...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 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.

Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.

Philosophy is regarded by many as inseparable from speculation. ... Philosophy has proceeded from speculation to science.

Common to the two geometries is only the general property of one-to-one correspondence, and the rule that this correspondence determines straight lines as shortest lines as well as their relations of intersection.

Occasionally one speaks... of signals or signal chains. It should be noted that the word signal means the transmission of signs and hence concerns the very principle of causal order.

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.