and it would be misleading to set schoolboys to verify propositions of geometry by measurement, if the space they are supposed to be studying had not this meaning.
I suspect that you are doubtful whether this abstraction of extensional relations quite fulfils your general idea of space; and, as a necessity of thought, you require something beyond. I do not think I need disturb that impression, provided you realise that it is not the properties of this more transcendental thing we are speaking of when we describe geometry as Euclidean or non-Euclidean.
Math. The view has been widely held that space is neither physical nor metaphysical, but conventional. Here is a passage from Poincaré's Science and Hypothesis, which describes this alternative idea of space:
"If Lobatchewsky's geometry is true, the parallax of a very distant star will be finite. If Riemann's is true, it will be negative. These are the results which seem within the reach of experiment, and it is hoped that astronomical observations may enable us to decide between the two geometries. But what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that everyone would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments."
Rel. Poincaré's brilliant exposition is a great help in understanding the problem now confronting us. He brings out the interdependence between geometrical laws and physical laws, which we have to bear in mind continually. We can add on to one set of laws that which we subtract from the other set. I admit that space is conventional—for that matter, the meaning of every word in the language is conventional. Moreover, we have actually arrived at the parting of the ways imagined by Poincaré, though the crucial experiment is not precisely the one he mentions. But I deliberately adopt the alternative, which, he takes for granted, everyone would consider less
