...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.

The philosopher of science is not much interested in the thought processes which lead to scientific discoveries he looks for a logical analysis of the completed theory, including the establishing its validity. That is, he is not interested in the context of discovery, but in the context of justification.

You see, there is no more purpose or meaning in the world than you put into it.

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.

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.

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.

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.

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.

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.

...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.

It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers.