Febbuaey 21, 1896.] 



SCIENCE. 



287 



•erence. But no one ever did or ever will count 

 a group of liorses, for instance, by first conceiv- 

 ing of an artificial * unit horse and then match- 

 ing it with each actual horse in turn — which 

 'measuring' the group of horses must mean if it 

 means anything. ' ' 



The whole point here is under what circum- 

 stances does one, not a mathematician or for 

 mathematical purposes, count a group of horses. 

 The answer is something of the following sort, it 

 seems to me : One counts when one wishes to 

 find out how many horses he has caught in a 

 day's hunt, whether the same number has been 

 driven back at night that were taken out in the 

 morning; how much money is to be got in sell- 

 ing them, it having been settled that each horse 

 is to fetch the same sum, etc. , etc. ; how one ranks 

 as a chieftain, or a soldier, compared with others, 

 etc., etc. In other words, one not having arrived 

 at the abstract interest of the mathematician (and 

 certainly the child to be educated has not) 

 counts only when there is some value to be as- 

 certained, and counts by setting off something 

 which, for present purposes is a sample unit of 

 value, e. g., a horse, then equating the total 

 value to the number of such units. Taking the 

 matter in its development then, (and not at the 

 stage of the mathematician when abstracts have 

 already become concretes) enumeration is al- 

 ways to define value, i. e. , to measure. 



If the book referred to did not recognize the 

 distinction between this sort of measuring and 

 the technical sort it should certainly be con- 

 demned. But one of the points emphasized is 

 that the former is an imperfect sort of measure- 

 ment; that we don't really know, e. g., what 

 the possession of 60 horses amounts to till we 

 know what one horse is worth, and so measur- 

 ing proper (measuring with measured units) is 

 substituted for mere counting, i. e., measuring 

 with undefined units of value. 



2. It is said that number is not ratio. If one 



* Whence and wherefore this artificial ? The point 

 to be proved involves nothing about an ' artificial' 

 unit, but only a unit of reference, and that surely a 

 horse is. But even if the term were relevant in the 

 argument the question would arise whether the use 

 of an artificial unit or of a measured unit is the es- 

 sence of technical measurement ; whether, indeed, a 

 foot is, psychologically, more artificial than a horse. 



is using ratio to denote a certain idea, and not 

 a technical abstraction of the mathematicians, I 

 do not see how this statement is to be reconciled 

 with Prof. Fine's own account of enumeration; 

 ' f To count a group of things on the fingers is 

 merely by assigning 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 corre- 

 spondence to the group of things.' " * And again, 

 "When we say of two groups of things that 

 they are equal numerically, we simply mean 

 that for each 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 cor- 

 respondence. ' ' What does the phrase italicized 

 mean, save the idea of ratio ? If this way of 

 stating it had only been known to me when the 

 book reviewed was written, I should gladly 

 have utilized it to indicate precisely the point 

 we were trying to make — the implicit presence 

 of the ratio idea in every number. 



Psychologically there is, of course, a differ- 

 ence in the mental attitude in recognizing a 

 thing as 'one,' as unity, as a whole, an indi- 

 vidual, and recognizing it as 'a one,' a unit. 

 The primary problem the educator has to face, 

 if he is to rationalize the teaching of arithmetic, 

 is the discovery of this difference. The answer 

 given is that ' one ' (qualitative individuality or 

 unity) becomes ' a one, ' a unit when it is 

 used to measure value; and that, in turn, the 

 need for this use arises when the thing is no 

 longer taken as an adequate end, but as a means 

 to be adjusted to some further end. E. g., 

 once more, when a man is wholly occupied in 

 riding or hunting, or feeding a horse, when 

 that absorbs his whole interest, he never takes 

 the numerical view; when he wants to know 

 how much of a horse owner he is, and how far 

 this horse contributes to that end, he neces- 

 sarily takes it. The question then is whether 

 ' one ' ever becomes ' a one, ' save as it is put 

 into a ' relation of one-to-one correspondence ? ' 



3. Prof. Fine remarks that ' the one postulate 

 of arithmetic is that distinct things exist. ' The 

 mathematician may perhaps be reminded that 

 this postulate is precisely one of the chief prob- 

 lems of the psychologist. Given a certain num- 



* Italics mine. 



