92 MYSTICISM AND LOGIC 



The notion of a limit, which is fundamental in the greater 

 part of higher mathematics, used to be defined by means 

 of quantity, as a term to which the terms of some series 

 approximate as nearly as we please. But nowadays the 

 limit is denned quite differently, and the series which it 

 limits may not approximate to it at all. This improve 

 ment also is due to Cantor, and it is one which has 

 revolutionised mathematics. Only order is now relevant 

 to limits. Thus, for instance, the smallest of the infinite 

 integers is the limit of the finite integers, though all 

 finite integers are at an infinite distance from it. The 

 study of different types of series is a general subject of 

 which the study of ordinal numbers (mentioned above) is 

 a special and very interesting branch. But the unavoid 

 able technicalities of this subject render it impossible to 

 explain to any but professed mathematicians. 



Geometry, like Arithmetic, has been subsumed, in 

 recent times, under the general study of order. It was 

 formerly supposed that Geometry was the study of the 

 nature of the space in which we live, and accordingly it 

 was urged, by those who held that what exists can only 

 be known empirically, that Geometry should really be 

 regarded as belonging to applied mathematics. But it 

 has gradually appeared, by the increase of non-Euclidean 

 systems, that Geometry throws no more light upon the 

 nature of space than Arithmetic throws upon the popula 

 tion of the United States. Geometry is a whole collection 

 of deductive sciences based on a corresponding collection 

 of sets of axioms. One set of axioms is Euclid s ; other 

 equally good sets of axioms lead to other results. Whether 

 Euclid s axioms are true, is a question as to which the 

 the pure mathematician is indifferent ; and, what is more, 

 it is a question which it is theoretically impossible to 

 answer with certainty in the affirmative. It might pos- 



