MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 471 



members of the " field " of K, other than the instants ot time, are to be considered as 

 lines taken as simple entities. Points are classes of these simple lines. But the 

 ordinary line of geometry which has parts and segments is a class of points, and so is 

 the ordinary plane of geometry. In Concept III., which is Leibnizian and monistic, 

 the points (perhaps " particles " is here a better word) move, and the straight lines 

 and planes disintegrate from instant to instant. In Concepts IV. and V. the points 

 similarly disintegrate. 



(ii) EXPLANATION OF SYMBOLISM. 



This explanation is only concerned with the general logical symbolism. The 

 special symbols which arise out of the ideas of the paper are defined in their proper 

 places. PEANO'S* chief symbols are used. The changes and developments from 

 PEANO, which will be found here, are due to RUSSELL and myself working in 

 collaboration for another purpose. It would be impossible to disentangle our various 

 contributions, t 



None of the reasoning of the paper is based upon any peculiarity of the symbolism. 

 It is used here only as an alternative form for enunciations, for the sake of its 

 conciseness and (above all) its precision. In the verbal enunciations precision has 

 been to some extent sacrificed to lucidity ; and the exact statement of what is meant 

 is always to be sought in the symbolic alternative form. The proofs have been 

 translated into words out of the symbolic form in which they were mostly elaborated. 



On D, =, c, e, =, = Df 



There are five copulas, namely, D, =, c, e, =. Here x^y means x implies y ; and 

 x == y means x implies y and y implies x; and xcy means all x's are y's; and xey 

 means x is a member of y ; and x, yen means x and y are members of u. Note that 

 xcy implies that x is a subclass of y ; a class will be said to contain a subclass and 

 to possess a member. Lastly, x = y means x is identical with y. Note that, if Df, 

 short for Definition, is placed at the end of the line, thus, 



x = y Df 



the symbols mean that x is defined to stand for y. In such a case y is some complex 

 of symbols, and x will be an abbreviated symbol standing for y. 



On (f>lx, (x).<f>!x, fax) . <l>\x, (x, y), fax, y), D, 



Propositioned Functions. <f>lx means x has the property <j>, where <j> is given 

 different forms corresponding to different properties; ftlx means x has the property 



* Cf. 'Notations de Logique Mathematique,' Turin, 1894; .and ' Formulaire Mathematique,' Turin, 

 1903. 



t See, however, RUSSELL'S articles, " Sur la Logique des Relations," ' Revue de Mathematiques,' 

 vol. VII., 1900-1901, Turin. 



