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SCIENCE 



[N. S. Vol. XLIV. No. 1134 



called this the section of functional theory, 

 because its ideas are essential to the con- 

 struction of logical denoting functions 

 which include as a special case ordinary 

 mathematical functions such as sine, log- 

 arithm, etc. In each of these three stages 

 it will be necessary gradually to introduce 

 an appropriate symbolism, if we are to pass 

 on to the fourth stage. 



The fourth logical section, the analytic 

 stage, is concerned with the investigation 

 of the properties of special logical con- 

 structions, that is, of classes and correla- 

 tions of special sorts. The whole of mathe- 

 matics is included here. So the section is 

 a large one. In fact, it is 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, com- 

 prising 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, in- 

 cluding 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 

 development seven special sorts of correla- 

 tions which are of peculiar interest. The 

 first sort comprises one-to-many, many-to- 

 one, and one-to-one correlations. The sec- 

 ond 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 relation, any 

 member of the field is either before or after 

 any other member. The third class com- 

 prises inductive relations, that is, correla- 

 tions on which the theory of mathematical 



induction depends. The fourth class com- 

 prises selective relations, which are re- 

 quired for the general theory of arithmetic 

 operations, and elsewhere. It is in connec- 

 tion with such relations that the famous 

 multiplicative axiom arises for considera- 

 tion. The fifth class comprises vector rela- 

 tions, from which the theory of quantity 

 arises. The sixth class comprises ratio re- 

 lations, which interconnect number and 

 quantity. The seventh class comprises 

 three-cornered and four-cornered 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 re- 

 member that the names are technical 

 names, meant, no doubt, to be suggestive, 

 but used in strictly defined senses. "We 

 have suffered much from critics who con- 

 sider it sufficient to criticize 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 mem- 

 ber, and this notion is defined without ap- 

 peal to the concept of the number one. 

 The notion of diversity is all that is 

 wanted. Thus the class a has only one 

 member, if (1) the class of values of x 

 which satisfies the propositional function, 



x is not a member of a, 

 is not the whole type of relevant values of 

 x, and (2) the propositional 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 pos- 

 sible for higher finite cardinal members. 

 Thus, step by step, the whole cycle of cur- 

 rent mathematical ideas is capable of log- 

 ical definition. The process is detailed and 

 laborious, and, like all science, knows noth- 



