468 DR. A. N. WHITEHEAD ON 



LEIBNIZ himself regarded the theory. This theory, though at present it is nominally 

 the prevailing one, has never been worked out in the form of a precise mathematical 

 concept. It is on this account criticized severely by BERTRAND RUSSELL (cf. loc. tit. 

 and 'Philosophy of LEIBNIZ,' Cambridge, 1900, p. 120), who, however, has gone 

 further than any of its upholders to give it mathematical precision. Of course, from 

 the point of view of this paper, we are not concerned with upholding or combatting 

 any theory of the material world. Our sole purpose is to exhibit concepts not 

 inconsistent with some, if not all, of the limited number of propositions at present 

 believed to be true concerning our sense-perceptions. 



Definition. Any concept of the material world which demands two classes of 

 objective reals will be called a Dualistic concept ; whereas a concept which demands 

 only one such class will be called a Monistic concept. 



The classical concept is dualistic ; Leibnizian concepts will be, in general, monistic 

 (cf. however Concept IV A.). OCCAM'S razor Entia non multiplicanda prseter 

 necessitatem formulates an instinctive preference for a monistic as against a 

 dualistic concept. Concept III. below is an example of a Leibnizian monistic concept. 

 The objective reals in it may be considered to represent either the particles or the 

 points of the classical concept. But they change their spatial relations. Perhaps 

 LEIBNIZ was restrained from assimilating his ideas more closely to Concept III. by a 

 prejudice against anything, so analogous to a point of space, moving a prejudice 

 which arises from confusing the classical dualistic concept with the monistic concepts. 

 It is of course essential that at least some members of the class of objective reals 

 should have different relations to each other at different instants. Otherwise we are 

 confronted with an unchanging world. Concept V. is another Leibnizian monistic 

 concept. 



The Time- Relation, In every concept a dyadic serial relation, having for its field 

 the instants of time and these only, is necessary. The properties of this Time- 

 Relation form the pure science of chronology. The time-relation is, in all concepts, a 

 serial relation ordinally similar to the serial relation which generates the series of 

 negative and positive real numbers.* This fact need not be further specified during 

 the successive consideration of the various concepts, nor need any of the propositions 

 of pure chronology be enunciated. 



Definition. The class of instants of time is always denoted by T in every concept. 



The Essential Relation. In every concept at least all the propositions of geometry 

 will be exhibited as properties of a single polyadic relation, here called the essential 

 relation. The field of the essential relation will consist, either of the whole class of 

 ultimate existents (e.g., in Concepts III., IVs. and V.), or of part of the class of 

 objective reals together with the instants of time (e.g., in Concept IVA.), or of the 

 whole class of objective reals (e.g., in Concept II.), or of part of the class of objective 



* For interesting reflections on this subject, influenced by the Kantian Philosophy and previous to the 

 modern " Logicization of Mathematics," cf. HAMILTON, ' Lectures on Quaternions,' preface. 



