334 ANlSfTJAL REPORT SMITHSONIAN INSTITUTION, 1961 



based solely on analogy. It is therefore necessary to make sure that 

 our arguments are based solely on our axioms, and that we do not 

 introduce tacitly assumptions brought over from Euclidean space. 

 Kigor of argument is therefore highly important in this sort of 

 mathematics; its role is to ensure that anything which we assert to 

 follow from the axioms of a structure holds for all structures obeying 

 those axioms and not merely for those familiar to our intuition. 



I would like to illustrate the variety of things which can share a 

 mathematical structure by discussing the example of a Boolean ring. 

 Now a ring means any set of objects which obeys the axioms A, B. 

 A Boolean ring is one which in addition satisfies the axioms : 



a+a=0, a^=a^ for all elements a 



The simplest structure which obeys the laws of a Boolean ring con- 

 sists of just two objects, and 1, with the usual law of multiplication 

 and the usual law of addition save that 1 + 1=0. An example of such 

 a ring is got by taking to mean the set of all even numbers, 1 to 

 mean the set of all odd numbers: then 1 + 1=0 means that the sum of 

 any two odd numbers is even. More complicated examples are : 



(1) Propositional logic. — The symbols of the algebra stand for 

 propositions, that is statements which are either true or false. If a 

 is a proposition, a=0 means that a is false, a=l that a is true. If a 

 and h are two propositions, ab is the proposition which says that both 

 a and h are true, a+h says that one but not both oi a+h are true. 

 Then 1+a says that a is false: for 1 is true, and not both of 1 and a 

 are true, a+a is always false: for either, neither, or both of a and a 

 are true; aa is the same as a. We have then, <2+a=0, a?=a; and the 

 other axioms of ring theory can be verified relatively easily. 



(2) Subsets of a set. — The symbols of the algebra stand for subsets 

 of a set, which is denoted by 1 ; stands for the empty set. If a and 

 h are two sets, ab is the set of objects common to a and h,a+b the set 

 of objects lying in just one of a,b. Again it is easy to verify the 

 axioms. 



(3) Electrical switching circuits. — The symbols of the algebra 

 stand for electrical circuits which involve switches. a=0 means that 

 the circuit a is always broken, a=l that it is always connected. If 

 two circuits are so arranged that they are always made or broken to- 

 gether, they are denoted by the same symbol ; if the one is always made 

 when the other is broken, one is denoted by a symbol «, say the other by 

 l+«. li a and h are two circuits in series, the circuit they form to- 

 gether is denoted by ab; if they are in parallel, the joint circuit is 

 a+b + ab. The laws of Boolean algebra are obeyed ; and the symbols 

 for a circuit give a means of working out how the circuit behaves. 



To sum up : The role of mathematics is to discover and investigate 

 structures which arise in our theoretical treatment of physical experi- 



