MATHEMATICS, FOUNDATIONS OF. No proposition of mathematics is considered to be established until it has been proved—that is to say, logically deduced from other propositions previously established. But obviously this process of proof must begin somewhere ; we must make some assumptions in order to start at all; and the problem arises, "What are the fundamental assumptions or axioms from which all the propositions of this subject can be deduced?" With regard, for instance, to Euclidean Geometry, this problem has been solved; it is found that all geometrical terms, such as circle or parallelogram, can be defined in terms of a few in definables, such as "point" and "straight line," and all the propo sitions of geometry can be deduced from a relatively small num ber of axioms about these indefinables, such as that through any two points passes one and only one straight line. When this has been done, we naturally want to discover whether these axioms are true. The answer to this question lies with the physicist; all that the mathematician can say is that if the axioms are true, then all the rest of geometry will be true also.
It is therefore clear that the mathematician asserts the propo sitions of geometry, not as absolute truths, but merely as implied by the axioms; and that, regarded as a branch of mathematics, geometry has no essential reference to physical space. For we can say, not only of physical points and planes but also of any classes of things which we may call points and planes, that if they obey the axioms of geometry, they obey the conclusions also.
So the mathematician regards geometry as simply tracing the consequences of certain axioms dealing with undefined terms, which are really variables in the ordinary mathematical sense, like x and y. And he demands of his axioms, not that they should be true on some particular physical interpretation of the undefined terms, but merely that they should be consistent with one another. If they were inconsistent this would probably appear from con tradictory consequences being deduced from them ; but although we had not as yet deduced any contradictory consequences, we could not therefore be sure that the assumptions were compatible with one another; for the latent contradiction might only be come manifest after more elaborate deductions. The only way
to provide positive proof is to find an interpretation of the undefined terms which will make the axioms true, since, if there are actual things for which they are true, the axioms must cer tainly be consistent. The things used for this purpose must not be taken from the physical world, or our proof would be subject to all the doubts and reservations of the experimental method. In fact, if the proof of consistency is to be a mathematical one, the entities which our undefined terms are interpreted to mean can only be taken from some other branch of mathematics. In the case of geometry we use the real numbers of algebra and analysis; for, if "point" be taken to mean ordered triad of numbers (x, y, z) and "plane" to mean set of such ordered triads satisfying a linear equation, and so on, it follows from the theory of real numbers that on this interpretation the axioms of geometry will be true, provided the theory of real numbers can be assumed.
Numbers.—We are thus thrown back to the theory of real numbers, for which we can again lay down axioms, using "real numbers" as a variable or undefined term meaning any things for which the axioms are true. If we proceed in this way we shall not, of course, have definite things called real numbers to use, as explained above, in showing the axioms of geometry consistent ; but we shall still be able to prove that if the axioms about real numbers are compatible, so are those of geometry; and our next step will be to investigate, for their own sake and for the sake of geometry, the consistency of the axioms about real numbers. This, in turn, we can establish by giving them a particular interpretation in terms of the rational numbers or fractions (the real numbers include also surds and other irrationals), and so indirectly in terms of the natural numbers o, 1, 2, 3 . . . etc. For the natural numbers, again, a system of axioms has been laid down by Peano, in which the undefined terms are "number," "zero" and "successor," and a proof may again be demanded of their compatibility. But now there is no simpler branch of mathe matics in which to interpret them; for with natural numbers we seem to have reached the most primitive mathematical material.