jANtTAEY 24, 1896.] 



SCIENCE. 



135 



group of things on the fingers is merely by as- 

 signing one of the fingers to each one of the 

 things to form a group of fingers which stand in 

 a relation of 'one-to-one correspondence' to the 

 group of things. And counting with numeral 

 words is not a whit more complex. The differ- > 

 ence is only that words instead of fingers are 

 attached to the things counted. But, the order 

 of the words being invariable, the last one used 

 in any act of counting is made to represent the 

 result, for which it serves as well as the group 

 of all that have been used would do. The 

 group of fingers or this final numeral word an- 

 swers as a register of the things by referring to 

 which one may keep account of them as a child 

 does of his marbles or pennies without remem- 

 bering them individually, and this is the sim- 

 plest and most immediate practical purpose that 

 counting serves. 



The number of things in any group of dis- 

 tinct things is simply that property of the 

 group which the group of fingers — or, it may 

 be, of marks or pebbles or numeral words — used 

 in counting it represents, the one property 

 which depends neither on the character of the 

 things, their order nor their grouping, but solely 

 on their distinctness. Gauss said with reason 

 that arithmetic is the pure science par excel- 

 lence. Even geometry and mechanics are 

 mixed sciences in so far as their reality is con- 

 ditioned by the correctness of the postulates 

 they make regarding the external world. But 

 the one postulate of arithmetic is that distinct 

 things exist. It is an immediate consequence 

 of this postulate that the result of counting a 

 group of such things is the same whatever the 

 arrangement or the character of the things, and 

 this is the essence of the number-concept. 



Counting, therefore, is not measuring and 

 number is not ratio. Pure number does not be- 

 long among the metrical, but among the non- 

 metrical mathematical concepts. The number 

 of things in a group is not its measure, but, as 

 Kronecker once said very happily, its ' inva- 

 riant,' being for the group in relation to all 

 transformations and substitutions what the dis- 

 criminant of a quantic, say, is for the quantic 

 in relation to linear transformations, unchange- 

 able. Nor are the notions of numerical equality 

 and greater and lesser inequality metrical. 



When we say of two groups of things that they 

 are equal numerically, we simply mean that for 

 each thing in the second there is one in the first 

 and for each thing in the first there is one in the 

 second, in other words that the groups may be 

 brought into a relation of one-to-one corre- 

 spondence, so that either one of them might be 

 taken instead of a group of fingers to represent 

 the other numerically. And when we say that 

 a first group is greater numerically than a second, 

 or that the second is less than the first, we mean 

 that for each thing in the second there is one in 

 the first, but not reciprocally one thing in the 

 second for each in the first. Instead of comparing 

 the groups directly we may count them sepa- 

 rately on the fingers, and by a comparison of the 

 results obtain the finger representation of the 

 numerical excess of the one group over the 

 other in case they are unequal. And this is all 

 that is meant when we say that by counting we 

 determine which of two groups is the larger 

 and by how much. 



It is therefore obvious, as for that matter our 

 authors themselves urge, that the rational 

 method of teaching a child the smaller numbers 

 is by presenting to him their most complete 

 symbols, corresponding groups of some one 

 kind of thing as blocks, marbles or dots. By 

 such aids he may be taught, with as great sound- 

 ness as concreteness, not only the numbers them- 

 selves and their simple relations, but the mean- 

 ing of addition, subtraction, multiplication and 

 division of integers and the ' laws ' which char- 

 acterize these operations. This accomplished, 

 he is ready to be taught notation and the addition 

 and multiplication tables and to be practised on 

 them until he has attained the art of quick and 

 accurate reckoning. ' Measuring with unde- 

 fined units ' is a fiction with which there is no 

 need to trouble him. For in however loose a 

 sense the word may be used, ' measuring ' at 

 least involves the conscious use of a unit of 

 reference. But no one ever did or ever will 

 count a group of horses, for instance, by first 

 conceiving of an artificial unit horse and then 

 matching it with each actual horse in turn — 

 which 'measuring' the group of horses must 

 mean if it means anything. A conception of 

 ' three ' which makes ' three horses ' mean in 

 the last analysis ' three times a fictitious unit 



