References

• Aberth, O., 1980, Computable Analysis, New York: McGraw-Hill. (Scholar)
• –––, 2001,Computable Calculus, New York: Academic Press. (Scholar)
• Bauer, A., 2005, “Realizability as the connection between computable and constructive Mathematics”, Lecture notes for a tutorial at a satellite seminar of CCA2005, Kyoto, Japan [available online]. (Scholar)
• Beeson, M., 1985, Foundations of Constructive Mathematics, Heidelberg: Springer Verlag. (Scholar)
• Berger, J., 2006, “The logical strength of the uniform continuity theorem”, in Logical Approaches to Computational Barriers, A. Beckmann, U. Berger, B. Löwe, and J. V. Tucker (eds.), Heidelberg: Springer Verlag. (Scholar)
• Berger, J., and Bridges, D.S., 2007, “A fan-theoretic equivalent of the antithesis of Specker’s theorem”, Proceedings of Royal Dutch Mathematical Society (Indagationes Mathematicae) (Indag. Math. N.S.), 18(2): 195–202. (Scholar)
• –––, 2009, “The fan theorem and positive-valued uniformly continuous functions on compact intervals”, New Zealand Journal of Mathematics, 38: 129–135. (Scholar)
• Berger, J., and Ishihara, H., 2005, “Brouwer’s fan theorem and unique existence in constructive analysis”, Mathematical Logic Quarterly, 51(4): 360–364. (Scholar)
• Berger, J., and Schuster, P.M., 2006, “Classifying Dini’s theorem”, Notre Dame Journal of Formal Logic, 47: 253–262. (Scholar)
• Berger, J., and Svindland, G., 2016, “A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov’s principle ”, Annals of Pure and Applied Logic, 167: 1161–1170. (Scholar)
• –––, 2016a, “Convexity and constructive infima”, Archive for Mathematical Logic, 55: 873–881. (Scholar)
• Bishop, E., 1967, Foundations of Constructive Analysis, New York: McGraw-Hill. (Scholar)
• –––, 1973, Schizophrenia in Contemporary Mathematics (American Mathematical Society Colloquium Lectures), Missoula: University of Montana; reprinted in Errett Bishop: Reflections on Him and His Research, American Mathematical Society Memoirs 39. (Scholar)
• Bishop, E. and Bridges, D., 1985, Constructive Analysis, (Grundlehren der mathematischen Wissenschaften, 279), Heidelberg: Springer Verlag. (Scholar)
• Bourbaki, N., 1984, Éléments d’histoire des mathématiques, Paris: Masson; English-language edition, Elements of the History of Mathematics, J. Meldrum (trans.), 2006, Berlin: Springer Verlag.
• Bridges, D.S., 1999, “Constructive methods in mathematical economics”, in Zeitschrift für Nationalökonomie (Supplement), 8: 1–21. (Scholar)
• Bridges, D.S., and Diener, H., 2007, “The pseudocompactness of [0, 1] is equivalent to the uniform continuity theorem”, Journal of Symbolic Logic, 72(4): 1379–1383. (Scholar)
• Bridges, D.S., and Richman, F., 1987, Varieties of Constructive Mathematics, London Mathematical Society Lecture Notes 97, Cambridge: Cambridge University Press. (Scholar)
• Bridges, D.S. and Vîță, L.S., 2006, Techniques of Constructive Analysis, Heidelberg: Springer Verlag. (Scholar)
• –––, 2011, Apartness and Uniformity—A Constructive Development, Heidelberg: Springer Verlag. (Scholar)
• Bridges, D.S., Ishihara, H., Rathjen, M.J., and Schwichtenberg, H. (eds), forthcoming, Handbook of Constructive Mathematics, Cambridge: Cambridge University Press. (Scholar)
• Brouwer, L.E.J., 1907, Over de Grondslagen der Wiskunde, Doctoral Thesis, University of Amsterdam; reprinted with additional material, D. van Dalen (ed.), by Matematisch Centrum, Amsterdam, 1981. (Scholar)
• –––, 1908, “De onbetrouwbaarheid der logische principes”, Tijdschrift voor Wijsbegeerte, 2: 152–158. (Scholar)
• –––, 1921, “Besitzt jede reelle Zahl eine Dezimalbruchentwicklung?”, Mathematische Annalen, 83: 201–210. (Scholar)
• –––, 1924, “Beweis, dass jede volle Funktion gleichmässig stetig ist”, Proceedings of Royal Dutch Mathematical Society, 27: 189–193. (Scholar)
• –––, 1924A, “Bemerkung zum Beweise der gleichmässigen Stetigkeit voller Funktionen”, Proceedings of Royal Dutch Mathematical Society, 27: 644–646. (Scholar)
• Cederquist, J., and Negri, S., 1996, “A constructive proof of the Heine-Borel covering theorem for formal reals”, in Types for Proofs and Programs (Lecture Notes in Computer Science, Volume 1158), 62–75, Berlin: Springer Verlag. (Scholar)
• Constable, R., et al., 1986, Implementing Mathematics with the Nuprl Proof Development System, Englewood Cliffs, NJ: Prentice-Hall. (Scholar)
• Coquand, T., 1992, “An intuitionistic proof of Tychonoff’s theorem”, Journal of Symbolic Logic, 57: 28–32. (Scholar)
• –––, 2009, “Space of valuations”, Annals of Pure and Applied Logic, 157: 97–109. (Scholar)
• –––, 2016, “Type Theory”, Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/entries/type-theory/>. (Scholar)
• Coquand, T., and Spitters, B., 2009, “Integrals and Valuations”, Journal of Logic and Analysis, 1(3): 1–22. (Scholar)
• Coquand, T., Sambin, G., Smith, J., and Valentini, S., 2003, “Inductively generated formal topologies”, Annals of Pure and Applied Logic, 124: 71–106. (Scholar)
• Curi, G., 2010, “On the existence of Stone-Čech compactification”, Journal of Symbolic Logic, 75: 1137–1146. (Scholar)
• Dent, J.E., 2013, Anti-Specker Properties in Constructive Reverse Mathematics, Ph.D. thesis, University of Canterbury, Christchurch, New Zealand. (Scholar)
• Diaconescu, R., 1975, “Axiom of choice and complementation”, Proceedings of the American Mathematical Society, 51: 176–178. (Scholar)
• Diener, H., 2008, Compactness under Constructive Scrutiny, Ph.D. thesis, Christchurch, New Zealand: University of Canterbury. (Scholar)
• –––, 2008a, “Generalising compactness”, Mathematical Logic Quarterly, 51(1): 49–57. (Scholar)
• –––, 2012,“Reclassifying the antithesis of Specker’s theorem”, Archive for Mathematical Logic, 51: 687–693. (Scholar)
• Diener, H., and Loeb, I., 2009, “Sequences of Real Functions on [0, 1] in Constructive Reverse Mathematics”, Annals of Pure and Applied Logic, 157(1): 50–61. (Scholar)
• Diener, H., and Lubarsky, R., 2013, “Separating the fan theorem and its weakenings”, in Logical Foundations of Computer Science (Lecture Notes in Computer Science, 7734), S. Artemov and A. Nerode (eds.), Berlin: Springer Verlag. (Scholar)
• Dummett, Michael, 1977 [2000], Elements of Intuitionism (Oxford Logic Guides, 39), Oxford: Clarendon Press, 1977; 2nd edition, 2000. [Page references are to the 2nd edition.] (Scholar)
• Ewald, W., 1996, From Kant to Hilbert: A Source Book in the Foundations of Mathematics (Volume 2), Oxford: Clarendon Press. (Scholar)
• Feferman, S, 1975, “A language and axioms for explicit mathematics”, in Algebra and Logit, J. N. Crossley (ed.), Heidelberg: Springer Verlag, pp. 87–139.
• –––, 1979, “Constructive theories of functions and classes”, in Logic Colloquium ‘78, M. Boffa, D. van Dalen, and K. McAloon (eds.), Amsterdam: North Holland, pp. 159–224. (Scholar)
• Friedman, H., 1975, “Some systems of second order arithmetic and their use”, in Proceedings of the 17th International Congress of Mathematicians, Vancouver, BC, 1974. (Scholar)
• –––, 1977, “Set theoretic foundations for constructive analysis”, Annals of Mathematics, 105 (1): 1–28. (Scholar)
• Goodman, N.D., and Myhill, J., 1978, “Choice Implies Excluded Middle”, Zeitschrift für Logik und Grundlagen der Mathematik, 24: 461. (Scholar)
• Hayashi, S., and Nakano, H., 1988, PX: A Computational Logic, Cambridge MA: MIT Press. (Scholar)
• Hendtlass, M., 2013, Constructing fixed points and economic equilibria, Ph.D. thesis, University of Leeds. (Scholar)
• Heyting, A., 1930, “Die formalen Regeln der intuitionistischen Logik”, Sitzungsberichte der Preussische Akadademie der Wissenschaften. Berlin, 42–56. (Scholar)
• –––, 1971, Intuitionism—an Introduction, 3rd edition, Amsterdam: North Holland. (Scholar)
• Hilbert, D., 1925, “Über das Unendliche”, Mathematische Annalen, 95: 161–190; translation, “On the Infinite”, by E. Putnam and G. Massey, in Philosophy of Mathematics: Selected Readings, P. Benacerraf and H. Putnam (eds.), Englewood Cliffs, NJ: Prentice Hall, 1964, 134–151. (Scholar)
• Hurewicz, W., 1958, Lectures on Ordinary Differential Equations, Cambridge, MA: MIT Press. (Scholar)
• Ishihara, H., 1992, “Continuity properties in constructive mathematics”, Journal of Symbolic Logic, 57 (2): 557–565. (Scholar)
• –––, 1994, “A constructive version of Banach’s inverse mapping theorem”, New Zealand Journal of Mathematics, 23: 71–75. (Scholar)
• –––, 2005, “Constructive Reverse mathematics: compactness properties”, in From Sets and Types to Analysis and Topology: Towards Practicable Foundations for Constructive Mathematics, L. Crosilla and P. Schuster (eds.), Oxford: The Clarendon Press. (Scholar)
• –––, 2006, “Reverse mathematics in Bishop’s constructive mathematics”, Philosophia Scientiae (Cahier Special), 6: 43–59. (Scholar)
• –––, 2013, “Relating Bishop’s function spaces to neighbourhood spaces”, Annals of Pure and Applied Logic, 164: 482–490. (Scholar)
• Johnstone, P.T., 1982, Stone Spaces, Cambridge: Cambridge University Press. (Scholar)
• –––, 2003, “The point of pointless topology”, Bulletin of the American Mathematical Society, 8: 41–53. (Scholar)
• Joyal, A., and Tierney, M., 1984, “An extension of the Galois theory of Grothendieck”, Memoirs of the American Mathematical Society, 309: 85 pp. (Scholar)
• Julian, W.H., and Richman, F., 1984, “A uniformly continuous function on [0, 1] that is everywhere different from its infimum”, Pacific Journal of Mathematics, 111: 333–340.
• Kushner, B., 1985, Lectures on Constructive Mathematical Analysis, Providence, RI: American Mathematical Society. (Scholar)
• Lietz, P., 2004, From Constructive Mathematics to Computable Analysis via the Realizability Interpretation, Dr. rer. nat. dissertation, Universität Darmstadt, Germany. (Scholar)
• Lietz, P., and Streicher, T., “Realizability models refuting Ishihara’s boundedness principle”, Annals of Pure and Applied Logic, 163(12): 1803–1807. (Scholar)
• Loeb, I., 2005, “Equivalents of the (Weak) Fan Theorem”, Annals of Pure and Applied Logic, 132: 51–66. (Scholar)
• Lombardi, H., Quitté, C., 2011, Algèbre Commutative. Méthodes constructives, Nanterre, France: Calvage et Mounet. (Scholar)
• Lorenzen, P., 1955, Einführung in die operative Logik und Mathematik (Grundlehren der mathematischen Wissenschaften, Volume 78), 2nd edition, 1969, Heidelberg: Springer. (Scholar)
• Lubarsky, R., and Diener, H., 2014, “Separating the Fan Theorem and Its Weakenings”, Journal of Symbolic Logic, 79(3): 792–813. (Scholar)
• Markov, A.A., 1954, Theory of Algorithms, Trudy Mat. Istituta imeni V.A. Steklova, 42, Moskva: Izdatel’stvo Akademii Nauk SSSR. (Scholar)
• Marquis, J.-P., “Category Theory”, 2015, The Stanford Encyclopedia of Philosophy (Winter 2015 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2015/entries/category-theory/>. (Scholar)
• Martin-Löf, P., 1968, Notes on Constructive Analysis, Stockholm: Almquist & Wiksell. (Scholar)
• –––, 1975, “An intuitionistic theory of types: predicative part”, in Logic Colloquium 1973, H.E. Rose and J.C. Shepherdson (eds.), Amsterdam: North-Holland. (Scholar)
• –––, 1980, “Constructive mathematics and computer programming”, in Proc. 6th. Int. Congress for Logic, Methodology and Philosophy of Science, L. Jonathan Cohen (ed.), Amsterdam: North-Holland. (Scholar)
• –––, 1984, Intuitionistic Type Theory, Notes by Giovanni Sambin of a series of lectures given in Padova, June 1980, Naples: Bibliopolis. (Scholar)
• –––, 2006, “100 years of Zermelo’s axiom of choice: what was the problem with it?”, The Computer Journal, 49(3): 345–350. (Scholar)
• Menger, K., 1940, “Topology without points”, Rice Institute Pamphlet, 27(1): 80–107 [available online]. (Scholar)
• Mines, R., Richman, F., and Ruitenburg, W., 1988, A Course in Constructive Algebra, Universitext, Heidelberg: Springer Verlag. (Scholar)
• Moerdijk, I., 1984, “Heine-Borel does not imply the fan theorem”, Journal of Symbolic Logic, 49(2): 514–519. (Scholar)
• Moore, G.H., 2013, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence, New York: Dover. (Scholar)
• Myhill, J., 1973, “Some Properties of Intuitionistic Zermelo-Fraenkel Set Theory”, in Cambridge Summer School in Mathematical Logic, A. Mathias and H. Rogers (eds.), Lecture Notes in Mathematics, 337, Heidelberg: Springer Verlag, 206–231. (Scholar)
• –––, 1975, “Constructive Set Theory”, Journal of Symbolic Logic, 40(3): 347–382. (Scholar)
• Naimpally, S., 2009, Proximity Approach to Problems in Topology and Analysis, Munich: Oldenbourg Verlag. (Scholar)
• Naimpally, S., and Warrack, B.D., 1970, Proximity Spaces (Cambridge Tracts in Math. and Math. Phys., Volume 59), Cambridge: Cambridge University Press. (Scholar)
• Nordström, B., Peterson, K., and Smith, J.M., 1990, “Programming in Martin-Löf’s Type Theory”, Oxford: Oxford University Press. (Scholar)
• Palmgren, E., 2007, “A constructive and functorial embedding of locally compact metric spaces into locales”, Topology and its Applications, 154: 1854–1880.
• –––, 2008, “Resolution of the uniform lower bound problem in constructive analysis”, Mathematical Logic Quarterly, 54: 65–69. (Scholar)
• –––, 2009, “From intuitionistic to formal topology: some remarks on the foundations of homotopy theory”, in: Logicism, Intuitionism and Formalism—what has become of them?, S. Lindström, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen (eds.), 237–253, Berlin: Springer Verlag. (Scholar)
• Petrakis, I., 2016, “A constructive function-theoretic approach to topological compactness”, in Logical Methods in Computer Science 2016, IEEE Computer Society Publications: 605–614.
• –––, 2016a, “The Urysohn Extension Theorem for Bishop Spaces”, in S. Artemov and A. Nerode (eds.), Symposium on Logical Foundations in Computer Science 2016 (Lecture Notes in Computer Science 9537), Berlin: Springer Verlag, 299–316. (Scholar)
• Picado, J., and Pultr, A., 2011, Frames and Locales: Topology without Points, Basel: Birkhäuser Verlag. (Scholar)
• Richman, F., 1983, “Church’s Thesis Without Tears”, Journal of Symbolic Logic, 48: 797–803. (Scholar)
• –––, 1990, “Intuitionism as generalization”, Philosophia Mathematica, 5: 124–128. (Scholar)
• –––, 1996, “Interview with a constructive mathematician”, Modern Logic, 6: 247–271. (Scholar)
• –––, 2000, “The Fundamental Theorem of Algebra: A Constructive Treatment Without Choice”, Pacific Journal of Mathematics, 196: 213–230. (Scholar)
• Riesz, F., 1908, “Stetigkeitsbegriff und abstrakte Mengenlehre”, Atti IV Congresso Internationale Matematica Roma II, 18–24. (Scholar)
• Sambin, G., 1987, “Intuitionistic formal spaces—a first communication”, in Mathematical Logic and its Applications, D. Skordev (ed.), 187–204, New York: Plenum Press. (Scholar)
• –––, forthcoming, The Basic Picture: Structures for Constructive Topology, Oxford: Oxford University Press. (Scholar)
• Sambin, G., and Smith, J. (eds.), 1998, Twenty Five Years of Constructive Type Theory, Oxford: Clarendon Press. (Scholar)
• Schuster, P.M., 2005, “What is continuity, constructively?”, Journal of Universal Computer Science, 11: 2076–2085 (Scholar)
• –––, 2006, “Formal Zariski topology: positivity and points”, Annals of Pure and Applied Logic, 137: 317–359. (Scholar)
• Schwichtenberg, H., 2009, “Program extraction in constructive analysis”, in Logicism, Intuitionism and Formalism—what has become of them?, S. Lindström, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen (eds.), Berlin: Springer Verlag, 255–275. (Scholar)
• Simpson, S.G., 1984, “Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations”, Journal of Symbolic Logic, 49(3): 783–802. (Scholar)
• –––, 2009, Subsystems of Second Order Arithmetic, second edition, Cambridge: Cambridge University Press. (Scholar)
• Specker, E., 1949, “Nicht konstruktiv beweisbare Sätze der Analysis”, Journal of Symbolic Logic, 14: 145–158. (Scholar)
• Steinke, T.A., 2011, Constructive Notions of Compactness in Apartness Spaces, M.Sc. Thesis, University of Canterbury, Christchurch, New Zealand. (Scholar)
• Troelstra, A.S., 1978, “Aspects of Constructive Mathematics”, in Handbook of Mathematical Logic, J. Barwise (ed.), Amsterdam: North-Holland. (Scholar)
• Troelstra, A.S., and van Dalen, D., 1988, Constructivism in Mathematics: An Introduction (two volumes), Amsterdam: North Holland. (Scholar)
• van Atten, M., 2004, On Brouwer, Belmont: Wadsworth/Thomson Learning. (Scholar)
• van Dalen, D., 1981, Brouwer’s Cambridge Lectures on Intuitionism, Cambridge: Cambridge University Press. (Scholar)
• –––, 1999, Mystic, Geometer and Intuitionist: The Life of L.E.J. Brouwer (Volume 1), Oxford: Clarendon Press. (Scholar)
• –––, 2005, Mystic, Geometer, and Intuitionist: The Life of L.E.J. Brouwer (Volume 2), Oxford: Clarendon Press. (Scholar)
• van Stigt, W.P., 1990, Brouwer’s Intuitionism, Amsterdam: North-Holland. (Scholar)
• Vickers, S., 2005, “Localic completion of generalized metric spaces I”, Theory and Applications of Categories, 14(15): 328–356. (Scholar)
• Waaldijk, F., 2005, On the foundations of constructive mathematics, Foundations of Science, 10(3): 249–324. (Scholar)
• Wallman, H., 1938, “Lattices of topological spaces”, Annals of Mathematics, 39: 112–126. (Scholar)
• Weihrauch, K., 2000, Computable Analysis (EATCS Texts in Theoretical Computer Science), Heidelberg: Springer Verlag. (Scholar)
• Weyl, H., 1946, “Mathematics and Logic”, American Mathematical Monthly, 53(1): 2–13. (Scholar)
• Whitehead, A.N., 1919, An Enquiry Concerning the Principles of Natural Knowledge, Cambridge: Cambridge University Press, second edition, 1925. (Scholar)

Related Literature

• Heijenoort, Jean van, 1967, From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press. (Scholar)
• Hilbert, David, 1928, “Die Grundlagen der Mathematik”, Hamburger Mathematische Einzelschriften 5, Teubner, Leipzig. Reprinted in English translation in van Heijenoort 1967. (Scholar)