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LOUIS COUTUBAT, De Vlnfini Mathematique. 115 appears forced, though peculiarly evident in the case of mathe- matical infinity, is strenuously denied throughout the work. His rational principles, of which the principle of continuity is the chief, are thus introduced more or less arbitrarily, and appear as dogmas rather than genuinely necessary axioms. The scoffer might even be tempted to identify his " reason " with common sense. Of this general criticism, we shall have abundant illustration in the second part of the work, which deals philosophically with the relation between number and quantity, and with the objections to infinite quantity. The first book, on number, begins with an excellent criticism of the empirical theory of number, by which M. Couturat means, not the much-refuted theory of Mill, but the formalist theory of Helmholtz and Kronecker. This theory defines numbers as mere signs with a fixed order of sequence, and en- deavours to deduce the properties of cardinal numbers from this definition of ordinals. Our author insists I think with complete success that we must begin with cardinal numbers, and that the attempts of Helmholtz and Kronecker involve either a vicious circle or a petitio. He then proceeds (book L, chap, iii.) to the rationalist theory of whole numbers one of the very few subjects in epistemology, I suppose, upon which there is at present a con- sensus among experts. Number requires only a logical and formal unit, created by thought : all the objects of a numbered collection must be regarded as units in this sense, and in so far identical, while the whole collection must, in turn, be regarded as a complex unity. The idea of unity is an a priori idea, and is never given by the data of sense. It is doubly involved in number, which is a " unity of a plurality of unities " (p. 361). l In the next chapter, after urging that number requires no schematism, as Kant had maintained for it is not successive enumeration, but simultaneous apprehension, which is needed (p. 354) our author begins to tread on more dangerous ground, by the consideration of infinite number. One would have supposed that the condition of being a completed whole, which he has urged as necessary to number, would have precluded the possibility of infinite number. But M. Couturat boldly contends that a collection is given as a whole, as soon as we have a law by which any required number of its members can be constructed, and from which no member is exempt. The conditions for a number of a collection may, there- fore, be satisfied, even if the collection is infinite, and successive enumeration of all its terms is impossible (p. 351). This is certainly the only hope of saving infinite number from contradic- tion, and M. Couturat has made the most of it. As he deals with the same topic more fully at a later stage (book iii.), I shall post- 1 " Units " would perhaps be a better word, but as there is only one word in French for unit and unity, the distinction has to be made by translation.