116 CRITICAL NOTICES : pone the criticism of this view till we have discussed book ii., which deals with quantity. The main contention of book ii. is, that quantity is as a priori as number, but is an entirely independent category. The idea of quantity, he says, is indefinable and irreducible : to define it as what can be increased or diminished involves an obvious circle (pp. 367-9). If quantity be indeed an independent category, we must of course agree that it cannot be defined in terms of other categories : but one could wish that M. Couturat had made more attempt to show that it is a category free from contradictions, and that there is some way of knowing about it without the help of number. Measurement, he admits, consists in the numerical expression of quantity, and thus introduces axioms due to the nature of number, and producing a conflict between number and quantity (pp. 404-5). But he thinks that the equality and addi- tion of quantities of the same kind can be effected without refer- ence to division into units, and therefore without dependence on number (p. 404). Chapters i. and ii. of this book give the axioms of equality and addition, which are declared a priori, and are stated in a form apparently free from reference to number. But they seem very insufficient for the foundation of a science of quantity. Thus he says, for example, that equality in general cannot be defined, but as soon as we have the idea of any kind of quantity, we have the idea of equal quantities in this kind, or rather of the same quantity in different objects (pp. 372-3). This seems vague, and it might be objected that we cannot have the idea of a kind of quantity until we know what we mean by equal quantities of this kind, since it is equality and inequality which constitute quantitative relations. Again he says (p. 389) that the sum of two quantities is a quantity of the same kind. If this axiom were really independent of division into units, it would hold equally of intensive quantities, but this is not the case. The sum of two temperatures, for example, is meaningless. He has also what he calls the axiom of the modulus (p. 399), according to which there exists a certain real quantity, called zero, such that, when added to any other quantity, it leaves that quantity un- changed. Having been told previously (p. 232) that a zero dis- tance is not nothing, but just as real as any other distance, we are not surprised at this axiom ; but it is a pity that no attempt is made to explain what a zero quantity is. Zero would seem to be about as non-existent as anything could be ; for it is defined as nothing but quantity, and further as containing no quantity. We are not told how to make something of this apparent non- entity ; and it is even supposed that objections to infinity will be silenced by the argument that the same objections apply to zero (p. 436). Zero is said to have a rational, not a logical, necessity (p. 402) ; without zero, it is said, measurement would be impossible. It seems strange to assign rational necessity to anything so grossly contradictory as mathematical zero, and its
