What we have said about definitions applies equally to statements of fact, of the sort which are going to constitute the body of our science. In the absence of simpler facts to cite as authority, we shall never be able to prove anything, however simple this may itself be; and in fact the simpler it be, the harder it is to find something simpler to underlie it. If we are to have a logical structure of any sort, we must begin by laying down certain terms which we shall not attempt to define, and certain statements which we shall not try to prove. Mathematics, physics, chemistry—in the large and in all their many minor fields—all these must start somewhere. Instead of deceiving ourselves as to the circumstances surrounding their start, we prefer to be quite frank in recognizing that they start where we decide to start them. If we don’t like one set of undefined terms as the foundation, by all means let us try another. But always we must have such a set.

The classical geometer sensed the difficulty of defining his first terms. But he supposed that he had met it when he defined these in words free of technical significance. “A point is that which has position without size” seemed to him an adequate definition, because “position” and “size” are words of the ordinary language with which we may all be assumed familiar. But today we feel that “position” and “size” represent ideas that are not necessarily more fundamental than those of “line” and “point,” and that such a definition begs the question. We get nowhere by replacing the undefined terms “point” and “line” and “plane,” which really everybody understands, by other undefined terms which nobody understands any better.

In handling the facts that it was inconvenient to prove, the classical geometer came closer to modern practice. He laid down at the beginning a few statements which he called “axioms,” and which he considered to be so self-evident that demonstration was superfluous. That the term “self-evident” left room for a vast amount of ambiguity appears to have escaped him altogether. His axioms were axioms solely because they were obviously true.

Laying the Foundation

The modern geometer falls in with Euclid when he writes an elementary text, satisfying the beginner’s demand for apparent rigor by defining point and line in some fashion. But when he addresses to his peers an effort to clarify the foundations of geometry to a further degree of rigor and lucidity than has ever before been attained, he meets these difficulties from another quarter. In the first place he is always in search of the utmost possible generality, for he has found this to be his most effective tool, enabling him as it does to make a single general statement take the place and do the work of many particular statements. The classical geometer attained generality of a sort, for all his statements were of any point or line or plane. But the modern geometer, confronted with a relation that holds among points or between points and lines, at once goes to speculating whether there are not other elements among or between which it holds. The classical geometer isn’t interested in this question at all, because he is seeking the absolute truth about the points and lines and planes which he sees as the elements of space; to him it is actually an object so to circumscribe his statements that they may by no possibility refer to anything other than these elements. Whereas the modern geometer feels that his primary concern is with the fabric of logical propositions that he is building up, and not at all with the elements about which those propositions revolve.

It is of obvious value if the mathematician can lay down a proposition true of points, lines and planes. But he would much rather lay down a proposition true at once of these and of numerous other things; for such a proposition will group more phenomena under a single principle. He feels that on pure scientific grounds there is quite as much interest in any one set of elements to which his proposition applies as there is in any other; that if any person is to confine his attention to the set that stands for the physicist’s space, that person ought to be the physicist, not the geometer. If he has produced a tool which the physicist can use, the physicist is welcome to use it; but the geometer cannot understand why, on that ground, he should be asked to confine his attention to the materials on which the physicist employs that tool.

It will be alleged that points and lines and planes lie in the mathematician’s domain, and that the other things to which his propositions may apply may not so lie—and especially that if he will not name them in advance he cannot expect that they will so lie. But the mathematician will not admit this. If mathematics is defined on narrow grounds as the science of number, even the point and line and plane may be excluded from its field. If any wider definition be sought—and of course one must be—there is just one definition that the mathematician will accept: Dr. Keyser’s statement that “mathematics is the art or science of rigorous thinking.”

The immediate concern of this science is the means of rigorous thinking—undefined terms and definitions, axioms and propositions. Its collateral concern is the things to which these may apply, the things which may be thought about rigorously—everything. But now the mathematician’s domain is so vastly extended that it becomes more than ever important for him to attain the utmost generality in all his pronouncements.

One barrier to such generalization is the very name “geometry,” with the restricted significance which its derivation and long usage carry. The geometer therefore must have it distinctly understood that for him “geometry” means simply the process of deducing a set of propositions from a set of undefined primitive terms and axioms; and that when he speaks of “a geometry” he means some particular set of propositions so deduced, together with the axioms, etc., on which they are based. If you take a new set of axioms you get a new geometry.