![]() Certain contents and methodological principles of mathematics, however, point towards areas beyond themselves, for example, synthetic projective geometry and the non-Euclidean geometries embedded in it, and, thus, these disciplines offer prospects of a mathematics of invigorative principles (mathematical physics and mathematics of living organisms). ![]() Its strong point in developing consciousness lies in its methodological purity and universality, its weak point is its limitations in describing sensory phenomena because of the special form of their contents. Steiner threw clear light on the possibilities and limits of mathematics. Rudolf Steiner was responsible for a new understanding of mathematics as preparatory training for spiritual comprehension as well as a methodological principle for the acquisition of knowledge in the natural sciences. On the other hand, because of this development, possibilities, which had not existed before to the same extent, were created for the training of pure thought and of supersensory imagination with the help of mathematics. Once that mathematics had become conscious of its spiritual (abstract) strength, it began a decisive campaign of conquest throughout all branches of knowledge. The integration of the non-Euclidean geometries into mathematics, which began in the sixties of the nineteenth century, played a part in the definitive separation of mathematical and especially geometrical abstraction from a way of looking at things determined by sensory perception. A short time later this faith extended only to mathematical contents and consequently made the mathematisation of the natural sciences and the rejection of romantic natural philosophy necessary in accordance with currents of thought prevailing at that time. ![]() The recognition at the beginning of the nineteenth century of the existence of a non-Euclidean geometry, thanks to Gauss, Lobachevskij and Bolyai, based on purely axiomatic (conceptual) considerations is considered as a symptom of faith in the truth content of pure thought, which was also expressed by German idealism though in a different way.
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