Goethe and Mathematics

From the book by Croce, one can clearly see how the way of thinking in the present day still prevents even outstanding minds from gaining the right access to Goethe's work. Among the various obstacles that arise for such minds, the misunderstanding of Goethe's relationship to mathematics is one of the most effective. From this it can be seen that Goethe had no skill in the treatment of mathematical problems. He himself admitted his inability in this respect sufficiently strongly. In his scientific works, therefore, one never finds the problems worked out in those areas in which a mathematical treatment is required by the nature of the subject.

Now, in the period that followed Goethe, mathematical treatment was regarded as essential for those parts of knowledge of nature that are considered to be truly exact. It was under the same impression that Kant had been under when he expressed the view that there is only as much real science in any knowledge as mathematics is contained in it. For this way of thinking, the rejection of Goethe's scientific approach is sealed from the outset.

But when it comes to assessing Goethe's relationship to mathematics, something quite different comes into play.

The study of mathematics gives a person a special position in relation to the penetration of the cognitive tasks themselves. In mathematical thinking, one deals with something that arises within the human soul. One does not look outwards, as in sensory experience, but builds up the content of thought purely within. And by thinking one's way from one mathematical structure to another, one does not have to rely on the evidence of the senses or of external experimentation, but remains entirely within one's inner soul life; one is dealing with an inner, conceptual view. One lives in the realm of the freely creative spirit.

Novalis, who was equally at home in the field of mathematics as in that of the free creative poetic imagination, saw in the former a perfect imaginative creation.

In more recent times, however, this trait has been denied in mathematics. It has been thought that this field of knowledge also borrows its truths from sensory observation, like an external experimental science, and that this fact is merely beyond human attention. It was only believed that one formed the mathematical forms oneself because one did not become aware of the borrowing from external observation. But this view has arisen only out of prejudice, which refuses to admit any free activity of the human mind. We are willing to accept scientific certainty only where we can rely on the statements of sense observation. And so, because the certainty of its truths cannot be denied, mathematics is also said to be a sense science.

Because in mathematics we live in the realm of the free creative spirit, its essence can be most clearly seen in inner self-knowledge. If one turns one's attention away from the structures that one works out in mathematical activity and back to that activity itself, one becomes fully aware of what one is doing. Then one lives in a kind of free creative spirituality.

One must only then summon up the flexibility of soul to extend the same creative inner activity that one unfolds in mathematics to other areas of inner experience. In this flexibility of soul lies the power to ascend to imaginative, inspired and intuitive knowledge, of which this weekly journal has often spoken.

In mathematics, every step one takes is inwardly transparent. One does not turn to the outside with the soul in order to determine the being of the other through the being of the one. One does, however, remain in a realm that, although created inwardly, relates to the external world through its own nature. Mathematics originates in the soul, but relates only to the non-spiritual. When the freely creative activity of the spirit ascends to the types of knowledge mentioned, however, one comes to grasp the soul itself and the realm of the world in which the soul lives.

Goethe's spiritual nature was such that he felt no need to cultivate mathematics himself. But his way of knowing was of a completely mathematical nature. He took in what concerned external nature through pure, refined observation, but then transformed it in his inner experience so that it became one with his soul, as is the case with freely created mathematical forms. Thus his thinking about nature became, in the most beautiful sense, a mathematical one. As a thinker of nature, Goethe was a mathematical spirit without being a mathematician.

He was just as open about his lack of knowledge of mathematics as he was about the mathematical direction of his way of looking at things. You can read about this in the essays that conclude his works on natural science under the title “On Natural Science in General. In this work he also stated that in all knowledge one must proceed as if one owed an account of one's findings to the strictest mathematician.

Through this direction of his quest for knowledge, Goethe was particularly predisposed to introducing a true scientific method of research into those scientific fields that cannot be determined by measure, number and weight because they are not quantitative but qualitative in nature. The opposing view wants to limit itself to what can be measured, counted and weighed, and leaves the qualitative as scientifically unattainable. It denies Goethe scientific validity because it does not see how he extends the rigor of research, which it demands where actual mathematics is applicable, to fields of knowledge where this is no longer the case.

Only when Goethe's methods of thought can be truly understood in this direction will it be possible to gain an unbiased judgment of the relationship between his knowledge and art. Only then will it be possible to see what the further development of his way of thinking can bring, both for art and for science.