Proof and perception
III
by
Michael Otte
Institut für Didaktik der Mathematik
Bielefeld, Germany
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We have seen so far (in parts I
and II)
that mathematical perception depends on
representations. This implies that mathematics
deals with intensional objects as the computer in
particular has reminded us makes us. For instance,
in Cabri-geometry
two triangles which seem to be completely the same
(or congruent) may behave differently when being
pulled around, because they have been constructed
in different ways (they are intensionally
different).
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Mathematics is interested in truths about
real objects and therefore is fundamentally interested in
extensions. There is, in fact, a systematic study of
extensions in set theory. Mathematics as well as set theory
depends on ostensive demonstration and indexicality and
thus, like any empirical science, once more on perception.
With respect to set theory, the axiom of extensionality
expresses these necessities. It leads us to consider higher
order entities, like predicates or functions or concepts, as
objects, that is, as sets. But concepts or ideas are also
and primarily mental instruments or schemes of action or
functions. In his paper on Russell's logic Gödel
accordingly proposed to introduce a version of the axiom of
extensionality for concepts, claiming that "no two different
properties belong to exactly the same things". And he
illustrates this proposal saying: "Two, for example, is the
notion under which fall all pairs and nothing else. There is
certainly more than one notion in the constructivistic sense
satisfying this condition, but there might be one common
'form' or 'nature' of all pairs" (Gödel in: P.A.
Schilpp (ed.) The Philosophy of Bertrand Russell, La Salle
1944, 138).
The notion of the set of all things or the
set of all truths and related notions appear strange,
contradictory in themselves or just wrongheaded. Piaget, for
example, rightly believes that "the set of all possibilities
is as antinomic as the set of all sets" and he thus
justifies the necessity of a genetic approach to learning
and epistemology. The 'possible' is a process and the same
holds true for notions like 'concept', 'idea' or 'meaning'.
These entities appear to be characterized by a
complementarity of process and function on the one side and
objective existence on the other.
About 30 years after the publication of
his essay on Russell, Gödel himself no longer believed
"that generally sameness of range is sufficient to exclude
the distinctness of two concepts" (see: Hao Wang, A Logical
Journey, The MIT Press 1996, 275). Gödel now no longer
believed that the range of applicability of a concept
generally forms a set. "Only concepts having the same
meaning (intension) would be identical", he now said. Ideas
or concepts seem entities, whose mode of being consists in
that they are universals and at the same time mere
collections of concrete instances of actions or
applications. They form, as was said processes, but
processes that establish their own internal constraints.
Thus the problem of meaning in mathematics
and science is inseparably linked with the status and the
role of theoretical ideas, concepts and hypostatic
abstractions. R. Thom, in his invited lecture to the Exeter
International Congress on Mathematics Education in 1972, put
the problem of meaning in central place. "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 1973, 202). And
Bruner in a similar vein asks, "What do we say to a young
child, asking if concepts like force or pressure really
exist?"
To develop a theory of meaning it seems
essential how we conceive of these generals or universals.
We could say that language is just an instrument of
communication, rather than of representation and that
therefore meaning is based on conventional rules. Human rote
learning is an example of a very rudimentary form of
cognitive activity. But normally it is accompanied by a
second-order phenomenon which we may call "learning to rote
learn". For any given subject, there is an improvement in
rote learning with successive sessions asymptotically
approaching a degree of skill, which varies from subject to
subject. This implies that sorts of intuitions or mental
representations of ideas, which help governing and steering
the activity, accompany even such types of algorithmic
activity. Secondly, universals or generals, if conceived
from the point of view of human activity are to be
understood in functional terms with related to certain
problems or applications.
A mathematical object, such as a
geometrical point, a number or a function, does not exist
independently of the totality of its possible
representations, but it is not to be confused with any
particular representation, either. It is a general that, as
was said, cannot as such be exhausted by any number of its
representations. An idea is not to be conceived as a
completely isolated and distinct entity in Platonic heaven,
but is on the other side not to be confused with any set of
intended applications. Primarily for the reasons Gödel
had enunciated, namely that the range of possible
applications is no definite set at all. Meanings are
generals in the sense of referring to an indefinite and
undetermined collection of possible applications. Second,
two predicates or concepts or functions (or functions of
functions) are to be considered as different even if they
apply to exactly the same class of objects because they
influence mental activity differently and may lead to
different developments.
Classical modern mathematics therefore, as
was said already, essentially deals with intensional objects
and this leads to the introduction of an infinite hierarchy
of ontological levels. This point of view is
anti-positivistic and anti-nominalistic in that it considers
concepts or ideas to be real, whereas anti-realism claims
that theoretical concepts are either unnecessary or at least
mere façon de parler (see: R.Tuomela, Theoretical
Concepts, Springer N.Y. 1973, 3).
The recursive and reflective nature of
mathematical method unfolds the complementarist character of
ideas. The topologist Salomon Bochner considers the
iteration of abstraction as of the distinctive feature of
the mathematics since the Scientific Revolution of the 17th
century.
"In Greek mathematics, whatever its originality
and reputation, symbolization ... did not advance beyond
a first stage, namely, beyond the process of
idealization, which is a process of abstraction from
direct actuality, ... However ... full-scale
symbolization is much more than mere idealization. It
involves, in particular, untrammeled escalation of
abstraction, that is, abstraction from abstraction,
abstraction from abstraction from abstraction, and so
forth; and, all importantly, the general abstract objects
thus arising, if viewed as instances of symbols, must be
eligible for the exercise of certain productive
manipulations and operations, if they are mathematically
meaningful. .... On the face of it, modern mathematics,
that is, mathematics of the 16th century and after, began
to undertake abstractions from possibility only in the
19th century; but effectively it did so from the outset"
(Bochner 1966, 18, 57).
The advent of the computer has enforced this trend.
Dijkstra, for instance, writes:
"Compared with the depth of the hierarchy of
concepts that are manipulated in programming, traditional
mathematics is almost a flat game, mostly played on a few
semantic levels, which, moreover, are thoroughly
familiar. The great depth of the conceptual hierarchy -
in itself a direct consequence of the unprecedented power
of the equipment - is one of the reasons why I consider
the advent of computers as a sharp discontinuity in our
intellectual history (E.W. Dijkstra, On a Cultural Gap,
The Math. Intelligencer Vol. 8, No. 1, 1986).
The realistic or rather complementarist attitude makes
sense from a dynamical viewpoint. By 'realistic' I do not
mean Platonist in the sense of Gödel, because I
consider the applications of an idea as essentially
belonging to it, although I appreciate Gödel's
anti-constructivism and anti-nominalism. An idea, I believe,
is simultaneously an entity in its own right as well as a
mental function or intellectual tool. This is what I want to
exemplify in the sequel.
Cognitive activity may, I believe, be
described as a system of means and objects and the dialectic
of means and objects may briefly be summarized as
follows:
- As in any other cognitive activity, object and means
of cognition are also linked in mathematical activity.
Mathematics cannot proceed in an exclusive orientation
towards universal, formal methods. This would in the last
instance amount to mathematical activity itself being
suited to mechanization and formalization. Mathematics,
too, forms specific concepts intended to help us
understand mathematical facts.
- Object and means are not only linked, but also stand
in opposition to one another. Objects or problems are
resistant to cognition. They do not produce the means to
their solutions out of themselves. Modern mathematics
even obtains its own dynamics in no small part from
applying theorems and methods which at first glance have
nothing to do with the problems at hand.
In this, we understand by "object" now any problem and by
"means" anything which seems appropriate to achieve
mediation between the subject and the object of cognition,
any idea which might help solving the problem and any
representation of that idea. Now, two different ideas may be
decisive in solving a particular problem and thus appear as
equivalent in this respect. Another problem may elucidate
their difference and may in turn itself be illuminated by
this difference. Fundamental ideas and theoretical concepts
are self-referential, that is they themselves, at least in
part, organize the process of their own deployment and
articulation. These ideas are what the development of an
entire theory is devoted to unravel and to explicate. In
mathematics to understand an idea or a concept means to
apply it and to develop a theory. These ideas are, however,
at the same time the beginning and the base of the
development. This means they have to be intuitively
impressive, must motivate and guide activity and orient
representation.
Two
examples:
A string around the
earth...
A circle rolls without
slipping...
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Reactions?
Remarks?
The reactions to the contribution of Michael Otte will
be
published in the May/June 99 Proof Newsletter
©
M. Otte 1999
Traduction
libre N. B.
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