Abstract
Paul Hertz was an outstanding German physicist, who also devoted himself to mathematical logic and wrote a series of papers that remained rather unnoticed, even if they influenced the development of proof theory and particularly Gentzen’s work. This chapter aims to examine Hertz’s logical theory placing it in its historical context and remarking its influence on Gentzen’s sequent calculus. The analysis of the formal structure of proofs was one of Hertz’s most important achievements and it can be regarded as an anticipation of a “theory of proofs” in the current sense. But also, it can be asserted that Hertz’s systems played the role of a bridge between traditional formal logic and Gentzen’s logical work. Hertz’s philosophical ideas concerning the nature of logic and its place in scientific knowledge will be also analysed in this chapter.
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- 1.
In his biographical article on Hertz, Bernays wrote “Diese Untersuchungen [of Hertz] sind Vorlüfer verschiedener neuerer Forschungen zur mathematischen Logik und Axiomatik, insbesondere hat G. Gentzens Sequenzenkalkul von den H.schen Betrachtungen über Satzsysteme seinen Ausgang genommen” ([5], p. 712), and in a chapter on sequent calculi, Bernays asserted: “... in der Hertz’schen Theorie der Satzsysteme ein gewiss bei weitem noch nicht hinsichtlich der möglichen Fragestellungen und Erkenntnisse ausgeschöpftes Forschungs-gebiet der Axiomatik und Logik vorliegt.” ([4], p. 5, footnote). Haskell Curry in a historical note of his textbook on mathematical logic stated: “For the present context it is worthwhile to point out that Gentzen was apparently influenced by Hertz [...] This throws some light on the role of “Schnitt” in the Gentzensystem”. ([7], p. 246 f.). Vittorio Michele Abrusci wrote an introductory chapter on Hertz’s logical work [1].
- 2.
Together with Moritz Schlick edited Hertz in 1921 the epistemological writings von Helmholtz. In some of these writings, problems concerning the philosophy and methodology of mathematics were discussed, especially in relation to the evaluation of non-Euclidean geometry. In these editions the following papers of von Helmholtz were included “über den Ursprung und die Bedeutung der geometrischen Axiome”, “über die Tatsachen, die der Geometrie zugrunde liegen”, “Zahlen und Messen” and “Die Tatsachen in der Wahrnehmung”.
- 3.
Further information about Hertz can be found in the Nachla\(\beta \) of Paul Bernays (ETH Zürich), the Nachla\(\beta \) of David Hilbert (Göttingen), biographical notes (not published) by Adriaan Rezus (Nijmegen), and above all in Hertz’s Nachla\(\beta \) located at Archives for Scientific Philosophy, University of Pittsburgh.
- 4.
“Unter Satz verstehen wir einen Inbegriff eines Komplexes, der auch nur aus einem ein einzigen Element bestehen kann und antecedens hei\(\beta \)t, und eines Elementes, das succedens hei\(\beta \)t.” ([10, p. 81)
- 5.
Tarski wrote: “The discussion in Hertz, P. (27) has some points of contact with the present exposition.” ([20], p. 62 fn 1).
- 6.
As a whole, Hertz’s conception on logical constants has certain similarities with the idea of logical constants as “puncuation marks”.
References
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Legris, J. (2014). Paul Hertz’s Systems of Propositions As a Proof-Theoretical Conception of Logic. In: Pereira, L., Haeusler, E., de Paiva, V. (eds) Advances in Natural Deduction. Trends in Logic, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7548-0_5
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