2o 4 SCIENTIFIC METHOD IN PHILOSOPHY 



shall say that the two collections are "similar." We 

 have just seen that two similar collections have the same 

 number of terms. This leads us to define the number 

 of a given collection as the class of all collections that are 

 similar to it ; that is to say, we set up the following 

 formal definition : 



" The number of terms in a given class ' is defined as 

 meaning " the class of all classes that are similar to the 

 given class." 



This definition, as Frege (expressing it in slightly 

 different terms) showed, yields the usual arithmetical pro- 

 perties of numbers. It is applicable equally to finite and 

 infinite numbers, and it does not require the admission 

 of some new and mysterious set of metaphysical entities. 

 It shows that it is not physical objects, but classes or the 

 general terms by which they are defined, of which 

 numbers can be asserted ; and it applies to o and i with- 

 out any of the difficulties which other theories find in 

 dealing with these two special cases. 



The above definition is sure to produce, at first sight, 

 a feeling of oddity, which is liable to cause a certain dis- 

 satisfaction. It defines the number 2, for instance, as the 

 class of all couples, and the number 3 as the class of all 

 triads. This does not seem to be what we have hitherto 

 been meaning when we spoke of 2 and 3, though it 

 would be difficult to say what we had been meaning. 

 The answer to a feeling cannot be a logical argument, 

 but nevertheless the answer in this case is not without 

 importance. In the first place, it will be found that when 

 an idea which has grown familiar as an unanalysed whole 

 is first resolved accurately into its component parts 

 which is what we do when we define it there is almost 

 always a feeling of unfamiliarity produced by the analysis, 

 which tends to cause a protest against the definition. In 



