ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



33 



less accidental. Its task is connected with something which is given 

 to it from without and which it cannot alter. It is much the same 

 with the science of history, which must contemplate the history of 

 mankind as it has actually occurred. Also zoology, botany, mineral- 

 ogy, geology, and chemistry work with given data. In order not 

 to become involved in futile speculations the last-mentioned sci- 

 ences are constantly and inevitably obliged to revert to observa- 

 tions by the senses. It is then their task to link together these in- 

 dividual observations by bonds of causality and in this way to 

 erect from the single stones an edifice, the view of which will render 

 it easier for limited human intelligence to comprehend nature. 

 Physics of all sciences stands nearest to mathematics in this respect, 

 because unlike the other sciences she is generally in need of only a 

 few observations in order to proceed deductively. But physics, 

 too, must resort to observations of nature, and could not do without 

 them for any length of time. 



Mathematics alone, after certain premises have been perma- 

 nently established, is able to pursue its course of development in- 

 dependently and unmindful of things outside of it. It can leave 

 entirely unnoticed, questions and influences emanating from the 

 outer world, and continue nevertheless its development. 



As regards geometry, the first beginnings of this science did 

 indeed take their origin in the requirements of practical life. But 

 it was not long before it freed itself from the restrictions of the prac- 

 tical art to which it owed its birth. Herodotus recounts that geom- 

 etry had its origin in Egypt where the inundations of the Nile ob- 

 literated the boundaries of the riparian estates, and by giving rise 

 to frequent disputes constantly compelled the inhabitants to compare 

 the areas of fields of different shapes. But with the early Greek 

 mathematicians, who were the heirs of the Egyptian art of measure- 

 ment, geometry appeared as a science which men pursued for its 

 own sake without a thought of how their intellectual discoveries 

 could be turned to practical account. 



Nevertheless, although the workers in the domain of pure math- 

 ematics are not stimulated by the thought that their researches are 

 likely to be of practical value, yet that result is still frequently real- 



