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Exactness as a constant, concepts in the change of mathematical reasoning. On analysis in the 19th and 20th centuries. (Exaktheit als Konstante, Begriffe im Wandel des mathematischens Denkens. Zur Analysis im 19. und 20. Jahrhundert.) (English) Zbl 0876.00012

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, August 16-23, 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 45-59 (1993).
Although Augustin Louis Cauchy’s infinitesimal calculus shows close resemblances to the modern way of dealing with this topic some of his theorems appear to be false, e.g., the theorem that a function is continuous at a place, if and only if it has a definite finite value at this place. The author criticizes the standard way in the historiography of mathematics to deal with the problem of “false theorems” in Cauchy. In what he calls the “inductive perspective”, i.e., the opinion that the development of science could be regarded as a process of continuous growth, these theorems are regarded simply as errors of Cauchy. In the alternative perspective of a rational reconstruction of science in terms of scientific research programs (Imre Lakatos) Cauchy’s theorems appear to be correct, but belonging to different research programs (some of these theorems are valid in non-standard analysis). The author criticizes that both perspectives judge the correctness of Cauchy’s mathematics from the perspective of modern mathematics; in both perspectives errors in Cauchy’s work appear, but there is no sufficient explanation of reasons for these errors.
The author opposes to these perspectives his attempt to approach “Cauchy’s real thinking” (p. 51). He demands that it is necessary to understand Cauchy’s mathematical language as a whole before it is possible to evaluate certain theorems. Cauchy’s basic concepts are quantity and value, connected by the relation “taking”. A quantity takes values. In this language a function is a special (dependent) quantity.
In the light of these foundations there are no mathematically significant errors in Cauchy’s thinking, but only some loose formulations.
Whereas the general inductive way of dealing with the history of mathematics (“Resultatismus”) focuses on the parts of history which are to some extent relevant for today’s mathematics, the author’s way of dealing with history (“Geneseologie”) is concentrated on what is changing in time, what was interesting in its time, but has lost its interest doday. “Resultatistic work misses the historical, geneseological work deals with what has become unimportant and useless” (p. 57). This resumée seems to overshoot the mark by far. In this opinion the history of mathematics appears to be an activity l’art pour l’art without any value for the science it is working on. There is, however, no need for a complete disjunction between the two approaches mentioned by the author. The “geneseological” approach could serve as a complement to and as a corrective for approaches basically interested in the pre-history of modern mathematics.
For the entire collection see [Zbl 0836.00022].

MSC:

00A30 Philosophy of mathematics
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
40-03 History of sequences, series, summability
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