ESSAYS 649 



that its conceptions emerge rom a vague state into a state in which their logical 

 character is more and more clearly defined, but in which development there is no 

 last stage. Some logicians think that they have discovered the last stage in 

 certain sciences, while nearly all writers of text-books strive to strike what they 

 consider a happy medium between a logically accurate exposition and the sort of 

 propositions which they imagine— sometimes on good grounds— that students find 

 it easy or beneficial to assimilate. A few writers of text-books have gone so far as 

 to mould their expositions on historical lines ; but few have the courage to lead a 

 student along the long path of the historical development of an idea, and reach 

 brevity by substituting their own notions of how such-and-such a stage of develop- 

 ment might be reached. They are like those would-be gods who acknowledge, 

 perhaps even explicitly in a footnote, that things were not so, but simply that if 

 they had the ordering of history things might have fallen out as they describe. 



These two qualities of schoolmasters can be found in almost every article on 

 mathematics in the Encyclopaedia Britannica, for example. One other character- 

 istic of most text-books is the deplorable way in which fundamental questions are 

 treated. One can imagine the sigh of relief that an author of one of these books 

 must give when he can at last give himself freely up to technical work ; when he 

 can leave the discussion of such things as integers and irrational numbers and 

 functions in general and write about Fourier's series or Bessel functions or 

 asymptotic solutions. No philosophical or even literary training is then required : 

 it is hardly necessary to use any other language than "we have," and then a line 

 of symbols ; " consider," and then a mention of what we are to consider ; then more 

 symbols ; and so on for a few hundred pages. But this avoidance of questions of 

 principle cannot altogether be urged against the article Mathematics in the 

 eleventh edition of the Encyclopcedia Britannica (191 1, 17, 878-83). This article 

 will now be analysed. 



If any one should want to find out how the mathematical articles in the 

 Encyclopcedia Britannica are arranged, he would naturally turn to the article 

 Mathematics. There would be nothing arbitrary in this selection : we need not 

 suppose that he, like the three sisters about whom the Dormouse told Alice, is 

 interested primarily only in things which begin with an M "such as mouse-traps, 

 and the moon, and memory, and muchness." The article in question seems to be 

 the central, co-ordinating one of all the mathematical articles in the Encyclopcedia. 



•It begins by referring to what is said to be the traditional definition of mathe- 

 matics as "the science of quantity," and gives reasons for rejecting as in- 

 adequate such a definition. The idea embodied in this definition seems, by the 

 way, to have been held by the late Prof. G. Chrystal when he wrote the article 

 Mathematics in the ninth edition of the Encyclopedia Britannica. These 

 reasons are afforded by Mr. Bertrand Russell's treatment of questions at the 

 foundations of mathematics, and the result is that " there is now no option but to 

 employ 'mathematics' in 'the general sense of the 'science concerned- with the 

 logical deduction of consequences from the general premisses of all reasoning ): 

 (p. 880)-. It is not necessary to do more than mention the uncritical attitude of the 

 article towards such things as the definitions of "real" numbers and integers 

 which were professedly defined by Russell by the same principle and yet which 

 were actually defined by him by different principles, or the reference (p. 880) to 

 the article Number as giving the arithmetic of infinite numbers, although it is 

 explicitly not dealt with in that article. These defects are merely small 

 incompetences and incorrectnesses ; larger incompetencies than these will also 

 be found to annoy a reader. 



