ESSAY-REVIEWS 671 



mathematical space, infinity, continuity, and serial order, has led Georg 

 Cantor, Dedekind, Frege, Peirce, Schroder, Peano, and others, to results 

 among which some found by Dr. Whitehead and Mr. Russell are probably 

 permanent ; and to which, in the second book under review, Mr. Russell 

 attempts to give a brightly written and popular introduction. 



By " mathematical philosophy " Mr. Russell apparently means what 

 other people have called " the philosophy of mathematics," and, on the 

 whole, the name used by these others seems preferable : there is nothing in 

 Mr. Russell's philosophy, apart from the applications, that strikes one as 

 particularly mathematical. Even the elaborate symbolism used in Dr. 

 Whitehead and Mr. Russell's large Principia Mathematica, which looks 

 something like the usual symbolism of mathematics though it has a different 

 purpose, is not used in the present volume, while a philosophy of mathe- 

 matics need not be mathematical any more than a philosophy of art need 

 be artistic. The purpose of the Russell-Whitehead symbolism — an attempt 

 to combine the advantages of those of Peano and Frege — is quite different 

 from that of mathematical symbolism. In fact, " mathematics is a study 

 which . . . may be pursued in either of two opposite directions. The 

 more familiar direction is constructive, towards gradually increasing com- 

 plexity ; from integers to fractions, real numbers, complex numbers ; from 

 addition and multiplication to differentiation and integration, and on to 

 higher mathematics. The other direction, which is less familiar, proceeds, 

 by analysing, to greater and greater abstractness and logical simplicity ; 

 instead of asking what can be defined and deduced from what is assumed to 

 begin with, we ask instead what more general ideas and principles can be 

 found, in terms of which what was our starting-point can be defined or 

 deduced. It is the fact of pursuing this opposite direction that characterises 

 mathematical philosophy as opposed to ordinary mathematics " (p. i). 



The book consists of eighteen chapters, which deal, in a somewhat popular 

 way, with the following subjects respectively : Peano 's derivation of the 

 whole theory of the natural numbers from three primitive ideas (" zero," 

 " number," and " successor "), and five primitive propositions in addition 

 to those of pure logic ; Frege's (1884) definition of number which was re- 

 discovered by Mr. Russell in 1901 ; definitions of all of Peano's primitive 

 ideas in logical terms, and considerations on finitude and " mathematical 

 induction " ; the essential characteristics of a relation which js to give rise 

 to " order," and a suggestion of a generalisation of serial relations (p. 41) ; 

 various kinds of relations ; ordinal " similarity " of relations, and " rela- 

 tion-numbers " ; logical definitions of rational, real, and complex numbers ; 

 the theory of transfinite cardinal numbers as it results from a combination 

 of the discoveries of Georg Cantor and Frege ; more general consideration 

 of infinite series, ordinal numbers, and Alephs ; the (ordinal) notions of 

 limit and continuity ; the same notions as applied to functions ; selections 

 and the " multiplicative axiom " ; Mr. Russell's " axiom of infinity " and 

 his theory of " logical types " ; the theory of deduction ; propositional func- 

 tions ; descriptions (the word " the " in the singular) ; classes (" the " in 

 the plural) ; and the relation of mathematics and logic. 



Since Mr. Russell and Dr. Whitehead began working (1900) at the reduc- 

 tion of mathematical conceptions to those of logic, they have steadily become 

 more and more symbolical, in the narrower sense, and less and less capable of 

 expressing matters about the principles of mathematics in ordinary language. 

 The present volume seems to mark a culmination of this incapacity. For 

 example, by a " proposition " is meant " a form of words which expresses 

 what is either true or false " (p. 155) ; and, on the same page, a " proposi- 

 tional function " is also said to be an " expression." But, on the very next 

 page (cf. pp. 157 ff.), Mr. Russell seems to find his restriction so awkward 

 that he speaks, as we all do, of a proposition as being true or false. This is 



