Logical Studies

Brouwer and Absolutely Unprovable Propositions

Document Type : Brief Research

Author

Morteza Moniri

Associate Professor, Department of Mathematics, Shahid Beheshti University

https://doi.org/10.30465/lsj.2024.49343.1474
Abstract
In this article, we discuss absolutely unprovable propositions from the point of view of Brouwerian intuitionism. According to Brouwer’s definition, a proposition is absolutely unprovable if the creative mind as an ideal mathematician has a proof that both the proposition itself and its negation are unprovable from a constructive point of view. Brouwer has shown that the existence of such propositions is impossible. In his book on Brouwer and Intuitionism, Mark van Atten has described and elaborated Brouwer’s short proof on this matter. The Persian translator of this book has reconstructed and explained this proof in two different ways. In this paper, we present a more appropriate reconstruction of Brouwer’s proof. In the meantime, we will deal with Gödel’s work in generalizing Brouwer’s result from propositional logic to first-order predicate logic. In addition, we will point out that such formalizations of intuitionistic ideas in the formal language of logic cannot do justice to Brouwer’s ideas.

Keywords

Intuitionism
Brouwer
Logic
Absolutely unprovable propositions
Subjects
(1) Van Atten, Mark (2008). *Brouwer's Philosophy*, translated by Mohammad Ardeshir, Tehran: Hermes.
2)  Gödel, Kurt (1986). Collected Works: Vol. 3: Unpublished Essays and Lectures, Edited by Solomon Feferman, Oxford, England and New York, NY, USA: Oxford University Press.
(3)  Hämeen-Anttila, Maria (2020).Gödel on intuitionism and constructive foundations of mathematics, University of Helsinki.
(4)  vanAtten, Mark &Sundholm, Göran (2017). “L.E.J. Brouwer's ‘Unreliability of the Logical Principles’: A New Translation, with an Introduction”, History and Philosophy of Logic, 38 (1), pp. 24-47.
(5)  vanAtten, Mark (2004). On Brouwer, Belmont, CA: Wadsworth/Thomson Learning.
Logical Studies
Volume 15, Issue 1
August 2024
Pages 137-146

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