The Rudolf Steiner Archive

12. Goethe and Mathematics

[ 1 ] One of the main obstacles to a fair appreciation of Goethe's significance for science is the prejudice that exists about his relationship to mathematics. This prejudice is twofold. Firstly, it is believed that Goethe was an enemy of this science and grossly misjudged its great importance for human cognition; and secondly, it is claimed that the poet excluded any mathematical treatment from the physical parts of natural science that he cultivated only because it was inconvenient to him, who enjoyed no culture in mathematics.

[ 2 ] With regard to the first point, however, it must be said that Goethe repeatedly expressed his admiration for mathematical science in such a decisive manner that there can be no question of any disdain for it. Indeed, he wants the whole of natural science to be imbued with the rigor that is characteristic of mathematics. "We have to learn from the mathematicians the recklessness of merely stringing the next thing together, or rather of deducing the next thing from the next thing, and even where we do not make use of a calculation, we must always go about our work as if we owed an account to the strictest geometrician." (Natw. Schr., 2nd vol., p. 19) "I heard myself accused of being an adversary, an enemy of mathematics in general, which no one can value more highly than I do.... ." [ibid. p. 45]

[ 3 ] As far as the second accusation is concerned, it is such that hardly anyone who has had an insight into Goethe's nature can seriously raise it. How often has Goethe not spoken out against the beginning of problematic natures that strive towards goals, regardless of whether they are within the limits of their abilities! And he himself was supposed to have transgressed this commandment, he was supposed to have set up scientific views, disregarding his inadequacy in mathematical matters? Goethe knew that the paths to the true are infinite, and that everyone can take the one that suits his abilities and reach his goal. "Every man must think in his own way: for he always finds on his way a truth, or a kind of truth, which helps him through life; only he must not let himself go; he must control himself.... .." ("Proverbs in Prose" [Natw., Schr., 4th vol., 2nd dept., p. 460]). "The least man can be complete if he moves within the limits of his abilities and skills; but even beautiful advantages are obscured, canceled and destroyed if that indispensable required evenness is missing..." [Ibid. p. 443]

[ 4 ] It would be ridiculous to claim that Goethe, in order to achieve anything at all, entered a field that lay outside his field of vision. It all depends on establishing what mathematics can achieve and where its application to natural science begins. Goethe really did make the most conscientious observations about this. When it comes to determining the limits of his productive power, the poet develops an ingenuity that is only surpassed by his ingenious profundity. We would particularly like to draw the attention of those who know nothing else to say about Goethe's scientific thinking than that he lacked a logically reflective way of thinking. The way in which Goethe defined the boundary between the scientific method he used and that of the mathematicians reveals a deep insight into the nature of mathematical science. He knew exactly what the basis of the certainty of mathematical theorems was; he had formed a clear idea of the relationship between mathematical and other natural laws.

[ 5 ] If a science is to have any cognitive value at all, it must open up a certain realm of reality for us. It must express some aspect of the content of the world. The way in which it does this forms the spirit of the science in question. Goethe had to know this spirit of mathematics in order to know what can and cannot be achieved in natural science without the help of the calculus. This is the point that matters. Goethe himself pointed this out in no uncertain terms. The way he does it reveals a deep insight into the nature of mathematics.

[ 6 ] We want to go into this nature in more detail. The subject of mathematics is size, that which allows for more or less. But size is not something that exists in itself. There is no thing in the wide range of human experience that is only size. In addition to other characteristics, every thing also has characteristics that can be determined by numbers. Since mathematics deals with magnitudes, it has no objects of experience that are complete in themselves, but only everything of them that can be measured or counted. It separates from things everything that can be subjected to the final operation. This gives it a whole world of abstractions within which it then works. It does not deal with things, but only with things insofar as they are quantities. It must admit that it only deals with one side of the real, and that the latter has many other sides over which it has no power. Mathematical judgments are not judgments that fully encompass real objects, but are only valid within the ideal world of abstractions, which we ourselves have conceptually separated from the latter as a side of reality. Mathematics abstracts size and number from things, establishes the entirely ideal relationships between sizes and numbers and thus floats in a pure world of thought. The things of reality, insofar as they are magnitude and number, then allow the application of mathematical truths. It is therefore a decided error to believe that one can grasp the whole of nature with mathematical judgments. Nature is not merely quantum; it is also quale, and mathematics has to do only with the former. The mathematical treatment and the purely qualitative treatment must work into each other's hands; they will meet in the thing of which they grasp each one side. Goethe describes this relationship with the words: "Mathematics, like dialectics, is an organ of the inner higher sense; in its practice it is an art like eloquence. For both, nothing has value but the form; they are indifferent to the content. Whether mathematics calculates pennies or guineas, rhetoric defends the true or the false, is completely the same to both..." ("Proverbs in Prose"; Natw. Schr., 4th vol., 2nd dept., p. 405). And "Entwurf einer Farbenlehre" 724 [ibid. 3rd vol., p. 277]: "Who does not confess that mathematics, as one of the most glorious human organs, is of great use to physics from one side?" In this realization, Goethe saw the possibility that a mind that enjoys no culture in mathematics can deal with physical problems. He must confine himself to the qualitative.