Page:Mind (New Series) Volume 6.djvu/129

LOUIS COUTUBAT, De Vlnfini Mathematique. 113 number. Infinite quantity, he urges, is given a priori, and does not stand or fall with infinite number. To maintain this thesis it is necessary to establish the independence of quantity, and this independence is, in fact if not in form, the chief theme of his work. The work is divided into two parts, of which the first, on the generalisation of number, is content to exhibit and analyse those results of mathematical science which bear on the question at issue ; while the second part, on number and quantity, adopts a critical and philosophical attitude, and endeavours to establish, by philosophical considerations, the legitimacy of the notions of in- finity and the continuum. Whichever may be the most valuable of these two parts, the second is the most interesting to the student of philosophy, and will be the more fully discussed here. . The first part is divided into four books : the first three deal with the generalisation of number as it appears in arithmetic, algebra and geometry respectively, while the fourth deals with mathematical infinity. The aim of the first three books is to ex- hibit the growing necessity and the diminishing conventionality of this generalisation in the three sciences in question. In arith- metic, whose primary subject-matter is positive integers alone, fractions, negative numbers and imaginary numbers can only be introduced in an arbitrary and conventional manner. We take two integers in a given order, regarded as forming a couple with certain arithmetical properties, and establish arbitrary definitions of the equality, addition, and multiplication of two such couples. According to the definitions chosen, one of the three kinds of generalised number results. 1 We cannot arithmetically introduce these generalisations in the ordinary way, for unity, in pure number, is indivisible, and such an expression as has no meaning except what we choose to assign to it. The arithmetical generalisation is, therefore, of necessity an arbitrary and apparently motiveless process, giving rise to symbols which are without arithmetical meaning, and are only subject by convention to arithmetical operations. Irrational numbers are still more difficult to introduce arithmeti- cally, and they first involve infinity. They express, in arithmetic, the mere absence of a number of any of the former kinds at certain points in the scale, and arise from the possibility of dividing all rational numbers, in an infinite number of ways, into two classes, such that any member of the first class is smaller than any member of the second, but no assignable member is the largest or smallest of the respective classes. There is thus a gap, which can only be filled in by a symbol expressing the absence of a rational number at the point in question. The algebraical generalisation, as M. Couturat calls it, is less formal, and proceeds from the desire to be always able to assign 1 For example, fractions are defined by the following conventions : (a, b) = (c, rf) if ad = be ; (a, b) + (c, d) = (ad + be, bd) ; (a, 6) x (c, d) = (ac, bd). 8