ESSAY-REVIEWS 



LOGIC AND MATHEMATICS, by the late P. E. B. Jourdain, M.A. : 

 on — 

 Lectures on the Philosophy .of Mathematics, by James Byrnie Shaw. 

 [Pp. viii + 206.] (Chicago and London : The Open Court Pub- 

 lishing Company, 19 18. Price 6s. net.) 



An Introduction to Mathematical Philosophy, by Bertrand Russell. 

 [Pp. viii + 208.] (London : George Allen and Unwin, Ltd. ; New 

 York : The Macmillan Company. Price 105. 6d. net.) 



The object of Prof. Shaw's Lectures " is to consider the whole field of mathe- 

 matics in a general way, so as to arrive at a clear understanding of exactly 

 what mathematics undertakes to do, and how far it accomplishes its pur- 

 pose ; to ascertain upon what presuppositions, if any, which are extra- 

 mathematical, the mathematician depends" (p. vi). With this end in 

 view, Prof. Shaw describes in some detail the subject-matter of mathematics 

 and the predominant characters of the objects with which mathematics 

 concerns itself. He finds that the objects it studies are numbers, figures, 

 arrangements, propositions and prepositional functions, operations, hyper- 

 numbers, processes leading to transmutations, and deductive systems ; the 

 chief characters of the objects are those of form or structure, in variance, 

 functionality, and inversion. But no one of these different kinds of objects 

 or principles — or even the sources or methods or regions of validity of 

 mathematics, which we shall mention afterwards — can furnish a satisfactory 

 definition of mathematics which would include the entire subject (pp. 154, 

 6-1 1). 



The second chapter discusses number and the arithmetisation of mathe- 

 matics. The five stages in the development of the idea of number are the 

 invention of integers, of fractions, of incommensurables, of what Prof. 

 Shaw calls the " ensemble " (p. 24), and of the very general ranges that 

 appear in the work of Frechet, Moore, and Volterra (p. 29). " For our pre- 

 sent purpose it is simply sufficient to cite these [last] investigations, in which 

 physical intuition is helpless, to prove our general thesis that mathematics 

 is a creation of the mind, and is not due to the generalisation of experiences 

 or to their analysis ; nor is it due to an innate form or mould which the 

 mind compels experience to assume ; but is the outcome of an evolution, 

 the determining factors of which are the creative ability of the mind and 

 the environment in which it finds the problems which it has to solve in some 

 manner and to some degree " (p. 30). Every one of the different branches 

 of mathematics leads to the same conclusion, as is shown in chapters iii-ix, 

 which are respectively on space and the geometrisation of mathematics — 

 this geometrisation being " that every problem of analysis has a geometrical 

 interpretation, and every problem of geometry may be formulated ana- 

 lytically " (pp. 38-9) ; arrangements of a group of objects, and mathematical 

 tactic ; logistic and the reduction of mathematics to logic (which has already 



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