RECENT ADVANCES IN SCIENCE 3 



R. Demos {ibid. 77-85), following up his reduction of nega- 

 tive propositions to descriptions of certain true propositions 

 in terms of their opposition to some other proposition (cf. 

 Science Progress, 191 7, 12, 192), applies an analogous reduc- 

 tion to modal propositions and propositions of practice. Demos 's 

 investigations are of great interest to those concerned with the 

 principles of mathematics, because they give examples of an 

 extension to certain propositions of B. Russell's theory of 

 descriptions. 



H. Lanz (Monist, 19 18, 28, 46-67) maintains that the 

 " actuality " which some mathematicians attribute to infinity 

 belongs " rather to the methodological than to the quantita- 

 tive character of infinite aggregates." 



W. Sierpinski (Compt. Rend. 19 16, 163, 688-91) has a note 

 on the part played in modern analysis by the axiom of Zer- 

 melo. 



L. E. J. Brouwer (Vers. Kon. Akad. van Wet. Amsterdam, 

 25, 1418-23) gives some addenda and corrections to his work 

 of 1907 on the principles of mathematics. Brouwer (ibid. 

 1424-6) gives two theorems in the theory of linear aggregates. 



C. Burs tin (Sitzungsber . der K. Akad. der Wiss. in Wien, 

 1915, 124 [Ha], 1187-202) considers the problem of splitting 

 up the number-continuum into an aggregate (which is of the 

 cardinal number of Cantor's second number-class) of every- 

 where dense aggregates, under the assumption of the exist- 

 ence of a well-ordered aggregate of the cardinal number re- 

 ferred to. Burstin also (ibid. 19 16, 125, 209-17), supposing 

 that the continuum can be well-ordered, shows that it can be 

 spilt up into an aggregate of c aggregates which are not mea- 

 surable in Lebesgue's sense. Here c is the cardinal number 

 of the continuum. 



M. Souslin (Compt. Rend. 19 17, 164, 88-91) has a note on 

 a definition of measurable (Baire) aggregates without trans- 

 finite numbers (cf. also N. Lusin, ibid. 91-6). 



G. Pal (Math, es phys. lapok, Budapest, 191 5, 24, 236-42) 

 proves that every continuous plane curve, considered as a 

 point-aggregate, is the projection of a spatial Jordan's curve. 



D. Konig (Math, es term. ert. Budapest, 1916, 34, 104-19) 

 reproduces the article on graphs and their application to the 

 theories of aggregates and determinants which appeared in 

 Math. Ann. (1916, 77, 455-65 ; Rev. sem. 25 [1], 278). 



