NA TURK 



273 



THURSDAY, AUGUST 31, 1911. 



THE FOUNDATIONS OF MATHEMATICS. 

 toincipia Mathematica. By Dr. A. N. Whitehead, 

 F.R.S.. and B. Russell, F.R.S. Vol. i. Pp. 

 SV + 666. (Cambridge: University Press, 1910.) 

 Price 255. net. 



THIS work contains some thousands of proposi- 

 tions, each, with its proof, expressed in a short- 

 hand so concise that if tiny were all expanded into 

 ordinary language, the room taken up would be ten 

 ■ties as large at leas! ; space, time, and mass are 

 not considered at all, and arithmetic is merely fore- 

 shadowed by the introduction of the symbols o, 1, 

 2. and j,.. How thru, il may be asked, can the 

 author-, pretend to be writing about mathematics? 

 The answer amounts to saying that for every branch 

 ot the tree of knowledge there is a corresponding 

 root, and every advance in climbing seems to compel 

 a similar advance in delving, just as the discovery 

 of non-Euclidean geometries led to the reconsidera- 

 tion of geometrical axioms, so Cantor's invention of 

 translinile numbers has reacted upon the theory of 

 elementary arithmetic, and hence upon the whole of 

 analysis and all its applications. 



Besides this, there has grown up a school of mathe- 

 maticians intensely interested in the logical side of 

 their subject. Indeed. this was inevitable as soon 

 as the primary distinction between ordinal and 

 cardinal number was fully grasped, and the nature 

 of the arithmetical continuum had been strictly de- 

 fined. The inquirer was driven back and back to 

 fcestions of order, and correspondence, and relations, 

 and classes, until he felt bound to construct a sym- 

 bolical logic fit to express the chain of deductions 

 he found latent in the most familiar processes of 

 arithmetic. This has led to an immense aggregation 

 of what may be called mathematical prolegomena ; 

 and with this the first volume of the '* Principia 

 Hathematica " is almost exclusively concerned. 



Thus the actual titles of it- two parts are " .Mathe- 

 matical Logic" and "Prolegomena to Cardinal 

 Arithmetic," and both are so elaborate that only a 

 nieagie account of them can be given in a review. 

 The theory of deduction is based upon seven assump- 

 tions, called primitive propositions, and upon the 

 notions of disjunction (p or ./) and implication (either 

 not-/> or q). In about thirty pages the authors 

 obtain the main results of the purely formal logic of 

 propositions. This is followed by a very interesting 

 section on "apparent variables," including the theory 

 of propositions of different orders. A real advance 

 seems to have been made here in the analysis of 

 vicious-circle fallacies, and false generalisations, 

 especially as they occur in mathematical reasoning. 

 It is pointed out that such phrases as "all proposi- 

 tions" or "all properties of x " are strictly meaning- 

 less, and a legitimate use of such terms is based 

 upon an axiom of reducibility (pp. 173-5) which is 

 stated in the form: "Any function of one argument 

 or of two is formally equivalent to a predicative func- 

 tion of the same argument or arguments," and its 

 NO. 2183, VOL. 87] 



main use is at the beginning of the calculus of 

 classes (p. 197). Whether this axiom is really 

 simpler than the introduction of "class" as a primi- 

 tive term seems debatable, but it does not matter 

 much for practical purposes. 



The next three sections deal with classes and rela- 

 tions, and introduce a large number of new symbols 

 and a long series of propositions. Fortunately here, 

 as elsewhere, each section is preceded by a summary, 

 giving the principal theorems; and, in fact, the 

 reader will find it helpful to go through all these sum- 

 maries (after the introduction) before attacking the 

 chapters in detail. 



Coming now to the more directly mathematical 

 part, we have first of all (p. 336) a discussion of unit 

 :s, which illustrates the subtleties of this new- 

 calculus. Thus it is found necessary to construct 

 a symbol for "the class of which the only member is 

 v," as distinguished from x itself. At first this seems 

 to be superfluous, but when we suppose .v to be a 

 class, we see that it is not. The next step is to 

 define the cardinal number 1 as the class of all unit 

 classes. Similarly the cardinal number 2 is defined 

 as the class of all couples (x, y) such that (.v, y) and 

 (v. x) are equivalent; and the ordinal number 2,. 

 .is the i lass of ordered couples (x, y) such that (y, x) 

 is different from (x, y). Besides these we_ have a 

 symbol 2 for the class of all relations consisting of a 

 single couple, including couples (x, x). Then we 

 have a series of theorems on subclasses, relative 

 types of classes, one-one and one-many relations, &c, 

 leading up to the fundamental notion of similarity 

 of classes which is the necessary basis of all arith- 

 metic proper. 



We next come to the difficult question of selections, 

 from relations and from classes of classes respec- 

 tively. 



"If k is a class of classes, then v is called a 

 selected .lass of k when m is formed by choosing one 

 term out of each member ol /,-." 



(It would perhaps be more precise to say "a class 

 selected from /,-," because m, as a class, is not generally 

 a member of fe.) Now at first sight it looks as if a 

 selected class could always be formed, but this is not 

 reallv obvious when fc is infinite, and, in fact, it has 

 not been proved in general. If it could be, it would 

 follow that every class can be well-ordered, and the 

 difficulty of asserting this in general can be seen 

 from a special case. Consider the aggregate of 

 colours, merely as sensations of my own ; how^ can 

 I order them", without importing some additional 

 foreign element, such as the time when I first be- 

 came conscious of a particular one, or its analysis 

 In a colour-box, or something of that sort? Besides 

 this, there is the logical difficulty of making an asser- 

 tion' about "every" class, for one reason because 

 assertions form a'class.' Hence the section (p. 561) 

 on the conditions for the existence of selections is 

 one of special interest : its most important bearing on 

 arithmetic is in the theory of multiplication. 



The final section is on inductive relations, especially 



1 This is undeniable, because "assertions do not form a class" is itself an 

 as-ertion, and only a formal, not a real contradiction of the above statement. 



