MOTIONLESS MOTION 291 



boundary of what contains.^ For modern science, as Cassirer 

 says, " Objective reality passes from place to change of 

 place. . . ." This fundamental opposition shows up again in 

 the analysis of the notion of " between," In V Physics, chapter 

 3, Aristotle defines the terms " together," " apart," " in con- 

 tact," " between," " in succession," " contiguous," and " con- 

 tinuous." In all but one of these terms the definition given 

 applies to mathematical objects as well as to things as they 

 exist in Nature. The one exception is the term " between." 

 " Between," he says, " is that which a changing thing, if it 

 changes continuously in a natural manner, naturally reaches 

 before it reaches that to which it changes last." The peculiar 

 nature of this term is well recognized by the modern phi- 

 losopher. Here is what Cassirer has to say about it: 



The evolution of modern mathematics has approached the ideal, 

 which Leibniz established for it, with growing consciousness and 

 success. Within pure geometry, this is shown most clearly in the 

 development of the general concept of space. The reduction of 

 metrical relation to projective realizes the thought of Leibniz that, 

 before space is defined as a quantum, it must be grasped in its 

 original qualitative peculiarity as an ' order of coexistence ' (ordre 

 des coexistences possibles) . The chain of harmonic constructions, 

 by which the points of projective space are generated, provides the 

 structure of this order, which owes its value and intelligibility to 

 the fact that it is not sensuously presented but is constructed by 

 thought through a succession of relational structures. ... In this 

 sense, modern geometry seeks to free a relation, such as the general 

 relation of ' between,' which at first seems to possess an irreducible 

 sensuous existence, from this restriction and to raise it to free logical 

 application. The meaning of this relation must be determined by 

 definite axioms of connection in abstraction from the changing 

 sensuous material of its presentation; for from these axioms alone 

 is gained the meaning in which it enters into mathematical 

 deduction.^ 



These opposed notions of " motion," " place," and " between " 

 arise from a fundamental difference in the respective notions 

 of Nature and the natural. 



^Physics IV, ch. 4, 212a 20. ° Substance and Function, ed. cit., pp. 91-92. 



