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Non euclidean geometry in geography9/1/2023 ![]() ![]() Despite the negative results of Godel, who showed that the more ambitious aims of the program could not be fulfilled, it was still dominant when I was taught mathematics in the 1970s. Mathematical formalism reached its high point with the Hilbert program in the early 20th Century. A set of axioms may be useful if the theorems it yields turn out to provide a good model for some real world phenomenon, but this is not a mathematical question (though it helps keep mathematicians in work). They are merely sentences in a formal language that can be combined and manipulated to form new sentences (theorems). The key point of formalism is that axioms like Euclid’s parallel postulate are neither true nor false. The discovery of non-Euclidean geometry led to the rise of formalism as the dominant philosophical approach in mathematics. Considered in this light, Euclidean plane geometry is the mathematical model associated with the Flat Earth theory. In non-Euclidean geometry, the interior angles of a triangle may add to more, or less, than 180 degrees.Įven worse for the rationalist program was the observation that the system of geometry (that is, “earth measurement”) most relevant to earth-dwellers is spherical geometry, in which straight lines are “great circles”, and in which the angles of a triangle add to more than 180 degrees. ![]() Precisely for that reason, the discovery, in the early 19th century of non-Euclidean geometries that did not satisfy Euclid’s requirement that parallel lines should never meet, was a huge blow to rationalism, from which it has never really recovered. The certainty of Euclidean geometry was, for centuries, the strongest argument for the rationalist that we could derive certain knowledge about the world. In an Internet discussion the other day, I was surprised to see the deductive certainty claimed by Mises presented as being similar to the “certainty” that the interior angles of a triangle add to 180 degrees. Among the most extreme versions of this is the “praxeological” economic methodology espoused by Mises and his followers, and also endorsed, in a more qualified fashion, by Hayek. One of the striking features of (propertarian) libertarianism, especially in the US, is its reliance on a priori arguments based on supposedly self-evident truths.
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