Logicism, Intuitionism, and Formalism

Sten Lindstrom is Professor of Philosophy at Umea University and has been a Research Fellow at the Swedish Collegium for Advanced Study (SCAS). He has published papers on intensional logic, belief revision and philosophy of language, and co-edited the books Logic, Action and Cognition: Essays in Philosophical Logic (Kluwer, 1997) and Collected Papers of Stig Kanger with Essays on his Life and Work, I-II (Kluwer, 2001). Erik Palmgren is Professor of Mathematics at Uppsala University. His research interests are mainly mathematical logic and the foundations of mathematics. He is presently working on the foundational programme of replacing impredicative constructions by inductive constructions in mathematics, with special emphasis on point-free topology and topos theory. Krister Segerberg is Emeritus Professor of Philosophy at Uppsala University and the University of Auckland. He is the author of papers in modal logic, the logic of action, belief revision and deontic logic, as well as the books An Essay in Classical Modal Logic (1971) and Classical Propositional Operators: An Exercise in the Foundations of Logic (1982). Viggo Stoltenberg-Hansen is professor of Mathematical Logic at Uppsala University. His main interests include computability and constructivity in mathematics.

• Introduction: The Three Foundational Programmes.- Introduction: The Three Foundational Programmes.- Logicism and Neo-Logicism.- Protocol Sentences for Lite Logicism.- Frege’s Context Principle and Reference to Natural Numbers.- The Measure of Scottish Neo-Logicism.- Natural Logicism via the Logic of Orderly Pairing.- Intuitionism and Constructive Mathematics.- A Constructive Version of the Lusin Separation Theorem.- Dini’s Theorem in the Light of Reverse Mathematics.- Journey into Apartness Space.- Relativization of Real Numbers to a Universe.- 100 Years of Zermelo’s Axiom of Choice: What was the Problem with It?.- Intuitionism and the Anti-Justification of Bivalence.- From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory.- Program Extraction in Constructive Analysis.- Brouwer’s Approximate Fixed-Point Theorem is Equivalent to Brouwer’s Fan Theorem.- Formalism.- “Gödel’s Modernism: On Set-Theoretic Incompleteness,” Revisited.- Tarski’s Practice and Philosophy: Between Formalism and Pragmatism.- The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory.- Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics.- Beyond Hilbert’s Reach?.- Hilbert and the Problem of Clarifying the Infinite.