ESSAYS 



MATHEMATICS IN AN ENCYCLOPEDIA (Philip E. B. 

 Jourdain, M.A.) 



A DISCUSSION of the nature and extent of articles in a proposed mathematics 

 dictionary has lately excited a good deal of interest in America. The Mathe- 

 matical Association of America has appointed a committee to investigate the 

 subject, and Dr. G. A. Miller (A mer. Math. Monthly, 1918, 25, 383-7; cf. 428) 

 has given an example of a proposed article dealing with the theory of groups. This 

 article has the conventional characteristics of an article for a dictionary. This is 

 not meant to imply that the article is not a good one of its kind : it is, in fact, a 

 yery thorough and exhaustive treatment of the things about which it means to give 

 information. But it seems to me that what it and most other dictionary articles 

 discuss is exactly what nobody really wants who is not bent on acquiring merely 

 the sort of knowledge that is required by examinations. Nearly all of Dr. Miller's 

 specimen article is devoted to definitions of "groups" in the general technical 

 meaning of the word, and particular qualities of groups. It is no uncommon 

 thing to read in an examination paper such a question as : " Define the terms 

 isomorphism and transitivity''' ; but such information, at least in this form, is not 

 required outside the examination room. Surely, an intelligent being who consults 

 an article on a science in an encyclopaedia does so from a wish to know what 

 such-and-such a department of knowledge is about and what has been done in it, 

 whether he does so for cultivating his own mind, or as a preliminary for cultivating 

 the minds of others, or for his own work of discovery. He does not want simply 

 to fill his memory with what certain words mean in the technical language of the 

 present : his aim would be to get a firm grasp of the principles underlying that par- 

 ticular subject, and to find out why certain notions were so important as to be fixed 

 by a name — such as " isomorphism," for instance. It is not the formal definitions 

 that we must seek in the first place : it is those more or less vague ideas which 

 have lost part of their vagueness so as to become apparently definable. Of course 

 we can never be quite certain that the definitions we may arrive at in some science 

 at a particular time are quite free from vagueness : a mathematician who is 

 nterested in the principles of his subject can find many instances of " definitions " 

 which were seriously given only a few years ago and which can now be seen not 

 to define at all. The theory that definitions should form the subject-matter of 

 articles in a dictionary or encyclopaedia, and a grea't part of the subject-matter 

 of text-books, is a theory held by those schoolmasters who think that examinations 

 are the goal of knowledge ; and we must always remember that professors are 

 only a better class of schoolmasters. 



Accordingly, we find other characteristics of text-books in encyclopaedia 

 articles written by or under the inspiration of such pedagogues. These articles very 

 often, if they contain anything but an incomplete or complete enumeration of 

 pseudo-definitions, are simply condensed text-books: the "bookwork" being 

 preserved in an abridged form and the "examples" omitted. Further, there is an 

 almost total neglect of the fact that a science is a living and growing organism ; 



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