36 ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



negative, we shall select and confine our remarks to a branch which 

 is commonly taken to be synonymous with mathematics, name^, to 

 arithmetic in the broadest sense of the word. 



The subject of inquiry in arithmetic is numbers and their com- 

 binations. On this account arithmetic is, of all sciences, most free 

 from what lies outside its boundaries. Perception by the senses is 

 necessary only in an extremely insignificant measure for the under- 

 standing of its definitions and premises. It is possible to acquaint 

 a person who lacks both sight and hearing with the fundamental 

 principles of arithmetic solely by the medium of "time." Such a 

 person needs only the sense of feeling. By slight excitations of his 

 skin, induced at equal or unequal intervals of time, he can be led to 

 the notion of differences of time and hence also to the notion of dif- 

 ferences of number. Uninfluenced by matter and force, independ- 

 ently, too, of the properties of geometrical magnitudes, arithmetic 

 could be conducted solely by its own intrinsic potencies to its high- 

 est goals, drawing deductively truth from truth, without a break. 



But what sort of a science should we arrive at by this method 

 of procedure? Nothing but a gigantic web of self-evident truths. 

 For, once we admit the first notions and premises to which a man 

 thus bereft of his senses can be led, we are compelled of necessity 

 also to admit the derivative results of arithmetic. If the beginnings 

 of arithmetic appear self-evident, the rest of it, too, bears this 

 character. Owing to this deductive character of arithmetic, and to 

 its exemption from influence from without, this science appears to 

 one person extremely attractive, while to another it appears ex- 

 tremely repulsive, according as each is constituted. Be that as it 

 may, however, a finished and complete science of this character 

 subserves no purpose in the comprehension of the world, or in the 

 advancement of civilisation. Hence, an arithmetic which heaps up 

 theorem on theorem with never a thought of how its results are to 

 be turned to practical account in the acquisition of knowledge in 

 other fields, resembles an inquisitive physician, who, taking up his 

 abode in a desert, should arrive there at momentous results in 

 bacteriology, but should bear them with him to his grave, without 

 their ever redounding to the benefit of humanity. The value of 



