material is so close that the mind does not pause upon the sign. The signs are intended only to stand for things and acts. But scientific terminology has an additional use. It is designed, as we have seen, not to stand for the things directly in their practical use in experience, but for the things placed in a cognitive system. Ultimately, of course, they denote the things of our common sense acquaintance. But immediately they do not designate them in their common context, but translated into terms of scientific inquiry. Atoms, molecules, chemical formulæ, the mathematical propositions in the study of physics—all these have primarily an intellectual value and only indirectly an empirical value. They represent instruments for the carrying on of science. As in the case of other tools, their significance can be learned only by use. We cannot procure understanding of their meaning by pointing to things, but only by pointing to their work when they are employed as part of the technique of knowledge.
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
