文档介绍：Appendix A 62
Models and Examples
The procedure followed in this work has been to start with some general, physically
based, ideas about space (the axioms) and deduce the properties of the geometry they
entail. It has not been to start with some known and familiar mathematical objects
and to inductively find axioms which describe the characteristics which they have in
common. Altho, of course, the physical ideas about space, and the Euclidean
geometry (that has been developing for millennia) which has substantially informed
those ideas, do originate in a process of induction from human experience.
This kind of procedure is open to the possibility that it may be vacuous. That is,
there may be nothing which satisfies the axioms; the axioms may be inconsistent
with each other. And the consequences so laboriously deduced may not be the
properties of anything. That this is not the case is established by the fact that
Euclidean geometry satisfies the axioms. That is, it is a mathematical model for
physical geometry. The axioms of physical geometry are, therefore, as consistent as
Euclidean geometry.
This being the case, it might be suggested that Euclidean geometry is the only thing
which satisfies the axioms and therefore physical geometry is not interesting. Either
this is true or it is not. If it is true then the axioms of physical geometry constitute a
new axiom system for Euclidean geometry. And that would be very interesting. If it
is not then there might be novel and interesting possibilities in physical geometry
worth exploring. The former is not the case, of course. This is demonstrated by the
existence of mathematical models of physical geometry different from the Euclidean
one. Non-Euclidean Riemannian geometry (. non-positively curved, simply
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Appendix A 63
 
connected Riemannian manifolds ) and the examples in this appendix provide such
demonstrations.
Apparently interesting possibilities in physical geometry might, in fac