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.
KeywordsPhysical Space Elliptic Geometry Euclidean Geometry Positive Curvature Hyperbolic Geometry
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- Euclid, The Elements.Google Scholar
- H. P. Manning, Introductory non-Euclidean Geometry, New York, 1963.Google Scholar
- R. Bonola, Non-Euclidean Geometry (trans. H. S. Carslow), New York, 1955.Google Scholar
- A. Einstein, The Foundation of the General Theory of Relativity ’ (originally published 1916) and ‘Cosmological Considerations on the General Theory of Relativity’ (originally published 1917) in A. Einstein et al., The Principle of Relativity (trans. W. Perrett and G. B. Jeffery), London, 1923.Google Scholar
- I. Kant, Critique of Pure Reason, Transcendental Aesthetic.Google Scholar
- H. Poincaré, Science and Hypothesis (originally published 1902), New York, 1952, part ii.Google Scholar
- H. Reichenbach, The Philosophy of Space and Time (originally published 1928) (trans. M. Reichenbach and J. Freund), New York, 1957, chapter 1.Google Scholar
- Rudolf Carnap, Philosophical Foundations of Physics, New York and London, 1966, part iii. (This gives a simple modern exposition of Reichenbach ‘s ‘conventionalist’ standpoint.)Google Scholar