THE PHILOSOPHY OF MATHEMATICS 235 



a lack of penetration as to what the authors quoted are really 

 driving at is a defect of the whole work; and even when the 

 passages quoted are wrong, they are generally at least as 

 satisfactory as the criticisms passed on them. 



I agree, however, with the main conclusion which seems 

 to follow from the critical examinations in the book — namely 

 that a thorough reconstruction of the philosophy of mathematics 

 was badly wanted at the end of the last century. The book 

 under review cannot by any stretch of imagination be said to 

 have supplied it. A slight sketch of the problem to be solved 

 will show how the doctrine of the variable has enabled a 

 satisfactory philosophy to be developed. 



In any consideration of the principles of mathematics two 

 distinct subjects should be kept separate — namely (i) the nature 

 of mathematical propositions considered in themselves apart 

 from any admixture of particular application and (2) the discus- 

 sion of the groups of particular facts which are special cases of 

 mathematical truths. These two subjects are respectively the 

 problem of the nature of pure mathematics, and that of the 

 applications of mathematics. That the addition of two and two 

 make four is a mathematical theorem ; that the dropping of two 

 apples into an empty basket and then of two more apples, with 

 none taken out, will leave four apples in the basket is a theorem 

 of applied mathematics. This distinction in its grosser forms 

 has been known and recognised for centuries, probably ever 

 since mathematics has been seriously studied. The recent 

 progress in the philosophy of mathematics has shown that it 

 cuts deeper and is of wider import than our predecessors 

 imagined. 



The problem of disengaging mathematics from its applica- 

 tions is not so easy as it looks. In popular thought it has been 

 done most completely in the arithmetic of integers, and with a 

 decreasing measure of success in the case of fractions, negative 

 numbers, real numbers and complex numbers. Geometry has 

 been the field of prolonged controversy over this point. The 

 necessity for a solution of the problem becomes urgent when 

 it is found that practically identical types of reasoning occur 

 in connection with widely different material. The mathe- 

 matician then seeks to disengage the abstract train of reasoning 

 between hypothesis and conclusion from the variety of materials 

 to which it can be applied. A fable will best illustrate the 



