I have read a thesis

Douglas Bertrand Marshall, Investigations into the Applicability of Geometry, Ph.D. Thesis, Harvard University (2011)


http://www.tc.umn.edu/~dmarshal/

where the author considers views of Aristotle, Galileo and Leibniz on the relationship between Geometry and Nature. The author considers the next challenges:

Challenging the Applicability of Geometry

p. 21 “[Protagorean Challenge] For all phi in Gamma, phi is a theorem of geometry, and when it comes to physical and material things, not phi.”

p. 25 “[No-Shapes Challenge] There are no geometric objects in nature. That is, there are in nature no points, lines, or surfaces which satisfy the axioms of geometry.”

p. 37 “[No-Structure Challenge] Nothing in nature is isomorphic either to Euclidean space, or to any Euclidean curve, or to any Euclidean surface.”

p. 41 “[No-Discrepancies Challenge] Given any natural item N and any geometric item G, there is no determinate or well-defined discrepancy between N and G.”

Evgenii
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http://blog.rudnyi.ru/2013/01/investigations-into-the-applicability-of-geometry.html

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