670 SCIENCE PROGRESS 



been noticed in Science Progress, 1918, 12, 541) ; operators and the 

 reduction of mathematics to algorithms — to purely arbitrary modes of com- 

 binations of operators (p. 95) ; hypernumbers and the reduction of mathe- 

 matics to algebra — to statements in terms of hypernumbers (p. 113) ; the 

 reduction of mathematics to " processes " — to modes of transition from a 

 given object or arrangement of objects to another object or arrangement of 

 objects (p. 117) ; and " the active exercise of the thinking function of mind " 

 (p. 122) as shown in systems of inferences. Chapters x-xiii find that mathe- 

 matics cannot be defined by reference to any particular one of its four dis- 

 tinct and independent principles : form (cf. pp. 9, 130.. 132, 135), invari- 

 ance (cf. pp. 9, 140-1), functionality (p. 146), inversion — the consideration 

 of problems leading to the creation of new mathematical entities, inversion 

 being the principle of " creating a class of objects that will satisfy certain 

 defining statements " (p. 152 ; cf. pp. 9, 149). The last three chapters are 

 inquiries into the source of the reality that is in mathematics, its method 

 of arriving at truth, and the realm of validity of its conclusions. 



On the whole it is, I think, impossible not to assent to Prof. Shaw's con- 

 clusion that it is impossible to reduce the whole of what most of us mean 

 by " mathematics " to any one of the categories marked off by him. It is 

 only by tacitly limiting the significance of the word " mathematics " to the 

 subject-matter alone that Russell and others came into conflict with Poin- 

 care, who tacitly supposed — like most of us — that " mathematics " denotes 

 something which includes certain mental activities, and which has thus a 

 life and a history and contains methods of discovery as well as logical sub- 

 ject-matter. But, while Prof. Shaw seems right in one of his main con- 

 tentions, I cannot help feeling that his distinction between a " statical " 

 and a "dynamical" part of mathematics (pp. 8, 117, 126) rests on an 

 analogy which is misleading : a transmutation, for example, can be defined 

 as a class of relations in as " statical " a way as an irrational number. Also 

 the point of the book is not, I think, made any clearer by the extraordinary 

 rhetoric (cf. pp. 1-8). For example, we read: "Could a powerful tele- 

 scope show us the antipodes, or could an electron wind its tortuous way 

 according to a law expressed by the Weierstrass non-differentiable func- 

 tion ?" (p. 3). 



On pp. 25-7 there is a mistaken attempt to discredit Cantor's proof that 

 there are more irrationals than rationals, by pretending that the proof 

 depends on the purely arbitrary assignment of an infinite set of coefficients. 

 Cantor's proof of 1892 explicitly avoided this arbitrariness, and, of course, 

 is not affected by the other " objection " (p. 27) that one and the same in- 

 finite collection can be arranged to correspond completely or not with one 

 another collection. What Cantor proved is that, where R is any one-one 

 correlation of a collection M with the collection of parts of M, there always 

 can be defined, as a function of R, a definite part of M to which no member 

 of M is correlated by R. 



If, indeed, Russell alone — as seems a mistake which Prof. Shaw shares 

 with many people — is to be credited with the theory that mathematics is 

 capable of a reduction to logical terms expressed by symbols of " logistic," 

 it is not difficult to understand the dislike which responsible mathematicians 

 have to this " logistic " in view of the frivolous and unedifying remarks on 

 truth and falsehood published by Russell in 1904, and quoted with deserved 

 scorn on pp. 162-3. It is hardly to be wondered at that no attempt should 

 have been made to settle clearly the merits of the dispute between Poincare 

 and Russell. 



In former days, such subjects as the nature of infinity and continuity 

 belonged to philosophy : now, thanks to some mathematicians of the last 

 half of the nineteenth century, they belong to mathematics. The logically 

 exact definition and investigation of such fundamental concepts as number, 



