...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.

The statement that although the past can be recorded, the future cannot, is translatable into the statistical statement: Isolated states of order are always postinteraction states, never preinteraction states.

...the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.

The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. ..."Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry.

We can... treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.

Absolute time would exist in a causal structure for which the concept indeterminate as to time order lends to a unique simultaneity, i.e., for which there is no finite interval of time between the departure and return of a first-signal...

We must... maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.