If error is corrected whenever it is recognized as such, the path of error is the path of truth.
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..
For the Lorentz transformation spatial measurements are also changed, because they are obtained relative to a moving system. In our example only time was transformed, while the distances between points at rest remained the same; the spatial coordinates, therefore, retain their identity.
This fact.. Proves that space measurements are reducible to time measurements. Time is therefore logically prior to space.
Light signals alone provide the metrical structure of the four-dimensional space-time continuum. The construction may be called light axioms.
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 order of betweenness does not depend on mutual distances.. Betweenness is purely a relational order.
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.
Once a definition of congruence is given, the choice of geometry is no longer in our hands; rather, the geometry is now an empirical fact.
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.
Why is Einstein's theory better than Lorentz's theory? It would be a mistake to argue that Einstein's theory gives an explanation of Michelson's experiment, since it does not do so. Michelson's experiment is simply taken over as an axiom.
.. 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.