The Rudolf Steiner Archive

12. Goethe and Mathematics

[ 1 ] Among the main obstacles standing in the way of a fair assessment of Goethe’s significance for science is the prejudice surrounding his relationship to mathematics. This prejudice is twofold. First, it is believed that Goethe was an enemy of this science and grossly underestimated its great importance for human knowledge; and second, it is claimed that the poet excluded any mathematical approach from the physical aspects of natural science that he studied solely because it was inconvenient for him, as he had no formal training in mathematics.

[ 2 ] As for the first point, it must be said that Goethe repeatedly expressed his admiration for mathematical science in such a decisive manner that there can be absolutely no question of him holding it in low regard. Indeed, he wanted the entire field of natural science to be imbued with the rigor that is characteristic of mathematics. “We have to learn from mathematicians the deliberateness of linking only one thing to the next—or rather, of deducing the next from the previous one—and even where we do not use any calculations, we must always proceed as if we were accountable to the most rigorous geometer...” (Natw. Schr., Vol. 2, p. 19) “I heard myself being accused as if I were an adversary, an enemy of mathematics in general, which yet no one can value more highly than I do... .” [Ibid., p. 45]

[ 3 ] As for the second accusation, it is of such a nature that hardly anyone who has gained insight into Goethe’s character can seriously level it. How often, after all, did Goethe speak out against the tendencies of problematic minds who strive toward goals without regard for whether they are thereby operating within the limits of their abilities! And was he himself supposed to have transgressed this principle—was he supposed to have put forward scientific views while disregarding his own inadequacy in mathematical matters? Goethe knew that there are infinitely many paths to the truth, and that anyone can follow the one that suits their abilities and still reach the goal. “Every person must think in their own way: for on their path they will always find a truth—or a kind of truth—that helps them through life; only they must not let themselves go; they must exercise self-control...” (“Prose Sayings” [Natw., Schr., Vol. 4, Part 2, p. 460]). “Even the humblest person can be complete if he remains within the limits of his abilities and skills; but even beautiful virtues are obscured, nullified, and destroyed when that indispensable balance is lacking...” [Ibid., p. 443]

[ 4 ] It would be ridiculous to claim that Goethe, in order to achieve anything at all, ventured into a field that lay outside his sphere of interest. It all comes down to determining what mathematics is meant to accomplish and where its application to the natural sciences begins. Goethe has indeed given this matter the most thorough consideration. When it comes to defining the limits of his creative power, the poet displays a sharpness of mind surpassed only by his genius and profundity. We would like to draw the attention of those, above all, who have nothing else to say about Goethe’s scientific thinking other than that he lacked a logical, reflective way of thinking. The way in which Goethe defined the boundary between the scientific method he employed and that of mathematicians reveals a deep insight into the nature of mathematical science. He knew exactly what the basis of the certainty of mathematical theorems is; he had formed a clear conception of the relationship between mathematics and the rest of the laws of nature.

[ 5 ] If a science is to have any value as a source of knowledge at all, it must open up a specific realm of reality to us. Some aspect of the content of the world must be expressed within it. The way in which it does so constitutes the spirit of the science in question. Goethe had to understand this spirit of mathematics in order to know what can and cannot be achieved in the natural sciences without the aid of calculus. This is the crucial point. Goethe himself pointed this out with the utmost clarity. The way he does so reveals a deep insight into the nature of mathematics..

[ 6 ] Let us examine this nature more closely. The subject matter of mathematics is magnitude—that which admits of more or less. Magnitude, however, is not something that exists in and of itself. Within the broad scope of human experience, there is no thing that is merely magnitude. In addition to other characteristics, every thing also has those that can be determined by numbers. Since mathematics deals with magnitudes, its subject matter is not objects of experience that are complete in themselves, but only those aspects of them that can be measured or counted. It separates from things everything that can be subjected to the ultimate operation. In this way, it acquires a whole world of abstractions within which it then operates. It does not deal with things as such, but only with things insofar as they are magnitudes. It must admit that it deals there only with one aspect of reality, and that the latter still has many other aspects over which it has no power. Mathematical judgments are not judgments that fully encompass real objects; rather, they are valid only within the ideal world of abstractions, which we ourselves have conceptually separated from reality as a aspect of reality. Mathematics abstracts magnitude and number from things, establishes entirely ideal relationships between magnitudes and numbers, and thus hovers in a pure world of thought. The things of reality, insofar as they are magnitude and number, then permit the application of mathematical truths. It is therefore a grave error to believe that one can grasp the entirety of nature through mathematical judgments. Nature is not merely a matter of quantity; it is also a matter of quality, and mathematics deals only with the former. The mathematical approach and the approach focused purely on the qualitative must work their way toward one another; they will meet in the thing of which each grasps one aspect . Goethe describes this relationship with the words: “Mathematics, like dialectics, is an organ of the inner, higher mind; in practice, it is an art like eloquence. For both, nothing has value except form; content is of no concern to them. Whether mathematics calculates pennies or guineas, or rhetoric defends the true or the false, is entirely the same to both…” (“Sayings in Prose”; Natural Writings, Vol. 4, Part 2, p. 405). And “Outline of a Theory of Colors” 724 [ibid., Vol. 3, p. 277]: “Who does not acknowledge that mathematics, as one of the most magnificent human faculties, has been of great benefit to physics from one perspective ?” In this insight, Goethe saw the possibility that a mind that finds no cultural fulfillment in mathematics can nevertheless engage with physical problems. It must limit itself to the qualitative.