No CrossRef data available.
Article contents
Connecting the Revolutionary with the Conventional: Rethinking the Differences between the Works of Brouwer, Heyting, and Weyl
Published online by Cambridge University Press: 15 December 2022
Abstract
Brouwer’s intuitionism was a far-reaching attempt to reform the foundations of mathematics. Although the mathematical community was reluctant to accept Brouwer’s work, its response to later-developed brands of intuitionism, such as those presented by Hermann Weyl and Arend Heyting, was different. The article accounts for this difference by analyzing the intuitionistic versions of Brouwer, Weyl, and Heyting in light of a two-tiered model of the body and image of mathematical knowledge. Such a perspective provides a richer account of each story and points to a possible connection between the community’s reaction and the changes each mathematician had proposed.
- Type
- Article
- Information
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of the Philosophy of Science Association
References
Barzin, Marcel, and Errera, Alfred. 1931. “Sur La Logique de M. Heyting.” L’Enseignement mathematique 30:248–50.Google Scholar
Barzin, Marcel, and Errera, Alfred. 1932a. “Note Sur La Logique de M. Heyting.” L’Enseignement mathematique 31:122–24.Google Scholar
Barzin, Marcel, and Errera, Alfred. 1932b. “Reponse a M. Heyting.” L’Enseignement mathematique 31:273–74.Google Scholar
Barzin, Marcel, and Errera, Alfred. 1933. “La Logique de M. Brouwer, Etat de La Question.” Bulletin Mathematique de la Societe Roumaine des Sciences 35:51–52.Google Scholar
Bell, John, and Korté, Herbert. 2015. “Hermann Weyl.” In The Stanford Encyclopedia of Philosophy, edited by Zalta, Edward N.. https://plato.stanford.edu/archives/win2016/entries/weyl/
Google Scholar
Brouwer, Luitzen Egbertus Jan. 1907. “On the Foundations of Mathematics.” PhD diss., University of Amsterdam.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1908. “The Unreliability of Logical Principles.” In L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics, edited by Heyting, Arend, 105–11. Amsterdam: North-Holland.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1912. “Intuitionism and Formalism.” In L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics, edited by Heyting, Arend, 123–38. Amsterdam: North-Holland.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1918. “Founding Set Theory Independently of the Principle of the Excluded Middle.” KNAW Verhandelingen 5:1–43.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1925. “Zur Begründung der intuitionistischen Mathematik I.” Mathematische Annalen 93:244–57.10.1007/BF01449963CrossRefGoogle Scholar
Brouwer, Luitzen Egbertus Jan. 1948. “Consciousness, Philosophy, and Mathematics.” In L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics, edited by Heyting, Arend, 480–94. Amsterdam: North-Holland.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1949. “The Non-Equivalence of the Constructive and the Negative Order Relation on the Continuum.” In L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics, edited by Heyting, Arend, 495–98. Amsterdam: North-Holland.Google Scholar
Brouwer, Luitzen Egbertus Jan. 1952. “Historical Background, Principles and Methods of Intuitionism.” In L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics, edited by Heyting, Arend, 508–15. Amsterdam: North-Holland.Google Scholar
Carson, Cathryn, Kojevnikov, Alexei, and Trischler, Helmuth. 2011. “The Forman Thesis: 40 Years After.” In Weimar Culture and Quantum Mechanics, edited by Carson, Cathryn, Kojevnikov, Alexei, and Trischler, Helmuth, 1–6. London: Imperial College Press/World Scientific.10.1142/7581CrossRefGoogle Scholar
Corry, Leo. 1989. “Linearity and Reflexivity in the Growth of Mathematical Knowledge.” Science in Context 3 (2):409–40.10.1017/S0269889700000880CrossRefGoogle Scholar
Corry, Leo. 2001. “Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics.” In Changing Images in Mathematics: From the French Revolution to the New Millennium, edited by Bottazzini, Umberto and Dahan-Dalmédico, Amy, 167–86. London: Harwood Academic Publishers.Google Scholar
Elkana, Yehuda. 1978. “Two-Tier Thinking: Philosophical Realism and Historical Relativism.” Social Studies of Science 8 (3):309–26.10.1177/030631277800800304CrossRefGoogle Scholar
Elkana, Yehuda. 1981. “A Programmatic Attempt at an Anthropology of Knowledge.” In Sciences and Cultures, Sociology of the Sciences Yearbook, edited by Mendelson, Everett and Elkana, Yehuda, 1–76. New York: Springer.Google Scholar
Feferman, Solomon. 1988. “The Development of Programs for the Foundations of Mathematics in the First Third of the 20th Century.” https://pdfs.semanticscholar.org/5a3a/112929a5a2a1f1fec4447cc584628918c0fb.pdf.Google Scholar
Forman, Paul. 1971. “Weimar Culture, Causality, and Quantum Theory, 1918–1927: Adaptation by German Physicists and Mathematicians to a Hostile Intellectual Environment.” Historical Studies in the Physical Sciences 3:1–115.10.2307/27757315CrossRefGoogle Scholar
Fourman, Michael. 1982. “Notions of Choice Sequences” In The L. E. J. Brouwer Centenary Symposium, edited by Troelstra, Anne Sjerp and van Dalen, Dirk, 91–105. Amsterdam: North-Holland.Google Scholar
Franchella, Miriam. 1995. “Like a Bee on a Windowpane: Heyting’s Reflections on Solipsism.” Synthese 105 (2):207–51.10.1007/BF01064219CrossRefGoogle Scholar
Gödel, Kurt. 1933. “On Intuitionistic Arithmetic and Number Theory.” In Collected Works: Volume I: Publications 1929–1936, edited by Feferman, Solomon, Dawson, John, Kleene, Stephen, Moore, Gregory, Solovay, Robert, and van Heijenoort, Jean, 286–95. Oxford: Oxford University Press.Google Scholar
Hesseling, Dennis E. 2003. Gnomes in the Fog: The Reception of Brouwer’s Intuitionism in the 1920s. Basel: Birkhu-ser Verlag.10.1007/978-3-0348-7989-7CrossRefGoogle Scholar
Heyting, Arend. 1930a. “Die Formalen Regeln Der Intuitionistischen Logik I.” In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, edited by Mancosu, Paolo, 311–27. Oxford: Oxford University Press.Google Scholar
Heyting, Arend. 1930b. “Die Formalen Regeln Der Intuitionistischen Logik II.” Sitzungsberichte der Preussischen Akademie der Wissenschaften: 57–71.Google Scholar
Heyting, Arend. 1930c. “Die Formalen Regeln Der Intuitionistischen Logik III.” Sitzungsberichte der Preussischen Akademie der Wissenschaften: 158–69.Google Scholar
Heyting, Arend. 1934. Mathematische Grundlagenforschung Intuitionismus Beweistheorie. Berlin: Springer-Verlag.Google Scholar
Heyting, Arend. 1959. “Some Remarks on Intuitionism.” In Constructivity in Mathematics, edited by Heyting, Arend, 67–71. Amsterdam: North-Holland.Google Scholar
Heyting, Arend. 1962. “After Thirty Years.” In Logic, Methodology and Philosophy of Science, edited by Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, 194–97. Stanford: Stanford University Press.Google Scholar
Heyting, Arend. 1974. “Intuitionistic Views on the Nature of Mathematics.” Synthese 27 (1/2):79–91.10.1007/BF00660890CrossRefGoogle Scholar
Heyting, Arend, editor. 1975. L. E. J. Brouwer Collected Works I—Philosophy and Foundations of Mathematics. Amsterdam: North-Holland
Google Scholar
Heyting, Arend. 1978. “History of the Foundations of Mathematics.” Nieuw Archief voor Wiskunde, 3rd Series 26 (1):1–21.Google Scholar
Iemhoff, Rosalie. 2008. “Intuitionism in the Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy, edited by Zalta, Edward N.. https://plato.stanford.edu/entries/intuitionism/
Google Scholar
Kleene, Stephen Cole, and Vesley, Richard. 1965. The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions. Amsterdam: North-Holland.Google Scholar
Kraft, P., and Kroes, P.. 1984. “Adaptation of Scientific Knowledge to an Intellectual Environment. Paul Forman’s ‘Weimar Culture, Causality, and Quantum Theory, 1918–1927’: Analysis and Review.” Centaurus 27 (1):76–99.10.1111/j.1600-0498.1984.tb00754.xCrossRefGoogle Scholar
Kreisel, Georg. 1968. “Lawless Sequences of Natural Numbers.” Compositio Mathematica 20:222–48.Google Scholar
Kreisel, Georg. 1987. “Gödel’s Excursions into Intuitionistic Logic.” In Gödel Remembered, edited by Weingartner, Paul and Schmetterer, Leopold, 67–170. Napoli: Bibliopolis.Google Scholar
Kreisel, Georg, and Troelstra, Anne Sjerp. 1970. “Formal Systems for Some Branches of Intuitionistic Analysis.” Annals of Mathematical Logic 1 (3):229–387.10.1016/0003-4843(70)90001-XCrossRefGoogle Scholar
Kripke, Saul. 2019. “Free Choice Sequences: A Temporal Interpretation Compatible with Acceptance of Classical Mathematics.” Indagationes Mathematicae 30 (3):492–99.10.1016/j.indag.2019.01.003CrossRefGoogle Scholar
Mancosu, Paolo, and Ryckman, Thomas. 2002. “Mathematics and Phenomenology: The Correspondence between O. Becker and H. Weyl.” Philosophia Mathematica 10 (2):130–202.10.1093/philmat/10.2.130CrossRefGoogle Scholar
Myhill, John. 1966. “Notes towards An Axiomatization of Intuitionistic Analysis.” Logique et Analyse 9 (35/36):280–97.Google Scholar
Niekus, Joop. 2005. “Individual Choice Sequences in the Work of L. E. J. Brouwer.” Philosophia Scientiæ 9 (S2): 217–32.10.4000/philosophiascientiae.392CrossRefGoogle Scholar
Norris, Christopher. 2000. Quantum Theory and the Flight from Realism: Philosophical Responses to Quantum Mechanics. London: Routledge.Google Scholar
Placek, Tomasz. 1999. Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism. Dordrecht: Springer.10.1007/978-94-015-9315-1CrossRefGoogle Scholar
Posy, Carl. 2005. “Intuitionism and Philosophy.” In The Oxford Handbook of Philosophy of Mathematics and Logic, edited by Shapiro, Stewart, 318–55. Oxford: Oxford University Press.10.1093/0195148770.003.0009CrossRefGoogle Scholar
Posy, Carl. 2008. “Brouwerian Infinity.” In One Hundred Years of Intuitionism (1907–2007), edited by van Atten, Mark, Boldini, Pascal, Bourdeau, Michel, and Heinzmann, Gerhard, 21–36. Basel: Birkhäuser.10.1007/978-3-7643-8653-5_2CrossRefGoogle Scholar
Schlote, Karl-Heinz. 2005. “B. L. van der Waerden, Moderne Algebra, First Edition (1930–1931).” In Landmark Writings in Western Mathematics 1640–1940, edited by Grattan-Guinness, Ivor, Corry, Leo, Guicciardini, Niccolo, Cooke, Roger, and Crépel, Pierre, 901–16. Amsterdam: Elsevier.10.1016/B978-044450871-3/50151-0CrossRefGoogle Scholar
Scholz, Erhard. 2000. “Herman Weyl on the Concept of Continuum.” In Proof Theory: History and Philosophical Significance, edited by Hendricks, Vincent, Pedersen, Stig Andur, and Jørgensen, Klaus Frovin, 195–220. Dordrecht: Kluwer.10.1007/978-94-017-2796-9_9CrossRefGoogle Scholar
Scholz, Erhard. 2005. “Philosophy as a Cultural Resource and Medium of Reflection for Hermann Weyl.” Revue de Synthèse 126 (2):331–51.10.1007/BF02965677CrossRefGoogle Scholar
Sieroka, Norman. 2009. “Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism.” Philosophia Scientiae 13 (2):85–96.10.4000/philosophiascientiae.295CrossRefGoogle Scholar
Sieroka, Norman. 2019. “Neighbourhoods and Intersubjectivity: Analogies between Weyl’s Analyses of the Continuum and Transcendental-Phenomenological Theories of Subjectivity.” In Weyl and the Problem of Space: From Science to Philosophy, edited by Bernard, Julien and Lobo, Carlos, 98–122. Switzerland: Springer.Google Scholar
Sundholm, Goran, and van Atten, Mark. 2008. “The Proper Explanation of Intuitionistic Logic: On Brouwer’s Demonstration of the Bar Theorem.” In One Hundred Years of Intuitionism (1907–2007), edited by van Atten, Mark, Boldini, Pascal, Bourdeau, Michel, and Heinzmann, Gerhard, 60–77. Basel: Birkhäuser.10.1007/978-3-7643-8653-5_5CrossRefGoogle Scholar
Troelstra, Anne Sjerp. 1983. “Analyzing Choice Sequences.” Journal of Philosophical Logic 12: 197–260.10.1007/BF00247189CrossRefGoogle Scholar
Troelstra, Anne Sjerp. 1991. “A History of Constructivism in the 20th Century.” University of Amsterdam, ITLI Prepublication Series ML-91-05.Google Scholar
van Atten, Mark. 2017. “The Development of Intuitionistic Logic.” In The Stanford Encyclopedia of Philosophy, edited by Zalta, Edward N.. https://plato.stanford.edu/entries/intuitionistic-logic-development/
Google Scholar
van Atten, Mark, and van Dalen, Dirk. 2002. “Arguments for the Continuity Principle.” Bulletin of Symbolic Logic 8 (3):329–74.10.2178/bsl/1182353892CrossRefGoogle Scholar
van Dalen, Dirk. 1978. “Brouwer: The Genesis of His Intuitionism.” Dialectica 32 (3–4):291–303.10.1111/j.1746-8361.1978.tb01318.xCrossRefGoogle Scholar
van Dalen, Dirk. 1995. “Hermann Weyl’s Intuitionistic Mathematics.” Bulletin of Symbolic Logic 1 (2):145–69.10.2307/421038CrossRefGoogle Scholar
van Dalen, Dirk. 2013. L. E. J. Brouwer—Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life. London: Springer-Verlag.10.1007/978-1-4471-4616-2CrossRefGoogle Scholar
van Stigt, Walter. 1990. Studies in the History and Philosophy of Mathematics: Brouwer’s Intuitionism. Amsterdam: North-Holland.Google Scholar
Verburgt, Lukas M. 2015. “‘A Terrible Piece of Bad Metaphysics’? Towards a History of Abstraction in Nineteenth and Early Twentieth-Century Probability Theory, Mathematics and Logic.” PhD diss., University of Amsterdam.Google Scholar
Weyl, Hermann. 1918. The Continuum: A Critical Examination of the Foundation of Analysis. Philadelphia: Thomas Jefferson University Press.Google Scholar
Weyl, Hermann. 1921. “On the New Foundational Crisis in Mathematics.” In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, edited by Mancosu, Paolo, 86–118. Oxford: Oxford University Press.Google Scholar
Weyl, Hermann. 1955. “Insight and Reflection.” In The Spirit and Uses of the Mathematical Sciences, edited by Saaty, Thomas and Weyl, Joachim, 281–301. New York: McGraw-Hill.Google Scholar