×

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

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].

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)

Biographic References:

Lorenzen, Paul
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lorenzen, P. (1954). Die Rolle der Logik in der Grundlagenkrisis der Analysis. In Applications scientifiques de la logique mathematique: Actes de \(2^{\text{e}}\) Colloque International Logique Mathematique, Paris 25-30 aout 1952, Institut Henri Poincaré(pp. 65-73). · Zbl 0057.24501
[2] Bernays, P. (1952). Review: “Über endliche Mengen” by Paul Lorenzen. Journal of Symbolic Logic, 17(4), 275-276. · doi:10.2307/2266627
[3] Coquand, T., & Neuwirth, S. (2017). An introduction to Lorenzen’s “Algebraic and logistic investigations on free lattices’ (1951)”. arXiv:1711.06139 [math.LO].
[4] Craig, W. (1957). Review: Einführung in die Operative Logik und Mathematik by Paul Lorenzen. Bulletin of the American Mathematical Society, 63(5), 316-320. · doi:10.1090/S0002-9904-1957-10127-X
[5] Ferreirós, J. (2007). Labyrinth of thought: A history of set theory and its role in modern mathematics. Basel: Birkhäuser. · Zbl 1119.03044
[6] Fraenkel, A. A., Bar-Hillel, Y., & Levy, A. (1973). Foundations of set theory(Vol. 67). Studies in logic and in the foundations of mathematics. Amsterdam: Elsevier. · Zbl 0248.02071
[7] Frey, G. (1957). Review: Paul Lorenzen: Einführung in die operative Logik und Mathematik. Zeitschrift für philosophische Forschung, 11, 631-632.
[8] Heyting, A. (1957). Review: Paul Lorenzen, Das Aktual-Unendliche in der Mathematik; Paul Lorenzen, Die Rolle der Logik in der Grundlagenkrisis der Analysis. Journal of Symbolic Logic, 22(4), 368. · doi:10.2307/2963932
[9] Linnebo, Ø., & Stewart, S. (2019). Actual and potential infinity. Noûs, 53(1), 160-191. · Zbl 1446.03013
[10] Lorenz, K. (2001). Basic objectives of dialogue logic in historical perspective. Synthese, 127, 255-263. · Zbl 0980.03031 · doi:10.1023/A:1010367416884
[11] Lorenzen, P. (1951a). Die Widerspruchsfreiheit der klassischen Analysis. Mathematische Zeitschrift, 54, 1-24. · Zbl 0042.01001 · doi:10.1007/BF01175131
[12] Lorenzen, P. (1951b). Konstruktive Begründung der Mathematik. Mathematische Zeitschrift, 53, 162-202. · Zbl 0041.34303 · doi:10.1007/BF01162411
[13] Lorenzen, P. (1951c). Maß und Integral in der konstruktiven Analysis. Mathematische Zeitschrift, 54, 275-290. · Zbl 0042.24702 · doi:10.1007/BF01574829
[14] Lorenzen, P. (1952a). Über den Mengenbegriff in der Topologie. Archiv der Mathematik, 3, 377-386. · Zbl 0048.40803 · doi:10.1007/BF01899377
[15] Lorenzen, P. (1952b). Über die Widerspruchsfreiheit des Unendlichkeitsbegriffes. Studium Generale, 5, 591-594. · Zbl 0048.00503
[16] Lorenzen, P. (1952c). Über endliche Mengen. Mathematische Annalen, 123, 331-338. · Zbl 0042.28001 · doi:10.1007/BF02054956
[17] Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin: Springer. · Zbl 0066.24802
[18] Lorenzen, P. (1956a). Die Fiktion der Überabzählbarkeit. In Proceedings of the International Congress of Mathematicians 1954, Amsterdam September 2 - September 9 (Vol. iii, pp. 273-279). North-Holland. · Zbl 0074.01301
[19] Lorenzen, P. (1956b). Über den, Operativismus. Unpublished document.
[20] Lorenzen, P. (1957). Das Aktual-Unendliche in der Mathematik. Philosophia Naturalis, 4(1), 1-11.
[21] Lorenzen, P. (1960). Constructive and axiomatic mathematics. Synthese, 12(1), 114-119. · Zbl 0147.24606 · doi:10.1007/BF00485536
[22] Lorenzen, P. (1987). The actual-infinite in mathematics. Translation by K. R. Pavlovic of Lorenzen (1957). In his Constructive philosophy(pp. 195-202). Amherst: University of Massachusetts Press.
[23] Niebergall, K.-G. (2004). Is ZF finitistically reducible? In G. Link (Ed.), One hundred years of Russell’s Paradox: Mathematics, logic, philosophy(pp. 153-180). Berlin, New York: Walter de Gruyter. · Zbl 1065.03037
[24] Schroeder-Heister, P. (2008). Lorenzen’s operative justification of intuitionistic logic. In M. van Atten, P. Boldini, M. Bourdeau, & G. Heinzmann (Eds.), One hundred years of intuitionism (1907-2007): The Cerisy conference(pp. 214-240). Basel: Birkhäuser.
[25] Sieg, W. (1999). Hilbert’s programs: 1917-1922. Bulletin of Symbolic Logic, 5(1), 1-44. · Zbl 0924.03002 · doi:10.2307/421139
[26] Stegmüller, W. 1958. Review: Paul Lorenzen, Einführung in die operative Logik und Mathematik. Philosophische Rundschau6(3/4), 161-182.
[27] Weyl, H. (1918). Das Kontinuum. Leipzig: Veit & Co · JFM 46.0056.11
[28] Skolem, T. (1957). Review: Paul Lorenzen, Einführung in die operative Logik und Mathematik. Journal of Symbolic Logic, 22(3), 289-290. · doi:10.2307/2963596
[29] Kahle, R., & Oitavem, I. (2020). Lorenzen between Gentzen and Schütte (In this volume, pp. 61-73).
[30] Coquand, T., & Neuwirth, S. (2020). Lorenzen’s proof of consistency for elementary number theory. History and Philosophy of Logic(to appear). https://doi.org/10.1080/01445340.2020.1752034.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.