I have read a thesis
Douglas Bertrand Marshall, Investigations into the Applicability of
Geometry, Ph.D. Thesis, Harvard University (2011)
where the author considers views of Aristotle, Galileo and Leibniz on
the relationship between Geometry and Nature. The author considers the
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.”
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