The Main Lines of Mathematics' 



By J. L. B. Cooper 



University College of South Wales and Monmouthshire 



Everyone knows some mathematics, yet few persons have an idea 

 of what the subject is about, even in the imprecise way in which they 

 know that physics is about matter or zoology about animals. Most 

 people either confuse mathematics with applied mathematics — and 

 let me explain, to avoid tliis confusion, that I am using the word 

 mathematics in its strict, and indeed its only logical, interpretation 

 to mean pure mathematics — or think that it is not about anything at 

 all, but it is a farrago of rules of calculation, such as one encounters 

 in elementary arithmetic, a rather distasteful preliminary to really 

 interesting pursuits such as keeping accounts or learning engineering. 

 This attitude develops into and is encouraged by some prevalent 

 theories of mathematics : the word "farrago," indeed, is one that I have 

 borrowed from Wittgenstein. Without doubt these theories have the 

 effect of laying an exclusive emphasis on the most uninteresting parts 

 of mathematics, manipulative teclmiques and the learning of notation ; 

 and one can hardly doubt that they contribute to the mechanistic 

 and rule-of -thumb methods of teaching the subject which are in evi- 

 dence in so many schools and are complained of in several recent 

 reports on secondary school mathematics. 



These theories have beliind them the authority of the most fashion- 

 able modern school of British pliilosophy, that of linguistic analysis, 

 and of its predecessor, logical positivism ; and with their backing have 

 penetrated all sorts of diverse fields, up to esthetics, literary criticism, 

 and the brains trust. So widely held are they that it would be just to 

 describe them as a folklore about our subject. 



The best-known phrase from this folklore or mythology of mathe- 

 matics is the sentence "Mathematics is a language." This is not, to be 

 exact, a tenet of any philosophical school. It is far too imprecise for 

 that. Even with all the possible twists one can give to the meanings 

 of the words "mathematics," "language," and "is," it is hard to find, 



* Reprinted, with extensive revisions, by permission from The Advancement of Science 

 (London), vol. 17, No. 70, March 1961. 



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