XIV. On mathematical concepts of the material world

Abstract

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 of 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. Poincaré 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 allowed of a simpler enunciation of physical laws. But in physical research so much depends upon a trained imaginative intuition, that it seems most unlikely that existing physicists would, in general, gain any advantage from deserting familiar habits of thought.