"Must we further recall the kinetic theory of gases, the facts explained by the breaking up of molecules into ions, the hypothesis suggested, for example, by Van der Waals by the view that an atom has an actual bulk? Must we point to a physical phenomenon of quite a different class, for example, to the coloring of the thin film forming the soap-bubbles which W. Thomson has taken as the measure of the size of a molecule?
"Such a résumé of results shows plainly that we cannot help the progress of science by blocking the path of theory and looking only at its positive aspects, that is to say, at the collection of facts that it explains. The value of a theory lies rather in the hypothesis which it can suggest, by means of the psychological representation of the symbols.
"We shall not draw from all this the conclusion that the atomic hypothesis ought to correspond to the extremely subtle sensations of a being resembling a perfected man. We shall not even reason about the possibility of those imaginary sensations, in so far as they are conceived simply as an extension of our own. But we shall repeat, in regard to the atomic theory, what an illustrious master is said to have remarked as to the unity of matter: if on first examination a fact seems possible which contradicts the atomic view of things, there is a strong probability that such a fact will be disproved by experience.
"Does not such a capacity for adaptation to facts, thus furnishing a model for them, perhaps denote the positive reality of a theory?"
And the above principles are as true of mathematical concepts as of chemical. Everywhere it is "capacity of adaptation to facts" that is the criterion of a branch of mathematics, except, of course, that in mathematics the facts are not always physical facts. Mathematics has successfully accomplished a generalization whereby its own methods furnish the material for higher generalizations. The imaginary number and the hyper-dimensional or non-Euclidian geometries may be absurd if measured by the standard of physical reality, but they nevertheless have something real about them in relation to certain mathematical processes on a lower level. There is no philosophic paradox about modern arithmetic or geometry, once it is recognized that they are merely abstractions of genuine features of simpler and more obviously practical manipulations that are clearly derived from the dealing of a human being with genuine realities.
In the light of these considerations, I cannot help feeling that the frequent attempts of mathematicians with a philosophical turn of mind, and philosophers who are dipping into mathematics, to derive geometrical entities from psychological considerations are quite mistaken, and are but another example of those traditional presuppositions of psychology which, Professor Dewey has pointed out (Jour. of Phil., Psy., and Sci. Meth., XI, No. 19, p. 508), were "bequeathed by seventeenth-century philosophy to psychology, instead of originating within psychology" ... that "were wished upon it by philosophy when it was as yet too immature to defend itself."
Henri Poincaré (Science and Hypothesis, Ch. IV, The Value of Science, Ch. IV) and Enriques (Problems of Science, Ch. IV, esp. B—The Psychological Acquisition of Geometrical Concepts) furnish two of the most familiar examples of this sort of philosophizing. Each isolates special senses, sight, touch, or motion, and tries to show how a being merely equipped with one or the other of these senses might arrive at geometrical conceptions which differ, of course, from space as represented by our familiar Euclidian geometry. Then comes the question of fusing these different sorts of experience into a single experience of which geometry may be an intelligible transcription. Enriques finds a parallel between the historical development and the psycho-genetic development of the postulates of geometry (loc. cit., p. 214 seq.). "The three groups of ideas that are connected with the concepts that serve as the basis for the theory of continuum (Analysis situs), of metrical, and of projective geometry, may be connected, as to their psychological origin, with three groups of sensations: with the general tactile-muscular sensations, with those of special touch, and of sight, respectively." Poincaré even evokes ancestral experience to make good his case (Sci. and Hyp., Ch. V, end). "It has often been said that if individual experience could not create geometry, the same is not true of ancestral experience. But what does that mean? Is it meant that we could not experimentally demonstrate Euclid's postulate, but that our ancestors have been able to do it? Not in the least. It is meant that by natural selection our mind has adapted itself to the conditions of the external world, that it has adopted the geometry most advantageous to the species: or in other words, the most convenient."
Now undoubtedly there may be a certain modicum of truth in these statements. As implied by the last quotation from Poincaré, the modern scientist can hardly doubt that the fact of the adaptation of our thinking to the world we live in is due to the fact that it is in that world that we evolved. As is implied by both writers, if one could limit human contact with the world to a particular form of sense response, thought about that world would take place in different terms from what it now does and would presumably be less efficient. But these admissions do not imply that any light is thrown upon the nature of mathematical entities by such abstractions. Russell (Scientific Method in Philosophy) is in the curious position of raising arithmetic to a purely logical status, but playing with geometry and sensation after the manner of Poincaré, to whom he gives somewhat grudging praise on this account.
The psychological methods upon which all such investigations are based are open to all sorts of criticisms. Chiefly, the conceptions on which they are based, even if correct, are only abstractions. There is not the least evidence for the existence of organisms with a single differentiated sense organ, nor the least evidence that there ever was such an organism. Indeed, according to modern accounts of the evolution of the nervous system (cf. G. H. Parker, Pop. Sci. Month., Feb., 1914) different senses have arisen through a gradual differentiation of a more general form of stimulus receptor, and consequently, the possibility of the detachment of special senses is the latter end of the series and not the first. But, however this may be, the mathematical concepts that we are studying have only been grasped by a highly developed organism, man, but they had already begun to be grasped by him in an early stage of his career before he had analyzed his experience and connected it with specific sense organs. It may of course be a pleasant exercise, if one likes that sort of thing, to assume with most psychologists certain elementary sensations, and then examine the amount of information each can give in the light of possible mathematical interpretations, but to do so is not to show that a being so scantily endowed would ever have acquired a geometry of the type in question, or any geometry at all. Inferences of the sort are in the same category with those from hypothetical children, that used to justify all theories of the pedagogue and psychologist, or from the economic man, that still, I fear, play too great a part in the world of social science.