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XIV. On Mathematical Concepts of the Material World. 



By A. N. WHITEHEAD, Sc.D., F.R.S., Felloiv of Trinity College, Cambridge. 



Keceived September 22, Head December 7, 1905. 



PREFACE. 



THE object of this memoir is to initiate the mathematical investigation of various 

 possible ways of conceiving the nature of the material world. In so far as its results 

 are worked out in precise mathematical detail, the memoir is concerned with the 

 possible relations to space of the ultimate entities which (in ordinary language) 

 constitute the "stuff" in space. An abstract logical statement of this limited 

 problem, in the form in which it is here conceived, is as follows : Given a set of 

 entities which form the field of a certain polyadic (i.e., many-termed) relation R, 

 what "axioms" satisfied by R have as their consequence, that the theorems ot 

 Euclidean geometry are the expression of certain properties of the field of R ? If the 

 set of entities are themselves to be the set of points of the Euclidean space, the 

 problem, thus set, narrows itself down to the problem of the axioms of Euclidean 

 geometry. The solution of this narrower problem of the axioms of geometry is 

 assumed (cf. Part II. , Concept I.) without proof in the form most convenient for this 

 wider investigation. But in Concepts III., IV., and V., the entities forming the field 

 of R are the " stuff," or part of the " stuff," constituting the moving material world. 

 POINCARE* has used language which might imply the belief that, with the proper 

 definitions, Euclidean geometry can be applied to express properties of the field 

 of any polyadic relation whatever. His context, however, suggests that his thesis 

 is, that in a certain sense (obvious to mathematicians) the Euclidean and certain 

 other geometries are interchangeable, so that, if one can be applied, then each of 

 the others can also be applied. Be that as it may, the problem, here discussed, is 

 to find various formulations of axioms concerning R, from which, with appropriate 

 definitions, the Euclidean geometry issues as expressing properties of the field of R. 

 In view of the existence of change in the material world, the investigation has to 

 be so conducted as to introduce, in its abstract form, the idea of time, and to provide 

 for the definition of velocity and acceleration. 



The general problem is here discussed purely for the sake of its logical (i.e., 

 mathematical) interest. It has an indirect bearing on philosophy by disentangling the 

 essentials of the idea of a material world from the accidents of one particular concept. 

 The problem might, in the future, have a direct bearing upon physical science if a 

 concept widely different from the prevailing concept could be elaborated, which 

 * Cf. ' La Science et 1'Hypothese,' chap. III., at the end. 



VOL. COV. A 400. 3 16.5.06 



