MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 467 



Definition. The complete class of those entities, which are members of the fields 

 of fundamental relations, is called the class of Ultimate Existents. This technical 

 name is adopted without prejudice to any philosophic solution of the question of the 

 true relation to existence of the material world as thus conceived. 



Every concept of the material world must include the idea of time. Time must be 

 composed of Instants (cf. BERTRAND EUSSELL, ' Mind,' N.S., vol. 10, No. 39). Thus 

 Instants of Time will be found to be included among the ultimate existents of every 

 concept. 



Definition. The class of ultimate existents, exclusive of the instants of time, will 

 be called the class of Objective Reals. 



The relation of a concept of the material world to some perceiving mind is not to 

 be part of the concept. Also we have no concern with the philosophic problem ol 

 the relation of any, or all, of these concepts to existence. 



In Geometry, as derived from the Greeks, the simple elements of space are points, 

 and the science is the study of the relations between points. Points occur as 

 members of the fields of these relations. Then matter (the ultimate " stuff" which 

 occupies space) in its final analysis, even if it is continuous, consists of entities, here 

 called particles, associated with the points by relations which are expressed by 

 saying that a particle occupies (or is at) a point. Thus matter merely occurs as one 

 portion of the field of this relation of occupation ; the other portion consists of points 

 of space and of instants of time. Thus " occupation " is a triadic relation holding in 

 each specific instance between a particle of matter, a point of space, and an instant of 

 time. According to this concept of a material world, which we will call the Classical 

 Concept, the class of ultimate existents is composed of three mutually exclusive 

 classes of entities, namely, points of space, particles of matter, and instants of time. 

 Corresponding to these classes of entities there exist the sciences of Geometry, of 

 Chronology, which may be defined as the theory of time considered as a one- 

 dimensional series ordinally similar to the series of real numbers, and of Dynamics. 

 There appears to be no science of matter apart from its relations to time and space. 



Opposed to the classical concept stands LEIBNIZ'S theory of the Relativity of Space. 

 This is not itself a concept of the material world, according to the narrow definition 

 here given. It is merely an indication of a possible type of concepts alternative to 

 the classical concept. It is not very obvious how to state this theory in the precise 

 nomenclature here adopted. The theory at least means that the points of space, as 

 conceived in the classical concept, are not to be taken among the objective reals. 

 But a wider view suggests that it is a protest against dividing the class of objective 

 reals into two parts, one part (the space of the classical concept) being the field of 

 fundamental relations which do not include instants of tune in their fields, and the 

 other part (the particles) only occurring in the fields of fundamental relations which 

 do include instants of time. In this sense it is a protest against exempting any part 

 of the universe from change. But it is not probable that this is the light in which 



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