628 SCIENTIFIC THOUGHT. 



for they mark more the attitude of the beholder than 

 the things which he regards. It is true that a very 

 small addition to our actual knowledge of facts, like the 

 sudden appearance of some characteristic feature in a 

 landscape, may sometimes entirely alter the whole aspect, 

 induce us to abandon our accustomed views, and call up 

 suddenly an unforeseen train of ideas ; in such a case, 

 perhaps, this insignificant discovery becomes historically 

 interesting, although it is mainly by the altered trains 

 of thought which it has evoked that it has become 

 important to us. 

 2. The difference of scientific knowledge and scientific 



Difference 



between thought is thus owing to the two factors which are 



thought and 



knowledge, involved the facts of science or nature on the one side 

 and the scientifically thinking mind on the other. Now 

 it might appear as if this difference vanished when we 

 approach the abstract science of mathematics, or at least 

 that of number ; for in numbering and counting we 

 have really only to do with a process of thought, and it 

 would seem as if the science of number were itself the 

 science of thought, or at least a portion of it. In fact, 

 the question arises, Is there any difference between 

 mathematical science and mathematical thought ? Some 

 considerations might induce us to think that there is 

 not. On the other side, I shall try to show in this 

 chapter that there is, and that the development of 

 mathematics during our period has brought this out 

 very clearly and prominently. 



Popular There is an opinion current among many thinking 



persons who have not occupied themselves with mathe- 



matics. matical science, though they may be very efficient in 



