Goethean Science
GA 1
12. Goethe and Mathematics
[ 1 ] 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.
[ 2 ] 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 ...”
[ 3 ] 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.)
[ 4 ] 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. [ 5 ] 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.
[ 6 ] 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.
12. Goethe und die Mathematik
[ 1 ] Zu den Haupthindernissen, die einer gerechten Würdigung von Goethes Bedeutung für die Wissenschaft entgegenstehen, gehört das Vorurteil, das über sein Verhältnis zur Mathematik besteht. Dieses Vorurteil ist ein doppeltes. Einmal glaubt man, Goethe sei ein Feind dieser Wissenschaft gewesen und habe ihre hohe Bedeutung für das menschliche Erkennen in arger Weise verkannt; und zweitens behauptet man, der Dichter habe jede mathematische Behandlungsweise aus den physikalischen Teilen der Naturlehre, die er gepflegt, nur deshalb ausgeschieden, weil sie ihm, der sich keiner Kultur in der Mathematik erfreute, unbequem war.
[ 2 ] Was den ersten Punkt betrifft, so ist dagegen zu sagen, daß Goethe wiederholt in so entschiedener Weise seiner Bewunderung der mathematischen Wissenschaft Ausdruck gegeben hat, daß von einer Geringschätzung derselben durchaus nicht die Rede sein kann. Ja, er will die gesamte Naturwissenschaft von jener Strenge durchdrungen wissen, die der Mathematik eigen ist. «Die Bedächtlichkeit, nur das Nächste ans Nächste zu reihen, oder vielmehr das Nächste aus dem Nächsten zu folgern, haben wir von den Mathematikern zu lernen, und selbst da, wo wir uns keiner Rechnung bedienen, müssen wir immer so zu Werke gehen, als wenn wir dem strengsten Geometer Rechenschaft zu geben schuldig wären..» (Natw. Schr.., 2. Bd.., S. 19) «Ich hörte mich anklagen, als sei ich ein Widersacher, ein Feind der Mathematik überhaupt, die doch niemand höher schätzen kann als ich... .» [Ebenda S. 45]
[ 3 ] Was den zweiten Vorwurf betrifft, so ist er ein solcher, daß ihn kaum jemand im Ernste erheben kann, der einen Einblick in Goethes Wesen getan hat. Wie oft hat sich denn nicht Goethe gegen das Beginnen problematischer Naturen ausgesprochen, die Zielen zustreben, unbekümmert darum, ob sie sich damit innerhalb der Grenzen ihrer Fähigkeiten bewegen! Und er selbst sollte dieses Gebot überschritten, er sollte naturwissenschaftliche Ansichten aufgestellt haben, mit Hinwegsetzung über seine Unzulänglichkeit in mathematischen Dingen? Goethe wußte, daß der Wege zum Wahren unendlich viele sind, und daß ein jeder jenen wandeln kann, der seinen Fähigkeiten gemäß ist, und er kommt ans Ziel. «Jeder Mensch muß nach seiner Weise denken: denn er findet auf seinem Wege immer ein Wahres, oder eine Art von Wahrem, die ihm durchs Leben hilft; nur er darf sich nicht gehen lassen; er muß sich kontrollieren.. ..» («Sprüche in Prosa» [Natw., Schr.., 4. Bd.., 2. Abt.., S. 460]). «Der geringste Mensch kann komplett sein, wenn er sich innerhalb der Grenzen seiner Fähigkeiten und Fertigkeiten bewegt; aber selbst schöne Vorzüge werden verdunkelt, aufgehoben und vernichtet, wenn jenes unerläßlich geforderte Ebenmaß abgeht..» [Ebenda S. 443]
[ 4 ] Es wäre lächerlich, wenn man behaupten wollte, Goethe habe, um überhaupt etwas zu leisten, sich auf ein Feld begeben, das außerhalb seines Gesichtskreises lag. Es kommt alles darauf an, festzustellen, was Mathematik zu leisten hat, und wo ihre Anwendung auf Naturwissenschaft beginnt. Darüber hat Goethe nun wirklich die gewissenhaftesten Betrachtungen angestellt. Der Dichter entwickelt da, wo es sich darum handelt, die Grenzen seiner produktiven Kraft zu bestimmen, einen Scharfsinn, der nur noch von seinem genialischen Tiefsinn übertroffen wird. Darauf möchten wir vor allem jene aufmerksam machen, die über Goethes wissenschaftliches Denken nichts anderes zu sagen wissen, als daß ihm die logischreflektierende Denkweise abging. Die Art, wie Goethe die Grenze zwischen der naturwissenschaftlichen Methode, die er anwendete, und jener der Mathematiker bestimmte, verrät eine tiefe Einsicht in die Natur der mathematischen Wissenschaft. Er wußte genau, welches der Grund der Gewißheit mathematischer Lehrsätze ist; er hatte sich eine klare Vorstellung darüber gebildet, in welchem Verhältnisse die mathematische zu der übrigen Naturgesetzlichkeit steht.
[ 5 ] Soll eine Wissenschaft überhaupt einen Erkenntniswert haben, so muß sie uns ein bestimmtes Wirklichkeitsgebiet erschließen. Es muß sich in ihr irgendeine Seite des Weltinhalts ausprägen. Die Art, wie sie das tut, bildet den Geist der betreffenden Wissenschaft. Diesen Geist der Mathematik mußte Goethe kennen, um zu wissen, was in der Naturwissenschaft ohne Hilfe des Kalküls zu erreichen ist, und was nicht. Hier liegt der Punkt, auf den es ankommt. Goethe selbst hat mit aller Bestimmtheit darauf hingewiesen. Die Art, wie er es tut, verrät eine tiefe Einsicht in die Natur des Mathematischen..
[ 6 ] Wir wollen auf diese Natur näher eingehen. Gegenstand der Mathematik ist die Größe, das, was ein Mehr oder Weniger zuläßt. Die Größe ist aber nichts an sich selbst Bestehendes. Es gibt im weiten Umkreise menschlicher Erfahrung kein Ding, das nur Größe ist. Neben anderen Merkmalen hat jedes Ding auch solche, die durch Zahlen zu bestimmen sind. Da die Mathematik sich mit Größen beschäftigt, hat sie zu ihrem Gegenstande keine in sich vollendeten Erfahrungsobjekte, sondern nur alles das von ihnen, was sich messen oder zählen läßt. Sie sondert alles, was sich der letzten Operation unterwerfen läßt, von den Dingen ab. So erhält sie eine ganze Welt von Abstraktionen, innerhalb welcher sie dann arbeitet. Sie hat es nicht mit Dingen zu tun, sondern nur mit Dingen, insofern sie Größen sind. Sie muß zugeben, daß sie da nur eine Seite des Wirklichen behandelt, und daß die letztere noch viele andere Seiten hat, über die sie keine Macht hat. Die mathematischen Urteile sind keine Urteile, die wirkliche Objekte voll umfassen, sondern sie haben nur innerhalb der ideellen Welt von Abstraktionen Gültigkeit, die wir selbst als eine Seite der Wirklichkeit von der letzteren begrifflich abgesondert haben. Die Mathematik abstrahiert die Größe und die Zahl von den Dingen, stellt die ganz ideellen Bezüge zwischen Größen und Zahlen her und schwebt so in einer reinen Gedankenwelt. Die Dinge der Wirklichkeit, insofern sie Größe und Zahl sind, erlauben dann die Anwendung der mathematischen Wahrheiten. Es ist also ein entschiedener Irrtum, zu glauben, daß man mit mathematischen Urteilen die Gesamtnatur erfassen könne. Die Natur ist eben nicht bloß Quantum; sie ist auch Quale, und die Mathematik hat es nur mit dem ersteren zu tun. Es müssen sich die mathematische Behandlung und die rein auf das Qualitative ausgehende in die Hände arbeiten; sie werden sich am Dinge, von dem sie jede eine Seite erfassen, begegnen. Goethe bezeichnet dieses Verhältnis mit den Worten: «Die Mathematik ist wie die Dialektik ein Organ des inneren höheren Sinnes; in der Ausübung ist sie eine Kunst wie die Beredsamkeit. Für beide hat nichts Wert als die Form; der Gehalt ist ihnen gleichgültig. Ob die Mathematik Pfennige oder Guineen berechne, die Rhetorik Wahres oder Falsches verteidige, ist beiden vollkommen gleich..» («Sprüche in Prosa»; Natw. Schr.., 4. Bd., 2. Abt.., S. 405). Und «Entwurf einer Farbenlehre» 724 [ebenda 3. Bd.., S. 277]: «Wer bekennt nicht, daß die Mathematik, als eines der herrlichsten menschlichen Organe, der Physik von einer Seite sehr vieles genutzt?» In dieser Erkenntnis sah Goethe die Möglichkeit, daß ein Geist, der sich in Mathematik keiner Kultur erfreut, sich mit physikalischen Problemen befassen kann. Er muß sich auf das Qualitative beschränken.
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.