740 



SCIENTIFIC THOUGHT. 



precision and consistency into the foundations of mathe- 

 matics, and everywhere to further the very necessary 

 process of critical sifting; but he denies that pure 

 logic can do all, and points to the valuable assistance 

 and suggestive power of geometrical construction and 

 representation. 1 Most of my readers will no doubt 

 agree with this view. Indeed the perusal of the fore- 

 going chapters must have produced on their minds the 

 conviction that, so far as the advance of science and also 

 of mathematics is concerned, it largely depends upon 

 the introduction of different aspects leading to different 

 courses of reasoning. The unification of all of these 

 into one consistent and uncontradictory scheme, though 

 it remains a pious hope and far-off ideal, has not been 

 the prominent work of the nineteenth century. Eather, 

 wherever it has been attempted it has had a narrowing 

 effect, and has resulted in a distinct curtailment of the 

 great and increasing resources of Scientific Thought. 



1 Prof. Klein summarises the 

 opinion which he holds as to the 

 present task of mathematical 

 science as follows : " Whilst I 

 everywhere demand the fullest 

 logical elaboration, I at the same 

 time emphasise that pari passu 

 with it the intuitive representation 

 of the subject should be furthered 

 in every possible . manner. Mathe- 

 matical developments which have 

 their origin in intuition cannot 

 count as a firm possession of science 

 unless they have been reduced to a 

 strict logical form. On the other 

 side, the abstract statement of 

 logical relations cannot satisfy us 

 until their importance for every 



form of representation has been 

 clearly demonstrated, so that we 

 recognise the manifold connections 

 in which the logical scheme stands 

 to other departments of knowledge 

 according to the field of application 

 which we select. I compare mathe- 

 matical science to a tree which 

 stretches its roots ever deeper into 

 the soil, and at the same time 

 expands its branches freely up- 

 wards. Are we to consider the 

 root or the branches as the more 

 important part ? The botanist will 

 tell us that the question is wrongly 

 put, and that the life of an organ- 

 ism consists in the interaction of 

 its various parts" (loc. cit., p. 91). 



