-dimensional variety and not of the usual three-dimensional type which defines the distribution of the Newtonian potential in the empty space around a gravitational mass of finite dimensions. However, the two-dimensional Laplace equation also gives the distribution of the potential around matter in special instances. Such is the case when we consider an attracting cylinder of uniform density, finite section and infinite length.
[138] In the particular case of the oscillating pendulum, we may restrict our attention to the real (i.e. non-imaginary) realm of the elliptical function considered. But the very existence of elliptical functions is dependent on the introduction of imaginary quantities.
[139] This is of course merely a figure of speech. It is not assumed that the molecules come into actual contact. Furthermore, it would be difficult to specify exactly how contact should be defined for molecules.
[140] Thus, Euler, when discussing absolute space and time, writes: “What is the essence of space and time is not important; but what is important is whether they are required for the statement of the law of inertia. If this law can only be fully and clearly explained by introducing the ideas of absolute space and absolute time, then the necessity for these ideas can be taken as proved.” Again, Riemann, when discussing the possible non-Euclideanism of space, maintains a similar attitude. We read: “It is conceivable that the measure relations of space in the infinitesimal are not in accordance with the assumptions of our [Euclidean] geometry, and, in fact, we should have to assume that they are not, if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”