Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T19:14:32.520Z Has data issue: false hasContentIssue false
Series:   Elements in the Philosophy of Mathematics

Mathematical Intuitionism

Published online by Cambridge University Press:  22 October 2020

Carl J. Posy
Carl J. Posy
Affiliation:
Hebrew University of Jerusalem

Summary

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
Get access

Keywords

Brouwerintuitionistic mathematicsintuitionistic logicphilosophy of mathematicsthe intuitionistic continuum
Type
Element
Information
Series: Elements in the Philosophy of Mathematics
Online ISBN: 9781108674485
Publisher: Cambridge University Press
Print publication: 12 November 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Bibliography

Aristotle, [1941] The Basic Works of Aristotle, McKeon, R (ed. and trans.), Random House, New York.Google Scholar
van Atten, M., [2002] On Brouwer, Thomson, Wadsworth, London.Google Scholar
van Atten, M.[2007] Brouwer Meets Husserl: On the Phenomenology of Choice Sequences, Springer, Dordrecht.Google Scholar
van Atten, M.[2018] ‘The Creating Subject, the Brouwer-Kripke Schema, and Infinite Proofs’, Indag. Math, 29, pp. 1565–636.Google Scholar
Van Atten, M., Boldini, P., Bourdeau, M., Heinzmann, G., (eds.) [2008] One Hundred Years of Intuitionism (1907–2007), van Atten, M., et. al. (eds.), Birkhäuser.Google Scholar
van Atten, M., and van Dalen, D. [2002] ‘Arguments for the Continuity Principle’, Bull. Symb. Logic, 8, pp. 329–47.Google Scholar
van Atten, M., van Dalen, D., and Tieszen, R. [2002] ‘Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum’, Philosophia Mathematica, 10, 2, pp. 203–26.Google Scholar
Auxier, R., Anderson, D., and Hahn, L. [2015] The Philosophy of Hilary Putnam, Open Court, Chicago.Google Scholar
Becker, O. [1927] Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene (Jahrbuch für Philosophie und phänomenologische Forschung), vol. 8, pp. 440–809.Google Scholar
Beeson, M. [1985] Foundations of Constructive Mathematics: Metamathematical Studies, Springer, Berlin.Google Scholar
Bell, J., [1988] Toposes and Local Set Theories, Oxford University Press, Oxford.Google Scholar
Bernays, P. [1930] ‘Die Philosophie der Mathematik und die Hilbertsche Beweistheorie’, in Bernays [1976], pp. 17–61.Google Scholar
Bernays, P.[1976] Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt.Google Scholar
Beth, E. [1947] ‘Semantical Considerations on Intuitionistic Mathematics’, Indag. Math., 9, 572–7.Google Scholar
Bezhanishvili, G., and Holliday, W. [2019] ‘A Semantic Hierarchy for Intuitionistic Logic’, Indag. Math., 30, pp. 403–69.Google Scholar
Bishop, E. [1967] Foundations of Constructive Analysis, McGraw Hill.Google Scholar
Boffa, M, van Dalen, D., and McAloon, M. (eds.) [1980] Logic Colloquium 78, North Holland, Amsterdam.Google Scholar
Borel, E. [1898] Leçons sur la Theorie des Fonctions, Paris, Gauthier-Villars.Google Scholar
Bridges, D, and Richmond, F., [1987] Varieties of Constructive Mathematics, Cambridge University Press, Cambridge.Google Scholar
Brouwer, L. E. J. [1975] Collected Works, 6 (abbreviated as CW) (Heyting, A, ed.), Amsterdam, North Holland Publishing Company.Google Scholar
Brouwer, L. E. J.[1907] Over de grondslagen der wiskunde, Dissertation, 1907, University of Amsterdam. (Translated as On the Foundations of Mathematics CW pp. 11–101.)Google Scholar
Brouwer, L. E. J.[1908] ‘Die möglichen Mächtigkeiten’, Atti del IV Congresso Internazional dei Matematici, Romo, 6–11 Aprile 1908. Rome, Academia dei Lincei, 569–71. CW, pp. 102–4.Google Scholar
Brouwer, L. E. J.[1908A] ‘De onbetrouwbaarheid der logische principes’, Tijdschrift voor Wijsbegeerte, 2, 152–8. CW, pp. 107–11.Google Scholar
Brouwer, L. E. J.[1912] Intuitionisme en formalisme, Amsterdam, translated (by A. Dresden) as ‘Intuitionism and Formalism’, Bull. Amer. Math Soc. 20 (1913), pp. 81–96. CW, pp. 123–38.Google Scholar
Brouwer, L. E. J.[1919] ‘Intuitionistische Mengenlehre’, Jber. Deutsch. Math. Verein, 28, 203–8, Proceedings Acad. Amsterdam, 23, pp. 949–54.Google Scholar
Brouwer, L. E. J.[1923] ‘Intuitionistische Zerlegung mathematischer Grundbegriffe’, Jahresbericht der deutschen Mathematiker Vereinigung, 33: 251–6. CW, pp. 275–80.Google Scholar
Brouwer, L. E. J.[1925] ‘Zur Begründung der intuitionistischen Mathematik I’, Mathematische Annalen, 93, pp. 244–77.Google Scholar
Brouwer, L. E. J.[1927]‘Über Definitionsbereiche von Funktionen’, Mathematische Annalen, 97, 1927: 60–75. CW, pp. 390–405.Google Scholar
Brouwer, L. E. J.[1927A] ‘Virtuelle Ordnung und unerweiterbare Ordnung’, J. Reine Angew. Math., 157, 255–7. CW, pp. 406–8.Google Scholar
Brouwer, L. E. J.[1930] ‘Die Struktur des Kontinuums’, Lecture delivered in Vienna, 14 March 1928. CW, pp. 429–40.Google Scholar
Brouwer, L. E. J.[1933] ‘Willen, Weten, Spreken’, Euclides, 9, pp. 177–93. Translated as ‘Will, Knowledge and Speech’, in van Stigt [1990], pp. 418–31.Google Scholar
Brouwer, L. E. J.[1942] ‘Zum freien Werden von Mengen und Funktionen’, Proceedings of the Acad. Amsterdam, 45, pp. 322–3(= Indag. Math., 4, pp. 107–8). CW, pp. 459–60.Google Scholar
Brouwer, L. E. J.[1948] ‘Consciousness, Philosophy and Mathematics’, Proceedings of the Tenth International Congress of Philosophy, Amsterdam, 3, pp. 1235–49. CW, pp. 480–94.Google Scholar
Brouwer, L. E. J.[1948A] ‘Essenteel negatieve eigenschappen’, Proceedings of the Acad. Amsterdam, 51 1948, pp. 963–4 (= Indag. Math., 10, pp. 322–3). Translated as ‘Essentially negative properties’, in CW, pp. 478–9.Google Scholar
Brouwer, L. E. J.[1949] ‘De non-aequivalentie van de constructieve en de negatieve orderelatie in het continuum’, Proceedings of the Acad. Amsterdam, 52, pp. 122–4(= Indag. Math., 11, pp. 37–9). Translated as ‘The non-equivalence of the constructive and the negative order relation on the continuum’. CW, pp. 495–6.Google Scholar
Brouwer, L. E. J.[1950] ‘Discours Final de M. Brouwer’, Les Methodes Formelles en Axiomatique, Colloques Internationaux du Centre National de la Recherche Scientifique, Paris, page 75. CW, p. 503.Google Scholar
Brouwer, L. E. J.[1952] ‘Historical Background, Principles and Methods of Intuitionism’, South African Journal of Science, 49:139–46. CW, pp. 508–15.Google Scholar
Brouwer, L. E. J.[1954] ‘Points and Spaces’, Canadian J. for Math. 6, pp. 1–17. CW, 522–40.Google Scholar
Brouwer, L. E. J.[1955] ‘The Effect of Intuitionism on Classical Algebra of Logic’, Proc. Royal Irish Academy, Section A, 57, pp. 113–16. CW 551–4.Google Scholar
Brouwer, L. E. J.[1981] Brouwer’s Cambridge Lectures on Intuitionism (van Dalen, D, ed.), Cambridge University Press.Google Scholar
Chatzidakis, Z., Koepke, P., and Pohlers, W. (eds.), [2006] Logic Colloquium ‘02 (Lecture Notes in Logic 27), Wellesley, A. K. Peters.Google Scholar
van Dalen, D. [1986] ‘Intuitionistic Logic’, in Handbook of Philosophical Logic, vol. 3, Gabbay, D and Guenther, F (eds.), D. Reidel Publishing Company, pp. 225–340.Google Scholar
van Dalen, D.[1997] ‘How Connected Is the Intuitionistic Continuum’, Journal of Symbolic Logic, 62, 4, pp. 1147–50.Google Scholar
van Dalen, D.[1999] ‘From Brouwerian Counter Examples to the Creating Subject’, Studia Logica, 62, pp. 305–14.Google Scholar
van Dalen, D.[1999a] Mystic, Geometer and Intuitionist: The Life of L.E.J. Brouwer. Volume 1, Oxford University Press.Google Scholar
van Dalen, D.[2005] Mystic, Geometer and Intuitionist: The Life of L.E.J. Brouwer. Volume 2, Oxford University Press.Google Scholar
van Dalen, D., and Troelstra, A. S. [1970] ‘Projections of Lawless Sequences’, in Kino, Myhill and Veslely [1970], pp. 163–86.Google Scholar
Del Santo, F., and Gisin, N. [2019] ‘Physics without Determinism: Alternative Interpretations of Classical Physics’, Physical Review A, 100 (062107)Google Scholar
Dewey, J. [1938] Logic: The Theory of Inquiry, New York, Holt.Google Scholar
Diaconescu, R. [1975] ‘Axiom of Choice and Complementation’, Proc. Amer. Math. Soc., 51, 176–8.Google Scholar
Dragalin, A. G. [1988] ‘Mathematical Intuitionism: Introduction to Proof Theory’, Translations of Mathematical Monographs, 67, Providence, American Mathematical Society.Google Scholar
Dummett, M. [1973] ‘The Philosophical Basis of Intuitionistic Logic’, in Logic Colloquium ’73, Rose, H. E., and Shepherdson, J. C. (eds.), Amsterdam, North Holland Publishing Company pp. 5–40, reprinted in Dummett [1978], pp. 215–47.Google Scholar
Dummett, M.[1975] ‘The Justification of Deduction’, Proceedings of the British Academy, 59, London, reprinted in Dummett [1978], pp. 290–318.Google Scholar
Dummett, M.[1978] Truth and Other Enigmas, Harvard University Press.Google Scholar
Dummett, M.[1978A] ‘Realism’ in [1978], pp. 145–65.Google Scholar
Dummett, M.[1991] The Logical Basis of Metaphysics, Harvard University Press.Google Scholar
Dummett, M.[1993] The Seas of Language, Oxford University Press.Google Scholar
Dummett, M.[2000] Elements of Intuitionism, 2nd ed., Oxford University Press. (First edition 1977).Google Scholar
Fraenkel, A. [1923] Einleitung in die Mengenlehere, 2nd ed., Berlin, Springer.Google Scholar
Gabbay, D., and Woods, J., eds. [2007] Handbook of the History of Logic, v. 8, Amsterdam, North Holland.Google Scholar
Gisin, N. [2019] ‘Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?, Erkenntnis, DOI http://doi.org/10.1007/s10670-019–00165-8Google Scholar
Gisin, N.[2019A] ‘Real Numbers Are the Hidden Variables of Classical Mechanics’, Quantum Studies: Mathematics and Foundations, DOI http://doi.org/10.1007/s40509-019–00211-8Google Scholar
Glivenko, V. [1929] ‘Sur quelques points de la logique de M. Brouwer’, Bulletin, Académie Royale de Belgique, 15, pp. 183–8.Google Scholar
Gödel, K. [1933] ‘Eine Interpretation des intuitionistischen Aussagenkalküls’, Ergebnisse eines mathematischen Kolloquiums, 4, pp. 39–40.Google Scholar
Gödel, K.[1933A] ‘Zur intuitionistischen Arithmetik und Zahlentheorie’, Ergebnisse eines mathematischen Kolloquiums, 4, 1933, pp. 34–8.Google Scholar
Gödel, K.[1958] ‘Über eine bisher noch nicht benutzte Erweiterung des finite Standpunktes’, Dialectica, 12, pp. 280–7.Google Scholar
Goodman, N., and Myhill, J. [1978] ‘Choice implies excluded middle’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 24, p. 461.Google Scholar
Griss, G. F. C. [1944] ‘Negatieloze intuitionistische wiskunde’, Verslagen. Akad. Amsterdam, 53, pp. 261–8.Google Scholar
Griss, G. F. C.[1946] ‘Negationless Intuitionistic Mathematics, I’, Proceedings of the Acad. Amsterdam, 49, pp. 1127–33 (= Indag. math., 8, pp. 675–81).Google Scholar
Griss, G. F. C.[1949] ‘Logique des mathématiques intuitionnistes sans negation’, Comptes Rendus Acad. Sci. Paris, 227, pp. 946–7.Google Scholar
Griss, G. F. C.[1950] ‘Negationless Intuitionistic Mathematics, II,’ Proceedings of the Acad. Amsterdam, 53, pp. 456–63 (= Indag. math., 12, pp. 108–15).Google Scholar
van Heijenoort, J. (ed.) [1967] From Frege to Gödel: A Source Book in Mathematical Logic, 1897–1931, Cambridge, MA, Harvard University Press.Google Scholar
Hesseling, D. E. [2003] Gnomes in the Fog: The Reception of Brouwer’s Intuitionism in the 1920’s, Basel, Birkhäuser Verlag.Google Scholar
Heyting, A. [1925] Intuitionistische Axiomatiek der Projectieve Meetkunde, Groningen, Noordhoff.Google Scholar
Heyting, A.[1930] ‘Die formalen Regeln der intuitionistischen Logik,’ Sitzungsberichte der preuszischen Akademie von Wissenschaften, phys. math. Kl., pp. 42–56.Google Scholar
Heyting, A.[1930A] ‘Die formalen Regeln der intuitionistischen Mathematik,’ Sitzungsberichte der preuszischen Akademie von Wissenschaften, phys. math. Kl., pp. 57–71, 158–169.Google Scholar
Heyting, A.[1966] Intuitionism: An Introduction, 2nd rev. ed., Amsterdam, North Holland Publishing Company. (First edition 1956, Third Edition 1971).Google Scholar
Heyting, A.[1967] Remarks on Kreisel [1967] in Lakatos [1967].Google Scholar
Heyting, A.[1969] Review of J. L. Destouches, ‘Sur la Mecanique Classique et l’Intuitionisme’, J. Symb. Logic, 34, p. 307.Google Scholar
Hilbert, D. [1923] ‘Die logischen Grundlagen der Mathematik,’ Mathematische Annalen, 88, 151–65.Google Scholar
Hilbert, D.[1926] ‘Über das Unendliche’, Mathematische Annalen 95, 161–90. Translated in van Heijenoort, (ed.) [1967], pp. 369–92.CrossRefGoogle Scholar
Hilbert, D., and Ackermann, W. [1928] Grundzüge der theoretischen Logik, 1st ed., Berlin, Springer.Google Scholar
van der Hoeven, G. F. [1981] Projections of Lawless Sequences, Ph.D. Thesis, Amsterdam, University of Amsterdam.Google Scholar
Husserl, E. [1913] ‘Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie’ in Jahrbuch für Philosophie und phänomenologischen Forschung, 1. Translated by W. Boyce-Gibson as Ideas: General Introduction to Pure Phenomenology, New York, Macmillan, 1931.Google Scholar
Husserl, E.[1948] Erfarhung und Urteil: Untersuchungen zur Genealogie der Logik, Langrebe, L (ed.), Hamburg, Classen & Goverts. Translated by J. Churchil and K. Ameriks as Experience and Judgment, Northwestern University Press, Evanston, 1973.Google Scholar
van Inwagen, P. [2012] ‘What Is an Ontological Category,’ in Novak, Novotny, Prokop, and Svoboda [2012], pp. 11–24.CrossRefGoogle Scholar
Kant, I. [1929] Immanuel Kant’s Critique of Pure Reason, N. K. Smith (trans.) London, Macmillan.Google Scholar
Kino, A., Myhill, J., and Vesley, R. E. (eds.), [1970] Intuitionism and Proof Theory, North Holland.Google Scholar
Kleene, S. C. [1945] ‘On the Interpretation of Intuitionistic Number Theory’, J. Symb. Logic, 10, pp. 109–24.Google Scholar
Kleene, S. C.[1952] Introduction to Metamathematics,Princeton, van Nostrand.Google Scholar
Kleene, S. C.[1973] ‘Realizability: A Retrospective Survey’, in Mathias and Rogers [1973].Google Scholar
Kleene, S. C., and Vesley, R. [1965] Foundations of Intuitionistic Mathematics, Amsterdam, North Holland.Google Scholar
Kolmogorov, A. N. [1932] ‘Zur Deutung der intuitionistischen Logik’, Mathematische Zeitschrift, 35, 58–65.Google Scholar
Kreisel, G. [1958] ‘A Remark on Free Choice Sequences and the Topological Interpretation’, Journal of Symbolic Logic, 23, 369–88.Google Scholar
Kreisel, G.[1967] ‘Informal Rigour and Completeness Proofs’, in Lakatos [1967], pp. 138–86.Google Scholar
Kreisel, G.[1968] ‘Lawless Sequences of Natural Numbers’, Compositio Mathematica, 20, 222–48.Google Scholar
Kreisel, G., and Troelstra, A. S. [1970] ‘Formal Systems for Some Branches of Intuitionistic Analysis’, Annals of Mathematical Logic, 1, 229–387.Google Scholar
Kretzman, N. (ed.) [1982] Infinity and Continuity in Ancient and Medieval Thought, Ithaca, Cornell University Press.Google Scholar
Kripke, S. A. [1965] ‘Semantical Analysis of Intuitionistic Logic, I,’ in Crossley, J and Dummett, M (eds.), Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium, Oxford, July 1963, North Holland, pp. 92–130.Google Scholar
Kripke, S. A.[2019] ‘ Free Choice Sequences: A Temporal Interpretation Compatible with Acceptance of Classical Mathematics’, Indag. Math., 30, pp. 492–9.CrossRefGoogle Scholar
Kushner, B. [1985] Lectures on Constructive Mathematical Analysis, Providence, AMS Publications.Google Scholar
Lakatos, I. (ed.), [1967] Problems in the Philosophy of Mathematics, Amsterdam, North Holland.Google Scholar
Lévy, P. [1927] ‘Logique classique, Logique brouwerienne et Logique mixte’, Académie Royale de Belgique, Bulletins de la Classe des Sciences, 5–13, pp. 256–66.Google Scholar
Lopez-Escobar, E. G. K. [1981] ‘Equivalence between Semantics for Intuitionism, I’, Journal of Symbolic Logic, 46, pp. 773–80.Google Scholar
Martin-Löf, P. [1984] Intuitionistic type theory, Naples, Bibliopolis.Google Scholar
Martino, E. [2018] Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics, Berlin, Spinger.Google Scholar
Mathias, A., and Rogers, H. (eds.) [1973] Cambridge Summer School in Mathematical Logic, August 1–21, 1971, Berlin, Springer.CrossRefGoogle Scholar
McCarty, D. C. [1984] Realizability and Recursive Mathematics, Department of Computer Science, Report CMU-CS-84–131, Pittsburgh, Carnegie Mellon University.Google Scholar
McCarty, D. C.[2005] ‘Intuitionism in Mathematics’, in Shapiro [2005], pp. 356–86.Google Scholar
McKinsey, J., and Tarski, A. [1948] ‘Some Theorems about the Sentential Calculi of Lewis and Heyting’, J. Symb. Logic, 13, pp. 1–15.Google Scholar
Miller, F. D. ‘Aristotle Against the Atomists’, in Kretzman, N. (ed.) [1982] pp. 37–86.Google Scholar
Myhill, J. [1967] ‘Notes Towards an Axiomatization of Intuitionistic Analysis,’ Logique et Analyse, 9, pp. 280–97.Google Scholar
Myhill, J.[1970] ‘Formal Systems of Intuitionistic Analysis, II’, in Kino, Myhill and Vesley [1970], pp. 151–62.Google Scholar
Myhill, J.[1973] ‘Some Properties of Intuitionistic Zermelo-Fraenkel Set Theory’, in Mathias and Rogers [1973], pp. 206–31.Google Scholar
Novak, L., Novotny, D., Prokop, S., and Svoboda, D. (eds.), [2012] Metaphysics: Aristotelian, Scholastic, Analytic, Frankfurt, Ontos Verlag (in cooperation with Studia Neoaristotelica).Google Scholar
Parsons, C. [2014] ‘The Kantian Legacy in Twentieth-Century Foundations of Mathematics’, in Parsons [2014A], pp. 11–39.Google Scholar
Parsons, C.[2014A] Philosophy of Mathematics in the Twentieth Century: Selected Essays, Cambridge, Harvard University Press.Google Scholar
Placek, T. [1999] Mathematical Intuitionism and Intersubjectivity, Kluwer.Google Scholar
Posy, C. J. [1976] ‘Varieties of Indeterminacy in the Theory of General Choice Sequences’, Journal of Philosophical Logic, 5, pp. 91–132.Google Scholar
Posy, C. J.[1977] ‘The Theory of Empirical Sequences,’ Journal of Philosophical Logic, 6, pp. 47–81.Google Scholar
Posy, C. J.[1980] ‘On Brouwer’s Definition of Unextendable Order’, History and Philosophy of Logic, 1, pp. 129–49.Google Scholar
Posy, C. J.[1982] ‘A Free IPC is a Natural Logic: Strong Completeness for Some Intuitionistic Free Logics’ (Topoi, v. 1 (1982), pp. 30–43; reprinted in J. K. Lambert (ed.), Philosophical Applications of Free Logic, Oxford University Press, 1991).Google Scholar
Posy, C. J.[1991] ‘Mathematics as a Transcendental Science’, in T. M. Seebohm, D. Follesdal, and J. N. Mohanty (eds.), [1991].Google Scholar
Posy, C. J.[2000] ‘Epistemology, Ontology and the Continuum’, in Grosholz and Breger, pp. 199–219.Google Scholar
Posy, C. J.[2005] ‘Intuitionism and Philosophy’, in Shapiro [2005], pp. 318–55Google Scholar
Posy, C. J.[2007] ‘Free Logics’, in Gabbay and Woods [2007], pp. 633–80.Google Scholar
Posy, C. J.[2008] “Brouwerian Infinity”, in van Atten, et. al. (eds.) [2008] pp. 21–36.Google Scholar
Posy, C. J.[2015] ‘Realism, Reference and Reason: Remarks on Putnam and Kant’, Auxier et al. [2015], pp. 565–98.Google Scholar
Posy, C. J.[Forthcoming] ‘Kant and Brouwer: Two Knights of the Finite’, in Posy and Rechter, [Forthcoming].Google Scholar
Posy, C., and Rechter, O. (eds.), [Forthcoming] Kant’s Philosophy of Mathematics: Vol. 2, Reception and Influence, Cambridge University Press.Google Scholar
Prawitz, D. [1965] Natural Deduction, Stockholm, Almqvist and Wiksell.Google Scholar
Rasiowa, H., and Sikorski, R. [1963] The Mathematics of Metamathematics, Warsaw, Panstwowe Wydawnictowo Naukow.Google Scholar
Rathjen, M. [2006] ‘Choice Principles in Constructive and Classical Set Theories’, in Chatzidakis et al., [2006] pp. 299–326.Google Scholar
Rogers, H., [1967] Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.Google Scholar
Seebohm, Th., Follesdal, D., and Mohanty, J. N. (eds.), [1991] Phenomenology and the Formal Sciences, the Center for Advanced Research in Phenomenology, and Kluwer Academic Publishers.Google Scholar
Shapiro, S. (ed.), [2005] The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press.Google Scholar
Specker, E. [1949] ‘Nicht konstrukiv bewiebare Sätze der Analysis’, Journal of Symbolic Logic, 14, 145–58.Google Scholar
Spector, C. [1962] ‘Provably Recursive Functionals of Analysis; a Consistency Proof of Analysis by an Extension of Principles Formulated in Current Intuitionistic Mathematics’, in Recursive Function Theory, Proc., Pure Mathematics, pp. 1–27.Google Scholar
van Stigt, W. [1990] Brouwer’s Intuitionism, Amsterdam, North Holland.Google Scholar
de Swart, H. C. M. [1976] ‘Another Intuitionistic Completeness Proof,’ Journal of Symbolic Logic, 41, 1976, pp. 644–62.Google Scholar
Takeuti, G. [1987] Proof Theory, 2nd ed., Amsterdam, North Holland.Google Scholar
Tennant, N. [1997] The Taming of the True, Oxford University Press.Google Scholar
Tennant, N.[2020] ‘Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman – Myhill Result, and Its Immediate Reception,’ Philosophia Mathematica, 28, pp. 139–171.Google Scholar
Tennant, N.[Forthcoming] ‘Does Choice Really Imply Excluded Middle? Part II: Historical, Philosophical, and Foundational Reflections on the Goodman – Myhill Result,’ Philosophia Mathematica, forthcoming.Google Scholar
Troelstra, A. S. [1969] Principles of Intuitionism, Lecture Notes in Mathematics, 95, Berlin, Springer.Google Scholar
Troelstra, A. S.[1977] Choice Sequences: A Chapter of Intuitionistic Mathematics, Oxford University Press.Google Scholar
Troelstra, A. S.[1983] ‘Analyzing Choice Sequences’, J. Phil. Logic, 12, 197–260.Google Scholar
Veldman, W. [1976] ‘An Intuitionistic Completeness Theorem for Intuitionistic Predicate Logic,’ Journal of Symbolic Logic 41.Google Scholar
Voorbraak, F. P. J. M. [1987] ‘Tensed Intuitionistic Logic’, Logic Group Preprint Series, No. 17, Philosophy Department, Utrecht, University of Utrecht.Google Scholar
Wavre, R., [1926] “Logique formelle et logique empiriste,” Revue de Métaphysique et de Morale, 33, pp. 65–75.Google Scholar
Weyl, H. [1917] Das Kontinuum, Kritische Untersuchungen über die Grundlagen der Analysis, Berlin, Teubner.Google Scholar
Weyl, H.[1921] ‘Über die neue Grundlagenkrise der Mathematik’, Mathematische Zeitschrift, 10, 39–79.Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mathematical Intuitionism
  • Carl J. Posy, Hebrew University of Jerusalem
  • Online ISBN: 9781108674485
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Mathematical Intuitionism
  • Carl J. Posy, Hebrew University of Jerusalem
  • Online ISBN: 9781108674485
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Mathematical Intuitionism
  • Carl J. Posy, Hebrew University of Jerusalem
  • Online ISBN: 9781108674485
Available formats
×