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# Computability and analysis: the legacy of Alan Turing

## Summary

§1. Introduction. For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid's Elements fall into two distinct categories: “problems,” which assert that a construction can be carried out to meet a given specification, and “theorems,” which assert that some property holds of a particular geometric configuration. For example, Proposition 10 of Book I reads “To bisect a given straight line.” Euclid's “proof” gives the construction, and ends with the (Greek equivalent of) Q.E.F., for quod erat faciendum, or “that which was to be done.” Proofs of theorems, in contrast, end with Q.E.D., for quod erat demonstrandum, or “that which was to be shown”; but even these typically involve the construction of auxiliary geometric objects in order to verify the claim.

Similarly, algebra was devoted to developing algorithms for solving equations. This outlook characterized the subject from its origins in ancient Egypt and Babylon, through the ninth century work of al-Khwarizmi, to the solutions to the quadratic and cubic equations in Cardano's Ars Magna of 1545, and to Lagrange's study of the quintic in his Réflexions sur la résolution algébrique des équations of 1770.

The theory of probability, which was born in an exchange of letters between Blaise Pascal and Pierre de Fermat in 1654 and developed further by Christian Huygens and Jakob Bernoulli, provided methods for calculating odds related to games of chance.

[1] , The failure in computable analysis of a classical existence theorem for differential equations, Proceedings of the American Mathematical Society, vol. 30 (1971), pp. 151-156.
[2] , Computable analysis, McGraw-Hill, New York, 1980.
[3] , , and , Noncomputable conditional distributions, Logic in Computer Science (LICS), 2011, IEEE Conference Presentations, Los Alamitos, California, 2011, pp. 107-116.
[4] , Proving strong normalization of CC by modifying realizability semantics, Types for proofs and programs (Nijmegen, 1993), Springer, Berlin, 1994, pp. 3-18.
[5] , Interpreting classical theories in constructive ones, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1785-1812.
[6] , Uncomputably noisy ergodic limits, Notre Dame Journal of Formal Logic, vol. 53 (2012), pp. 347-350.
[7] and , Gödel's functional (“Dialectica”) interpretation, Handbook of proof theory (, editor), North-Holland, Amsterdam, 1998, pp. 337-405.
[8] , , and , Local stability of ergodic averages, Transactions of the American Mathematical Society, vol. 362 (2010), pp. 261-288.
[9] and , Fundamental notions of analysis in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 139 (2006), pp. 138-184.
[10] , Die Nichtkonstruktivität des Brouwerschen Fixpunktsatzes, Archiv für Mathematische Logik und Grundlagenforschung, vol. 25 (1985), pp. 183-188.
[11] and , Sur les fonctions calculables, Annales de la Société Polonaise de Mathématique, vol. 16 (1937), p. 223.
[12] , The realizability approach to computable analysis and topology, Ph.D. thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 2000.
[13] , , and , Turing's unpublished algorithm for normal numbers, Theoretical Computer Science, vol. 377 (2007), pp. 126-138.
[14] , Foundations of constructive mathematics, Springer, Berlin, 1985.
[15] , , , , and , A constructive version of Birkhoffs ergodic theorem for Martin-Löf random points, Information and Computation, vol. 210 (2012), pp. 21-30.
[16] , Foundations of constructive analysis, McGraw-Hill, New York, 1967.
[17] and , Constructive analysis, Springer, Berlin, 1985.
[18] , Le calcul des intégrales définies, Journal de Mathematiques Pures et Appliquees. Serie 6, vol. 8 (1912), pp. 159-210.
[19] , La théorie de la mesure et al théorie de l'integration, Leçons sur la théorie des fonctions, Gauthier-Villars, Paris, 1950, pp. 214-256.
[20] , Notions of probabilistic computability on represented spaces, Journal of Universal Computer Science, vol. 14 (2008), pp. 956-995.
[21] , Effective Borel measurability and deducibility of functions, Mathematical Logic Quarterly, vol. 51 (2005), pp. 19-44.
[22] and , Effective choice and boundedness principles in computable analysis, The Bulletin of Symbolic Logic, vol. 17 (2011), pp. 73-117.
[23] , Weihrauch degrees, omniscience principles and weak computability, The Journal of Symbolic Logic, vol. 76 (2011), pp. 143-176.
[24] , , and , The Bolzano–Weierstrass theorem is the jump of weak Kőnig's lemma, Annals of Pure and Applied Logic, vol. 163 (2012), pp. 623-655.
[25] , , and , A tutorial on computable analysis, New computational paradigms: Changing conceptions of what is computable (, , and , editors), Springer, New York, 2008, pp. 425-491.
[26] , , and , Randomness and differentiability, submitted; preliminary version at http://arxiv.org/abs/1104.4465.
[27] , Parabolic Julia sets are polynomial time computable, Nonlinearity, vol. 19 (2006), pp. 1383-1401.
[28] and , Computing over the reals: Foundations for scientific computing, Notices of the American Mathematical Society, vol. 53 (2006), pp. 318-329.
[29] and , Non-computable Julia sets, Journal of the American Mathematical Society, vol. 19 (2006), pp. 551-578.
[30] and , Computability of Julia sets, Springer, Berlin, 2008.
[31] and , Varieties of constructive mathematics, Cambridge University Press, Cambridge, 1987.
[32] , Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossen Dritten. Erster Teil: Allgemeine Mengenlehre, Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, vol. 12 (1918), no. 5, reprinted in [35], pp. 150-190.
[33] , Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossen Dritten. Zweiter Teil: Theorie der Punkmengen, Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, vol. 12 (1919), no. 7, reprinted in [35], pp. 191-221.
[34] , Über Definitionsbereiche von Funktionen, Mathematische Annalen, vol. 97 (1927), no. 1, pp. 60-75.
[35] , Collected works, volume I: Philosophy and foundations of mathematics, (, editor), North-Holland Publishing Company, Amsterdam, 1975.
[36] , Algorithmic operators in constructive complete separable metric spaces, Doklady Akademii Nauk, vol. 128 (1959), pp. 49-52, (in Russian).
[37] and , Index sets in computable analysis, Theoretical Computer Science, vol. 219 (1999), pp. 111-150.
[38] , Effectively closed sets and graphs of computable real functions, Theoretical Computer Science, vol. 284 (2002), pp. 279-318.
[39] , Index sets for computable differential equations, Mathematical Logic Quarterly, vol. 50 (2004), pp. 329-344.
[40] , A formulation of the simple theory of types, The Journal of Symbolic Logic, vol. 5 (1940), pp. 56-68.
[41] and , A new method of establishing conservativity of classical systems over their intuitionistic version, Mathematical Structures in Computer Science, vol. 9 (1999), pp. 323-333.
[42] and , The calculus of constructions, Information and Computation, vol. 76 (1988), pp. 95-120.
[43] and , Inductively defined types, COLOG-88 (Tallinn, 1988), Springer, Berlin, 1990, pp. 50-66.
[44] and , A proof of strong normalisation using domain theory, Logical Methods in Computer Science, vol. 3 (2007), no. 4.
[45] , Functionality in combinatory logic, Proceedings of the National Academy of Sciences, vol. 20 (1934), pp. 584-590.
[46] and , Combinatory logic, vol. I, North-Holland, Amsterdam, 1958.
[47] , The universal computer: The road from Leibniz to Turing, W. W. Norton, New York, 2000.
[48] , Constructive pseudonumbers, Commentationes Mathematicae Universitatis Carolinae, vol. 16 (1975), pp. 315-331, (in Russian).
[49] , The differentiability of constructive functions of weakly bounded variation on pseudo numbers, Commentationes Mathematicae Universitatis Carolinae, vol. 16 (1975), pp. 583-599, (in Russian).
[50] and , Algorithmic randomness and complexity, Springer, New York, 2010.
[51] , Mathematical intuitionism: Introduction to proof theory, American Mathematical Society, Providence, Rhode Island, 1988.
[52] , Elements of intuitionism, second ed., Oxford University Press, New York, 2000.
[53] , A computable approach to measure and integration theory, Information and Computation, vol. 207 (2009), pp. 642-659.
[54] and , A domain-theoretic approach to computability on the real line, Theoretical Computer Science, vol. 210 (1999), pp. 73-98.
[55] , Kronecker's views on the foundations of mathematics, The history of modern mathematics ( and , editors), Academic Press, San Diego, 1989, pp. 67-77.
[56] , Kronecker's fundamental theorem of general arithmetic, Episodes in the history of modern algebra (1800–1950), American Mathematical Society, Providence, Rhode Island, 2007, pp. 107-116.
[57] , Kronecker's algorithmic mathematics, Mathematical Intelligencer, vol. 31 (2009), pp. 11-14.
[58] (editor), From Kant to Hilbert: A source book in the foundations of mathematics, vol. 1 and 2, Oxford University Press, Oxford, 1996.
[59] , , , and , Martin-Löf random points satisfy Birkhoff's ergodic theorem for effectively closed sets, Proceedings of the American Mathematical Society, vol. 140 (2012), pp. 3623-3628.
[60] and , Randomness and non-ergodic systems, arXiv:1206.2682.
[61] , , , and , Algorithmic aspects of Lipschitz functions, preprint.
[62] and , Computable de Finetti measures, Annals of Pure and Applied Logic, vol. 163 (2012), pp. 530-546.
[63] , Grundgesetze der Arithmetik, Band I, Hermann Pohle, Jena, 1893.
[64] , On the computational complexity of maximization and integration, Advances in Mathematics, vol. 53 (1984), pp. 80-98.
[65] , Classically and intuitionistically provable functions, Higher set theory ( and , editors), Springer, Berlin, 1978, pp. 21-27.
[66] , Uniform test of algorithmic randomness over a general space, Theoretical Computer Science, vol. 341 (2005), pp. 91-137.
[67] , , and , Randomness on computable probability spaces—a dynamical point of view, Theory of Computing Systems, vol. 48 (2011), pp. 465-485.
[68] , , and , A constructive Borel–Cantelli lemma: constructing orbits with required statistical properties, Theoretical Computer Science, vol. 410 (2009), pp. 2207-2222.
[69] , , and , Effective symbolic dynamics, random points, statistical behavior, complexity and entropy, Information and Computation, vol. 208 (2010), pp. 23-41.
[70] , Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112 (1936), pp. 493–465, translated as The consistency of elementary number theory in [71], pp. 132–213.
[71] , Collected works, (, editor), North-Holland, Amsterdam, 1969.
[72] , Alan Turing and the foundations of computable analysis, The Bulletin of Symbolic Logic, vol. 17 (2011), pp. 394-430.
[73] and , How incomputable is the separable Hahn–Banach theorem?, Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 393-425.
[74] , Une extension de l'interprétation de Gödel à l'analyse, et son application à l'élimination des coupures dans l'analyse et la théorie des types, Proceedings of the second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 63-92.
[75] , , and , Proofs and types, Cambridge University Press, 1989.
[76] , Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 34-38, translated by and as On intuitionistic arithmetic and number theory in [217], reprinted in [78] (1933e),pp. 287-295.
[77] , Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280-287, reprinted with English translation in Feferman et al., (editors), Kurt Gödel: Collected Works, volume 2, Oxford University Press, New York, 1990, pp. 241-251.
[78] , Collected works, ( et al., editors), vol. I, Oxford University Press, New York, 1986.
[79] , Recursive number theory: A development of recursive arithmetic in a logic-free equation calculus, North-Holland, Amsterdam, 1957.
[80] , Recursive analysis, North-Holland, Amsterdam, 1961.
[81] , A formulae-as-type notion of control, Proceedings of the 17th ACM SIGPLAN–SIGACT symposium on Principles of Programming Languages (POPL '90), Association for Computing Machinery, New York, 1990, pp. 47-58.
[82] , Computable functionals, Fundamenta Mathematicae, vol. 42 (1955), pp. 168-202.
[83] , On the definition of computable functionals, Fundamenta Mathematicae, vol. 42 (1955), pp. 232-239.
[84] , On the definitions of computable real continuous functions, Fundamenta Mathematicae, vol. 44 (1957), pp. 61-71.
[85] , Some approaches to constructive analysis, Constructivity in mathematics (, editor), North-Holland, Amsterdam, 1959, pp. 43-61.
[86] , Ein Kriterium für die Annahme des Maximums in der Berechenbaren Analysis, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 193-196.
[87] , Konstruktive Darstellungen reeller Zahlen und Folgen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 24 (1978), pp. 365-374.
[88] , Konstruktive Darstellungen in topologischen Raumen mit rekursiver Basis, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 565-576.
[89] , Effectivity andeffective continuity offunctions between computable metric spaces, Combinatorics, complexity, and logic (, , , , and , editors), Springer, Singapore, 1997.
[90] , An effective Riemann Mapping Theorem, Theoretical Computer Science, vol. 219 (1999), pp. 225-265.
[91] , A real number structure that is effectively categorical, Mathematical Logic Quarterly, vol. 45 (1999), pp. 147-182.
[92] , A Banach–Mazur computable but not Markov computable function on the computable real numbers, Annals of Pure and Applied Logic, vol. 132 (2005), pp. 227-246.
[93] and , Random elements in effective topological spaces with measure, Information and Computation, vol. 181 (2003), pp. 32-56.
[94] , Die formalen Regeln der intuitionistischen Logik. I, II, III, Sitzungsberichte der königlich-preussischen Akademie der Wissenschaften. Berlin, (1930), pp. 42–56, 57–71, and 158-169.
[95] , Intuitionism: An introduction, North-Holland, Amsterdam, 1956.
[96] and , Grundlagen der Mathematik, vol. 1, Springer, Berlin, 1934, vol. 32, 1939.
[97] , The formulae-as-types notion of construction, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism, Academic Press, London, 1980, pp. 480-490.
[98] and , Computability of probability measures and Martin-Löfrandomness over metric spaces, Information and Computation, vol. 207 (2009), pp. 830-847.
[99] , , and , Computability of the Radon–Nikodym derivative, Models of computation in context (, , , and , editors), Springer, Berlin, 2011, pp. 132-141.
[100] , Computability of the Radon–Nikodym derivative, Computability, vol. 1 (2012), pp. 3-13.
[101] , The effective topos, The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), North-Holland, Amsterdam, 1982, pp. 165-216.
[102] and , A blend of methods of recursion theory and topology, Annals of Pure and Applied Logic, vol. 124 (2003), pp. 141-178.
[103] and , Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 125-151.
[104] , Recursive and nonextendible functions over the reals; filter foundation for recursive analysis, II, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 87-110.
[105] , Lipschitz continuous ordinary differential equations are polynomial-space complete, Computational Complexity, vol. 19 (2010), pp. 305-332.
[106] and , Complexity theory for operators in analysis, Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC '10), Association for Computing Machinery, New York, 2010, pp. 495-502.
[107] , Berechenbare Analysis, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 265-303.
[108] , Konstruktive Analysis, Deutscher Verlag der Wissenschaften, Berlin, 1961.
[109] , Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41-73.
[110] , On the interpretation of intuitionistic number theory, The Journal of Symbolic Logic, vol. 10 (1945), pp. 109-124.
[111] , Introduction to metamathematics, D. Van Nostrand Company, New York, 1952.
[112] and , The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam, 1965.
[113] , The maximum value problem and NP real numbers, Journal of Computer and Systems Sciences, vol. 24 (1982), pp. 15-35.
[114] , Some negative results on the computational complexity of total variation and differentiation, Information and Control, vol. 53 (1982), pp. 21-31.
[115] , On the computational complexity of ordinary differential equations, Information and Control, vol. 58 (1983), pp. 157-194.
[116] , On the computational complexity of integral equations, Annals of Pure and Applied Logic, vol. 58 (1992), pp. 201-228.
[117] and , Computational complexity of real functions, Theoretical Computer Science, vol. 20 (1982), pp. 323-352.
[118] , Computing power series in polynomial time, Advances in Applied Mathematics, vol. 9 (1988), pp. 40-50.
[119] , Applied proof theory: Proof interpretations and their use in mathematics, Springer, Berlin, 2008.
[120] , On the interpretation of non-finitist proofs II. Interpretation of number theory. Applications, The Journal of Symbolic Logic, vol. 17 (1952), pp. 43-58.
[121] , Some elementary inequalities, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 55 (1952), pp. 334-338, Indagationes Mathematicae, vol. 14.
[122] and , Ensembles récursivement mesurables et ensembles recursivement ouverts et fermés, Comptes Rendus de l'Académie des Sciences. Paris, vol. 245 (1957), pp. 1106-1109.
[123] , , and , Fonctionnelles récursivement définissables et fonctionnelles récursives, Comptes Rendus de l'Académie des Sciences. Paris, vol. 245 (1957), pp. 399-402.
[124] , Partial recursive functionals and effective operations, Constructivity in mathematics (, editor), North-Holland, Amsterdam, 1959, pp. 290-297.
[125] and , Theory of representations, Theoretical Computer Science, vol. 38 (1985), pp. 35-53.
[126] , Compactness in constructive analysis revisited, Annals of Pure and Applied Logic, vol. 36 (1987), pp. 29-38.
[127] , Typed lambda-calculus in classical Zermelo–Frænkel set theory, Archive for Mathematical Logic, vol. 40 (2001), pp. 189-205.
[128] , Grundzüge einer arithmetischen Theorie der algebraischen Grössen, Riemer, Berlin, 1882, also published in Journal für reine und angewandte Mathematik, vol. 92 (1882), pp. 1–122, and [132], vol. II, pp. 237–387.
[129] , Ein Fundamentalsatz der allgemeinen Arithmetik, Journal für die reine und angewandte Mathematik, vol. 100 (1887), pp. 490-510, reprinted in [132], vol. IIIa, pp. 209–240.
[130] , Über den Zahlbegriff, Philosophische Aufsätze, Eduard Zeller zu seinem fünfzigjährigen Doctorjubiläum gewidmet, Fues, Leipzig, 1887, pp. 261-274, reprinted in [132], vol. IIIa, pp. 249–274. Translated as On the concept of number by William Ewald in [58], vol. 2, pp. 947–955.
[131] , Vorlesungen über Zahlentheorie, (, editor), Teubner, Leipzig, 1901, republished by Springer, Berlin, 1978.
[132] , Werke, (, editor), vol. 1–5, Chelsea Publishing Company, New York, 1968.
[133] and , Demuth's path to randomness, Proceedings of the 2012 international Workshop on Theoretical Computer Science: Computation, physics and beyond (WTCS 2012), Springer, Berlin, 2012, pp. 159-173.
[134] , Lectures on constructive mathematical analysis, American Mathematical Society, Providence, RI, 1984, translated from the Russian by .
[135] , The constructive mathematics of A. A. Markov, American Mathematical Monthly, vol. 113 (2006), pp. 559-566.
[136] , Classes récursivement fermés et fonctions majorantes, Comptes Rendus de l'Académie des Sciences. Paris, vol. 240 (1955), pp. 716-718.
[137] , Extension de la notion de fonction réicursive aux fonctions d'une ou plusieurs variables réelles I–III, Comptes Rendus de l'Académie des Sciences. Paris, vol. 240 and 241 (1955), pp. 2478-2480 and 13–14 and 151–153.
[138] , Remarques sur les opérateurs récursifs et sur les fonctions récursives d'une variable réelle, Comptes Rendus de l'Académie des Sciences. Paris, vol. 241 (1955), pp. 1250-1252.
[139] , Les ensembles récursivement ouverts ou fermés, et leurs applications à l'Analyse récursive, Comptes Rendus de l'Academie des Sciences. Paris, vol. 245 (1957), pp. 1040-1043.
[140] , Quelques propriétés d'analyse récursive, >Comptes Rendus de l'Académie des Sciences. Paris, vol. 244 (1957), pp. 838–840 and 996-997.
[141] , Les ensembles récursivement ouverts ou férmes, et leurs applications à l'Analyse récursive, Comptes Rendus de l'Académie des Sciences. Paris, vol. 246 (1958), pp. 28-31.
[142] , Sur les possibilites d'extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles, Le raisonnement en mathematiques et en sciences, Editions du Centre National de la Recherche Scientifique, Paris, 1958, pp. 67-75.
[143] , Quelques procédés de definition en topologie recursive, Constructivity in mathematics (, editor), North-Holland, Amsterdam, 1959, pp. 129-158.
[144] , From constructive mathematics to computable analysis via the realizability interpretation, Ph.D. thesis, Fachbereich Mathematik, TU Darmstadt, Darmstadt, 2004.
[145] and , Sheaves in geometry and logic, Springer, New York, 1994.
[146] , On the continuity of constructive functions, Uspekhi Matematicheskikh Nauk, vol. 9 (1954), pp. 226-230, (in Russian).
[147] , On constructive functions, Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 52 (1958), pp. 315-348, (in Russian, English translation in American Mathematical Society Translations Series 2, vol. 29, 1963).
[148] , The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602-619.
[149] , Notes on constructive mathematics, Almqvist and Wiksell, Stockholm, 1970.
[150] , An intuitionistic theory of types: predicative part, Logic Colloquium '73 ( and , editors), North-Holland, Amsterdam, 1973.
[151] and , Computable analysis and Blaschke products, Proceedings of the American Mathematical Society, vol. 136 (2008), pp. 321-332.
[152] , Computable analysis, Razprawy Matematyczne, Warsaw, 1963.
[153] , Uniformly computable aspects of inner functions: estimation and factorization, Mathematical Logic Quarterly, vol. 54 (2008), pp. 508-518.
[154] , A uniformly computable implicit function theorem, Mathematical Logic Quarterly, vol. 54 (2008), pp. 272-279.
[155] and , The introduction of non-recursive methods into mathematics, The L. E. J. Brouwer Centenary Symposium ( and , editors), North-Holland, Amsterdam, 1982, pp. 319-335.
[156] , , and , Recursive limits on the Hahn–Banach theorem, Errett Bishop: Reflections on him and his research (, editor), American Mathematical Society, 1985.
[157] , Pi-0-1 classes in computable analysis and topology, Ph.D. thesis, Cornell University, Ithaca, USA, 2002.
[158] , Recursive metric spaces, Fundamenta Mathematicae, vol. 55 (1964), pp. 215-238.
[159] , On computable sequences, Fundamenta Mathematicae, vol. 44 (1957), pp. 37-51.
[160] , An evaluation semantics for classical proofs, Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, 1991, pp. 96-107.
[161] , A recursive function defined on a compact interval and having a continuous derivative that is not recursive, Michigan Mathematical Journal, vol. 18 (1971), pp. 97-98.
[162] , Recursive functions and intuitionistic number theory, Transactions of the American Mathematical Society, vol. 61 (1947), pp. 307-368.
[163] , Computability and randomness, Oxford University Press, New York, 2009.
[164] , A constructive mapping of the square onto itself displacing every constructive point, Doklady Akademii Nauk, vol. 152 (1963), pp. 55-58, (in Russian) Translated in: Soviet Mathematics Doklady, vol. 4 (1963), pp. 1253–1256.
[165] , λμ-calculus: an algorithmic interpretation of classical natural deduction, Logic programming and automated reasoning (St. Petersburg, 1992), Springer, Berlin, 1992.
[166] , A computational aspect of the lebesgue differentiation theorem, Journal of Logic and Analysis, vol. 1 (2009), pp. 1-15.
[167] , , and , Schnorr randomness and the Lebesgue differentiation theorem, Proceedings of the American Mathematical Society, (to appear).
[168] , How incomputable is finding Nash equilibria?, Journal of Universal Computer Science, vol. 16 (2010), pp. 2686-2710.
[169] , On the (semi)lattices induced by continuous reducibilities, Mathematical Logic Quarterly, vol. 56 (2010), pp. 488-502.
[170] , The emperor's new mind. concerning computers, minds and the laws of physics, Oxford University Press, New York, 1989.
[171] , Rekursive Funktionen, Akademischer Verlag, Budapest, 1951.
[172] and , On a simple definition of computable functions of a real variable, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 1-19.
[173] and , The wave equation with computable initial data whose unique solution is nowhere computable, Mathematical Logic Quarterly, vol. 43 (1997), pp. 499-509.
[174] and , A computable ordinary differential equation which possesses no computable solution, Annals of Mathematical Logic, vol. 17 (1979), pp. 61-90.
[175] and , The wave equation with computable initial data such that its unique solution is not computable, Advances in Mathematics, vol. 39 (1981), pp. 215-239.
[176] and , Noncomputability in analysis and physics: a complete determination of the class of noncomputable linear operators, Advances in Mathematics, vol. 48 (1983), pp. 44-74.
[177] and , Computability in analysis and physics, Springer, Berlin, 1989.
[178] and , The computational complexity ofsome Julia sets, Proceedings of the 35th Annual ACM Symposium on Theory of Computing (, editor), Association for Computing Machinery, New York, 2003, pp. 177-185.
[179] and , On the hierarchy and extension of monotonically computable real numbers, Journal of Complexity, vol. 19 (2003), pp. 672-691.
[180] and , A hierarchy of Turing degrees of divergence bounded computable real numbers, Journal of Complexity, vol. 22 (2006), pp. 818-826.
[181] , Recursive real numbers, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 784-791.
[182] , Review of “Peter, R., Rekursive Funktionen”, The Journal of Symbolic Logic, vol. 16 (1951), pp. 280-282.
[183] and , Singular coverings andnon-uniform notions of closed set computability, Mathematical Logic Quarterly, vol. 54 (2008), pp. 545-560.
[184] and , Principia mathematica, vol. 1, Cambridge University Press, 1910, vol. 2, 1912; vol. 3, 1913.
[185] , Algorithmic randomness, martingales, and differentiability, in preparation.
[186] , Komplexität von Algorithmen mit Anwendung auf die Analysis, Archiv für Mathematische Logik und Grundlagenforschung, vol. 14 (1971), pp. 54-68.
[187] , Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer, Berlin, 1971.
[188] , Numerik analytischer Funktionen und Komplexität, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 92 (1990), pp. 1-20.
[189] , Topological spaces allowing type 2 complexity theory, Computability and complexity in analysis ( and , editors), Fern Universität, Hagen, September 1995, pp. 41-53.
[190] , Fast online multiplication of real numbers, STACS 97 ( and , editors), Springer, Berlin, 1997, pp. 81-92.
[191] , Online computations of differentiable functions, Theoretical Computer Science, vol. 219 (1999), pp. 331-345.
[192] , Extended admissibility, Theoretical Computer Science, vol. 284 (2002), pp. 519-538.
[193] , Spaces allowing type-2 complexity theory revisited, Mathematical Logic Quarterly, vol. 50 (2004), pp. 443-459.
[194] , Admissible representations for probability measures, Mathematical Logic Quarterly, vol. 53 (2007), pp. 431-445.
[195] , Outline of a mathematical theory of computation, Technical monograph prg-2, Oxford University, Oxford, November 1970.
[196] , The pointwise ergodic theorem in subsystems of second-order arithmetic, The Journal of Symbolic Logic, vol. 72 (2007), pp. 45-66.
[197] , Subsystems of second order arithmetic, second ed., Cambridge University Press, Cambridge, 2009.
[198] , Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich, Skrifter, Norske Videnskaps-Akademi i Oslo, Matematisk-Naturvidenskapelig Klasse, vol. 6, J. Dybwad, Oslo, 1923, pp. 1-38, translated in [217], pp. 302–333.
[199] and , Lectures on the Curry–Howard isomorphism, Elsevier, Amsterdam, 2006.
[200] , Nicht konstruktiv beweisbare Sätze der Analysis, The Journal of Symbolic Logic, vol. 14 (1949), pp. 145-158.
[201] , Der Satz vom Maximum in der rekursiven Analysis, Constructivity in mathematics (, editor), North-Holland, Amsterdam, 1959, pp. 254-265.
[202] , Logos, Logic, and Logistiké, History and philosophy of moden mathematics ( and , editors), University of Minnesota, Minneapolis, 1988, pp. 238-259.
[203] , Intensional interpretations of functionals of finite type, I, The Journal of Symbolic Logic, vol. 32 (1967), pp. 198-212.
[204] , Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, Berlin, 1973.
[205] , Realizability, Handbook of proof theory (, editor), North-Holland, Amsterdam, 1998, pp. 407-473.
[206] and , Constructivism in mathematics: An introduction, North-Holland, Amsterdam, 1988.
[207] , On computable numbers, with an application to the “Entscheidungsproblem”, Proceedings of the London Mathematical Society, vol. 42 (1936), pp. 230-265.
[208] , On computable numbers, with an application to the “Entscheidungsproblem”. A correction, Proceedings of the London Mathematical Society, vol. 43 (1937), pp. 544-546.
[209] , Finite approximations to Lie groups, Annals of Mathematics, vol. 39 (1938), pp. 105-111.
[210] , Systems of logic based on ordinals, Proceedings of the London Mathematical Society. Series 2, vol. 45 (1939), pp. 161-228.
[211] , A method for the calculation of the zeta-function, Proceedings of the London Mathematical Society. Series 2, vol. 48 (1943), pp. 180-197.
[212] , Rounding-off errors in matrix processes, Quarterly Journal of Mechanics and Applied Mathematics, vol. 1 (1948), pp. 287-308.
[213] , Some calculations of the Riemann zeta-function, Proceedings of the London Mathematical Society. Series 3, vol. 3 (1953), pp. 99-117.
[214] and , Basic developments connected with the concept of algorithm and with its applications in mathematics, Algorithms in modern mathematics and computer science ( and , editors), Springer, Berlin, 1981.
[215] , On Brouwer, Wadsworth/Thomson Learning, Belmont, CA, 2004.
[216] , Luitzen Egbertus Jan Brouwer, The Stanford encyclopedia of philosophy (, editor), summer 2011 ed., 2011.
[217] , From Frege to Gödel: A sourcebook in mathematical logic, 1879–1931, Harvard University Press, Cambridge, 1967.
[218] , Die formalistische Grundlegung der Mathematik, Erkenntnis, vol. 2 (1931), 116-121, translated by and as The formalist foundations of mathematics (P. Benacerraf and H. Putnam, editors), Philosophy of Mathematics: Selected Readings, 2nd edition, Cambridge University Press, Cambridge, 1983, pp. 61-65.
[219] , Ergodic convergence in probability, and an ergodic theorem for individual random sequences, Teoriya Veroyatnostei i ee Primeneniya, vol. 42 (1997), pp. 35-50.
[220] , Ergodic theorems for individual random sequences, Theoretical Computer Science, vol. 207 (1998), pp. 343-361.
[221] , Computability, Springer, Berlin, 1987.
[222] , Computability on the probability measures on the Borel sets of the unit interval, Theoretical Computer Science, vol. 219 (1999), pp. 421-437.
[223] , Computable analysis, Springer, Berlin, 2000.
[224] , Computational complexity on computable metric spaces, Mathematical Logic Quarterly, vol. 49 (2003), pp. 3-21.
[225] and , Representations of the real numbers and of the open subsets of the set of real numbers, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 247-260.
[226] and , Is wave propagation computable or can wave computers beat the Turing machine?, Proceedings of the London Mathematical Society, vol. 85 (2002), pp. 312-332.
[227] , Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines, Theoretical Computer Science, vol. 332 (2005), pp. 337-366.
[228] , An algorithm for computing fundamental solutions, SIAM Journal on Computing, vol. 35 (2006), pp. 1283-1294.
[229] , Computing Schrödinger propagators on Type-2 Turing machines, Journal of Complexity, vol. 22 (2006), pp. 918-935.
[230] , Computable analysis of the abstract Cauchy problem in a Banach space and its applications I, Mathematical Logic Quarterly, vol. 53 (2007), pp. 511-531.
[231] , Time, communication, and the nervous system, Annals of the New York Academy of Sciences, vol. 50 (1948), pp. 197-220.
[232] (editor), Leibniz: Selections, Charles Scribner's Sons, New York, 1951.
[233] and , A computable version of the Daniell-Stone theorem on integration and linear functionals, Theoretical Computer Science, vol. 359 (2006), pp. 28-42.
[234] , , and , Effective properties of sets and functions in metric spaces with computability structure, Theoretical Computer Science, vol. 219 (1999), pp. 467-486.
[235] , Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 65-78.
[236] , Lebesgue convergence theorems and reverse mathematics, Mathematical Logic Quarterly, vol. 40 (1994), pp. 1-13.
[237] and , Measure theory and weak König's lemma, Archive for Mathematical Logic, vol. 30 (1990), pp. 171-180.
[238] , Disproof of some theorems of classical analysis in constructive analysis, Uspekhi Matematicheskikh Nauk, vol. 10 (1955), pp. 209-210, (in Russian).
[239] and , The arithmetical hierarchy of real numbers, Mathematical Logic Quarterly, vol. 47 (2001), pp. 51-65.
[240] and , Computability theory of generalized functions, Journal of the Association for Computing Machinery, vol. 50 (2003), pp. 469-505.
[241] , Computable real-valued functions on recursive open and closed subsets of Euclidean space, Mathematical Logic Quarterly, vol. 42 (1996), pp. 379-409.
[242] , Real hypercomputation and continuity, Theory of Computing Systems, vol. 41 (2007), pp. 177-206.