Michael Otte

The ideas of Hermann Grassmann in the context of the mathematical and philosophical tradition since Leibniz

of Leibniz’s ideas. This disagreement provides a new opportunity to present the issue within the framework of broader mathematical and philosophical concerns. What had in fact changed was that the ontological foundation of classical epistemology was no longer valid during the 19th century. The conviction that thinking directly understands being itself, no longer existed. Scientific thinking now either is committed to positivistic empiricism, or insists on the “theory ladenness” of observation as well as of intuition, while at the same time trying Die einzige, ausdriickliche Anerkennung von mathematischer Seite erhielt Hermann Grassmann (1809-1877), der, wie wir heute sagen konnen, sowohl in der Sprachwissenschaft wie in der Mathematik Aul3ergewohnliches geleistet hat, von der Fiirstlich Jablonowskischen Gesellschaft fur sein Werk “Geometrische Analyse gekniipft an die von Leibniz erfundene geometrische Charakteristik” im Rahmen einer Preisaufgabe dieser Gesellschaft zum angegebenen Thema. In diesem Werk entwickelt Grassmann Ideen zu einem mit den geometrischen Objekten direkt operierenden Kalktil, die auf Leibniz zurtickgehen. Leibniz war sowohl mit der euklidischen wie cartesischen Behandlung der Geometrie unzufrieden und suchte einen Zeichenkalkiil, mit dessen Hilfe “die Analyse wirklich zu Ende gefiihrt werden konnte’ ’ . J. Echeverrfa hat ktirzlich als erster die allgemein akzeptierte Auffassung kritisiert, da8 Grassmanns Werk als legitime Entwicklung und Fortfiihrung der Leibnizschen Ideen betrachtet werden kann. Die dadurch aufgetretene Meinungsverschiedenheit bietet eine willkommene Gelegenheit, die Problematik in ihrem weiteren mathematischen und philosophischen Kontext aufs neue zu analysieren. Tatsachlich ist Leibniz und Grassmann das Interesse an Fragen der Ontologie gemeinsam, und das grenzt sie beispielsweise gegemiber der vorherrschenden positivistischen Tradition des 18. und 19.

Mathematical Knowledge and the Problem of Proof.

Every proof is faced with the requirement of proving that the proof is correct, and the proof of the correctness of the proof again meets the same requirement and the proof of the correctness of the correctness of the proof also, etc. In order to escape from an infinite regress into which one is led one has to come down with a purely algorithmic criterion for correctness or to claim that thinking is identical with its subject matter. Whence the preference of number and more generally of conceptualism in pure mathematics. Conceptualism is a kind of nominalism that does not give a realist understanding of mathematics (note that Platonism is not an opponent of nominalism as some seem to believe). The paper presents some examples and reflections intending to hint at the role of formal thought in the process of knowledge growth. It argues that there is no division of labor according to which certain modes of human cognition are associated with certain tasks and certain cognitive roles exclusively. In this connection, the paper claims that the subject matter of mathematical activity is represented within the system of activity by many different means. Mathematics differs in fact from logic in as much as a principle of heterogeneity or of flexible ‘means-objects-relationships’ is valid. Formalization in contrast brings forward a principle of homogeneity — that like follows like. Every subject matter requires principles homogeneous with itself. The paper tries to draw some conclusions from this difference with respect to the role of formalization within human cognitive development.

Does Mathematics Have Objects? in what Sense?

ION The topologist Salomon Bochner considers the iteration of abstraction as the distinctive feature of the mathematics since the Scientific Revolution of the 17th century. “In Greek mathematics, whatever its originality and DOES MATHEMATICS HAVE OBJECTS? IN WHAT SENSE? 199 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). In a similar vein Peirce writes: “One extremely important grade of thinking about thought, which my logical analyses have shown to be one of the chief, if not the chief, explanation of the power of mathematical reasoning, is a stock topic of ridicule among the wits. This operation is performed when something, that one has thought about any subject, is itself made a subject of thought” (NEM IV, 49). In this way even the means and conditions of thought become an object of it. A predicative or attributive use of some concept is transformed into a referential use in order to incorporate the entity thus synthesized into new relational structures. The above mentioned example of the introduction of the imaginary numbers provides a case in point. At first after having been introduced to generalize certain algebraic operations, these “numbers” seemed the paradigmatic model of an artificial invention, whilst the subsequent history of complex functiontheory would tend to provide this invention with the characteristics of something indubitably objective. In all necessary reasoning, Peirce continues “the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations” (CP 5.162). Piaget, as we have seen, entertains a similar view of mathematical generalization but misses the fact that perception and observation will necessarily play a role throughout the process. In the discussion of the so-called fundamental theorem of algebra, it is said, that Lagrange had tacitly and implicitly used the intermediate value theorem for continuous functions to give a proof of this theorem. In actual fact Lagrange did nothing but provide algorithms for attaining approximate solutions of algebraic equations. It was Cauchy who read the intermediate value theorem for continuous functions into Lagrange’s argument, trying

Das Prinzip der Kontinuität

ZusammenfassungDer Text handelt von der Entwicklung des Prinzips der Kontinuität und von seiner Rolle in der Entwicklung der Mathematik. Diese Entwicklung wird dabei einer Dynamik zugeschrieben, die auf der Wechselwirkung des „Prinzips der Kontinuität” mit seinem Gegenpol beruht, den man das „Prinzip der Identität” (Cassirer) nennen könnte. Beispielweise war Aristoteles einerseits derjenige, der das Prinzip der Kontinuität in die Naturgeschichte eingeführt hat. Auf der anderen Seite gilt Aristoteles wohl zumeist und vor allem als der große Vertreter einer Logik, welche auf der Annahme der Möglichkeit klarer Unterteilungen und strenger Klassifikationen beruht. In der Neuzeit ist Leibniz’ Philosophie zu nennen, wenn es um die Entwicklung des Prinzips der Kontinuität in seiner Wechselwirkung mit dem Prinzip der Identität geht. Heute erscheint das Prinzip der Kontinuität in der Mathematik im Begriff der stetigen Funktion. Im Funktionsbegriff drückt sich dann jene Komplementarität in der mathematischen Denkweise, die man sich am Gegensatz von Kontinuierlichem und Diskretem vergegenwärtigt, sehr deutlich aus.

Theorie, Computer und Bildung

The advent of the computer has intensified the interest in fundamental epistemological and pedagogical questions. Traditionally, the content side of mathematics education was considered only in connection with the logical and cognitive aspects of teaching, the consequences both for human self — understanding in general and from a social and psychological point of view remaining untreated. Confronted with the computer, we begin to realize that our interactions with objects are generally also interactions with each other, and that our view of knowledge is generally also a perspective on ourselves.The paper is divided into several sections: 1. Knowledge and the knowing subject, 2. Theory and school, 3. Sonya’s solution of the „rabbit - and - chicken” problem, 4. Mechanism and complementarity, 5. Context of cognitive tools, 6. Conclusion.