15. Spiritual Science, Natural Science and Technology

Rudolf Steiner: Dearly beloved attendees, who can really prove the Pythagorean theorem? The Pythagorean theorem cannot be proved by drawing a right-angled triangle on the blackboard and then using one of the usual methods to prove it. That is only one illustration of the proof. The point is that anyone who wants to prove the Pythagorean theorem is forced to have what can be constructed mathematically in their inner vision - even if only in the inner vision of the geometric spatial vision. So imagine a consciousness that did not have this spatial vision. He would not have before him the substantial element of that Pythagorean proposition, and it would be quite senseless to prove the Pythagorean theorem. We can only prove the Pythagorean theorem by having before us the substantial element of the conception and shaping of space. The moment we ascend to another form of consciousness, something else is added to the ordinary view of space. [...] The point, then, when it comes to the Pythagorean theorem, if it is to be proved, is that this view of space must underlie it. But for this it is first necessary to find one's way, as it were, into this new configuration of consciousness. But as long as one has no conception of the configuration of space, one cannot arrive at the observation that leads to the proof of the Pythagorean theorem. And one believes that the results of spiritual scientific research cannot be proved in the same way only as long as one has not yet made the transition from ordinary consciousness to the experiencing consciousness that I have described.