' PRESIDENTIATi ADDRESS. 301 



correlations of special sorts. The whole of mathematics is included here. So 

 the section is a large one. In fact, it i.s mathematics, neither more nor less. 

 But it includes an analysis of mathematical ideas not hitherto included in the 

 scope of that science, nor, indeed, contemplated at all. The essence of this 

 stage is construction. It is by means of suitable constructions that the great 

 framework of applied mathematics, comprising the theories of number, quantity, 

 time, and space, is elaborated. 



It is impossible even in brief outline to explain how mathematics is 

 developed from the concepts of class and correlation, including many-cornered 

 correlations, which are established in the third section. I can only allude to 

 the headings of the process which is fully developed in the work, ' Mathematica 

 Principia,' by Mr. Russell and myself. There are in this process of develop- 

 ment seven special sorts of correlations which are of peculiar interest. The 

 first sort comprises one-to-many, many-to-one, and one-to-one correlations. The 

 second sort comprises serial relations, that is, correlations by which the members 

 of some field are arranged in a serial order, so that, in the sense defined by the 

 lelation, any member of the field is either before or after any other member. 

 The third class comprises inductive relations, that is, correlations on which the 

 theory of mathematical induction depends. The fourth class comprises selec- 

 tive relations, -which are required for the general theory of arithmetic operations, 

 and elsewhere. It is in connection with such relations that the famous multipli- 

 cative axiom arises for consideration. The fifth class comprises vector relations, 

 from which the theory of quantity arises. The sixth class comprises ratio 

 relations, which interconnect number and quantity. The seventh class com- 

 prises three-cornered and four-oornei'ed relations which occur in Geometry. 



A bare enumeration of technical names, such as the above, is not very 

 illuminating, though it may help to a comprehension of the demarcations of the 

 subject. Please remember that the names are technical names, meant, no doubt, 

 to be suggestive, but u.sed in strictly defined senses. We have suffered much 

 from critics who consider it sufficient to criticise our procedure on the slender 

 basis of a knowledge of the dictionary meanings of such terms. For example, 

 a one-to-one correlation depends on the notion of a class with only one member, 

 and this notion is defined wnthout appeal to the concept of the number one. 

 The notion of diversity is all that i.<; wanted. Thus tlie class a has only one 

 jnember, if (1) tlie class of values of x which satisfies the projxjsitional 

 function, 



X is not a member of a, 



is not the. whole type of relevant values of x, and (2) the propositioiial function, 



X and y are members of a, and x is diverse from y, 



is false, whatever be the values of x and y in the relevant type. 



Analogous procedures are obviously possible for higher finite cardinal mem- 

 bers. Thus, step by step, the whole cycle of current mathematical ideas is 

 capable of logical definition. The process is detailed and laborious, and, like 

 all science, knows nothing of a royal road of airy phrases. Tlie essence of the 

 process is, first to construct the notion in terms of the forms of propositions, 

 that is, in terms of the relevant prepositional functions, and secondly to prove 

 the fundamental truths which hold about the notion by reference to the results 

 obtained in the algebraic section of logic. 



It will be seen that in this process the whole apparatus of special indefinable 

 mathematical concepts, and special a priori mathematical premises, respecting 

 number, quantity, and space, has vanished. Mathematics is merely an apparatus 

 for analysing the deductions which can be drawn from any particular premises, 

 supplied by common sense, or by more refined scientific observation, so far as 

 these deductions depend on the forms of the propositions. Propositions of 

 certain forms are continually occurring in thought. Our existing mathematics is 

 the analysis of deductions, which concern those forms and in some way are 

 important, either from practical utility or theoretical interest. Here I am 

 speaking of the science as it in fact exists. A theoretical definition of mathe- 

 matics must include in its scope any deductions depending on the mere forms 



