672 SCIENCE PROGRESS 



gross carelessness ; it is like the habit of those mathematicians — whom 

 nowadays we would not call precise — who used to say that a " function " is 

 an " expression." These careless people were rightly corrected by Frege, 

 and it is inexcusable for any modern writer on the logical principles of 

 mathematics to revert to pre-Frege errors and confusions. Many years ago 

 Mr. Russell himself, in Mind for 1906, criticised this error in H. MacColl ; 

 when Whitehead and Russell's Principia Mathematica began to appear in 

 1910, Mr. Russell acknowledged the justice of the present writer's strictures 

 in the Cambridge Review on a crop of similar errors, but said that they were 

 all due to Dr. Whitehead ; now Mr. Russell himself repeats (for at least the 

 third time) some of these errors. 



The book does not contain much that is new except disavowals. There 

 is a disavowal (in the note on p. 203) of a fundamental characteristic of Dr. 

 Whitehead and Mr. Russell's Principia Mathematica which, it may be neces- 

 sary to add, imitates a book of Newton in title, but in title alone. And 

 there is a disavowal of the method which Mr. Russell used in his Principles 

 of Mathematics of 1903 to show that there are an infinity of things in the 

 world of mathematics. This " proof " of 1903 was, of course, fundamental 

 to the whole attitude Mr. Russell then adopted towards the question of 

 infinity ; but " the first thing to observe is that there is an air of hocus- 

 pocus about it [this argument] : something reminds one of the conjurer who 

 brings things out of the hat" (p. 135). The argument relied on forming 

 the complete set of individuals, classes, classes of classes, etc., but now 

 Mr. Russell's " theory of types " forbids us to put together individuals, 

 classes, etc., which are of different " logical types." So Mr. Russell parts 

 company with his former self — and also, it may be mentioned, with Plato 

 (p. 138). It is apparently what Mr. Russell calls " a robust sense of reality " 

 (pp. 135, 170 ; cf. p. 169) that has led him to this theory. Indeed, Mr. 

 Russell remarks with some approval that " the reader will feel convinced 

 that it is impossible to manufacture an infinite collection out of a finite col- 

 lection of individuals, though he may be unable to say where the flaw is in 

 the above construction " (p. 135). But Mr. Russell's process for preserving 

 part at least of the numbers of ordinary mathematicians strikes one as quite 

 as much of a conjuring trick as his former method. In the first place, of 

 individuals there may be only a finite number or even none at all (p. 134). 

 Then we have another type (classes of individuals) ; then another type 

 (classes of classes), and so on. We manufacture all these types out of classes 

 or classes of classes ... of no individuals, and get any finite number we 

 wish, however large, by simply going up the hierarchy of " logical types." 

 But, then, " classes " are (p. 182) said to be fictions, even if there are indi- 

 viduals, so that it is difficult to see how numbers at all can be preserved 

 if they are only to belong to fictions, or fictions of fictions, or so on — perhaps 

 even the more so as there may not be any individuals to give rise to such 

 fictions. The case as regards individuals comes to a very lame conclusion. 

 Mr. Russell is finally reduced to defining " individuals " (p. 142) by means 

 of the kind of symbols by which they are symbolised, in order to avoid a 

 plunge into metaphysics. This would be about as satisfactory as to define 

 a species of animal by the name we may give to it, so that the species might 

 vary if one had a cold in the head. The curious belief in the power of the 

 word is also shown when a proposition is defined, as has already been re- 

 marked, as a " form of words," even though Mr. Russell finds it convenient 

 to speak of propositions as true or false. Mr. Russell has sometimes enthu- 

 siastically praised Humpty-Dumpty's way of treating words, but it seems 

 that in his later works he has fallen far behind Humpty-Dumpty. 



To sum up, if some theory of " logical types " is necessary, most mathe- 

 maticians would be grateful if the grounds were stated, not as Mr. Russell 

 has stated them here, but convincingly, undogmatically, and freed from 



