Page:Mind (New Series) Volume 6.djvu/130

 114 CRITICAL NOTICES : roots to equations of the first or second degree. Thus equations of the first degree, to be always soluble, require negative numbers and fractions ; equations of the second degree require irrational and imaginary numbers. But the algebraical generalisation is unable to obtain the so-called transcendental numbers (e and TT for instance), which are not roots of any algebraical equation of finite degree. These mark the distinction between quantity and number, and point to the former to justify the generalisation of number. Proceeding to the geometrical generalisation, M. Couturat points out that, even in algebra, there would be no need to demand that equations should always have a root, if these equations were not the statement of real problems arising outside algebra. Such problems occur in geometry, where first we find the true interpretation, as quantities, of the symbols which, from a numerical point of view, appeared to indicate impossible problems. All varieties of number, including transcendental and imaginary numbers, find here their justification and their motive. In all this there is, from one standpoint, nothing to criticise : it is a clear and lucid account of the motives which have led mathe- maticians to abandon the restriction to positive integers which the pure idea of number would seem to impose. When the unit, divisible and possessed of qualities, is substituted for the abstract unity of arithmetic, all the apparatus of mathematical analysis inevitably arises. But the unit itself is not thereby freed from difficulties, and the contradictions latent in the quantitative unit appear to be inadequately realised by our author. Of this we have evidence in his fourth book, on mathematical infinity, where he argues that infinite quantities are actually given, and enforces his contention by considering the intersection of two lines which gradually become parallel. These have no intersection at a finite distance, but they cannot, he says, suddenly cease to have an intersection, for such a breach of continuity, though not logically contradictory, would be irrational (p. 216). This is the first application of a somewhat dubious principle, borrowed from Cournot, and emphatically stated in the preface (p. x.), according to which philosophy has to choose, between several equally logical alternatives, the most rational one. The use of this principle, w r hich is frequent, seems to me not very happy, and indeed may be employed to cloak what is really intellectual capitulation. M. Couturat appears to be an idealist, and on critical occasions appeals to the reason as against the under- standing (see e.ij. pp. 537, 565). But the function of reason, in the opinion of most idealists, is not to pronounce between alternatives which logic leaves undecided, but rather to find a logically possible alternative where the understanding finds none. This view of reason seems never to occur to our author, and the possibility of contradictions in results to which the understanding