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.

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.

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 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 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 essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization. ... The separation of relevant from irrelevant factors is the beginning of knowledge.

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.