ESSAY-REVIEWS 673 



appeals — which have been shown to cut both ways — to a " robust sense of 



reality." 



Mr. Russell assures us (p. v) that he has made " the utmost endeavour to 

 avoid dogmatism on such questions as are still open to serious doubt." 

 However, on p. 117, he asserts that the " multiplicative axiom . . . can 

 be enunciated, but not proved, in terms of logic." If Mr. Russell's boast 

 (p. 167) — which seems to mean that his love for truth caused him to rise 

 superior to the discomforts of a prison — is not humbug, it is impossible to 

 account for the fact that the proof of this axiom in Science Progress (1918, 

 13, 178, 299-304) should be overlooked. Dr. Whitehead and Mr. Russell 

 had both seen this proof-— an amended form of an earlier attempt which 

 was defective through a mistaken belief that a certain essential link, which 

 happened to be known to the author of the above paper, had been already 

 supplied by F. Hartogs, and that therefore it was not necessary again to 

 supply this link — in July 191 8 — while this book was yet unwritten, or, at 

 least, while it was yet in manuscript. The criticisms made on both Dr. 

 Whitehead and Mr. Russell were obviously based on inattention, and were 

 naturally superficial : there was a total disregard of the fact that an un- 

 ambiguous rule was provided for arranging the " chains " of M " in classes 

 of direct continuations," in such a way that there cannot be a chain of M 

 which continues all of the chains of any particular class of direct continua- 

 tions ; and also that, for each class of direct continuations, it is obviously 

 possible to conclude that if the class has chains respectively of all types 

 less than a, for example, M has a chain of type a>. In most cases there 

 would be no grounds to find fault with either Dr. Whitehead or Mr. Russell 

 for ignoring a particular piece of work ; but in this case the work was a 

 serious attempt, against which there have been advanced no sound or even 

 intelligent criticisms, which would make superfluous large parts of Dr. 

 Whitehead and Mr. Russell's own Principia Mathematica, and would also 

 give a necessary foundation, which neither of the authors mentioned could 

 discover, for Dr. Whitehead's chief results on the arithmetic of cardinal 

 numbers. It cannot, then, it seems, be denied that Mr. Russell has allowed 

 other motives to interfere with his love of truth. Mr. Russell once remarked 

 that he loved his country, but loved the truth still more : paradoxical as it 

 may seem in view of recent action under the Defence of the Realm Act, it is, 

 perhaps, the case that Mr. Russell loves his country more than he loves the 

 truth. 



That limits and continuity of functions can be defined purely ordinally, 

 whereas they are usually defined in terms involving number, is stated 

 (p. 107) to have been shown by Dr. Whitehead. The facts of the case are 

 these : In 1902 the present writer communicated to Dr. Whitehead a defini- 

 tion of a continuous function which had certain very great advantages for 

 purely ordinal treatment. In his book of 1903 Mr. Russell remarked that 

 continuity of functions necessarily involves number ; and so the present 

 writer developed his purely ordinal treatment of continuous functions in 

 one of the best-known mathematical journals — Crelle's Journal for 1905. 

 Copies of this paper were sent to Dr. Whitehead and Mr. Russell ; and up 

 to 1912 the present writer published much, extending the scope of purely 

 ordinal conceptions in various other parts of mathematics. In 1913 Dr. 

 Whitehead and Mr. Russell published a purely ordinal definition of a con- 

 tinuous function, without mentioning any other work. This definition was 

 practically the same as the one published by the present writer eight years 

 previously, and is the definition referred to by Mr. Russell in the above 

 passage. The matter is not of great importance except that it throws doubt 

 on the good faith of Dr. Whitehead and Mr. Russell. 



Lastly, on pp. 203-5 there are conclusions as to the logical nature of 

 mathematics which would provoke a smile. This is mainly owing to the 



