156 MYSTICISM AND LOGIC 



that it left the existence of irrationals merely optative, 

 and for this reason the stricter methods of the present 

 day no longer tolerate such a definition. We now define 

 an irrational number as a certain class of ratios, thus 

 constructing it logically by means of ratios, instead of 

 arriving at it by a doubtful inference from them. Take 

 again the case of cardinal numbers. Two equally 

 numerous collections appear to have something in 

 common : this something is supposed to be their car- 

 dinal number. But so long as the cardinal number is 

 inferred from the collections, not constructed in terms 

 of them, its existence must remain in doubt, unless in 

 virtue of a metaphysical postulate ad hoc. By defining 

 the cardinal number of a given collection as the class of 

 all equally numerous collections, we avoid the necessity 

 of this metaphysical postulate, and thereby remove a 

 needless element of doubt from the philosophy of arith- 

 metic. A similar method, as I have shown elsewhere, 

 can be applied to classes themselves, which need not be 

 supposed to have any metaphysical reality, but can be 

 regarded as symbolically constructed fictions. 



The method by which the construction proceeds is 

 closely analogous in these and all similar cases. Given a 

 set of propositions nominally dealing with the supposed 

 inferred entities, we observe the properties which are 

 required of the supposed entities in order to make these 

 propositions true. By dint of a little logical ingenuity, 

 we then construct some logical function of less hypo- 

 thetical entities which has the requisite properties. This 

 constructed function we substitute for the supposed in- 

 ferred entities, and thereby obtain a new and less doubtful 

 interpretation of the body of propositions in question. 

 This method, so fruitful in the philosophy of mathematics, 

 will be found equally applicable in the philosophy of 



