Page:Mind (New Series) Volume 6.djvu/134

 118 CRITICAL NOTICES : The third book consists of an answer to the usual objections to infinity, which is given in the form of a dialogue between a finitist and an infinitist. The proofs that infinite number is impossible, he says, all rest on one of two fallacies : (1) that infinity is the largest of all numbers, and (2) that all infinities are equal. Both these are false, he says, and the difficulties to which they have given rise are all resolved by Cantor's transfinite numbers (p. 455). The actual infinity of the series of natural numbers is not contra- dictory, he says, for it results from their law of formation, by which they are given as a totality. 1 That the resulting infinity may reveal a latent contradiction in the law of formation itself, appears not to occur to him. Thus when the finitist objects that an infinite collection can never be really given or really a whole, the infinitist replies that, in that case, number itself must be contradictory (p. 471) a conclusion from which, one would think, a bold disputant would scarcely shrink. But infinite quantity, he says, is even simpler than infinite number, for it does not neces- sarily consist of a collection of units. It is difficult to see in what sense a quantity is infinite, unless it is compared with some unit, of which an infinite number would be required to reproduce it. But M. Couturat holds that an infinite quantity can be given to begin with as a totality, and that measurement can be effected otherwise than by the addition of elements (p. 483). How the latter is possible, he does not explain ; integration, which he gives as an example, is essentially an addition of elements. The possi- bility of the former, one would think, is sufficiently disproved by the stock argument, that mathematical infinity consists essentially in the absence of totality, the absence of completed synthesis. But infinite quantity, as M. Couturat says, depends on continua, and these, like most of the fundamentals in his work, he regards as given by reason, not by logic (pp. 497-8). For one equipped only with logic, it is impossible to follow into this fortress of reason, where the shafts of logic cannot penetrate. In the last book, conclusions are drawn from the previous dis- cussions. Beginning with number, it is pointed out that number, like the concept, depends on generalisation and abstraction ; it is, indeed, the other face of the concept, the extension corresponding to a given intension. But since it requires that its constituents should each be united, and all be similar, its application to reality is never wholly legitimate, for similarity and unity seem to exist in inverse ratio. The latter is best found in organisms, the former in homogeneous continua. All this is excellent, like what was said on number in book ii. ; but it is a pity that a similar process of criticism is not applied to continuous quantity, which stands at 1 There is some inconsistency on this point. Sometimes it is argued that all the natural numbers are given by successive addition of unity, sometimes, with Cantor, that another principle of formation is also re- quired. Compare, e.g., pp. 364, 465.