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Penrose is a physicist. He sees everything from a physicist's perspective.
Most mathematicians would disagree that either Euclidean Geometry OR Number
Theory is a "physical theory".
There is, in fact, no reason to think that "physical" space IS Euclidean.
Euclidean Geometry seems "natural" and physical space appears Euclidean because
Euclidean Geometry is the simplest geometry.
Likewise, from a purely physical point of view even the "natural numbers"
require the ability to distinguish between two "identical" objects. I suppose
it may be easy to distinguish between two elephants standing side by side but
what about two electrons or two slime molds?
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