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Dilemmas. He writes:
. . . we seem to find clashes between the things that scientists tell us about our
furniture, clothes and limbs and the things that we tell about them. We are apt to
express these felt rivalries by saying that the world whose parts and members are
described by scientists is different from the world whose parts and members we
describe ourselves, and yet, since there can be only one world, one of these seem-
ing worlds must be a dummy-world. (Ryle, 1964, p. 68)
In fact, theorists do not describe chairs, clothes, or limbs at all, as we
wanted to say by describing the different principles of individuation. And
Ryle therefore concludes that if the feuds between science and common
knowledge are to be dissolved, their dissolution can come only "from
drawing uncompromising contrasts between their businesses" (p. 81). It is a
better policy to remind people "how different and independent their trades
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MICHAEL OTTE AND FALK SEEGER
actually are" than to pretend that all "are really fellow-workers in some joint
but unobvious missionary enterprise" (p. 81).
The conlusion Ryle proposes is difficult to maintain in a society that as a
whole has been transformed into a "laboratory" for complex technologies.
To point to the fact of a radical division of labor that prevails in our soci-
eties is no more sufficient when the society as a whole irrevocably and
completely depends on science and technology, and that demands that ev-
erybody be educated scientifically to a certain degree. Application of
knowledge is a sociohistorical process that is more strongly influenced by
knowledge about humans than by knowledge about objects. Nowhere is the
technologically or scientifically manufacturable taken as a guideline for ac-
tion. Political or social considerations always interfere.
Our picture of science can now be sketched more completely. As science
is a social system too, it inherits the dichotomies that beset society. For in-
stance, it is not as purely objective as might appear so far. It must seem al-
most obvious that much of the dynamics and orientation of theoretical
knowledge is governed by the self-image and the desires or wishes of the
cognitive subject, by that which it considers as relevant. Otherwise, discon-
tinuities and revolutions in the history of science could not be explained and
would even remain unthinkable. In this manner, normative and objective as-
pects of science become inseparably entangled, and human interactions with
objective reality take different forms in analogy to different forms of social
interactions.
Positivist science in general tends to ignore such involvements and bases
its activities on a strict separation between subject and object as well as on
the assumption of an independent but knowable reality. It thereby excludes
the problems of knowledge application from its proper concern, too.
The sciences begin with the distinction between subject and object, or
their activity is based on it, but they are not aware of this fact. They do not
see what they assume operatively. They operate with existing things, but do
not concern themselves with the essence or with the reality of this existence.
Being is, as Kant said, no real predicate of logic (Kant, 1787, B 626).
Essence or existence, however, are important categories for the dynamics of
the learning process, as this process is at the same time a process of
developing the subject or the personality. From this, it can be concluded that
the self-image of science may not be appropriate for being introduced into
its reasoning.
Epistemologically, recent centuries were under the sign of nominalism.
The evolution of industrial capitalism was accompanied by a state of mind
that understood the mental process as overcoming a limiting philosophy
having medieval roots. This has led to the idea that there is complete free-
dom in forming concepts. Only after humanity, as it is said, took the liberty
of creating its own concepts according to its own goals did reasoning be-
come, on the one hand, a means toward any purpose, and, on the other hand,
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was divested of objective meaning. It became disengaged and instrumental
at the same time. Shedding scholastics as a transition to nominalism!
Nominalism prevails with Augustine’s "discovery of subjectivism," which
shifts the aporias of history and of mechanical motion into the subject (cf.
von Weizsäcker, 1971, p. 435).
With regard to mathematics, this philosophy seems to come to a head
with Georg Cantor’s repeatedly quoted, world-famous statement of 1883:
"The essence of mathematics lies precisely in its freedom;" the other half of
the quote which deals with the metaphysical constraints on applied mathe-
matics always being suppressed and forgotten.
Again, this double process that liberated knowledge from its situational
context and the individual from its metaphysical religious and social bounds
depends very much on the development of literacy and printed media. In the
limit of idealization, all of mathematics can be regarded as a collection of
grammatically correct potential texts. And this kind of knowledge requires a
new kind of mind, a mind that could achieve knowledge without social in-
volvement, the mind of the rational, detached individual.
6. THE PROBLEM OF MEANING
On the one hand, general compulsory education, as institutionalized in our
schools, has always been dependent of this kind of knowledge. It has always
relied on the theoretization of knowledge, despite the fact that this theo-
retization permanently causes practical pedagogical troubles and difficulties.
Only the complexity of theoretical generalizations, which are determined
objectively as well as socially, makes it possible to link individual develop-
ment to the complex possibilities of the "real world:" Theoretization opens
up a universe of experience that is rich enough to allow a very great variety
of members of society to participate and to develop and simultaneously
keeps all the conflicting immediate demands and wishes at a distance leav-
ing a certain necessary autonomy to the school. On the other hand, this theo-
retization and abstractness of knowledge causes a fundamental problem in
education, namely, the problem of meaning.
The real problem which confronts mathematics teaching is not that of rigor, but
the problem of the development of "meaning," of the "existence" of mathematical
objects. (Thom, 1972, p. 202)
This fundamental problem of mathematics teaching in the framework of
compulsory general education comes from the antiempiricist nature of theo-
retical knowledge in particular. Theoretical terms are neither concrete ob-
jects nor properties of such objects and are not names of empirical objects
either. The content of a theoretical term consists of relationships between
things and not of things (their properties, etc.) themselves. J. Bruner has
stated this problem to be the problem of the indirect nature of scientific
knowledge.
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MICHAEL OTTE AND FALK SEEGER
[Scientific concepts] are inferences we draw from certain regularities in our
observations. This is all very familiar to us…. To a young student, who is used
to thinking of things that either exist or do not exist, it is hard to tell the truth in
answer to his question whether pressure really exists. (Bruner, 1960, p. 69)
Stating the fundamental problem of mathematical education in such a way
by no means expresses a danger of forgetting the pedagogical, organiza-
tional, social, or psychological implications of education and schooling. On
the contrary, formulating the central issue of didactics in this way implies,
for instance, that a direct teaching of scientific concepts is impossible. There
is no simple way of relaying a theoretical concept from teacher to student.
Concepts cannot be handed over like concrete objects, because they are not
such things nor their names.
According to how we have explained this problem of meaning, it seems
obvious that this problem is not a technical problem but a problem of the
development of the individual. And this question of subjective development
is not to be restricted to the cognitive aspects either but implies a develop-
ment of personality. This links mathematics education to philosophy and
history. The reference to history adds, or better, makes aware of, a philo-
sophical element to scientific reasoning, which refers precisely to the role
the subject’s self-understanding has.
In a certain sense, the difficulty is contained in Leibniz’ principle of the
identity of indiscernibles, which was mentioned already. Now Bertrand
Russell has pointed out that it follows from the analysis of this principle that
a subject or a substance is either nothing else but the sum of its properties,
thus losing its subject character, or that it cannot be defined at all. From this,
Russell concludes that Leibniz’ principle makes no sense. As opposed to
that, we believe that Russell’s analysis expresses a profound philosophical
difficulty. This problem has become salient in our time, in particular, in the
subject-machine problem, but it has more or less explicitly played a key role
in the development of fundamental concepts of mathematics since the 17th
and the 18th centuries. Among these, the concept of (mechanical) motion
would have to be named first, and, correspondingly, the concept of function
(cf. Bibler, 1967). But also in the philosophy of psychology, it has played a
major role, as reflected in Gestalt psychology’s statement that the whole is
more than the sum of its parts (cf., e.g., Wertheimer, 1925/1967).
In the present debate on the question "Can computers think?" the result is
that the (human) subject can neither be identified with the totality of his or
her presently accessible properties, for, otherwise, she or he could be simu-
lated on a suitably programmed computer, nor that the subject can be con-
ceived of as a substance beyond all his or her properties, because, otherwise
she or he would be inaccessible to his or her own self-reflection. In this
case, the human subject would lose his or her subject character, as human
cognition differs from mere information by the fact that the subject not only
knows but knows that she or he knows. Knowledge and metaknowledge are
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HUMAN SUBJECT IN HISTORY
inseparably connected in human cognition. Thus, it may not be a solution to
argue, that the "whole is one its parts" (Minsky, 1988; Minsky & Papert,
1973).
It may thus be said that certain current problems with science and tech-
nology (the problem of meaning, the man-machine problem, the problem of
interpretation of quantum physics, etc.) have got nominalism into difficul-
ties, difficulties that have to do with the subject’s self-image, and that all the
aporias and dualisms, the polarity of the finite and the infinite, the para-
doxes of motion, of time, and of the present, and so forth are manifested
mainly in the subject, indeed in the cognizing subject’s self-image as a
(potentially) universal and actually always limited being. As a consequence,
we conclude that thinking has an existential import and that theory can be
understood as a "mode of life."
Now mathematics education has to do with the relationship between
mathematics and the human subject, and requires that these problems are
dealt with in the sense that an effort at historical reconstruction is made,
which must also make use of philosophical insights that were present before
the rise of individualism and so-called subjective turn. In this sense, von
Weizsäcker writes:
I would not have turned to studying classical philosophy if I had not encountered
inconsistencies in the conceptual traditions of modem physics and humanities and
in modern philosophy which I could only hope to understand by going back to
their historical sources. To me, the great steps of progress in modern times like
the emergence of the exact natural sciences, the shaping of subjectivity and the
growth of a historical consciousness seem to have been paid for by certain con-
strictions to the questions raised and the concepts formed. Whoever inquires at
one spot into the problems of modern times, say, into the foundations of physics,
will rediscover the very same structures which have already been discovered by
the Greek philosophers, although from another perspective. (von Weizsäcker,
1992, p. 440; our translation, M.O./F.S.)
This quotation may not be understood as a plea to recur in mathematics ed-
ucation to Greek philosophy as a general didactical strategy, even though
this might prove worthwhile at varying occasions. It should be read as a plea
to teach mathematics as a historically grown subject in the sense in which
Foucault put it that "real science recognizes its own history without feeling
attacked" (Martin, 1988, p. 12).
If we try to condense what has been said so far about the parallelism be-
tween the principle of identity in mathematics and the process of identity
formation of the human subject, it becomes quite clear that the contribution
of mathematics education to the self cannot be seen as residing in changes
of the self as a substance. Focusing instead on the relational aspects and on
the processes of becoming, we find the relation of the self to time to be of
utmost importance (cf. Brockmeier, 1991). The self has to be understood as
intimately connected to processes that develop in time, identity being what
remains constant in the flux of time. The development of the self, then, is
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better conceived of as a "way," a "journey along the road," very much in
analogy to what happens to the protagonist in the classical Bildungsroman
(cf. Bakhtin, 1981). Using the metaphor of the equivalence or identity of the
two sides of an equation, the aim of mathematics education cannot be to ar-
rive at the shortest possible way to the conclusion that A equals A . We argue
that this "minimal loop" approach (cf. Churchman, 1968) is closely con-
nected to a formalist view in mathematics education that largely excludes a
historical perspective on the becoming of mathematical concepts and theo-
ries. A "maximal loop" approach to furthering the development of mathe-
matics as part of the development of the self instead seeks to bring the sub-
ject into contact with as many different perspectives on mathematics as
possible along the "way." We have given some reasons above why this
should necessarily also be a "journey into the past," as we have to learn
from history how the subject itself is involved in the processes of construct-
ing meaning.
7. CONCLUDING REMARKS
We have tried to underline above that the perspectivity of knowledge is a
necessary by-product of literacy and literal culture. If one takes a look now
at the conditions that frame a realization of the above deliberations in the
mathematics classroom, it becomes clear that it has to be taken into account
that learning in classrooms is mostly an outcome of an oral discourse as part
of an oral culture. The development of meaning, thus, cannot be seen only
in the decontextualization, in the liberation from the concrete situation that
was made possible through literacy. Constructing meaning in classroom
learning is a result of contextualization and situatedness that is typical for
the discourse in schools. The linearization and individualization of literal
thinking (Havelock, 1986) has to be complemented by the orality of the
classroom, by conversation and discussion, which all put the subject in rela-
tion to other subjects and make her or him experience that their own per-
spective is only one among different possible ones.
The importance of a historical perspective extends well beyond the stu-
dents’ discovery that similar problems existed a long time ago and that their
obsolescence seems unwarranted. In the course of a historical study, the
process of constructing meaning conies into focus. In this way, it is con-
ceivable that substantial and functional thinking are not only steps in a pro-
cess of evolution that culminates in functional thinking as having the most
general claim to truth and objectivity. The different modes of thinking will
rather be understood as resulting from a certain worldview, and it becomes
clear that the universal claim of our own worldview is only a relative one.
The relation between universality and particularity is a key to an under-
standing of the role of the human subject. The epistemological situation of
the subject has been styled above by a potential universality and an actual