The geometer will, if you insist, go on calling his undefined terms by the familiar names “point,” “line,” “plane.” But you must distinctly understand that this is a concession to usage, and that you are not for a moment to restrict the application of his statements in any way. He would much prefer, however, to be allowed new names for his elements, to say “We start with three elements of different sorts, which we assume to exist, and to which we attach the names A, B and C—or if you prefer, primary, secondary and tertiary elements—or yet again, names possessing no intrinsic significance at all, such as ching, chang and chung.” He will then lay down whatever statements he requires to serve the purposes of the ancient axioms, all of these referring to some one or more of his elements. Then he is ready for the serious business of proving that, all his hypotheses being granted, his elements A, B and C, or I, II and III, or ching, chang and chung, are subject to this and that and the other propositions.
The objection will be urged that the mathematician who does all this usurps the place of the logician. A little reflection will show this not to be the case. The logician in fact occupies the same position with reference to the geometer that the geometer occupies with reference to the physicist, the chemist, the arithmetician, the engineer, or anybody else whose primary interest lies with some particular set of elements to which the geometer’s system applies. The mathematician is the tool-maker of all science, but he does not make his own tools—these the logician supplies. The logician in turn never descends to the actual practice of rigorous thinking, save as he must necessarily do this in laying down the general procedures which govern rigorous thinking. He is interested in processes, not in their application. He tells us that if a proposition is true its converse may be true or false or ambiguous, but its contrapositive is always true, while its negative is always false. But he never, from a particular proposition “If A is B then C is D,” draws the particular contrapositive inference “If C is not D then A is not B.” That is the mathematician’s business.
The Rôle of Geometry
The mathematician is the quantity-production man of science. In his absence, the worker in each narrower field where the elements under discussion take particular concrete forms could work out, for himself, the propositions of the logical structure that applies to those elements. But it would then be found that the engineer had duplicated the work of the physicist, and so for many other cases; for the whole trend of modern science is toward showing that the same background of principles lies at the root of all things. So the mathematician develops the fabric of propositions that follows from this, that and the other group of assumptions, and does this without in the least concerning himself as to the nature of the elements of which these propositions may be true. He knows only that they are true for any elements of which his assumptions are true, and that is all he needs to know. Whenever the worker in some particular field finds that a certain group of the geometer’s assumptions is true for his elements, the geometry of those elements is ready at hand for him to use.
Now it is all right purposely to avoid knowing what it is that we are talking about, so that the names of these things shall constitute mere blank forms which may be filled in, when and if we wish, by the names of any things in the universe of which our “axioms” turn out to be true. But what about these axioms themselves? When we lay them down in ignorance of the identity of the elements to which they may eventually apply, they cannot by any possibility be “self-evident.” We may, at pleasure, accept as self-evident a statement about points and lines and planes; or one about electrons, centimeters and seconds; or one about integers, fractions, and irrational numbers; or one about any other concrete thing or things whatever. But we cannot accept as self-evident a statement about chings, changs and chungs. So we must base our “axioms” on some other ground than this; and our modern geometer has his ground ready and waiting. He accepts his axioms on the ground that it pleases him to do so. To avoid all suggestion that they are supposed to be self-evident, or even necessarily true, he drops the term “axiom” and substitutes for it the more color-less word “postulate.” A postulate is merely something that we agreed to accept, for the time being, as a basis of further argument. If it turns out to be true, or if we can find circumstances under which and elements to which it applies, any conclusions which we deduce from it by trustworthy processes are valid within the same limitations. And the propositions which tell us that, if our postulates are true, such and such conclusions are true—they, too, are valid, but without any reservation at all!
Perhaps an illustration of just what this means will not be out of place. Let it be admitted, as a postulate, that
is greater, by 1, than
