least as much in need of it. As regards this conception, M. Couturat points out that it is not given by sensible experience, since sensation can never reach an accuracy sufficient to exclude discreteness. Thus irrationals suffice to prove that quantity is not empirical; it is in fact, he says, a priori and due to reason. With the justification of quantity comes the justification of infinity, which reason, he says, does not obtain by successive synthesis, but sees from the start (p. 565). From this conclusion, he goes on to Kant's antinomies, whose theses, since the realised infinite is not impossible, he declares to be all false (p. 567). The antitheses, therefore, he regards as true, and Kant's objection to them, he says, is due to the schematism of the categories, which led him to regard quantitative synthesis as necessarily successive, while every quantity, in fact, is given first as a whole, not as a synthesis of parts.
It seems to me that this argument, as well as the whole argument against the "finitists " throughout the w T ork, rests on a misapprehension of their position. That infinity follows necessarily from certain premisses—e.g., the reality of space and time as something more than relations—must be admitted; that infinity is useful and unobjectionable in mathematics, is by this time almost self-evident; but that mathematical infinity is philosophically valid might, I imagine, be met by two converging lines of argument. The first and more usual argument would urge the contradictions of infinity, which M. Couturat, I think, has not succeeded in disproving; the second might urge that, in all the cases where infinity is unavoidable, there has been some undue hypostatising of relations, which makes the attainment of a completed substantive whole impossible. These lines of argument, however, can be only suggested within the space of a review.
Finally, I wish to urge the very solid merits of the work, to which it is difficult, in the course of detailed criticism, to do justice. The position taken up is the only one from which infinite quantity can be philosophically defended, and the main thesis is carefully and consistently worked out. To take up an unpopular cause is always praiseworthy, and almost always useful; in the case of M. Couturat it is certainly both, and his book is likely long to remain the classic advocate of mathematical infinity.
