i 4 4 SCIENCE PROGRESS 



There is also an interest in combining with the discussion of these definitions an 

 attempt to understand how the notion of number arose. Mr. Russell does not 

 deal directly with the origin of numbers ; but, though his immediate concern is 

 with logical definition, his discussion has obvious points of contact with the mental 

 route that led men to conceive and employ numbers. It will, therefore, be 

 convenient and profitable to begin with Francis Galton's description of the 

 Damara who was bartering sticks of tobacco for sheep. 



The rate of exchange was two sticks of tobacco for each sheep. A sheep was 

 driven out and two sticks of tobacco laid down ; then a second sheep was sent to 

 join the first, a third to join them, and so on. Two sticks were tendered in payment 

 as each sheep joined the group. In the phraseology of Bertrand Russell and his 

 confreres, the Damara established a one-one relation between a collection of sheep 

 and a collection of sticks of tobacco in which two sticks composed one item : to each 

 sheep two sticks corresponded. Counting would have been quicker, but would 

 have arrived at the same result that the bargain had been rightly concluded. 

 " In actual fact," writes Mr. Russell, "it is simpler logically to find out whether 

 two collections have the same number of terms than it is to define what that 

 number is. An illustration will make this clear. If there were no polygamy or 

 polyandry in the world, it is clear that the number of husbands living at any 

 moment would be exactly the same as the number of wives. We do not need a 

 census to assure us of this. . . . The relation of husband to wife is what is 

 called one-one." The Damara's method of bargaining, with its plentiful parallels 

 in primitive trading, shows that it is not only "simpler logically," but more obvious 

 in practice to realise that two collections have the same number of terms than to 

 assign a number to the collection, or realise that there are such things as numbers. 

 The Damara takes us one step into the tail of Mr. Russell's first definition by 

 introducing to us the conception of similar classes. There was a one-one relation 

 between the sheep and the tobacco-stick couples : every sheep was correlated 

 with one pair and every pair with one sheep. " Two classes are said to be 

 'similar' when there is a one-one relation which correlates the terms of one class 

 each with one term of the other class." Under perfect monogamy husbands and 

 wives are similar classes, and so were the sheep bought by the Damara and the 

 pairs of sticks of tobacco he paid for them. This similarity of classes is a wider 

 conception than number, for it can be used without numbering or without the 

 ability to number. It is also more fundamental than number, for it is easily 

 perceived that, in counting a collection, a one-one relation is established between 



the items and the numbers i, 2, 3, 4 n. If the Damara had counted his 



sheep and his sticks of tobacco he would still have been busy with one-one 

 relations and similar classes. For simplification of study, suppose him to halve his 

 collection of tobacco sticks by making each pair into one : one stick is now the 

 price of one sheep. By counting ten sheep he establishes similarity between 

 the collection and the numbers 1 to 10 ; by counting ten sticks of 1 tobacco he 

 establishes similarity between the same set of numbers and the tobacco collection. 

 Since the sheep collection and the tobacco collection are similar to the same 

 collection of numbers they are similar to one another, and the Damara knows, or 

 would know if he could count, that his bargain has been rightly concluded. Thus, 

 numbering appears to be essentially an extension of the more fundamental 

 similarity of classes, and numbers to be a vast collection from which standard 

 classes can be selected to place in relations of similarity with other classes. 



There was, it is reasonable to suppose, a process of transition between the 

 method of the Damara and the use of numbers proper. Digital enumeration, if 



