The Geometry of Space

Abstract

LET us begin discussion of the question of the geometry of Space by making the well known distinction between pure and physical geometry. The basic element of pure geometry is the point, a term usually left undefined. Which other terms are undefined will depend on the way in which the geometry is axiomatised; but ‘distance’ and the relation of being ‘collinear with’ are often also undefined. Other terms such as ‘line’ and ‘plane’ may be defined in terms of the undefined terms, axioms set up and theorems proved therefrom. We can then give any interpretation we like to this axiomatic system, and if the axioms are true, their consequences, the theorems, will be true also. Different sets of axioms or different interpretations of the axioms lead to different ‘spaces’ in the metaphorical sense of the term. Thus a Hausdorff space means a collection of ‘points’, the relations between which satisfy Hausdorff’s axioms, while ‘momentum space’ is a collection of ‘points’, each ‘point’ representing a different possible state of momentum of a particle.