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the phrase be allowed, probably contained the real germ of number. The notion 

 and use of numbers so short-circuits mental method that their actual function 

 and nature are obscured. This concealment of the nature and function of numbers 

 beneath their effectiveness is one reason why Bertrand Russell's definition seems, 

 at first sight, so cumbersome and so elaborate. An attempt to follow the genesis 

 of the notion and use of numbers enables us to appreciate the definition by a 

 better understanding of what number really is. Digital enumeration is not 

 necessarily counting, and was not counting, in our sense, when it was first used. 

 A class or collection of fingers was observed to be similar to two other classes 

 which were then known to be similar to one another. If three fingers corre- 

 sponded, one by one, to three turnips, and also to three apples, there were as 

 many apples as turnips. We incline to consider this process to be one of 

 numbering because our understanding of the nature of number is confused by our 

 tendency, natural but erroneous, to think of numbers as properties of the col- 

 lections or classes they denote : Locke fell headlong into this error when he 

 described numbers as qualities inseparable from bodies. The natural numbers 

 1, 2, 3, .... , compose a vast collection from which classes can be selected related 

 by similarity to classes that are numbered ; a collection of five apples is similar to 

 the collection i, 2, 3, 4, 5, since there is a one-one relation between the members 

 of the two classes. The number five, used to designate the group of apples, 

 expresses this relation. All classes of five are denoted by this number, be they 

 pennies or cauliflowers or miscellaneous objects, because their items stand in a 

 one-one relation to the numbers from one to five. Five is not the class of turnips, 

 or of whatever items it contains, nor is it a property of that class ; it is a class, the 

 class containing the numbers 1 to 5, to which the class of turnips, or whatever 

 items it contains, is similar, or to which it stands in a one-one relation. All 

 collections of five objects are in a common class because they are all similar to 

 the class 1, 2, 3, 4, 5, a common class that may be defined by the common 

 property of similarity to the first five natural numbers. Similarly, all classes of N 



objects form a common class because they are similar to the class 1, 2, 3, N. 



A number, then, is a standard class, whatever its inmost metaphysical nature may 

 be, and it is used to denote classes that are similar to it. Also numbers are 

 permanent classes, permanent standards : this is one reason why the three fingers 

 used to compare the three apples with the three turnips do not, in the first 

 instance, constitute a number. 



Digital groupings— three fingers, or six fingers, and so on — were at first 

 temporary numbers, so to speak ; they gave rise to real or permanent numbers 

 when they were themselves recognised as examples of those puzzling entities. 

 Familiarity gradually detached particular groupings of fingers, those of three and 

 four for example, from the classes of threes and fours, as standards of reference. 

 The ideas of three and four proceeded from the continuation of this process as the 

 finger groupings became signs of what we call numbers — collections to which the 

 finger groupings were themselves similar. Stones, sticks, etc., might be used as 

 supplemental fingers, or instead of them. By the use of written signs and by the 

 use of names, numbers were firmly established as concepts. One by one the 

 lower numbers settled into the human conceptual system and provided the basis 

 for the formation of the arithmetical series that contains an unlimited supply of 

 numbers. 



The vertical strokes still retained in the Roman Numerals are a remnant of 

 the earliest form of numeration — one stroke for one, two strokes for two, etc. The 

 stroke notation may represent the primary digital enumeration — strokes repre- 



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