THE PHILOSOPHY OF MATHEMATICS 237 



and even now in many cases, mathematicians who considered 

 this question were driven to an extreme formalism. They laid 

 the emphasis on the rules adopted conventionally to produce 

 equivalent collocations of symbols. This formalism was always 

 — and rightly — severely criticised by those who were philo- 

 sophers first and mathematicians afterwards, if at all. But 

 perhaps such critics did not alwa3'S understand that an important 

 problem remained which the mathematicians were endeavouring 

 to solve. 



But meanwhile a notable discovery has been made by the 

 joint and partially independent work of three men : Frege, 

 Peano and Bertrand Russell — a German, an Italian and an 

 Englishman — by which a flood of light has been let in upon 

 the whole question, so that the problem has received its solu- 

 tion in all essentials. The discovery is that of the generalised 

 conception of the variable and of its essential presence in all 

 mathematical reasoning. This discovery empties mathematics 

 of everything but its logic. For the future mathematics is logic, 

 whereas according to the old formalism mathematics is logic 

 plus conventions as to marks, and according to the older 

 tradition mathematics is logic applied to the domains of number, 

 quantity and space. 



A very cursory examination of one of the many ways in 

 which the abstract science of Geometry can be presented will 

 form the best explanation of the position of the doctrine of the 

 variable in the philosophy of mathematics. Instead of thinking 

 of the class of straight lines of actual space, we start by 

 considering any class k whose members are also classes. We 

 may, if we like, call the members of k the straight lines ; but 

 this is a detail of nomenclature and does not alter the fact that 

 K is any class whose members are classes. 



Then, in the place of the old axioms of Geometry which 

 were stated as true of physical space, certain propositions 

 involving /c are considered^ not stated as true but merely 

 enunciated for inspection. In this new sense we will call them 

 the " axioms," as a convenient short title. For example three 

 such " axioms " are : 



" If X is a member of /c, A, is a class with at least three 

 members;" 



" If X and /x are distinct members of k, A, and ^l cannot 

 possess more than one common member, if any ; " 



16 



