MAIN LINES OF MATHEMATICS — COOPER 331 



to do SO would be to treat mathematics as a sort of game, like a chemical 

 system with any random combination of elements allowed: and it 

 would make the possibility of applying mathematics an insoluble 

 problem. We cannot take the subject seriously without a conviction 

 that any contradictions there may be are peripheral and remediable. 

 Such a conviction cannot, again, be based solely on an appeal to 

 empirical experience, since, for example, mathematics deals with sets 

 of indefinitely large magnitude and with infinite sets, and these cannot, 

 as such, be part of empirical experience. It is because of this that 

 empiricist philosophers try to explain mathematics away, in the ways 

 I have described. I do not wish to involve myself in metaphysical 

 knots ; but it seems reasonable to say that the source of our convictions 

 about mathematics must arise from a correspondence between the 

 terms of mathematical theories and some of the things which our minds 

 either bring to or find in our experience of the external world, or create 

 by generalization, abstraction, and extrapolation beyond experience. 

 I shall leave this thorny subject and go on to discuss what it is that 

 mathematics deals with. 



The subject of any mathematical theory is a mathematical structure; 

 and by a mathematical structure I mean a set of objects for which some 

 defined relations exist between its elements, or between sets of its ele- 

 ments or even between sets of sets of elements. 



Most of the mathematical structures we encounter are complex, in 

 the sense that they are combinations of structures with fewer relations. 

 Complex structures can be seen as being built up out of elementary 

 structures : I am using the word elementary in the sense of logically 

 simple, not the pedagogic sense. There are two main types of ele- 

 mentary structure in mathematics, which between them appear to 

 cover all the cases which arise in modern mathematics : 



(1) Algehraic structures^ m which the defined relations are between 

 finite numbers of elements. 



(2) Topological structures^ in which the relations are between pairs 

 of sets, or between individual elements and sets, or between individual 

 elements and sets of sets. 



I can best show what this means by first taking a familiar complex 

 structure, the real numbers, and explaining the elementary structures 

 which it involves. The real numbers have three elementary algebraic 

 structures : 



(1) The structure of addition : three numbers a, 5, c may be related 

 by the rule a+h=G. 



(2) The structure of multiplication : three numbers may be related 

 by db = G. 



(3) The structure of order: two numbers a, h may be related by 

 a>'b. 



