VI. The Present State of Philosophy and Science

I do not want to point out that the determinations of neither one nor the other of the two geometries are correct with the results of the visionary research. They are only proof that thought operations can lead beyond the area that immediately encompasses our space. But it must be said that if one understands the implications of these geometries, one can imagine that there are completely different factual connections than in the sensory world. For the latter is ultimately expressed in the formulas of geometry. If different formulas apply to a world than those of Euclidean geometry, then this world is a different world from ours. And we can say: with Riemannian and Lobachevskian geometry, the geometer's yearning to go beyond the world of the senses, to grasp something intellectually that does not lie in the realm of the sensual world at all, is fulfilled. That is why these non-Euclidean geometries are symptomatically significant for our century.