Decemher 31, 1009] 



SCIENCE 



959 



perfonuable is to perfrom it. Here noth- 

 ing can succeed except success. Happily 

 the pi'oeedure in question need not be ap- 

 plied to all mathematical concepts and 

 theorems but only to those— and they are 

 not so numerous— upon which, it is ad- 

 mitted, the remainder rest. Well, an ex- 

 amination of the volumes of the lievista 

 di Matematica and of its continuation, the 

 Uevuc dc Mathenmtiques, will show that 

 the principal mathematical branches have 

 been successfully subjected to the treat- 

 ment in question, with reference, however, 

 to primitive-systems differing- somewhat 

 from that above given. As for the latter 

 system, its adequacy to the demands of 

 the thesis has been shown by Russell in his 

 "Principles" with approximate complete- 

 ness and with as much rigor as discourse, 

 mainly non-s.ymbolie, can be reasonably 

 expected to attain. If, as is to be expected, 

 new branches of mathematics shall arise in 

 the days to come, though we can not be ab- 

 solutely certain, we may confidently ex- 

 pect that they will be congruous with 

 existing doctrines and will not demand a 

 radical change in foundations. 



Process of Testing the Thesis Illustrated. 

 — The little time that remains to me for 

 this address, I shall devote to illustrating 

 by means of a few cardinal examples, the 

 procedure by which the thesis is .iustified. 

 And I shall begin with the concept of 

 cardinal miniber. Before defining cardinal 

 number of a class, we define what is meant 

 by sameness of cardinal number, or, better, 

 what is meant by saj'ing this class and that 

 have the same cardinal number. Two 

 classes a aud h are said to have the same 

 cardinal number when there is a biuniforra 

 relation, or, as we commonly phrase it, a 

 one-one coiTclation between them. A 

 slight change in the statement is necessary 

 to prove suitable for zero. Then the car- 

 dinal number of a class a is defined to be 

 the class whose terms are the classes having 



each of them, according to the preceding 

 definition, the same cardinal number as a. 

 Thus with each ela.ss is associated a defi- 

 nite cardinal number. That of the null- 

 class is named zero and denoted by ; that 

 of a singular class is called 07ie and de- 

 noted by 1. Addition of cardinals is de- 

 finable in terms of logical addition of 

 classes: if a and b be two disjoint classes 

 having respectively the numbers a and j3, 

 the sum a + /3 is the number of the logical 

 sum (a class) a -j-b oi. a and 6. If a and 

 b are singular classes, the cardinal of their 

 sum may be named tivo and denoted by the 

 symbol 2, in which case 1 + 1 = 2; and so 

 on. Multiplication of cardinals is also de- 

 fined in purely logical tenns. This is done 

 by means of the concept (due to White- 

 head) of multiplicative class, which is itself 

 given in terms of logical constants: k be- 

 ing a class of disjoint classes, the multi- 

 plicative class of k is the class of all the 

 classes each of which contains one and but 

 one term of each class in k. Then the 

 product of the cardinal numbers of the 

 classes in k is defined to be the cardinal 

 number of the multiplicative class of A-. 

 As multiplication and addition in class 

 logic are commutative, associative and dis- 

 tributive, it readily follows that these 

 laws are valid for cardinal numbere. In 

 the manner indicated the entire theory of 

 cardinals can be established. And thus it 

 appears— to refer again to an example be- 

 fore cited- that the foundation assumed 

 by Weierstrass for the theory of the real 

 variable is itself underlaid by a basis in 

 pure logic. 



It is noteworthy that the foregoing con- 

 cept of cardinal is independent of the (as 

 yet undefined) notion called order and that 

 it equally comprises both finite and infinite 

 cardinals, the distinction of finite and in- 

 finite being this: the cardinal number of a 

 class a is infinite or finite according as a 

 is or is not such that there is a class b com- 



