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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