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XLIX. Notices respecting New Books. 



Lectures on the Philosophy of Mathematics. By James Byrnie 

 Shaw. Chicago-London : The Open Court Publishing Co., 

 1918. Pp. viii + 206. Price 6s. net. 



T~\E,. SHAW, without entering very deeply into the subject, has 

 ^-J undoubtedly succeeded in producing an extremely stimu- 

 lating and suggestive little book, which every mathematician and 

 physicist will find very useful. The author divides the subject- 

 matter of mathematics into eight " divisions " thus : Static mathe- 

 matics containing four divisions, viz. 1. Numbers, leading to 

 arithmetic; 2. Figures (geometry); 3. Arrangements, leading 

 to tactic ; and 4. Propositions (logistic), and Dynamic mathe- 

 matics consisting of 5. Operators, leading to the corresponding 

 calculus; 6. Hypernumbers (algebra); 7. Processes (transmu- 

 tations) ; and 8. Systems, leading to general inference. These 

 divisions are treated in as many chapters, II. to IX., the first 

 chapter being dedicated to generalities about the meaning of the 

 Philosophy of Mathematics. Independently of these divisions 

 the author considers the "central principles" of mathematics to 

 be classifiable under four heads : Form, In variance, Functionality, 

 and Ideality or " Inversion." JEach of these central principles 

 appears in each of the said eight divisions. The four principles 

 are treated in Chapters X. to XIII. Three more chapters, on the 

 Sources of Mathematical Reality, on the Methods and the Validity 

 of Mathematics, together with a Table exhibiting the said divisions 

 and principles, complete this work. General references aud 

 special quotations enhance its value. A very welcome feature 

 of the book is the emphasis put on the creative role of the 

 mathematician. The only spot on the sunny surface of this 

 beautiful and carefully chiselled work is a slip of the pen on 

 page 42: "similar figures exist" in Lobatchevskyan geometry. 

 This will be read, of course, " do not exist." Dr. Shaw's book 

 may be warmly recommended to all interested in pure or applied 

 mathematics. 



