XII
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 naturalscientific 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 naturalscientific 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.
