Septembee 23, 1910] 



SCIENCE 



391 



much to facilitate the attainment of a 

 wider outlook by directing the attention 

 of their students to the more general and 

 less technical aspects of the various parts 

 of the subject, and especially by the intro- 

 duction into the courses of instruction of 

 more of the historical elements than has 

 hitherto been usual. 



All attempts to characterize the domain 

 of mathematics by means of a formal defi- 

 nition which shall not only be complete, 

 but which shall also rigidly mark oE that 

 domain from the adjacent provinces of 

 formal logic, on the one side, and of phys- 

 ical science, on the other side, are almost 

 certain to meet with but doubtful success; 

 such success as they may attain will prob- 

 ably be only transient, in view of the 

 power which the science has always shown 

 of constantly extending its borders in un- 

 foreseen directions. Such definitions, many 

 of which have been advanced, are apt to 

 err by excess or defect, and often contain 

 distinct traces of the personal predilections 

 of those who formulate them. There was 

 a time when it would have been a tolerably 

 suiBcient description of pure mathematics 

 to say that its subject-matter consisted of 

 magnitude and geometrical form. Such a 

 description of it would be wholly inade- 

 quate at the present day. Some of the 

 most important branches of modern math- 

 hematics, such as the theory of groups, and 

 universal algebra, are concerned, in their 

 abstract forms, neither with magnitude 

 nor with number, nor with geometrical 

 form. That great modern development, 

 projective geometry, has been so formu- 

 lated as to be independent of all metric 

 considerations. Indeed the tendency of 

 mathematicians under the influence of the 

 movement known as the arithmetization 

 of analysis, a movement which has become 

 a dominant one in the last few decades, is 

 to banish altogether the notion of measur- 



able quantity as a conception necessary to 

 pure mathematics; number, in the ex- 

 tended meaning it has attained, taking its 

 place. Measurement is regarded as one of 

 the applications, but as no part of the basis, 

 of mathematical analysis. Perhaps the 

 least inadequate description of the general 

 scope of modern pure mathematics — I will 

 2iot call it a definition — would be to say 

 that it deals with form, in a very general 

 sense of the term; this would include 

 algebraic form, geometrical form, func- 

 tional relationship, the relations of order 

 in any ordered set of entities such as num- 

 bers, and the analysis of the peculiarities 

 of form of groups of operations. A strong 

 tendency is manifested in many of the re- 

 cent definitions to break down the line of 

 demarcation which was formerly supposed 

 to separate mathematics from formal logic ; 

 the rise and development of symbolic logic 

 has no doubt emphasized this tendency. 

 Thus mathematics has been described by 

 the eminent American mathematician and 

 logician B. Peirce as "the science which 

 draws necessary conclusions," a pretty 

 complete identification of mathematics 

 with logical procedure in general. A defi- 

 nition which appears to identify all 

 mathematics with the Mengenlehre, or 

 theory of aggregates, has been given by E. 

 Papperitz: "The subject-matter of pure 

 mathematics consists of the relations that 

 can be established between any objects of 

 thought when we regard those objects as 

 contained in an ordered manifold; the law 

 of order of this manifold must be subject 

 to our choice." The form of definition 

 -which illustrates most strikingly the tend- 

 encies of the modern school of logistic is 

 one given by Mr. Bertrand Russell. I re- 

 produce it here, in order to show how wide 

 is the chasm between the modes of expres- 

 sion of adherents of this school and those 

 of mathematicians under the influence of 



