NECESSARY TRUTHS— EFFECTS OF EXPERIENCE. 653 



ositions. These latter we shall consider farther on. As 

 regards the mathematical judgments, they are all * rational 

 propositions ' in the sense defined on p. 644, for they express 

 results of comparison and nothing more. The mathemati- 

 cal sciences deal with similarities and equalities exclusively, 

 and not with coexistences and sequences. Hence they have, 

 in the first instance, no connection with the order of ex- 

 perience. The comparisons of mathematics are between 

 numbers and extensive magnitudes, giving rise to arith- 

 metic and geometry respectively. 



Number seems to signify primarily the strokes of our 

 attention in discriminating things. These strokes remain 

 in the memory in groups, large or small, and the groups can 

 be compared. The discrimination is, as we know, psycho- 

 logically facilitated by the mobility of the thing as a total 

 (p. 173). But within each thing we discriminate parts ; so 

 that the number of things which any one given phenome- 

 non may be depends in the last instance on our way of 

 taking it. A globe is one, if undivided ; two, if composed 

 of hemispheres. A sand-heap is one thing, or twenty 

 thousand things, as we may choose to count it. We amuse 

 ourselves by the counting of mere strokes, to form rhythms, 

 and these we compare and name. Little by little in our 

 minds the number-series is formed. This, like all lists of 

 terms in which there is a direction of serial increase, car- 

 ries with it the sense of those mediate relations between its 

 terms which we expressed by the axiom " the more than the 

 more is more than the less." That axiom seems, in fact, 

 only a way of stating that the terms do form an increasing 

 series. But, in addition to this, we are aware of certain 

 other relations among our strokes of counting. We may 

 interrupt them where we like, and go on again. All the 

 while we feel that the interruption does not alter the strokes 

 themselves. We may count 12 straight through ; or count 

 7 and pause, and then count 5, but still the strokes will be 

 the same. We thus distiuguisli between our acts of count- 

 ing and those of interrupting or grouping, as between an 

 unchanged matter and an operation of mere shufiling per- 

 formed on it. The matter is the original units or strokes; 



