We have now seen how, from the physiological construction of our mental apparatus, the process of concept formation and the experience of concept connections are the basis of the whole of mental life. The laws of the mutual interaction of the most general or elementary concepts operated in the formation of the concepts, thing, group, co-ordination. Here were found the fundamentals of logic or the science of concepts. A special process of abstraction yielded the concept of number, and with it the corresponding field of mathematics, arithmetic, algebra, and the theory of numbers.
By means of the second fundamental fact of physiology, the threshold, another elementary fact was explained, that of continuity. The co-ordination of individual things under the influence of this concept was expanded into the co-ordination of continuous phenomena-series, and yielded the correspondingly more general concept of the function. From the application of the number concept to continuous things, the idea of measurement resulted. In mathematics the concept of continuity led to higher analysis and the theory of functions. Finally, the concept of continuity proved to be an inexhaustible aid for the extension of scientific knowledge and for the formulation of natural laws in mathematical form.
[PART III]
THE PHYSICAL SCIENCES
42. General.
In the formal sciences we began the specialization of the object from the most general concept of thing conceivable, possessing no other characteristic attribute than its capability of being distinguished from other things; and we carried the specialization so far that we could follow in its movements an object definite as to time and space. This object, to be sure, was defined only in that it occupied a definite space, and accordingly had a definite form. As a matter of fact, the spacial thing of geometry and phoronomy reveals no further attributes.
It is here that the physical sciences enter into their dominion one after the other, and fill the empty space of the geometric thing with definite attributes. These are the secondary qualities of Locke, of which he assumed that they do not belong so much to the bodies themselves as that they merely appear to us so on account of the nature of our human sense organs. Now that our knowledge concerning the nature of those properties as well as the structure of our sense organs is much more thorough, we have more definite ideas also of the subjective part of the corresponding experiences, and in a large measure are able to separate it from the objective part.
All properties which physical bodies in contradistinction to geometric bodies possess can be traced back to a fundamental concept, which, in conjunction with the concepts explained in the former chapter, serves to characterize and distinguish the physical structure. For example, the fact that we can distinguish cubes of equal size but of different material, different temperature, and different luminosity, can be traced back always and entirely to the different kinds of energy acting in the geometric space in question. The concept of energy, therefore, plays approximately the same rôle in the physical sciences as the concept of thing in the formal sciences, and the essentials of this new field of science are the comprehensive knowledge and development of this concept. Because of its great importance it has long been known and applied in individual forms. But the systematization of the physical sciences relative to energy is a matter of only recent date.