THE FUNDAMENTAL CONCEPTIONS AND METHODS OF 



MATHEMATICS 



BY PROFESSOR MAXIME BQCHER 



[Maxima Bocher, Professor of Mathematics, Harvard University, b. August 28, 

 1867, Boston, Mass. A.B. Harvard, 1888; Ph.D. Gottingen, 1891. In- 

 structor, Assistant Professor and Professor, Harvard University, 1891-. 

 Fellow of the American Academy. Author of Ueber die Reihenentwickel- 

 ungen der Potentialtheorie; and various papers on mathematics.] 



I. Old and New Definitions of Mathematics 



I AM going to ask you to spend a few minutes with me in consider- 

 ing the question: what is mathematics? In doing this I do not propose 

 to lay down dogmatically a precise definition ; but rather, after hav- 

 ing pointed out the inadequacy of traditional views, to determine 

 what characteristics are common to the most varied parts of mathe- 

 matics but are not shared by other sciences, and to show how this 

 opens the way to two or three definitions of mathematics, any one of 

 which is fairly satisfactory. Although this is, after all, merely a dis- 

 cussion of the meaning to be attached to a name, I do not think that 

 it is unfruitful, since its aim is to bring unity into the fundamental 

 conceptions of the science with which we are concerned. If any of 

 you, however, should regard such a discussion of the meaning of words 

 as devoid of any deeper significance, I will ask you to regard this 

 question as merely a bond by means of which I have found it con- 

 venient to unite what I have to say on the fundamental conceptions 

 and methods of what, with or without definition, we all of us agree 

 to call mathematics. 



The old idea that mathematics is the science of quantity, or that 

 it is the science of space and number, or indeed that it can be charac- 

 terized by any enumeration of several more or less heterogeneous 

 objects of study, has pretty well passed away among those mathe- 

 maticians who have given any thought to the question of what 

 mathematics really is. Such definitions, which might have been 

 intelligently defended at the beginning of the nineteenth century, 

 became obviously inadequate as subjects like projective geometry, 

 the algebra of logic, and the theory of abstract groups were de- 

 veloped; for none of these has any necessary relation to quantity 

 (at least in any ordinary understanding of that word), and the last 

 two have no relation to space. It is true that such examples have 

 had little effect on the more or less orthodox followers of Kant, 

 who regard mathematics as concerned with those conceptions which 



