As a further illustration of the elusiveness of absolute shape and size, Poincaré asks us to conceive of a hollow spherical volume placed anywhere in space, and to assume that the temperature in the sphere decreases progressively from the centre, becoming absolute zero at the surface. He assumes that this hollow sphere is peopled by imaginary beings whose bodies expand and contract with the temperature and that all material bodies in the sphere behave in a similar manner. If we should supplement these suppositions by assuming that the refractive index of the medium in the sphere’s interior varies in a certain definite way, the rays of light in this hypothetical world would describe circles.
This closed universe would of course appear infinite to its inhabitants, since as they proceeded from the centre to the surface their bodies would grow smaller, their steps shorter, so that it would be impossible for them to reach its boundary however long they walked. The geometricians of this imaginary world would feel justified in proceeding exactly as we have done ourselves. They would define as remaining congruent when displaced, hence as rigid, those bodies which appeared to them to remain the same wherever they carried them. Owing to the paths devised for the light rays and to the sameness in the reduction of the sizes of all objects as the centre was left behind, the expanding and contracting bodies of this universe would present all the characteristics of rigidity. On conducting measurements with their rigid rods the hypothetical beings would obtain non-Euclidean results, their entire world would appear to them as non-Euclidean, and non-Euclidean geometry would be as inevitable to them as Euclidean geometry is to the average layman. Some Kant among the hypothetical beings would surely arise and explain that non-Euclidean space was the a priori form of pure sensibility, transcending reason and experience. Then eventually some great mathematician would come along, sweep all those cobwebs aside, and prove that there existed other perfectly consistent types of geometries and that the ingrained preference of his fellow citizens for non-Euclidean geometry was due to the dictates of common experience and constituted by no means the a priori form of pure sensibility.
It is not merely in its philosophical aspect that Poincaré’s illustration is interesting. The major point is the following: The hypothetical beings would be just as much entitled to assert that space was non-Euclidean as we are to assert that it is Euclidean. It is true that if we could look into their world we should say that they had a wrong understanding of measurement, and were totally in error when they assumed that their bodies were rigid, since we could see them getting smaller and smaller as they neared the surface. But we must not forget that the imaginary beings, in turn, could they but view our Euclidean bodies, would return the compliment and accuse us of having a wrong understanding of rigidity and measurement.
From all this it follows that by a mere variation in physical conditions the same space would be considered non-Euclidean or Euclidean. Obviously, by reason of this contradiction, space itself can have nothing to do with the problem; the type of space which physicists are discussing reduces therefore to a relational synthesis of physical results. Space itself remains amorphous.
Poincaré develops analogous arguments when he discusses the parallax observations conducted on the rays of starlight. Euclidean geometry, for instance, regarded purely as a system of measurement, is from a mathematical point of view the simplest type of geometry for the same reason that a monosyllable is simpler than a polysyllable. It is therefore obviously to our interest to retain it if possible. Of course if, as in the hypothetical world discussed previously, material bodies behaved like non-Euclidean solids and if light rays followed appropriate courses, we should have to abandon Euclidean geometry for reasons of practical convenience. But since, in the world we live in, our habitual solids behave to a high order of approximation as do Euclidean solids, our preference for Euclidean geometry seems perfectly legitimate even from the standpoint of physics.
Suppose now that the parallaxes of the very distant stars turned out to be negative: would Euclidean geometry and Euclidean space have to be abandoned? As Poincaré points out, this would by no means be necessary or even advisable. It is true that if we assumed, as is the custom, that rays of starlight follow geodesics through space, negative parallaxes would imply a trace of Riemannianism in space; but the primary point to decide is, “How do we know that rays of light follow geodesics?” Obviously this is capable neither of proof nor of disproof. An empirical proof that such a contention was correct or incorrect would be possible only were we to know beforehand how the geodesics of space were situated, for then we could determine by observation whether rays of light followed them or not. But how could we establish the way the geodesics lie unless we were already apprised of this geometry which we now proposed to determine? Obviously, our procedure would be circular. Can we at least assume that rays of light must inevitably follow geodesics? Would any other assumption be impossible? Certainly not. A denial of the assumption would modify our understanding of optical phenomena; but what if it did? We could always get out of the difficulty by varying the laws of optical transmission, and still retain Euclidean geometry. In other words, the geometry the physicist credits to space is contingent on his acceptance of a number of physical laws; and by varying these laws in an appropriate way he could still account for observed facts and credit corresponding types of geometry to space. Since all these various systems of physical laws would account for the facts of experience, how can we ever hope to decide which one of these systems corresponds to reality? And under the circumstances, what use is there in discussing the real geometry of space? All we can discuss is expediency.
In other words, Poincaré, by divorcing space from its material content, geometry from physics, places space and its geometry beyond the control of experiment; so that there is really nothing left for the physicist to argue about.
As a matter of fact, the entire conflict is more apparent than real; it centres round the meaning we wish to ascribe to the word reality, whether we conceive of reality in the pragmatic scientific sense or in the metaphysical sense as embodying “true being.” For the theoretical physicist, “reality” means that hypothesis which will permit him to co-ordinate natural phenomena with the maximum of simplicity. He knows of no other test for reality and would probably evince very little interest in the unattainable reality of the metaphysician. If, therefore, a co-ordination of the facts of experience presents greater simplicity when he assumes space to be Euclidean or non-Euclidean, then space is Euclidean or non-Euclidean in spite of the fact that phenomena might just as well have been co-ordinated (though in a less simple way) had some other hypothesis been selected.
Riemann expresses the selective rôle of the criterion of simplicity when he writes:
“Nevertheless it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”[14]