92 SCIENCE PROGRESS 



calculus are not reviewed in this paper, though the author will 

 shortly deal with them in a paper to be published elsewhere. 



Logic and Principles of Mathematics. — The question as to 

 what Zeno really meant to prove or disprove by his arguments 

 on motion has excited some interest in modern times on account 

 of the enormously important logical questions about infinity 

 and continuity which were first raised by these arguments. 

 Philip E. B. Jourdain (Mind, 1916, 25, 42) attempts to give a 

 connected account of Zeno's purpose, which account is some- 

 what different from the well-known interpretation of Paul 

 Tannery ; and it is interesting to read how the modern theory 

 of aggregates permits us to disprove a plausible " proof " that 

 a flying arrow does not move even in a continuous space. 



The strange views of some eminent mathematicians, such 

 as Poincare, Schoenflies, and many others, in their treatment 

 of the logical difficulties which underlie questions of funda- 

 mental importance in mathematics are reviewed, among many 

 other developments and views of Russell's philosophy, by 

 Jourdain (Monist, 1916,26, 24). At bottom these views simply 

 reduce to the method of not mentioning the difficulties, and 

 we thus have an instance of a curious logical weakness of 

 very many mathematicians. 



The views of the late Julius Konig of Budapest on these 

 questions were fairly fully given to the world in his Nene 

 Grundlagen der Logik, Arithmetik und Mengenlehre (Leipzig, 

 1 914), and an account of this book is given by G. Vivanti 

 (Boll, di bibl. e st. delle sci. mat. 1916, 18, 37). 



Frege's treatment of psychological logic in his Grundgesetze 

 is translated in the Monist (191 6, 26, 182). It is a necessary 

 preliminary to his logical views, which are of such deep im- 

 portance to the principles of mathematics. An example of how 

 mathematicians are led nowadays to logic is afforded by the 

 perception by reflecting ones among them that verbs express 

 generalised functions. Such glimpses of the entities with which 

 thought deals is not only of great logical and philosophical 

 importance ; it has even shown itself to be of importance in 

 the technical development of mathematics. 



There have been recently published two papers dealing with 

 the formulation of branches of mathematics by axioms. In 

 the first place, Dr. R. L. Moore (Bull. Amer. Math. Soc. 1916, 

 22, 225) shows that a certain non-metrical pseudo-Archimedean 



