NECESSARY TRUTHS— EFFECTS OF EXPERIENCE. 655 



equal to each other ; tliose which the same thing refuses to 

 arouse are those which we deem unequaL 



This is as clear a restatement as I can make of Mill's 

 doctrine.* And its failure is written upon its front. Woe to 

 arithmetic, were such the only grounds for its validity! 

 The same real things are countable in numberless ways, 

 and pass from one numerical form, not only to its equiva- 

 lent (as Mill implies), but to its other, as the sport of 

 physical accidents or of our mode of attending may de- 

 cide. How could our notion that one and one are eternally 

 and necessarily two ever maintain itself in a world where 

 every time we add one drop of water to another we get not 

 two but one again ? in a world where every time we add a 

 drop to a crumb of quicklime we get a dozen or more ? — 

 had it no better warrant than such experiences ? At most 

 we could then say that one and one are usually two. 

 Our arithmetical propositions would never have the con- 

 fident tone which they now possess. That confident 

 -tone is due to the fact that they deal with abstract and ideal 

 numbers exclusively. Wliat we mean by one plus one 

 IS two ; we make two out of it ; and it would mean two 

 still even in a world where physically (according to a 

 <5onceit of Mill's) a third thing was engendered every time 

 one thing came together with another. We are mas- 

 ters of our meanings, and discriminate between the things 

 Mve mean and our waj^s of taking them, between our strokes 

 of numeration themselves, and our bundlings and separat- 

 ings thereof. 



Mill ought not only to have said, " All things are uum- 

 "bered." He ought, in order to jorove his point, to have 

 shown that they are unequivocally numbered, which they no- 

 toriously are not. Only the abstract numbers themselves are 

 unequivocal, only those which we create mentally and hold 

 fast to as ideal objects always the same. A concrete natural 

 thing can always be numbered in a great variety of ways. 

 *' We need only conceive a thing di^dded into four equal 

 parts (and all things may be conceived as so divided)," as 



*For the original statements, cf. J. S. Mill's Logic, bk. ii. chap, vl 

 §§3, 3 ; and bk. iii. chap. xxiv. § 5. 



