Hoppe Bungles Geometry

Jonathan Catalan quotes Hoppe defending Euclidean geometry:

“Recognizing knowledge as praxeologically constrained explains why the empiricist-formalist view [of geometry] 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 is relating something to a standard."

"Without standards, there is no measurement; and there is no measurement, then, 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 which 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 that it must simultaneously be regarded as an empirically meaningful discipline, because it is not only the very precondition for any empirical spatial description, it is also the precondition for any active orientation in space.”

What a muddle! First of all, the main non-Euclidean geometry's differ from Euclidean geometry on the question of, given a line any point lying on the same plane, how many lines through the point never meet the given line? In hyperbolic geometry the answer is an infinite number, in Euclidean geometry, the answer is one, and in elliptic geometry, zero. But if the curvature of space in the non-Euclidean geometries is taking place on a large enough scale, it will make no difference to humans actions whatsoever.

To see how a "mistaken" geometry can still be the basis for successful action, think of laying out fields on the surface of the earth. People proceed as if they were doing so on a plane where, for instance, the Pythagorean theorem applies. But, of course, they are really doing so on the surface of the sphere. However, at the scale we operate, treating that surface as a plane is plenty good enough. There is no reason why an analogous situation couldn't hold in relationship to three-dimensional space.

And what of this notion of standards that can't be falsified by measurement? Of course this is true, but so what? What we might, in fact, do is to find a different standard that makes better sense of the world. For instance, let us say we were taking a column of mercury as our standard for measuring length. We would find that on cold days everything was a lot longer than it was on hot days. If we switched our standard to a steel bar, we would find that the length of most objects settled down a good bit, while the column of mercury now is understood to be growing longer and shorter. We might decide we liked the new standard better, as it made things a lot simpler.


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