Fallacious Application of "non-Euclidean" to Physical Space
Physicists sometimes talk about space—by which they of course mean physical space—as Euclidean or non-Euclidean. The problem with this way of speaking is that geometry is timeless. It cannot really apply to physical space.
Notice that if there's one lesson that Einstein's relativity has taught us, it is that space is intrinsically temporal: you can't have one without the other, which is why the combination in the relativistic context is usually called spacetime. Our measurements of length are always in time.
To see this point more clearly (more clearly at least if you are a physicist; I make no guarantees for others), think of the the Minkowski diagram, which plots time on the vertical axis and position on the horizontal (looking at a diagram may be helpful). (It also applies to general relativity with flat spacetimes, that is, regions far from masses.) Light rays are marked at 45-degree angles that divide the plane into four quadrants; the "light cone" consists of the north and south quadrants. There are time-like intervals (points that that lie within the "light cone," that is, that are separated enough in time that they can connect causally) and space-like intervals (points outside the light-cone, that is separated so far in space that they cannot connect causally).
To test whether space is Euclidean, one would have to set out measuring rods in the present, in other words, along the space-like interval parallel to the position (horizontal) axis. (And then test whether parallel lines remain parallel, or else either converge or diverge....)
But relativity has shown us that what one considers the present depends on one's state of motion: the ordering of events is not absolute, there is no unambiguous or absolute "present". On the Minkowski diagram in the frame of a primary, stationary observer, the "present" of a second, moving observer appears as an x' axis tilted obliquely to the x axis.
The assumption of what we usually mean by "length measurement" is that one measures both ends at once (as de Koninck points out, in contradistinction from Maritain, there is no absolute notion of length apart from an observer situated in space and time). Length measurements that are simultaneous in one frame are not simultaneous in another. Because of the relativity of simultaneity, "at once," and thus length measurement, becomes tied to the relative states of motion of the measurer and the object measured.
As we have seen, there is no unambiguous "now"; so the application or denial of the qualifier "Euclidean" to physical space confuses physics for pure mathematics (the error of Descartes). It presumes some sort of a timeless frame for making length measurements and it is precisely the existence of such an absolute frame that relativity denies.
This argument occurred to me when reading Vincent Smith, and was corroborated by de Koninck writing about Eddington.
Vincent Edward Smith, Philosophical Physics (New York: Harper & Brothers, 1950), 355.
Charles de Koninck, The writings of Charles de Koninck, vol 1, ed. & trans. Ralph McInerny (Notre Dame, Ind. : University of Notre Dame Press, 2008), 147-158.
