空間と幾何学 (完)

Abstract

The concept of space is not simple, but complex. It is composed of three element-concepts : extension, qualitative relation, and quantitative relation. The aim of this article is to investigate some philosophical issues about space and geometry from this point of view. I am concerned with three kinds of space and geometry : abstract space and abstract geometry, intuitively apprehensible space (for short intuitive space) and intuitive geometry, and empirical space and empirical geometry. Both of the second and third geometry are models of the first. The three element-concepts of the first geometry, of course, have not any concrete meaning, but although their meaning is indefinite, its scope is defined implicitly by the axioms of the geometry. Concerning this space and geometry, I examine characteristics of an axiomatic system, giving an example of Hilbert's axiomatic system of Euclidean geometry and comparing it with that of Euclid himself. Some people, empiricists in particular, may be opposed to assuming the existence of the second space and geometry, intuitive space and intuitive geometry, but if we are prepared to recognize physical space having extension, this assuming is inevitable. Hence those who reject this assumption, like Whitehead and Russell, have to construct their physical space from data given directly to their sense-experience, but this logically constructed space has not extension at all. The three element-concepts of intuitive space and intuitive geometry are intuitively apprehensible extension, topological relation, and metric relation determined by conventions. I characterize the intuitive space and intuitive geometry in detail from these three aspects. For example, intuitive space differs from visual space with respect to topological relation, that is to say, the former is divisible infinitely, while the latter has minimum extension (quanta of sense-percepts or Berkeley's and Hume's minimum sensibile). Accordingly we cannot construct non-trivial exact geometry in visual space. Giving another example, I show that intuitive space is neither Euclidean nor nonEuclidean in the sense that we are able to construct both Euclidean geometry and non-Euclidean geometries in that space. This conclusion depends mainly on the third respect above mentioned. Finally I add that statements of intuitive geometry are not analytic, but synthetic. Concerning the third space, empirical space, I distinguish physical space from perceptual space. My first issue about this third space is to elucidate the relation between physical space and perceptual space. This problem, however, depends on that of empirical interpretation of geometry, so that I must first solve the latter. My physical space with extension is inferred from perceptual space by the use of intuitive space. For I take the trichotomy of space and geometry instead of Whitehead's and Russell's dichotomy, which does not approve of intuitive space and intuitive geometry. I clarify the relation between this physical space and perceptual space. The three element-concepts of physical space and physical geometry are empirical extension, topological or situational relation, and metric relation determined by a procedure of measurement. I examine the issues about measurement of physical space in detail, and conclude that metrics of physical space are also, like intuitive space, conventional. They are not intrinsic, but extrinsic to physical space. From this results the important thesis of the Riemann-Poincaré conventionalism that A. Grünbaum made clear. Lastly I introduce and appraise briefly Grünbaum's critical exposition of three other sorts of conventionalism.