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 LOUIS COUTURAT, De I'Infini Mathematique. 117 necessity for measurement seems at best doubtful. Temperature, for example, has a purely fictitious zero, and yet is measured by the thermometer. The fact is, I think, that quantitative zero is a limit necessarily arising out of the infinite divisibility of extensive quantities, and that M. Couturat's axioms are only applicable to extensive quantities. In this way division into units is surrepti- tiously introduced ; but by immediately declaring every axiom in turn a priori, without analysis of its grounds, the necessity of reference to number is concealed. 1 There are several other points in this book which call for criticism. On p. 416, the axiom of continuity is given in the form : "If all the quantities of a kind can be divided into two classes, such that all the quantities of the one are smaller (or larger) than all the quantities of the other, there exists a quantity of this kind which represents this mode of division, and which is at once larger than all the quantities of the inferior class, and smaller than all the quantities of the superior class ". Against this definition a dilemma may be presented : cither the word all is not to be taken as including the quantity which lies between the two classes, in which case the definition applies equally to discreta, e.g., the series of natural numbers ; or the word all is to be taken rigidly, in which case there is a palpable contradiction in supposing another quantity of the same kind, which does not belong to either of the classes into w 7 hich all the quantities of the kind have been divided. That such a contra- diction must necessarily hold of continua, I have no wish to deny ; but it seems bold to say, of an axiom involving this con- tradiction, that though indemonstrable, " it possesses an evidence almost equal to that of analytical judgments " (p. 416), and that its justification is not logical but rational (p. 554). Again, in discussing the definition of number as the ratio of quantities of the same kind, he defends the definition from the vicious circle which it appears to involve, by declaring that ratio is a funda- mental idea not susceptible of definition (p. 426). In view of the obvious method of defining ratio by means of number, this almost reminds one of Mill's "final inexplicability ". Finally he says that incommensurable quantities must have a ratio, for otherwise two continuous variables would have a ratio one moment and none the next (p. 428), and that to identify unity with the quantitative unit is impossible, since it either supposes unity divisible or the unit indivisible. He does not seem to realise that the possibility of continuous variation, and the impossibility of indivisible units, are the very points which a champion of quantity, as against number, has to establish. 1 On one of these axioms, the axiom that of two magnitudes of the same kind one must be the larger, there is a rather serious inconsist- ency ; it is denied as regards imaginaries on p. 46, and affirmed to be a priori on p. 384.