“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. Recognizing knowledge as praxeologically constrained explains why the empiricist-formalist view is incorrect and why the empirical success of Euclidean geometry is no mere accident. Spatial knowledge is also included in the meaning of action. Action is the employment of a physical body in space. Without acting there could be no knowledge of spatial relations and no measurement. Measuring relates something to a standard. Without standards, there is no measurement, and there is no measurement which could ever falsify the standard. Evidently, the ultimate standard must be provided by the norms underlying the construction of bodily movements in space and the construction of measurement instruments by means of one’s body and in accordance with the principles of spatial constructions embodied in it. Euclidean geometry, as again Paul Lorenzen in particular has explained, is no more and no less than the reconstruction of the ideal norms underlying our construction of such homogeneous basic forms as points, lines, planes and distances which are in a more or less perfect but always perfectible way incorporated or realized in even our most primitive instruments of spatial measurements such as a measuring rod. Naturally, these norms and normative implications cannot be falsified by the result of any empirical measurement. On the contrary, their cognitive validity is substantiated by the fact that it is they that make physical measurements in space possible. Any actual measurement must already presuppose the validity of the norms leading to the construction of one’s measurement standards. It is in this sense that geometry is an a priori science and must simultaneously be regarded as an empirically meaningful discipline because it is not only the very precondition for any empirical spatial description, but it is also the precondition for any active orientation in space.” (Hoppe 2006: 287–288).I have already dealt with the argument 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” in my post here. The fact that Euclidean geometry is highly useful in engineering and construction does not refute the epistemological status of applied geometry as synthetic a posteriori. And 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.
The second substantive point that Hoppe makes is that “Euclidean geometry … is no more and no less than the reconstruction of the ideal norms underlying our construction of such homogeneous basic forms as points, lines, planes and distances.” But geometry in that sense is analytic a priori and to assert that pure geometry cannot be refuted by experience is to assert a truth that even empiricists agree with.
Thirdly, the fact that we make measurements of real space – even non-Euclidean space – using Euclidean geometry and tools constructed with Euclidean geometry as a basis does not prove that Euclidean geometry is a universally and necessarily true theory of real space throughout the universe known a priori.
For example, a two-valued classical logic can be used to show that certain events described by quantum mechanics are not strictly subject to that same classical logic: rather, a non-classical logic is required for the quantum world. But the fact that we use two-valued classical logic as a basis for this does not prove that classical logic is a universally and necessarily true logic throughout all levels of reality in the universe. All that is presupposed is that classical logic is valid and sound in a specific domain: that of the ordinary macroscopic world that human beings inhabit. But even its validity in that domain must be ultimately judged a contingent fact about the universe.
Next, even though scientific instruments are constructed with Euclidean geometry (see Hoppe 2006: 288, n. 23), this does not prove what Hoppe thinks it does. That such instruments are indeed very useful is explained by the fact that Euclidean geometry is an approximation of the geometry of space in a limited domain: that is, a domain where there is only slight curvature of space in small dimensions and involving bodies of relatively small mass and negligible acceleration. But this is a contingent and a posteriori fact about the universe, not a necessary and a priori one.
We could in fact design and construct the same, or even better, tools or scientific instruments using non-Euclidean geometry, but the use of Euclidean geometry provides an easier shortcut because Euclidean geometry is a good approximation of limited small areas of space within the universe – a universe which nevertheless has a non-Euclidean geometry.
Rudolf Carnap explains:
“Our instruments occupy such tiny parts of space that the question of how our space deviates from Euclidean geometry does not enter into their construction. Consider, for example, a surveyor’s instrument for measuring angles. It contains a circle divided into 360 equal parts, but it is such a small circle that, even if space deviated from the Euclidean to a degree that Gauss hoped he could measure (a much greater degree than the deviation in relativity theory), it would still have no effect on the construction of this circle. In small regions of space, Euclidean geometry would still hold with very high approximation. This is sometimes expressed by saying that non-Euclidean space has a Euclidean structure in small environments. From a strict mathematical standpoint, it is a matter of a limit. The smaller the region of space, the closer its structure gets to the Euclidean. But our laboratory instruments occupy such minute portions of space that we can completely disregard any influence non-Euclidean space might have on their construction.” (Carnap 1966: 149).But all these observations are empirical discoveries, not a priori truths.
Carnap, Rudolf. 1966. Philosophical Foundations of Physics: An Introduction to the Philosophy of Science (ed. Martin Gardner). Basic Books, New York and London.
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.