## Goethean Science

GA 1

### 12. Goethe and Mathematics

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.