The Principles of Mathematics (1903)

§ 149

Among the traditional problems of mathematical philosophy, few are more important than the relation of quantity to number. Opinion as to this relation has undergone many revolutions. Euclid, as is evident from his definitions of ratio and proportion, and indeed from his whole procedure, was not persuaded of the applicability of numbers to spatial magnitudes. When Des Cartes and Vieta, by the introduction of co-ordinate Geometry, made this applicability a fundamental postulate of their systems, a new method was founded, which, however fruitful of results, involved, like most mathematical advances of the seventeenth century, a diminution of logical precision and a loss in subtlety of distinction. What was meant by measurement, and whether all spatial magnitudes were subsceptible of a numerical measure, were questions for whose decision, until very lately, the necessary mathematical instrument was lacking; and even now much remains to be done before a complete answer can be given. The view prevailed that number and quantity were the objects of mathematical investigation, and that the two were so similar as not to require careful separation. Thus number was applied to quantities without any hesitation, and conversely, where existing numbers were found inadequate to measurement, new ones were created on the sole ground that every quantity must have a numerical measure.(§ 149 ¶ 1)

All this is now happily changed. Two different lines of argument, conducted in the main by different men, have laid the foundations both for large generalizations, and for thorough accuracy in detail. On the one hand, Weierstrass, Dedekind, Cantor, and their followers, have pointed out that, if irrational numbers are to be significantly employed as measures of quantitative fractions, they must be defined without reference to quantity; and the same men who showed the necessity of such a definition have supplied the want which they had created. In this way, during the last thirty or forty years, a new subject, which has added quite immeasurably to theoretical correctness, has been created, which may legitimately be called Arithmetic; for, starting with integers, it succeeds in defining whatever else it requires—rationals, limits, irrationals, continuity, and so on. It results that, for all Algebra and Analysis, it is unnecessary to assume any material beyond the integers, which, as we have seen, can themselves be defined in logical terms. It is this science, far more than non-Euclidean Geometry, that is really fatal to the Kantian theory of à priori intuitions as the basis of mathematics. Continuity and irrationals were formerly the strongholds of the school who may be called the intuitionists, but these strongholds are theirs no longer. Arithmetic has grown so as to include all that can strictly be called pure in the traditional mathematics.(§ 149 ¶ 2)