432 SCIENCE PROGRESS 



As a result of Rignano's articles referred to in Science 

 Progress of July 191 5 (10, 116), Giuseppe Peano (Scientia, 

 191 5, 18, 165) has treated, in an interesting article, the function 

 of symbols in mathematics. The part played by symbolism 

 in arithmetic, algebra, and the geometry of vectors is first 

 treated, and stress is laid upon the fact that symbolism 

 provides a means both of shortening work and, what is much 

 more important, of giving a new classification of ideas. In 

 arithmetic this may be illustrated by the Hindu-Arabic 

 system of numeration, and in the symbols of algebraic opera- 

 tions by the fact that the verbal equivalents of formulae involve 

 words which stand also for ideas not used in those formulae. 

 In the evolution of algebraic symbolism there are three 

 stages: (1) Ordinal language; (2) the technical language of 

 Euclid, in which there is a one-to-one correspondence between 

 words and ideas ; (3) the abbreviation of the words of this 

 technical language, which began about the sixteenth century 

 and was nearly complete in the system used by Newton. 

 Mathematical logic or the algebra of logic was the last to appear, 

 but is in no way inferior to the above symbolisms. In any 

 mathematical reasoning there are specially mathematical and 

 general logical ideas ; mathematical logic represents the latter 

 by symbols and finds that the ideas are subject to the rules of 

 a calculus very like algebra. This was Leibniz's and Boole's 

 discovery. Against those who consider mathematical logic as 

 a science in itself, Rignano's criticisms are quite just, but not 

 against those who consider it as an instrument for solving 

 mathematical problems which resist the ordinary methods. 

 It seems to the reviewer that this is rather an unnecessarilv 

 restricted view to take of mathematical logic, and that the 

 proper reply to Rignano is that, until comparatively lately, 

 symbolism in mathematics and logic had simply the end of 

 helping reasoning by brevity or analogy or by showing the 

 mechanical character of logical deductions (of course this has 

 no psychological implications), but that a great part of what 

 modern mathematical logic does is to increase our subtlety by 

 emphasising differences in reasoning instead of analogies. 



Prof. E. B. Van Vleck (Bull. Amer. Math. Soc. 191 5, 21, 

 321) gives a most interesting address summarising the part 

 played by the theory of sets of points in geometry and the 

 closely connected subject of dynamical trajectories. Up to 



