Among the main hindrances standing in the way of a just evaluation of Goethe's significance for science belongs the preconception that exists about his relationship to mathematics. This preconception is twofold. Firstly, one believes that Goethe was an enemy of this science and failed in the worst way to recognize its great significance for human knowing; and secondly, one maintains that the poet excluded any mathematical approach from the physical parts of the natural science pursued by him only because the mathematical approach was uncomfortable to him, as he had benefited from no training in mathematics.
As regards the first point, one can say in refutation of it that Goethe repeatedly gave expression to his admiration for the science of mathematics in such a decisive manner that there can be absolutely no question of his attaching little value to it. In fact, he wants to be sure that all natural science is permeated by that strictness which is characteristic of mathematics. “We must learn from the mathematicians to take care to place next to each other only the elements that are closest to each other, or rather to deduce from each the elements closest to it, and even where we use no calculations, we must always proceed as though obliged to render account to the strictest geometrician.” “I heard myself accused of being an opponent, an enemy, of mathematics altogether, which no one, after all, can value more highly than I do ...”
As regards the second criticism: it is of such a kind that hardly anyone who has once looked into Goethe's nature could raise it seriously. How often has Goethe spoken out against the undertakings of problematical people who strive for goals without bothering about whether, in doing so, they are keeping within the bounds of their abilities! And he himself should have violated this precept, he should have set up natural-scientific views, ignoring his insufficiencies in mathematical things! Goethe knew that the paths to what is true are infinitely many, and that each person can travel the one most in accordance with his abilities, and will arrive at his goal. “Every human being must think in his own way: for he will always find something true along his path, or a kind of truth that will help him through life; but he must not just let himself go; he must control himself ...” (Aphorisms in Prose). “The least of men can be complete if he is active within the limits of his abilities and skills; but even good qualities become obscured, cancelled out, and destroyed if that absolutely essential proportion is lost.” (Ibid.)
It would be ludicrous for someone to assert that Goethe would go into an area lying outside his field of vision in order to accomplish anything at all. Everything depends upon establishing what task mathematics has and where its application to natural science begins. Now Goethe did actually undertake the most conscientious study of this. Where it is a question of determining the limits of his productive powers, the poet develops a sharpness of understanding surpassed only by his genius' depth of understanding. We would especially like to make those people aware of this who have nothing else to say about Goethe's scientific thinking than that he lacked a logical, reflective way of thinking. The manner in which Goethe established the boundary between the natural-scientific method he employed and that of the mathematicians reveals a deep insight into the nature of the science of mathematics. He knew exactly what the basis is for the certainty of mathematical theorems; he had formed a clear picture for himself of the relationship in which mathematical lawfulness stands with respect to the lawfulness of the rest of nature. If a science is to have any value at all as knowledge, it must open up for us a particular region of reality. Some aspect or other of the world content must manifest itself in it. The way in which it does this constitutes the spirit of a particular science. Goethe had to recognize the spirit of mathematics in order to know what can be attained in natural science without the help of computation and what cannot. This is the point that really matters. Goethe himself indicated this with great decisiveness. The way he does this reveals a deep insight into the nature of the mathematical.
Let us examine this nature more closely. Mathematics deals with magnitude, with that which allows of a more or less. Magnitude, however, is not something existing in itself. In the broad scope of human experience there is nothing that is only magnitude. Along with its other characteristics, each thing also has some that are determined by numbers. Since mathematics concerns itself with magnitudes, what it studies are not objects of experience complete in themselves, but rather only everything about them that can be measured or counted. It separates off from things everything that can be subjected to this latter operation. It thus acquires a whole world of abstractions within which it then works. It does not have to do with things, but only with things insofar as they are magnitudes. It must admit that here it is dealing only with one aspect of what is real, and that reality has yet many other aspects over which mathematics has no power. Mathematical judgments are not judgments that fully encompass real objects, but rather are valid only within the ideal world of abstractions that we ourselves have conceptually separated off from the objects as one aspect of reality. Mathematics abstracts magnitude and number from things, establishes the completely ideal relationships between magnitudes and numbers, and hovers in this way in a pure world of thoughts. The things of reality, insofar as they are magnitude and number, allow one then to apply mathematical truths. It is therefore definitely an error to believe that one could grasp the whole of nature with mathematical judgments. Nature, in fact, is not merely quantity; it is also quality, and mathematics has to do only with the first. The mathematical approach and the approach that deals purely with what is qualitative must work hand in hand; they will meet in the thing, of which they each grasp one aspect. Goethe characterizes this relationship with the words: “Mathematics, like dialectics, is an organ of the inner, higher sense; its practice is an art, like oratory. For both, nothing is of value except the form; the content is a matter of indifference to them. It is all the same to them whether mathematics is calculating in pennies or dollars or whether rhetoric is defending something true or false.” (Aphorisms in Prose) And, from Sketch of a Colour Theory: “Who does not acknowledge that mathematics is one of the most splendid organs of man, is from one aspect very useful to physics?” In this recognition, Goethe saw the possibility that a mind which does not have the benefit of a mathematical training can still occupy itself with physical problems. Such a mind must limit itself to what is qualitative.