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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). |
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.
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Two
examples: |
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The reactions to the contribution of Michael Otte will
be
published in the May/June 99 Proof Newsletter
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