Even the circle, square, etc., of geometry exhibit a difference from the squares and circles of familiar acquaintance, and the further one proceeds in mathematical science the greater the remoteness from the everyday empirical thing. Qualities which do not count for the pursuit of knowledge about spatial relations are left out; those which are important for this purpose are accentuated. If one carries his study far enough, he will find even the properties which are significant for spatial knowledge giving way to those which facilitate knowledge of other things—perhaps a knowledge of the general relations of number. There will be nothing in the conceptual definitions even to suggest spatial form, size, or direction. This does not mean that they are unreal mental inventions, but it indicates that direct physical qualities have been transmuted into tools for a special end—the end of intellectual organization. In every machine the primary state of material has been modified by subordinating it to use for a purpose. Not the stuff in its original form but in its adaptation to an end is important. No one would have a knowledge of a machine who could enumerate all the materials entering into its structure, but only he who knew their uses and could tell why they are employed as they are. In like fashion one has a knowledge of mathematical conceptions only when he sees the problems in which they function and their specific utility in dealing with these problems. “Knowing” the definitions, rules, formulae, etc., is like knowing the names of parts of a machine without knowing what they do. In one case, as in the other, the meaning, or intellectual content, is what the element accomplishes in the system of which it is a member.
II
Science and Social Progress
Assuming that the development of the direct knowledge gained in occupations of social interest is carried to a perfected logical form, the question arises as to its place in experience. In general, the reply is that science marks the emancipation of mind from devotion to customary purposes and makes possible the systematic pursuit of new ends. It is the agency of progress in action. Progress is sometimes thought of as consisting in getting nearer to ends already sought. But this is a minor form of progress, for it requires only improvement of the means of action or technical advance. More important modes of progress consist in enriching prior purposes and in forming new ones. Desires are not a fixed quantity, nor does progress mean only an increased amount of satisfaction. With increased culture and new mastery of nature, new desires, demands for new qualities of satisfaction, show themselves, for intelligence perceives new possibilities of action. This projection of new possibilities leads to search for new means of execution, and progress takes place; while the discovery of objects not already used leads to suggestion of new ends.
That science is the chief means of perfecting control of means of action is witnessed by the great crop of inventions which followed intellectual command of the secrets of nature. The wonderful transformation of production and distribution known as the industrial revolution is the fruit of experimental science. Railways, steamboats, electric motors, telephone and telegraph, automobiles, aeroplanes and dirigibles are conspicuous evidences of the application of science in life. But none of them would be of much importance without the thousands of less sensational inventions by means of which natural science has been rendered tributary to our daily life.
It must be admitted that to a considerable extent the progress thus procured has been only technical: it has provided more efficient means for satisfying preexistent desires, rather than modified the quality of human purposes. There is, for example, no modern civilization which is the equal of Greek culture in all respects. Science is still too recent to have been absorbed into imaginative and emotional disposition. Men move more swiftly and surely to the realization of their ends, but their ends too largely remain what they were prior to scientific enlightenment. This fact places upon education the responsibility of using science in a way to modify the habitual attitude of imagination and feeling, not leave it just an extension of our physical arms and legs.
The advance of science has already modified men’s thoughts of the purposes and goods of life to a sufficient extent to give some idea of the nature of this responsibility and the ways of meeting it. Science taking effect in human activity has broken down physical barriers which formerly separated men; it has immensely widened the area of intercourse. It has brought about interdependence of interests on an enormous scale. It has brought with it an established conviction of the possibility of control of nature in the interests of mankind and thus has led men to look to the future, instead of the past. The coincidence of the ideal of progress with the advance of science is not a mere
