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  • Print publication year: 2011
  • Online publication date: October 2011

History of constructivism in the 20th century


§1. Introduction. In this survey of the history of constructivism. more space has been devoted to early developments (up till ca. 1965) than to the work of the later decades. Not only because most of the concepts and general insights have emerged before 1965, but also for practical reasons: much of the work since 1965 is of a too technical and complicated nature to be described adequately within the limits of this article.

Constructivism is a point of view (or an attitude) concerning the methods and objects of mathematics which is normative: not only does it interpret existing mathematics according to certain principles, but it also rejects methods and results not conforming to such principles as unfounded or speculative (the rejection is not always absolute, but sometimes only a matter of degree: a decided preference for constructive concepts and methods). In this sense the various forms of constructivism are all ‘ideological’ in character.

Constructivism as a specific viewpoint emerges in the final quarter of the 19th century, and may be regarded as a reaction to the rapidly increasing use of highly abstract concepts and methods of proof in mathematics, a trend exemplified by the works of R.Dedekind and G. Cantor.

The mathematics before the last quarter of the 19th century is, from the viewpoint of today, in the main constructive, with the notable exception of geometry, where proof by contradiction was commonly accepted and widely employed.

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M., Barzin and A., Errera [1927], Sur la logique deM. Brouwer, des Lettres et des Beaux Arts de Belgique. Bulletin de la Classe des Sciences, Cinquième Série, vol. 13, Academie Royale des Sciences, pp. 56–71.
M. J., Beeson [1985], Foundations of Constructive Mathematics, Springer-Verlag, Berlin.
E. W., Beth [1956], Semantic construction of intuitionistic logic, Koninklijke Nederlandse Akademie van Wetenschappen. Afdeling Letterkunde: Mededelingen. Nieuwe Reeks, vol. 19.
E. W., Beth [1959, 1965 2], The Foundations of Mathematics, North-Holland, Amsterdam.
E., Bishop [1967], Foundations of Constructive Analysis, McGraw-Hill, New York.
E., Borel [1914 2, 1928 3, 1959 4], Leçons sur la Théorie des Fonctions, Gauthier-Villars, Paris.
L. E. J., Brouwer [1905], Leven, Kunst, Mystiek (Life, Art and Mysticism), Waltman, Delft, (Dutch).
L. E. J., Brouwer [1907], Over de Grondslagen der Wiskunde (on the Foundations of Mathematics), Maas en van Suchtelen, Amsterdam, (Dutch).
L. E. J., Brouwer [1908], Over de onbetrouwbaarheid der logische principes (on the unreliability of the principles of logic), Tijdschrift voor Wijsbegeerte, vol. 2, pp. 152–158, (Dutch) English translation: (Brouwer 1975, pp. 107–111).
L. E. J., Brouwer [1919], Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten II: Theorie der Punktmengen, Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen. Tweede Reeks. Afdeling Natuurkunde, vol. 12/7. (Also [Brouwer 1975], pp. 191–221.)
L. E. J., Brouwer [1924a], Beweis dass jede volle Funktion gleichmässig stetig ist, Koninklijke Akademie van Wetenschappen. Proceedings of the Section of Sciences, vol. 27, pp. 189–193. (Also [Brouwer 1975], pp. 274–280.)
L. E. J., Brouwer [1924b], Intuitionistische Zerlegung mathematischer Grundbegriffe, Jahresbericht der Deutsche Mathematiker-Vereinigung, vol. 33, pp. 241–256. (Also [Brouwer 1975], pp. 429–440.)
L. E. J., Brouwer [1927], Über Definitionsbereiche von Funktionen, Mathematische Annalen, vol. 97, pp. 60–75. (Also [Brouwer 1975], pp. 390–405.)
L. E. J., Brouwer [1930], Die Struktur des Kontinuums, Gistel, Wien. (Also [Brouwer 1975], pp. 429–440.)
L. E. J., Brouwer [1949], Consciousness, philosophy andmathematics, Proceedings of the Tenth International Congress in Philosophy, August 1948, Vol. 1 (E. W., Beth, H. J., Pos, and H. J. A., Hollak, editors), North-Holland, Amsterdam, pp. 1235–1249. (Also [Brouwer 1975], pp. 480–494.)
L. E. J., Brouwer [1975], Collected Works, Vol. 1, North-Holland, Amsterdam, edited by A., Heyting.
S., Buss [1986], Bounded Arithmetic, Studies in Proof Theory. Lecture Notes, vol. 3, Bibliopolis, Naples.
M. A. E., Dummett [1977], Elements of Intuitionism, Clarendon Press, Oxford.
Y. L., Ershov [1972], La théorie des énumérations, Actes du Congrès International des Mathématiciens 1970, Vol. 1 (M., Berger, J., Dieudonné, J., Leroy, J.-L., Lions, and M. P., Malliavin, editors), Gauthier-Villars, Paris, pp. 223–227.
S., Feferman [1975], A language and axioms for explicit mathematics, Algebra and Logic, Springer, Berlin, pp. 87–139.
S., Feferman [1979], Constructive theories of functions and classes, Logic Colloquium '78 (M., Boffa, D. van, Dalen, and K., McAloon, editors), North-Holland, Amsterdam, pp. 159–224.
S., Feferman [1988a], Hilbert's program relativized: proof-theoretical and foundational reductions, The Journal of Symbolic Logic, vol. 53, no. 2, pp. 364–384.
S., Feferman [1988b], Weyl vindicated: ‘DasKontinuum’ 70 years later, Temi e Prospettive della Logica e della Filosofia Contemporanee I, Cooperativa Libraria Universitaria Editrice Bologna, Bologna, pp. 59–93.
H., Friedman [1977], Set-theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105, pp. 1–28.
G., Gentzen [1935], Untersuchungen Über das logische Schließen. II, Mathematische Zeitschrift, vol. 39, no. 1, pp. 405–431.
J.-Y., Girard [1987], Linear logic, Theoretical Computer Science, vol. 50, no. 1, p. 101.
R. L., Goodstein [1957], Recursive Number Theory, North-Holland, Amsterdam.
R. L., Goodstein [1959], Recursive Analysis, North-Holland, Amsterdam.
A., Heyting [1930], Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch Mathematische Klasse, Die formalen Regeln der intuitionistischen Mathematik, Ibidem pp. 57–71, 158–169, pp. 158–169.
A., Heyting [1934], Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie, Springer-Verlag, Berlin.
A., Heyting [1956, 1966 2, 1971 3], Intuitionism. An introduction, North-Holland,Amsterdam.
D., Hilbert and P., Bernays [1934, 1968 2], Grundlagen der Mathematik. I, Springer-Verlag, Berlin.
J. M. E., Hyland [1982], The effective topos, The L.E.J. Brouwer Centenary Symposium (A. S., Troelstra and D. van, Dalen, editors), North-Holland, Amsterdam, pp. 165–216.
S. C., Kleene [1945], On the interpretation of intuitionistic number theory, The Journal of Symbolic Logic, vol. 10, pp. 109–124.
S. C., Kleene and R. E., Vesley [1965], The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, North-Holland, Amsterdam.
A. N., Kolmogorov [1932], Zur deutung der intuitionistischen Logik, Mathematische Zeitschrift, vol. 35, pp. 58–65.
G., Kreisel [1968], Lawless sequences of natural numbers, Compositio Mathematica, vol. 20, pp. 222–248 (1968).
G., Kreisel and A. S., Troelstra [1970], Formal systems for some branches of intuitionistic analysis, Annals of Pure and Applied Logic, vol. 1, pp. 229–387.
S., Kripke [1965], Semantical analysis of intuitionistic logic, Formal Systems and Recursive Functions (J., Crossley and M. A. E., Dummett, editors), North-Holland, Amsterdam, pp. 92–130.
J., Lambeck [1972], Deductive systems and categories III, Toposes, Algebraic Geometry and Logic (F. W., Lawvere, editor), Springer, Berlin, pp. 57–82.
J., Lambek and P. J., Scott [1986], Introduction toHigherOrderCategorical Logic, Cambridge Studies in Advanced Mathematics, vol. 7, Cambridge University Press, Cambridge.
F. W., Lawvere [1971], Quantifiers and Sheaves, Proceedings of the International Congress of Mathematicians, Nice 1971, Gauthier-Villars, Paris, pp. 1506–1511.
P., Lorenzen [1955], Einf Ührung in die operative Logik und Mathematik, Springer-Verlag, Berlin.
P., Lorenzen [1965], Differential und Integral. Eine konstruktive Einführung in die klassische Analysis, Akademische Verlagsgesellschaft, Wiesbaden.
H., Luckhardt [1989], Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken, The Journal of Symbolic Logic, vol. 54, no. 1, pp. 234–263.
G., Metakides and A., Nerode [1979], Effective content of field theory, Annals of Mathematial Logic, vol. 17, no. 3, pp. 289–320.
J. J. A., Mooij [1966], La Philosophie des Mathématiques de Henri Poincaré, Gauthier-Villars, Paris.
G. H. M, üller [1987], Ω-Bibliography of Mathematical Logic, Springer-Verlag, Berlin, edited by G. H., Müller in collaboration with W., Lenski, Volume VI. Proof theory, Constructive Mathematics, edited by J. E., Kister, D., van Dalen, and A. S., Troelstra.
R. J., Parikh [1971], Existence and feasibility in arithmetic, The Journal of Symbolic Logic, vol. 36, pp. 494–508.
H., Poincaré [1902], Science et Hypothèse, Flammarion, Paris.
H., Poincaré [1905], La Valeur de la Science, Flammarion, Paris.
H., Poincaré [1908], Science et Méthode, Flammarion, Paris.
H., Poincaré [1913], Dernières Pensées, Flammarion, Paris.
M. B., Pour-El and J. I., Richards [1989], Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.
H., Rasiowa and R., Sikorski [1963], The Mathematics of Metamathematics, Państwowe Wydawnictwo Naukowe, Warsaw.
J., Richard [1905], Les principes des mathématiques et le problème des ensembles, Revue génénerale des sciences pures et appliquées, vol. 16, p. [541, Also in Acta Mathematica 30, 1906, pp. 295–296.
N. A., Shanin [1958], O konstruktiviom ponimanii matematicheskikh suzhdenij (on the constructive interpretation of mathematical judgments), Trudy Ordena Lenina Matematicheskogo Instituta imeni V.A. Steklova, Akademiya Nauk SSSR 52, (Russian) Translation: American Mathematical Society Translations, Series 2, 23, pp.108–189, pp. 226–311.
S., Simpson [1988], Partial realizations of Hilbert's Program, The Journal of Symbolic Logic, vol. 53, no. 2, pp. 349–363.
Th., Skolem [1923], Begründung der elementaren Arithmetik durch die rekurrirende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichen Ausdehnungsbereich, Videnskapsselskapets Skrifter, I.Matematisk-naturvidenskabelig klasse, vol. 6, pp. 1–38, Translation (van Heijenoort 1967, pp.302–333).
C. A., Smoryński [1973], Applications of Kripke models, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis (A. S., Troelstra, editor), Springer, Berlin, pp. 324–391.
E., Specker [1949], Nicht konstruktiv beweisbare Sätze der Analysis, The Journal of Symbolic Logic, vol. 14, pp. 145–158.
W. W., Tait [1981], Finitism, The Journal of Philosophy, vol. 78, pp. 524–546.
A. S., Troelstra [1977], Choice Sequences, Clarendon Press, Oxford, A Chapter of Intuition- istic Mathematics.
A. S., Troelstra [1983], Analyzing choice sequences, The Journal of Philosophical Logic, vol. 12, pp. 197–260.
A. S., Troelstra and D., van Dalen [1988], Constructivism in Mathematics. Vol. II, North-Holland, Amsterdam, An introduction.
A. M., Turing [1937], On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42, pp. 230–265, Corrections, Ibidem 43, pp. 544–546.
D., van Dalen [1990], The war of the frogs and the mice, or the crisis of the Mathematische Annalen, The Mathematical Intelligencer, vol. 12, no. 4, pp. 17–31.
J., van Heijenoort [1967], From Frege to Gödel, Harvard University Press, Cambridge.
H., Weyl [1918], Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig.