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Radu, Mircea

A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann. (English) Zbl 1039.01007

Hist. Math. 30, No. 3, 341-377 (2003).
The author states as the aim of this paper “to reconstruct and examine the significance of Hölder’s arguments against the axiomatization of arithmetic” (p.345) based in particular on Otto Hölder’s 1892 review of Robert Graßmann’s book “Die Zahlenlehre oder Arithmetik” [Verlag von R. Graßmann, Stettin (1891), see also JFM 23.0158.01]. For this purpose Graßmann’s formal approach to arithmetic is sketched as developed together with his brother Hermann Günther Graßmann and based on a general theory of forms. The structure of mathematics (Formenlehre) with its most general discipline, the theory of magnitudes (Größenlehre), is given by the laws governing the abstract binary algebraic operations in the different mathematical disciplines.
In his review, Hölder, following a Kantian line of thought, is most interested in questions of mathematical ontology, in particular the existence of arithmetical objects. He gives two interpretations of Graßmann’s approach, an analytic and a synthetic one, especially stressing the weaknesses of the analytic side. Against Graßmann’s preference for the formal side of mathematics, “Hölder defended the intuitionist position according to which the foundation of pure mathematics is represented by the faculty of the mind of constructing infinite sequences whose mathematical description can be found in arithmetic, and, as a result, he refused to reduce mathematical existence to consistency” (p.376).
Reviewer: Volker Peckhaus (Paderborn)

MSC:

01A55 History of mathematics in the 19th century
01A72 Schools of mathematics
03-03 History of mathematical logic and foundations

Keywords:

arithmetic; abstract algebra; formalism; intuitionism; axiomatics; Grundsatz; Grundformel; Größenlehre

Citations:

JFM 23.0158.01
PDFBibTeX XMLCite
\textit{M. Radu}, Hist. Math. 30, No. 3, 341--377 (2003; Zbl 1039.01007)
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References:

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