Lecture III

Try to sense the tremendous difference between this mathematical feeling bound to human experience, and the bare, bleak mathematical space layout of analytical geometry, which establishes a point somewhere in abstract space, draws three coordinating axes at right angles to each other and thus isolates this thought-out space scheme from all living experience. But man has in fact torn this thought-out spatial diagram out of his own inner life. So, if we are to understand the origin of the later mathematical way of thinking that was taken over by science, if we are to correctly comprehend its self-sufficient presentation of structures, we must trace it back to the self-experienced mathematics of a bygone age. Mathematics in former times was something completely different. What was once present in a sort of dream-like experience of three-dimensionality and then became abstracted, exists today completely in the unconscious. As a matter of fact, man even now produced mathematics from his own three-dimensionality. But the way in which he derives this outline of space from his experiences of inward orientation is completely unconscious. None of this rises into consciousness except the finished spatial diagram. The same is true of all completed mathematical structures. They have all been severed from their roots. I chose the example of the space scheme, but I could just as well mention any other mathematical category taken from algebra or arithmetic. They are nothing but schemata drawn from immediate human experience and raised into abstraction.