“Further, the old rationalist claims that Euclidean geometry is a priori yet incorporates empirical knowledge about space becomes supported, too, in view of our insight into the praxeological constraints on knowledge. Since the discovery of non-Euclidean geometries and in particular since Einstein’s relativistic theory of gravitation, the prevailing position regarding geometry is once again empiricist and formalist. It conceives of geometry as either being part of empirical, a posteriori physics, or as being empirically meaningless formalisms. That geometry is either mere play or forever subject to empirical testing seems to be irreconcilable with the fact that Euclidean geometry is the foundation of engineering and construction, and that nobody in those fields ever thinks of such propositions as only hypothetically true.” (Hoppe 2006: 287–288).On the contrary, that Euclidean geometry is highly useful in certain areas does not refute the epistemological status of applied geometry as synthetic a posteriori. And pure geometry remains analytic a priori and necessarily true and known a priori only when it is understood as a pure, non-empirical theory.
For how do human beings know that non-Euclidean, curved Riemannian geometry is a better theory of space-time, and that Euclidean geometry is actually only a useful approximation applicable to a certain domain? The answer is: empirically, not a priori.
And the fact that people in “engineering and construction” might never think of Euclidean geometry as “only hypothetically true” or a mere approximation is irrelevant: it commits an appeal to invalid authority.
Hoppe, Hans-Hermann. 2006. The Economics and Ethics of Private Property: Studies in Political Economy and Philosophy (2nd edn.), Ludwig von Mises Institute, Auburn, Ala.