MATHEMATICS AND METAPHYSICIANS 93 



sibly be shown, by very careful measurements, that 

 Euclid s axioms are false ; but no measurements could 

 ever assure us (owing to the errors of observation) that 

 they are exactly true. Thus the geometer leaves to the 

 man of science to decide, as best he may, what axioms are 

 most nearly true in the actual world. The geometer 

 takes any set of axioms that seem interesting, and 

 deduces their consequences. What defines Geometry, 

 in this sense, is that the axioms must give rise to a series 

 of more than one dimension . And it is thus that Geometry 

 becomes a department in the study of order. 



In Geometry, as in other parts of mathematics, Peano 

 and his disciples have done work of the very greatest 

 merit as regards principles. Formerly, it was held by 

 philosophers and mathematicians alike that the proofs in 

 Geometry depended on the figure ; nowadays, this is 

 known to be false. In the best books there are no figures 

 at all. The reasoning proceeds by the strict rules of 

 formal logic from a set of axioms laid down to begin with. 

 If a figure is used, all sorts of things seem obviously to 

 follow, which no formal reasoning can prove from the 

 explicit axioms, and which, as a matter of fact, are only 

 accepted because they are obvious. By banishing the 

 figure, it becomes possible to discover all the axioms that 

 are needed ; and in this way all sorts of possibilities, 

 which would have otherwise remained undetected, are 

 brought to light. 



One great advance, from the point of view of correct 

 ness, has been made by introducing points as they are 

 required, and not starting, as was formerly done, by 

 assuming the whole of space. This method is due partly 

 to Peano, partly to another Italian named Fano. To 

 those unaccustomed to it, it has an air of somewhat 

 wilful pedantry. In this way, we begin with the following 



