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82 A. E. TAYLOR : Mr. Schiller's implied disapproval of my supposed zeal for con- troversy is however, of course, a very trivial and merely personal affair. My real concern is with the rest of his complaint. " Even he," he goes on, " prudently refrains from trying to illustrate how ' between two doctrines which are, so far as their consequences in practice are concerned, indistinguishable, there may yet be all the difference between proved truth and demonstrable contradiction'." To convince Mr. Schiller that the paucity of illustrations was due more to regard for the editor's space than to any doubt as to the existence of such illustrations, I take this opportunity of offering an example or two of what I meant. It is, as is well known, a disputed point in arithmetical theory -whether the concept of the cardinal number of a group of objects is logically derivative from that of their serial order, and consequently a result of the operation of counting, as in the doctrine of Helm- holtz and Dadekind (the theorie empirique of M. Couturat), or logic- ally independent both of the notion of order and of the operation of counting (Couturat's theorie rationaliste). Now, inasmuch as in all the applications of numerical theory to practical measurement and computation the numbers with which we have to work are always finite, and in the case of finite numbers, owing to the so- called law of the " invariance of number," there is always a one-to- one correspondence between the cardinal number of a group and the ordinal number of its last term, the practical J consequences of the two theories are indistinguishable. It is only when we come to deal with those objects of pure " theory," the transfinite cardinals and ordinals, for which this correspondence no longer holds good, that the marked logical advantages of the last-named doctrine be- come visible. Yet it may fairly be claimed that both theories can- not possibly be true, and, moreover, that neither theory is, as on Mr. Schiller's premisses it should be, " meaningless," while many students of M. Couturat's work, De I'Infini MatJiematique, would, I suspect, not refuse to call one of them "proved truth," and the other "demonstrable contradiction". My second example shall be the pair of contradictory propositions, " the hundredth digit to the right of the decimal point in the ex- pression of TT in the denay scale of notation is a 9," " the hundredth digit to the right of the decimal point in the said expression is not a 9 ". No one, I imagine, will pretend that it is ever necessary, or even advisable, in the practical applications of geometry to make writer. What Prof. James says of Helmholtz (Principles of Psychology, ii., p. 278), that " his genius moves most securely when it keeps close to particular facts," whereas his more speculative views " in spite of many beauties " are " vacillating and obscure," seems to me no less true of Prof. James himself. At any rate, I do not see that what Prof. James may say without offence of Helmholtz becomes a reason for indignation when said by some one else of Prof. James. 1 As always, by practice, I understand the origination by individuals of changes in the temporal order of events, and by practical consequences, consequences consisting in such changes.