To the last query this important observation should be added: Mathematicians have clearly understood, at least since Poincare, that question about the possibility of non-Euclidean geometries, and hence as well about the Fifth Postulate, is entirely bound up with what is referred to as the “metric” of geometric space. But the metric is entirely a function of how one part of space may be taken to represent another as quantitatively equivalent.
Therefore those who take representation as a purely “logical” function, in the sense of the analytic philosophers, also take the difference between Euclidean and non-Euclidean geometries as not a difference between the true and the false, but between alternative ways of setting the real under an intelligible representation, which it is up to us to create. (This was Poincare’s view, for instance.)
Such a view as this is the beginning of the road to the view that mathematics itself is simply a derivation from “logic.” Anyone who holds that quantity is something real must therefore find it unsatisfactory. But on the other hand, the idea that representation lies at the foundation of metrical reality seems far too adequate as a founding principle to simply dismiss out of hand. The question then seem to be whether this representation has a real foundation, as opposed to a merely “logical” one.