# Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

# Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

Shows that the use of real numbers in geometry can be accounted for by the conservativeness of mathematics, without assuming the truth of the theory of real numbers. Focuses on Hilbert’s axiomatization of Euclidean geometry, which, since it doesn’t involve real numbers, shows that real numbers are theoretically dispensable in geometry. Discusses Hilbert’s representation and uniqueness theorems. Shows how the representation theorem explains the utility of real numbers in geometric reasoning (without requiring that the theory of real numbers be true), while the uniqueness theorem establishes that the axiomatization without numbers has certain quite desirable properties.

*Keywords:*
application of mathematics, conservativeness, geometry, Hilbert, mathematical truth, real numbers, representation and uniqueness theorems

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