AN INTRODUCTION TO KANT'S AESTHETICS

136 TWO CHALLENGES
one has to wonder whether one might not equally well say that all poetry is
written down already somewhere in a Platonic realm of poetry.
There is, then, no unique way to solve problems in mathematics that could
count as “the natural path of enquiry and reflection in accordance with rules.”
Similarly, the expression “in accordance with rules” (nach Regeln) is problematic.
It is rather vague and can be misleading. Rules do not prescribe how they have
to be used. (The German word “nach” suggests an even stronger connection
between reflection and rules than does the English expression “in accordance
with.”) There is always much freedom in how and where to make use of them,
when and what to take, let, and apply to what. Rules even change. After all, there
is a history of mathematics. The old rules usually do not turn out to be false.
They simply apply to an older part of mathematics that is not being considered
any more, that has been modified, or that has been integrated into another area
of mathematics.
It is for these reasons that Newton cannot, contrary to what Kant wrote, make
“entirely intuitive” all “the steps that he had to take” (see quote above), simply
because he did not really have to take them and because other steps were possible,
too. Just think of Leibniz’s infinitesimal calculus as an alternative to
Newton’s. The steps are “intuitive” in the sense that we can reproduce them, that
they do not violate other mathematical rules, that they make sense, and that they
are part of a theory that works as far as we can see at the time and within a
certain framework. The steps are intuitive in this sense, but not in an absolute
sense, as if no other steps were possible. Moreover, a new theory often makes us
look further and enlarge that framework, as was the case with relativity theory,
for instance. The concept of what is “entirely intuitive” changes if we adopt a
Kuhnian or Wittgensteinian perspective, making us more aware of historical,
social, and cultural aspects.
Of course, in many respects mathematics is not like poetry. There are strict
rules in mathematics that should never be violated, such as the rule of non-contradiction.
So there is more freedom in poetry. And mathematics is more abstract
than poetry, because you can sometimes change a line here and there and still
say that it is the same proof, which is hardly so in poetry. There we would tend
to say that changing a line is making another poem. But nevertheless, there are
creative elements in mathematics that Kant might have underestimated. We will
come back to this.
From the mid-1780s on we find several passages that relate rules to learnability
and taste and at the same time separate all these aspects from genius and the
arts:
If I derive rules from other rules that are already known, this is a case of talent but
not of genius. (Anthropology Lecture Notes Mrongovius, 1784/85, XXV 1310)