RECENT ADVANCES IN SCIENCE 193 



in for the purpose of reconciling the two definitions of finite 

 classes, is treated. 



R.J. Kortmulder has published (De logische grondslagen der 

 wiskunde, Amsterdam, 191 6) his doctorial thesis on the logical 

 and philosophical principles of mathematics, more particularly 

 arithmetic. Finite and transfinite numbers, both cardinal and 

 ordinal, are considered. With this may be compared a paper 

 by Kortmulder with the same title in Wiskundig Tijdschrift, 

 191 5-6, 12, 214-7). 



K. Vorovka (Casopis pro pestovdni mathematiky a fysiky, 

 Prague, 1914, 43, 154-62) writes on Henri Poincare's judgment 

 on the relations of mathematics to logic. 



A. Korselt {Jahresber. der Deutschen Mathematiker-Vereini~ 

 gung, 1916,25, 132-8) has a paper professing to give a solution of 

 the paradoxes of Burali-Forti, Bernstein, Russell, and Richard. 



F. Enriques {Rev. de Metaphys. et de Morale, 191 7, 24, 149- 

 64) considers the question of whether there is any validity in 

 some work on the infinite in mathematics. " Suppose that 

 there is potentially given by thought an infinity of objects, the 

 question is, is there any reason to consider as logically defined a 

 new object of thought which expresses the totality or the limit, 

 even when the objects spoken of are not constructed with 

 reference to a concept of that kind which we suppose given 

 a priori." This kind of " realism," which is chiefly due in a 

 mathematical respect to Georg Cantor " and of which the 

 philosopher B. Russell has developed in the widest sense the 

 philosophical consequences," rests on the fundamental principle 

 that " every infinity of objects virtually defined can be con- 

 sidered as a totality forming a class and constituting a new 

 logical object." Modern work on mathematical logic has 

 shown this principle to be in error in certain cases, and Enriques 

 points out this defect of new realism. It seems to the reviewer, 

 however, that this criticism wholly neglects the essential fact 

 that since 1905 Russell has explicitly abandoned the supposition 

 that there are such things as classes, and that a great part of 

 Cantor's theory can be expressed without supposing that there 

 are such things as classes at all. 



Also the theory of Russell that we need not assume that 

 there are any such things as " classes " is quite neglected in a 

 paper by D. Mirimanoff (U Enseignement math. 191 7, 19, 37-52). 

 The solution suggested in this paper of the contradictions of 



