MATHEMATICS 39 



dense, and certainly in no subject is pride in that ignorance 

 more conspicuous." 



I shall endeavor to point out that there is no reason for 

 this prejudice, and that there are many reasons why popular 

 ignorance of mathematics ought to be decreased. 



A Scientific Definition of Mathematics 



A definition of mathematics which comes much nearer em- 

 bracing all that is included is one that was given originally 

 by Benjamin Peirce and quoted by Professor Bocher in his 

 address before the Scientists at the St. Louis World's Fair. 

 He said that "Mathematics is the science which draws neces- 

 sary conclusions." To be sure, that defintion without qualifi- 

 cation will lead to very great misunderstanding. It is only 

 when it is properly understood that this definition seems to 

 give a grasp of the entire field. 



Indeed, mathematics is not formal logic, as might per- 

 haps be inferred by a misunderstanding of the preceding 

 definition. Formal logic seeks to study the rules of reasoning 

 for their own sake ; mathematics accepts these rules without 

 further question, and seeks to apply them. Formal logic is 

 concerned with the study of the methods of drawing conclu- 

 sions ; mathematics deals with the very act of drawing those 

 conclusions. Professor Bocher himself proceeded to explain 

 that the processes of mathematics consisted in drawing these 

 necessary conclusions from certain original assumptions taken 

 from the world as we find it, or in general from the results 

 of experiment by various scientists in their sciences ; that then 

 the supreme tests of the value of the whole process consists 

 in seeing whether the conclusions also agree with the evidence 

 of nature and of science. If they do, we say that the process 

 has been useful, and we say that it constitutes a branch of 

 mathematics. Elaborations and extensions of theories thus be- 

 gun are also recognized as mathematics. You will instantly 

 think of many examples of this process, some of which I shall 



