RECENT ADVANCES IN SCIENCE 7 



and an extract from the second chapter, on the logical relations 

 of concepts and propositions,' is given in the Rev. de Metaphys. 

 (191 7, 24, 15-58). In this work the trace of Frege's ideas is 

 almost as marked as that of Russell's, but although functions 

 (which Couturat strangely says are " expressions ") of the 

 " second order " (functions of functions) are mentioned, it is 

 added that " these functions give rise to certain difficulties and 

 paradoxes which require attentive study and special precautions, 

 but we have not to deal with them in this elementary treatise " 

 (p. 16). Again (p. 28) the axiom of Frege which allows us to 

 pass from two equal classes to the equivalence of the preposi- 

 tional functions (such as x e a) which correspond to them, is 

 mentioned as giving rise to the difficulties discovered by Russell 

 when it is applied to functions of the second order. Here again 

 further developments are explicitly dismissed. 



A translation of those parts of Frege's Grundgesetze which 

 deal with function, concept, class, and relation is given in the 

 Monist (191 7, 27, 114-27) ; and with this it is interesting to 

 compare a previous account (ibid. 191 6, 26, 415-27) of Dede- 

 kind's work on logic and arithmetic. 



Philip E. B. Jourdain (Scientia, 191 7, 21, 1-12) attempts 

 to point out the function of symbolism in mathematical logic — - 

 a question discussed by Rignano and Peano in 191 5 (cf. Science 

 Progress, 10, 116 and 432). Jourdain also (Monist, 1917, 27, 

 142-51) discusses the " entity " of a number or other logical 

 thing as distinguished from its " existence." 



N. Wiener (Trans. Amer. Math. Soc. 191 7, 18, 65-72) con- 

 siders certain formal in variances in Boolean algebras ; and E. V. 

 Huntington (Proc. Nat. Acad. Sci., Washington, D.C., 1916, 

 2, No. 11) gives a set of five independent postulates for cyclic 

 order. 



With regard to the paper by F. Hartogs before mentioned in 

 Science Progress (191 7, 11, 453), the essential point in what is 

 proved is that it is always possible to construct a well-ordered 

 aggregate such that its cardinal number is greater than that of 

 any given aggregate M. From this follows that, if we assume 

 that all aggregates are comparable, any aggregate M can be 

 well-ordered. 



Arithmetic and Algebra. — W. Hope- Jones (Math. Gaz. 191 7, 

 9, 5-8) applies the principles of the theory of probability 

 to approximations in arithmetic, and decides, for example, that 



