Judgment, Constructive Mathematics, and Intuitionism

AbstractIn the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic (Notable exceptions are the works of Schroeder-Heister 2008; Coquand and Neuwirth 2017; Kahle and Oitavem 2020.), and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen (1957), was not a major motivation for operative logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity.

Download Full-text

G. É. Minc. Poditor térmoν ν kvantornyh pravilah konstruktiνnogo isčisléniá predikátov. Isslédovaniá po konstruktivnoj matématiké i matématičéskoj logiké, I, edited by A. O. Slisénko, Zapiski Naučnyh Séminarov Léningradskogo Otdéléniá Matématičéskogo Instituta im. V. A. Stéklova AN SSSR (LOMI), vol. 4, Moscow1967, pp. 112–122. - G. E. Mints. Choice of terms in quantifier rules of constructive predicate calculus. English translation of the preceding. Studies in constructive mathematics and mathematical logic, Part I, edited by A. O. Slisenko, Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, vol. 4, Consultants Bureau, New York1969, pp. 43–46.

Journal of Symbolic Logic ◽

10.2307/2269980 ◽

1971 ◽

Vol 36 (3) ◽

pp. 525-525

Author(s):

J. van Heijenoort

Keyword(s):

Mathematical Logic ◽

English Translation ◽

Mathematical Institute ◽

Steklov Mathematical Institute ◽

Predicate Calculus ◽

Constructive Mathematics

Download Full-text

SEPARATING THE FAN THEOREM AND ITS WEAKENINGS

Journal of Symbolic Logic ◽

10.1017/jsl.2014.9 ◽

2014 ◽

Vol 79 (3) ◽

pp. 792-813 ◽

Cited By ~ 2

Author(s):

ROBERT S. LUBARSKY ◽

HANNES DIENER

Keyword(s):

Constructive Mathematics ◽

Kripke Models ◽

Fan Theorem

AbstractVarieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

Download Full-text