September 3, 1903] 



MATU^R'E. 



4U 



with, is a remarkable event, and the fact that the 

 criticism, pertinent and lucid as it is, of the work of j 

 (lie g-reat Continental thinkers is adverse on many ! 

 tundamental points should claim for it the patient 

 roiisideration of both classes of students. We 

 quote : — 



" The distinction of philosophy and mathematics is 

 broadly one of point of view : mathematics is con- 

 structive and deductive, philosophy is critical, and in 

 a certain impersonal sense controversial. Wherever 

 we have deductive reasoning, we have mathematics ; 

 but the principles of deduction, the recognition of in- 

 definable entities, and the distinguishing between 

 ■irh entities, are the business of philosophy." 



In answer to the question, " what is mathematics? " 

 \\ f are told that 



" Pure Mathematics is the class of all propositions 

 of the form ' p implies q ' where p and q are pro- 

 positions containing one or more variables, the same 

 in the two propositions, and neither p nor q contains 

 any constants except logical constants." 



These logical constants are defined in terms of the 

 ' fundamental concepts which mathematics accepts as 

 indefinable ; the philosophical discussion of the latter 

 occupies part i. of this volume. The remaining six 

 parts are devoted to the establishment of the rriain 

 thesis, that what is ordinarily known as mathematics' 

 is deducible from these fundamental concepts by purely 

 logical processes. This, of course, necessitates 'a 

 philosophical account of the processes which are 

 admissible ; the carrying out of the deductions in their 

 most abstract and rigorous form lies in the prdvihce' 

 of symbolic logic, and is reserved for the' second 

 volume. 



The mathematical reader is recommended in the 

 preface to pass over some of the more philosophical 

 portions and begin at part iv., on *' Order." We do 

 not endorse this recommendation, for the exact estab- 

 lishment of the notion of order is one of the most- 

 tedious pieces of work that the mathematical philo- 

 s.opher has to do; besides, many of the preceding 

 chapters are not only extremely interesting in them- 

 selves, but absolutely essential to a correct appreciation 

 of the science of arithmetic subsequently developed. 

 For example, a number will be found to be defined as 



class. 



Concerning the notion of class, some slight criticism 

 may not be inappropriate. The distinction between 

 class, class-concept, and concept of class, which is of 

 fundamental importance to exact thinking, is made 

 admirably clear, but the same cannot be said of what 

 is necessary > to constitute a class. A class may be 

 defined either extensionally, by an enumeration of its 

 terms, or intensionally, by the concept which denotes 

 its terms. The former method seems applicable only 

 to finite classes; we cannot agree with the author that 

 it is logically, though not practically, applicable to 

 infinite classes, unless some meaning is attached to 

 the word " enumeration " different from what is 

 ordinarily understood. On the other hand, the latter 

 method implies that a class is defined by a predicate, 

 I and contains those terms of which the predicate is 

 predicable ; but this leads to an apparent contradiction 

 which Mr. Kussell has discovered ; for consider the 

 NO. 1766, VOL. 68] 



predicates which are not predicable of themselves, for 

 example, humanity, which is not human; "not pre- 

 dicable of itself" seems to be a predicate defining a 

 class of predicates, yet to suppose that this defining, 

 predicate either is, or is. not, contained in that class, 

 leads to a contradiction. A similar contradiction is 

 reached when we consider the class whose terms 

 are all the classes,; each of which does not constitute 

 as one a term of itself as many ; for in attempting to 

 form this class, at any stage the terms already 

 obtained constitute a class which must be included as 

 a new term, and so on. This may be compared with 

 the attempt to sum a numerical series each of whose 

 terms is the sum of all the preceding terms ; the com- 

 parison does not completely explain the paradox, but 

 suggests that a distinction should be made among m- 

 finite classes somewhat like that between convergence 

 and divergence. 



Leaving the logical side of the subject, w-e come 

 to the first mathematical idea to be defined, that of 

 number. It <\-as formerly supposed that the notions 

 of " I " and " +1 " were fundamental, and that from 

 them all other numbers could be defined. In the pre- 

 sent work the number of terms in a class is defined, 

 in a manner slightly differing from Peano's, as the 

 class of all classes similar to the given class. 

 Similarity depends on a one-one relation, which can 

 be defined without reference to "number, and indicates 

 by Mr. Russell's " principle of abstraction " the pos- 

 session of a common property which may be called the 

 number. \'arious reasons are given for preferring 

 this definition, one of the chief being the inclusion of 

 the infinite numbers introduced by Cantor. 

 ' Part iJi. deals with quantity and magnitude,' between 

 which a subtle distinction is drawn, and contains an 

 introduction to the problems of infinity and continiiity, 

 which are to be more fully discussed in part v. Part 

 iv. develops the difficult theory of order and Dedekind's 

 theory of integers. The next part is necessarily based 

 largely on the work of Cantor. To readers un- 

 acquainted with the '■ Mengenlehre," the introduction 

 of transfinite numbers must appear rather startling, 

 but this is perhaps partly due to aii unusual weakness 

 in the English language. It must be remembered 

 that by a trjyisfinite carding! number is meant a certain 

 kind of infiniteness of aggregate, the same number 

 belonging to different aggregates which are similar 

 in the preceding sense; and a, transfinite ordinal 

 number is another name for a type of infinite -series, .or 

 of generating relation. • . , - ' 



In the chapters on real numbers and irrationals, jve 

 approach controversial ground. The particular object 

 which the arithmetisers of mathematics have here in 

 view is to complete the series of rational hiimbers W 

 the introduction, without any appeal to' intuition, of 

 other numbers, so as to satisfy the abstract 'definition' 

 of continuity. One consequence of this will'' be that 

 it will then be possible to assign a real number to 

 every point on a straight line. Three great thinkers 

 — Dedckind, Weierstrass and Cantor — have done this, 

 making their definitions of an irrational number de- 

 pend upon the theory of limits. '' Their methods'are 

 j explained and criticised, the chief objection beihg that 

 .^i'i .•'.7/;.:': J .orf 



