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Teachers, that the philosophical faculties of the universities were to be given
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 415-429.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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responsibility for the scientific education of future secondary school teachers.
Up to that point, the task of these faculties had been to provide a general edu-
cation for the students of the higher faculties of theology, medicine, and law.
On the other hand, the Edict Concerning the Students Entering the
Universities of June 25th 1812, for the first time made a firm distinction be-
tween schools and universities, and the minimal mathematical knowledge re-
quired of all students who were to enter the universities was defined (cf.
Jahnke, 1990b, chap. 1).
At the beginning of the 19th century, there was no pedagogy or didactics
taught at the universities. The discourse about education, the shaping of syl-
labi, and the educational value of the various subjects was held immediately
in the public sphere, as far as such a sphere existed, and was poorly struc-
tured, open, and reflected the manifold interests concerned. This constellation
resulted in a certain immediacy of the discourse with all its strengths and
weaknesses. Authors involved were prominent intellectuals in the history of
German thought (such as W. von Humboldt, F. D. Schleiermacher, J. F.
Herbart, J. G. Fichte, F. W. J. Schelling, G. W. F. Hegel) as well as per-
sons who are now forgotten. There were contributions of varying quality.
Insights still valid stand beside utterances of trivial emptiness.
The impact of catchwords, however, should not be underestimated. In
practical affairs, they are frequently more decisive than sophisticated concep-
tions. They serve as points of crystallization that unify the thinking of many
people, although (and just because) these people may mean very different
things by these words. The practical meaning of the catchwords can be read
off from the actual activity. Their theoretical background can be reconstructed
from the writings of certain authors. What we find there is valid, of course,
only for the author concerned, but it is of general validity insofar as it reveals
the possible meanings of these catchwords within their historical context and
gives access to the intellectual universe of the time. The influence of a peri-
od's culture is thus quite essentially mediated by its language, and whoever
wants to analyze the influence of this culture must analyze its language.
2. CULTURAL FOUNDATIONS OF NEOHUMANIST
EDUCATIONAL PHILOSOPHY
The "Suvern Syllabus" of 1812, referred to as the "constitutional document
of the new Gymnasium" (Paulsen, 1897, translated), dryly says that "devel-
oping organic reasoning" is the condition for penetrating science (Mushacke,
1858, p. 231, translated). What is meant by organic reasoning is not said,
and in order to understand this, one must recur to the general philosophy of
science and cultural context of the period. First, it will have to be remembered
that there was a general mental revolution at the turn from the 18th to the 19th
century, which may be referred to as overcoming the mechanistic worldview.
The mechanistic theories of the rationalist era were supplemented by concepts
reflecting the historical character of nature and humanity, and they tried to
CULTURAL INFLUENCES: A HISTORICAL CASE
416

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HANS NIELS JAHNKE
grasp the specificity of organic beings. The idea of organism, which origi-
nated in biology, was extended metaphorically to other realms and manifesta-
tions of life. This expressed the profound conviction that all spheres of life
are holistic. Just as an organism is not composed additively of its elements,
because the elements cannot exist alone and separately, science is no sum of
isolated insights, but rather a holistic theoretical entity. Organic reasoning is
thus characterized by the attempt to grasp the holistic character of the objects
and by the fact that it is holistic as reasoning itself, that is, develops from its
own conditions and tries to understand a thing from itself. In an analogous
way, ethics and art can also be understood holistically.
The ideas about education and instruction belonged to this field of thought.
Persons are themselves holistic, they cannot be educated by adding a certain
knowledge to them from without, but they must develop themselves from
within. This is why Selbsttätigkeit (self-activity) was the guiding concept of
the neohumanist-idealist pedagogy of the period. In a narrower sense, this
pedagogy was based on a certain "transfer hypothesis." This hypothesis
again refers to the holistic character of education, saying that to become edu-
cated human beings, persons must, in their own development, have had at
least once the experience of getting totally involved with a problem and cop-
ing with it productively. Only persons who have seen at least in one particular
field that there are things that are holistic and have their own laws will be in a
position to assess what it means not just to adhere to a number of rules in
their own life, but to have the inner freedom to act.
It is clear that such a conception has nothing to do with transfer hypotheses
according to which mathematics trains logical reasoning. Rather, logical rea-
soning and the ability to classify things according to external characteristics,
the so-called intellectual-mechanical abilities, were considered to be a subor-
dinate prestage to "organic thinking." Only after the holistic and organicist
ideas of the Humboldtian era had been dismissed under the supremacy of a
scienticist school of thought in the second half of the 19th century, did the
equation "formal education = training of logical reasoning" emerge.
3. PURE MATHEMATICS AND THE NEOHUMANIST VIEW OF THE
RELATION OF THEORY AND PRACTICE
In the context of this educational philosophy, the Zeitgeist placed particular
emphasis on the classical languages of Greek and Latin. Mathematics, by no
means as a matter of course, also played an important role within the neohu-
manist reform of the Gymnasium. The Süvern syllabus prescribed six math-
ematics lessons per week for each grade. This was certainly not only a con-
sequence of the neohumanist ideal of education, but also due to the model of
France and the École Polytechnique. A fact specific for Germany, however,
was that educational value was accorded only to pure mathematics, the syl-
labus intentionally neglecting everyday practical applications. In the neohu-
manist program, mathematics was highly esteemed as a theoretical, (pure)
417

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and a systematic science (in the sense of the idea of organism). Mathematics
was deemed to be of educational value because it was understood to be a dis-
cipline of theoretical reasoning that unfolds from its own conditions.
From the very outset, the emphasis on pure mathematics and the negative
attitude toward everyday practical applications played an important role. In
this section, I shall look for contemporary justifications for this esteem for
pure mathematics, which is one-sided in our eyes today, analyzing in the next
section how this orientation was actually made to prevail in school, and what
was its role in further developments.
Already in Wilhelm von Humboldt's writings on the organization of edu-
cation, there is an emphasis on pure mathematics in the few quotes in which
he speaks of mathematics at all. Education was to be developed so as to en-
sure:
. . . that understanding, knowledge, and intellectual creativity become fascinating
not by external circumstances, but rather by its internal precision, harmony, and
beauty. It is primarily mathematics that must be used for this purpose, starting
with the very first exercises of the faculty of thinking. (Humboldt, 1810/1964b,
p. 261, translated)
In the Latvian Syllabus, he expressed himself against the tendency
. . . of distancing oneself from the possibility of future scientific activity and
considering only mondane life . . . . Why, for example, should mathematics be
taught according to Wirth, and not according to Euclid, Lorenz, or another rigor-
ous mathematician? Any suitable mind, and most are suitable, is able to exercise
mathematical rigor, even without extensive education; and if, because of the lack
of specialized schools, it is considered necessary to integrate more applications
into general education, this can be done particularly toward the end of schooling.
However, the pure should be left pure. Even in the field of numbers, I do not fa-
vor too many applications to carolins, ducats, and the like. (Humboldt,
1809/1964a, p. 194, translated)
This is a definite position taken against everyday practical applications. It can
be also seen, however, that Humboldt shows a willingness to compromise.
Statements of quite similar kind can also be found among mathematics teach-
ers of the period.
Propositions of this kind seem to express an idealistic and romanticist
worldview in which there is no place for problems of practical and, in par-
ticular, technical applications of science. This may well be true for some au-
thors of the time (although not for Humboldt). Nevertheless, it can be shown
that such views express a reasonable and realistic view of the relationship be-
tween theory and practice, which may also claim to be relevant for educa-
tional reflections in the computer age. To make this evident, I have a docu-
ment that is indeed historically unique. It is from the mathematician August
Leopold Crelle (1780-1855). Crelle is well-known as the founder of the