332 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1961 



The structures are linked by a variety of laws : thus 



A. a + 5 = 5 + a a+ {h + c) = {a+l) + c 



ab = l)a a{hc) = {ab)c 



a{h + G) =db + aG 



There is an element 0, such that a+0=a for all a. For any a there is 

 an element — a, such that a+ { — a)=0. 



B. There is an element 1 such that a.l = a for all a. 



C. For each a=0 there is an element 5 such that al) = l. 



In addition, the real numbers have a topological structure, which is 

 what is involved whenever we talk about such matters as limits, con- 

 vergence and so on. The typical relationship which describes the 

 topology of the real numbers is that of neighborhood : 



A neighborhood of a real number a is any set which includes an 

 interval with a as midpoint. 



If we consider geometry, again, we are once more dealing with a 

 complex structure. The structures involved in geometry are mostly 

 algebraic: they involve finite sets of objects, in relations such as inci- 

 dence — a point being on a plane, a plane passing through a line. It is 

 only when we come to consider differential properties — tangency, 

 curvature of curves and surfaces — that topological structures are 

 brought in. 



The characteristic differences between classical mathematics — say 

 that of a century ago — and that of modern mathematics is that classical 

 mathematics dealt preeminently with complex structures, modern 

 mathematics with less complex ones. 



The reason for this is very practical. If we discuss a complex struc- 

 ture that structure may be so tightly specified by the numerous rela- 

 tions which define it that, roughly speaking, only one example of the 

 structure exists: or, to be more exact, if we have two sets of objects 

 which both have that structure, then they are exact pictures of one 

 another, as one Euclidean plane is an exact picture of the other : the 

 objects in the two sets can be made to correspond univocally so that 

 all the relations are transferred by the correspondence. For instance, 

 anything which obeys the sets of laws A, B, C, and in addition has the 

 order structure of the real numbers is an exact picture, in this sense, 

 of the real numbers. On the other hand, a less complex structure may 

 have as examples vastly different things. Thus, any set of objects 

 which obeys the laws A and B is called a commutative ring with unit. 

 Any deductions we make which are based exclusively on A and B will 

 hold for any such ring. Now, the polynomials form such a ring : so do 

 the functions of a real variable ; and consequently if we restrict our- 

 selves to the axioms for a ring, our conclusions will be valid for a wide 

 variety of mathematical objects, whereas conclusions based on the en- 



