Book · January 994 citations 110 reads 2,264 authors

Journal für die reine und angewandte Mathematik, the first mathematical peri-
odical in Germany. He worked for 20 years as a technician in the Prussian
administration of public construction and also participated in constructing the
CULTURAL INFLUENCES: A HISTORICAL CASE
418

---

HANS NIELS JAHNKE
Prussian railways. Besides, he was very interested in mathematics and in
mathematics instruction. In 1828, he was nominated advisor for mathematics
within the Prussian ministry of education (cf. Eccarius, 1974).
In this quality, he gave several statements about the educational value of
mathematics, the most pronounced in the preface to his Enzyklopädische
Darstellung der Theorie der Zahlen (Crelle, 1845), saying that the use of
mathematics is twofold, first for immediate application, second for training
the ability to think. Upon weighing these two purposes against one another, it
would show that in most practical problems it is difficult to apply mathemat-
ics because they are too complicated. Frequently, the application of mathe-
matics would even lead to serious mistakes and errors, because people rely
on its rigor and certainty without considering the complexity of the condi-
tions. Quite indubitable, however, was the use of mathematics to exercise the
ability to think, and this was also the indispensable condition for the direct
application of mathematics.
419
Only after a mathematical spirit has been awakened by assiduously exercising
judgment by means of mathematics (without regard for applications), and only
then, one may quite boldly count on the uses of mathematics in applications.
Mere knowledge in mathematics, intended for applications . . . is not sufficient
for appropriate applications, but the guiding principle must be the mathematical
spirit, the mathematical way of thinking. Only he who tackles applications on
this basis will err less easily, for he will first of all examine what mathematics
can properly achieve, and where and how the tool can be usefully applied
. . . . Hence it is quite right that mathematics be exercised as much as possible
in schools . . . at first without any consideration of applications in common life.
(Crelle, 1845, pp. IX-X, translated).
With the notion of the faculty of judgment, Crelle referred to Kant. In the
Critique of Pure Reason, Kant had defined this by stating that a rule does not
say by itself to which cases it can be applied, and that, therefore, the applica-
tion of a rule requires a faculty of judgment.
In general, Crelle's position can be described as follows. On the one hand,
there is the direct application of science, in which scientific laws are applied
to practical problems by specialization. According to Crelle, however, this is
only rarely possible. In most cases, science does not provide rules that can be
applied directly. Nevertheless, also in these situations, science may play a