The Enigma Of The Universe : Space: Euclidean or Non-Euclidean?

Published: 26.09.2014

Thus, upto the end of the nineteenth century, the various forms of non-Euclidean geometry were developed. But the question as to which doctrine-the Euclidean, or some kind of non-Euclidean-is true, that is to say, is the geometry of the actual universe, was not settled.

Here it is necessary to distinguish between pure or mathe­matical geometry and physical geometry. The statements of pure geometry hold logically, but they deal only with abstract structures and say nothing about physical space. Physical geometry describes the structure of physical space: it is a part of physics.[1]

Now, Euclidean and non-Euclidean geometries are mathematical geometries and propose different geometrical properties for the space. The question as to which of them is physical geometry, is an important one.

For instance, if we draw a triangle, one of whose vertices is at the earth, one is somewhere in the Great Nebula in Andromeda, and one is in the spiral Nebula in Conces Venatici, will this triangle be Euclidean or non-Euclidean? That is to say, will the sum of its three angles be axactly equal to 180°, or greater than 180° or less than 180°?

As we have seen, Newton thought that the Euclidean geometry is the physical geometry. When the above question was put to famous mathematician Poincare, he said that it was merely a matter of convenience. According to Poincare, the question can not be settled only by empirical observations. He maintained that we might adopt any of the possible forms of geometry, if we made the suitable modifications of the physical laws. To ask, which is the true geometry, is, in his view, as absurd as to ask whether the old or the metric system is the true one. Still he asserted that the physicists would always choose the Euclidean structure because of its simplicity.[2]

It follows from the above view of Poincare that in order to describe the mutual spatial relations of terrestrial objects, we have a choice of many different systems, any one of which will serve the purpose. The same is true of three-dimensional space, which has not in itself any definite structure or geometrical properties: essentially it is mere threefold continuity and nothing more, and the geometry to be used for the description of local relations is arbitrary and at the disposal of the mathematician. It would therefore seem to be impossible to tell by observation whether space is Euclidean or non-Euclidean. Also, since we can always map non-Euclidean space on Euclidean space, we can always assume Euclidean geometry to be the geometry of actual space, provided we make the requisite changes in our physical laws. It is purely a question of convenience, whether we prefer to have an easily intelligible geometry with complicated physical laws, or a less intelligible geometry with simple physical laws.[3]

But this does not mean that geometry is independent of experience. For the validity of the statements of the physical geometry is to be established empirically - as it has to be in any other part of physics - after rules for measuring the magnitudes involved, especially length, have been stated.[4] Though the physicist is free to choose the rules for measuring length, after this choice is made, the question of the geometrical structure of physical space becomes empirical; it is to be answered on the basis of the results of experiments. Alternatively, the physicist may freely choose the structure of physical space; but then he must adjust the rules of measurement in view of the observational facts.[5]

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Sources
Title: The Enigma Of The Universe Publisher: JVB University Ladnun English Edition: 2010 HN4U Online Edition: 2014

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Some texts contain  footnotes  and  glossary  entries. To distinguish between them, the links have different colors.
  1. Eddington
  2. Euclid
  3. Newton
  4. Poincare
  5. Science
  6. Space
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