Conceptions of infinity and set in Lorenzen’s operationist system. (English) Zbl 1494.03011

Antos, Carolin

Heinzmann, Gerhard (ed.) et al., Paul Lorenzen – mathematician and logician. Contributions presented at the workshop, Konstanz, Germany, March 8–9, 2018. Cham: Springer. Log. Epistemol. Unity Sci. 51, 23-46 (2021).

The author deals with P. Lorenzen’s early work on operative logic and mathematics preceding his later dialogical logic. She challenges the standard view that operative mathematics was motivated by Lorenzen’s rejection of the actual infinite. She shows, however, that there was a shift in Lorenzen’s treatment of infinity. Whereas his early writings on operative mathematics in the early 1950s were thought to serve as alternative to set theory and connected to the debate on countable vs. uncountable sets, his later writings in the second half of the decade shift towards the distinction between potential and actual infinity.
The author imbeds Lorenzen’s early contributions to foundations into contemporary debates. She sees a connection to D. Hilbert’s axiomatic programme, but stresses Lorenzen’s alternative to the metaaxiomatical justification of axiomatic systems consisting in a justification of protological principles (p. 27). In Section 3, the author deals with Lorenzen’s elimination of the classical notion of set, and finally discusses his shift towards a rejection of the actual infinite. As to the reasons for this shift, the author can only speculate. Lorenzen had been appointed a professor of mathematics at the University of Bonn in 1952. In 1956, he was called on a professorship for philosophy at the University of Kiel. One reason for the shift might have been that he later wanted to address a wider, also philosophical audience.
For the entire collection see [Zbl 1470.03010].

Reviewer: Volker Peckhaus (Paderborn)

MSC:

03A05	Philosophical and critical aspects of logic and foundations
00A30	Philosophy of mathematics
01A60	History of mathematics in the 20th century
03-03	History of mathematical logic and foundations
03B20	Subsystems of classical logic (including intuitionistic logic)
03B30	Foundations of classical theories (including reverse mathematics)

Keywords:

countable sets; uncountable sets; actual infinity; potential infinity; operative mathematics; operative logic; protologic; foundational crisis; formalism; intuitionism; constructivism

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