Mathematics, foundations of

References and further reading

	Ayer, A. (1959) Logical Positivism, New York: Free Press. (Includes influential expressions of the positivists’ conception of mathematics.)
	Barwise, J. (1977) Handbook of Mathematical Logic, Amsterdam: North Holland. (Useful collection of survey articles on many areas of mathematical logic and foundational studies.)
	Beeson, M.J. (1985) Foundations of Constructive Mathematics, Berlin: Springer. (Useful survey of recent logical and mathematical developments in constructive mathematics.)
	Benacerraf, P. (1965) ‘What Numbers Could Not Be’, Philosophical Review 74: 47–73. (Structuralist argument that arithmetic is not a science concerning particular objects – the numbers – but rather a science elaborating the structure that all arithmetical progressions share in common. Dedekind argued for such a view a century earlier.)
	Benacerraf, P. (1973) ‘Mathematical Truth’, Journal of Philosophy 70: 661–680; repr. in Benacerraf and Putnam (1964), 403–420. (Argues that existing philosophies of mathematics are subject to a dilemma – they cannot give a satisfactory account of both mathematical truth and mathematical knowledge.)
	Benacerraf, P. and Putnam, H. (1964) Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edn, 1983. (Collection of influential papers in the philosophy of mathematics.)
	Bernays, P. (1950) ‘Mathematische Existenz und Widerspruchsfreiheit’, repr. in Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. (A kind of structuralist argument.)
	Bishop, E. (1967) Foundations of Constructive Analysis, New York: McGraw-Hill. (Influential development of analysis from a constructive vantage somewhat more restrictive than that of intuitionism.)
	Bolyai, J. (1832) ‘Science of Absolute Space’, trans. G.B. Halsted, in R. Bonola (ed.) Non-Euclidean Geometry: A Critical and Historical Survey of its Development, New York: Dover, 1955. (Celebrated essay including the first published description of a consistent non-Euclidean geometry.)
	Bridges, D. (1987) Varieties of Constructive Mathematics, Cambridge: Cambridge University Press. (Mathematical survey of recent developments in various constructivist approaches to mathematics. Requires some background and some knowledge of classical mathematics.)
	Brouwer, L.E.J. (1905) Leven, Kunst, en Mystiek, Delft; excerpts trans. ‘Life, Art and Mysticism’, in Collected Works, vol. 1, ed. A. Heyting, Amsterdam: North Holland, 1975; unabridged trans. W. van Stigt, Notre Dame Journal of Formal Logic 37 (3), 1996. (Early statement of various of Brouwer’s philosophical views.)
	Brouwer, L.E.J. (1907) Over de Grondslagen der Wiskunde, Amsterdam and Leipzig; trans. ‘On the Foundations of Mathematics’, in Collected Works, vol. 1, ed. A. Heyting, Amsterdam: North Holland, 1975. (Brouwer’s doctoral dissertation. First sustained presentation of his intuitionist philosophy.)
	Brouwer, L.E.J. (1913) ‘Intuitionism and Formalism’, Bulletin of the American Mathematical Society 20: 81–96; repr. in Benacerraf and Putnam (1964), 77–89. (Attempt by Brouwer to distinguish his intuitionist position from a position he labels ‘formalism’. Includes an early statement of his antipathy to logical reasoning.)
	Brouwer, L.E.J. (1955) ‘The Effect of Intuitionism on Classical Algebra of Logic’, Proceedings of the Royal Irish Academy §A 57: 113–116; repr. in Collected Works, vol. 1, ed. A. Heyting, Amsterdam: North Holland. (Perhaps the best and clearest statement of his view separating mathematical from logical reasoning, including intuitionistically valid logical reasoning.)
	Brouwer, L.E.J. (1981) Brouwer’s Cambridge Lectures on Intuitionism, ed. D. van Dalen, Cambridge: Cambridge University Press. (Late statement of Brouwer’s intuitionist views, taken from lectures given 1946–51.)
	Carnap, R. (1931) ‘Die logizistische Grundlegung der Mathematik’, Erkenntnis 2: 91–105; trans. E. Putnam and G. Massey, ‘The Logicist Foundations of Mathematics’, in Benacerraf and Putnam (1964), 41–52. (Statement of the type of logicist view that became popular among the logical empiricists of the 1930s and 1940s.)
	Chihara, C. (1990) Constructibility and Mathematical Existence, Oxford: Oxford University Press. (Argues that classical mathematics does not require commitment either to such linguistic objects as open sentences or to abstract objects (concepts) to which they might be taken to refer.)
	Dedekind, R. (1888) Was sind und was sollen die Zahlen?, Braunschweig: Vieweg; trans. W.W. Beman (1901), ‘The Nature and Meaning of Numbers’, in Essays on the Theory of Numbers, New York: Dover, 1963. (§73 includes an interesting statement of Dedekind’s philosophical conception of the axiomatic method and its relationship to the older genetic method.)
	Detlefsen, M. (1986) Hilbert’s Program: An Essay on Mathematical Instrumentalism, Boston, MA, and Dordrecht: Reidel. (Examination of the arguments of Hilbert’s programme using Gödel’s incompleteness theorems, especially the second.)
	Detlefsen, M. (1990a) ‘On an Alleged Refutation of Hilbert’s Program using Gödel’s First Incompleteness Theorem’, Journal of Philosophical Logic 19: 343–377. (Critical evaluation of proposed arguments against Hilbert’s programme using Gödel’s first incompleteness theorem.)
	Detlefsen, M. (1990b) ‘Brouwerian Intuitionism’, Mind 99: 501–534. (Investigation of Brouwer’s view of logical reasoning and the distinction that he and Poincaré believed to exist between mathematical and logical reasoning.)
	Detlefsen, M. (1996) ‘Philosophy of Mathematics in the 20th Century’, in Routledge History of Philosophy, vol. 9, Philosophy of Science, Logic and Mathematics in the 20th Century, London: Routledge. (Survey of the main currents of foundational thinking in the twentieth century.)
	Dummett, M.A.E. (1973) ‘The Philosophical Basis of Intuitionist Logic’, in H.E. Rose and J.C. Shepherdson (eds) Proceedings of the Logic Colloquium, Bristol, July 1973, Amsterdam: North Holland, 1975, 5–40; repr. in Truth and Other Enigmas, London: Duckworth, 1978, 215–247; and in Benacerraf and Putnam (1964), 97–129. (Argues that an adequate account of the meanings of mathematical propositions reveals the logic of mathematics to be intuitionistic logic. The view of meaning proposed is like that of the later Wittgenstein in that it equates the meaning of a proposition with its canonical usage; takes the canonical usage of a sentence in mathematics to consist in the role it plays in giving proofs.)
	Dummett, M.A.E. (1977) Elements of Intuitionism, Oxford: Clarendon Press, 1990. (Further development of the argument set out in Dummett (1973). Also gives a basic presentation of various notions of intuitionistic mathematics (for example, choice sequences and spreads) and intuitionistic logic and its metatheory.)
	Feferman, S. (1988) ‘Hilbert’s Program Relativized: Proof-Theoretical and Foundational Reductions’, Journal of Symbolic Logic 53: 364–384. (Argues that finitary methods are not the only epistemically gainful methods for proving consistency and that significant partial realizations of Hilbert’s programme can be realized by exploiting this fact.)
	Field, H. (1980) Science Without Numbers, Princeton, NJ: Princeton University Press. (Defence of the view that mathematical knowledge is, at least in large part, logical knowledge. Also argues, contra Quine and Putnam, that realism with respect to physics does not entail realism with respect to mathematics.)
	Field, H. (1984) ‘Is Mathematical Knowledge Just Logical Knowledge?’, Philosophical Review 93: 509–552. (Argues that mathematical knowledge is predominantly logical in character.)
	Frege, G. (1873) ‘Über eine geometrische Darstellung der imaginären Gebilde in der Ebene’, doctoral dissertation, University of Göttingen; repr. in I. Angelleli (ed.) Kleine Schriften, Hildesheim, 1967; partial trans. in Collected Papers on Mathematics, Logic and Philosophy, ed. B. McGuinness, Oxford: Blackwell, 1984. (An attempt to vindicate the ultimately Kantian view that, at bottom, geometric laws – even those concerning imaginary points, lines, and so on – are all based on intuition, by showing how to replace imaginary items with real, intuitable items.)
	Frege, G. (1874) ‘Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen’, Habilitationsschrift, University of Jena; trans. in Collected Papers on Mathematics, Logic and Philosophy, ed. B. McGuinness, Oxford: Blackwell, 1984. (Includes a statement of the basic precept of Frege’s logicism; namely, that there can be no intuition of so pervasive and abstract a concept as that of magnitude or number.)
	Frege, G. (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: Pohle; trans. ‘ Begriffsschrift, a Formula Language, Modelled Upon That of Arithmetic, for Pure Thought’, in J. van Heijenoort (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967, 1–82. (Presentation of Frege’s symbolic scheme for logic and mathematics. He hoped that its employment would foster rigour in the conduct of proof and thereby allow for a clearer view of the basic laws upon which arithmetic is founded.)
	Frege, G. (1884) Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau: Koebner; trans. J.L. Austin, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Oxford: Blackwell, and Evanston, IL: Northwestern University Press, 2nd edn, 1980. (First full-length presentation of Frege’s logicist view of arithmetic; focuses on its philosophical basis.)
	Frege, G. (1891) Funktion und Begriff, Jena: Pohle; trans. P.T. Geach, ‘Function and Concept’, in Collected Papers on Mathematics, Logic and Philosophy, ed. B. McGuinness, Oxford: Blackwell, 1984. (Basic presentation of Frege’s functional conception of propositions.)
	Gauss, K. (1817), in Briefwechsel mit H.W.M. Olbers, Hildesheim: Olms, 1976; letter to Bessel (1929), in Werke, Leipzig: Teubner, 1863–1903, vol. 8, 200. (Includes statements of Gauss’ view concerning the ‘external’ character of geometry versus the ‘internal’ character of arithmetic.)
	Gillies, D. (1982) Frege, Dedekind and Peano on the Foundations of Arithmetic, Assen: Van Gorcum. (Useful discussion of some of the similarities and differences between Frege, Dedekind and Peano.)
	Gödel, K. (1947) ‘What is Cantor’s Continuum Problem?’, American Mathematical Monthly 54: 515–525; revised version repr. in Benacerraf and Putnam (1964). (Basic expression of Gödel’s realist and non-empiricist views of mathematics.)
	Gödel, K. (1951) ‘Some Basic Theorems on the Foundations of Mathematics and Their Implications’, in Collected Works, vol. 3, ed. S. Feferman, J.W. Dawson Jr, W. Goldfarb, C.D. Parsons and R.M. Solovay, New York and Oxford: Oxford University Press, 1995. (Edited text of Gödel’s 1951 Josiah Gibbs Lecture. Argues for a Platonist conception and against a conventionalist conception of mathematics. Also argues for a non-mechanist view of the human mind. Useful introductory essay by George Boolos.)
	Hahn, H. (1933) Einheitswissenschaft, vol. 2, Logik, Mathematik und Naturerkennen, ed. R. Carnap and H. Hahn; trans. ‘Logic, Mathematics and Knowledge of Nature’, in A. Ayer (ed.) Logical Positivism, New York: Free Press, 1959. (Influential statement of the type of logicist view of mathematics popular among the positivists.)
	Heijenoort, J. van (1967) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press. (Collection of basic papers in mathematical logic and the foundations of mathematics. Useful forewords and bibliography.)
	Hellman, G. (1989) Mathematics Without Numbers, Oxford: Oxford University Press. (Detailed development of a non-Platonist account of mathematics.)
	Heyting, A. (1930) ‘Die formale Regeln der intuitionistischen Logik’ (The Formal Rules of Intuitionistic Logic), Sitzungsberichte der preußischen Akademie von Wissenschaften, physikalisch-mathematische Klasse 42–56. (First symbolic formalization of intuitionist logic.)
	Heyting, A. (1956) Intuitionism: An Introduction, Amsterdam: North Holland; 3rd revised edn, 1971. (Introduction to philosophical considerations concerning intuitionism and a survey of intuitionist formalizations of logic and various parts of mathematics.)
	Hilbert, D. (1900) ‘Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris 1900’, Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse: 253–97; trans. ‘Mathematical Problems’, Bulletin of the American Mathematical Society 8: 437–479, 1902. (Interesting remarks concerning the nature of mathematical truth and the axiomatic method.)
	Hilbert, D. (1926) ‘Über das Unendliche’, Mathematische Annalen 95: 161–190; trans. ‘On the Infinite’, in J. van Heijenoort (ed.) From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press, 1967. (Mature statement of basic philosophical precepts of Hilbert’s programme. Emphasizes the relationship with Kant’s general critical epistemology.)
	Hilbert, D. (1928) ‘Die Grundlagen der Mathematik’, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6: 65–85; trans. ‘The Foundations of Mathematics’, in J. van Heijenoort (ed.) From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967. (Repeats some of the material covered in Hilbert (1926); also includes an important clarificatory remark concerning the character of Hilbert’s ‘formalism’ (1967: 475).)
	Hilbert, D. (1930) ‘Naturerkennen und Logik’, Die Naturwissenschaften 18: 959–963; repr. in Gesammelte Abhandlungen, vol. 3, Berlin: Springer, 1935. (Perhaps the fullest statement of Hilbert’s philosophical ideas; develops his view of the a priori and refines his ‘formalism’.)
	Hilbert, D. (1935) Gesammelte Abhandlungen, vol. 3, Berlin: Springer; repr. New York: Chelsea, 1965. (Collection of some of the most important papers in Hilbert’s foundational corpus.)
	Hodes, H. (1984) ‘Logicism and the Ontological Commitments of Arithmetic’, Journal of Philosophy 81: 123–149. (Argues that arithmetical claims can be translated into a second-order logic in which the second-order variables range over functions and concepts.)
	Hodges, W. (1993) Model Theory, Cambridge: Cambridge University Press. (Thorough and up-to-date survey of model theory. Requires some background in logic.)
	Kant, I. (1781/1787) Critique of Pure Reason, trans. N. Kemp Smith, London: Macmillan, 1929. (Includes the classic statement of Kant’s views in the philosophy of mathematics. The ‘A’ and ‘B’ prefixes to sources refer to pages of the 1781 and 1787 German editions respectively.)
	Kant, I. (1783) Prolegomena to any Future Metaphysics that Shall Come Forth as Scientific, trans. L. White Beck, Indianapolis, IN, and New York: Bobbs-Merrill, 1950. (Includes an abbreviated statement of the views on the nature of mathematics given in Kant (1781/1787).)
	Kitcher, P. (1983) The Nature of Mathematical Knowledge, Oxford: Oxford University Press. (Develops an empiricist conception of mathematics which emphasizes its similarities to natural science.)
	Kreisel, G. (1958) ‘Hilbert’s Programme’, Dialectica 12: 346–372; revised version in Benacerraf and Putnam (1964). (Discussion of the possible significance of a Hilbert-type programme in light of Gödel’s theorems and other more recent work in proof theory.)
	Lobachevskii, N.I. (1829) ‘Geometrische Untersuchungen zur Theorie der Parallellinien’, Berlin: Fincke, 1840; trans. G.B. Halsted, ‘Geometrical Researches in the Theory of Parallels’, in R. Bonola (ed.) Non-Euclidean Geometry: A Critical and Historical Survey of its Development, New York: Dover, 1955. (Elaboration of Lobachevskii’s famous work in non-Euclidean geometry.)
	Macintyre, A. (1977) ‘Model Completeness’, in J. Barwise (ed.) Handbook of Mathematical Logic, Amsterdam: North Holland. (Survey of basic results and methods linking Robinson’s notion of model completeness to various questions in algebra.)
	Maddy, P. (1980) ‘Perception and Mathematical Intuition’, Philosophical Review 84: 163–196. (Argues that we can perceive certain kinds of sets.)
	Maddy, P. (1990) Realism in Mathematics, Oxford: Oxford University Press. (Attempt to show how realism in mathematics can be combined with a naturalistic mathematical epistemology.)
	Parsons, C. (1980) ‘Mathematical Intuition’, Proceedings of the Aristotelian Society 80: 145–168. (Criticism of Quine in which it is argued that the Quinean view cannot account for an important difference in the kind of evidentness displayed by elementary mathematical truths on the one hand, and hypotheses of theoretical physics on the other.)
	Parsons, C. (1983) ‘Quine on the Philosophy of Mathematics’, in Mathematics in Philosophy, Ithaca, NY: Cornell University Press. (Critical examination of Quine’s philosophical ideas on mathematics.)
	Parsons, C. (1990) ‘The Structuralist View of Mathematical Objects’, Synthese 84: 303–346. (Argues that structuralism does not offer an adequate account of certain very elementary mathematical objects, such as geometric figures.)
	Peano, G. (1889) Arithmetices principia, nova methodo exposita, Turin: Bocca; partial trans. ‘The Principles of Arithmetic’, in J. van Heijenoort (1967), 83–97. (Early attempt by Peano to produce a symbolic formulation of arithmetic.)
	Poincaré, H. (1902) La Science et l’hypothèse, Paris: Flammarion; trans. in G.B. Halsted (ed.) The Foundations of Science, Lancaster, PA: The Science Press, 1946. (Chapter 1 presents Poincaré’s basically Kantian conception of mathematical reasoning.)
	Poincaré, H. (1905) Le valeur de la science, Paris: Flammarion; trans. The Value of Science, New York: Dover, 1958; and trans. in G.B. Halsted (ed.) The Foundations of Science, Lancaster, PA: The Science Press, 1946. (Part 1 presents Poincaré’s basically intuitionist view of arithmetic, his so-called conventionalist view of geometry and his Kantian conception of mathematical inference or reasoning.)
	Poincaré, H. (1908) Science et méthode, Paris: Flammarion; trans. in G.B. Halsted (ed.) The Foundations of Science, Lancaster, PA: The Science Press, 1946. (Book 2 gives a statement of Poincaré’s view of logic and its place in mathematics.)
	Putnam, H. (1967) ‘Mathematics Without Foundations’, Journal of Philosophy 64: 5–22. (Argues that existence claims can be seen as asserting the possible (as opposed to the actual) existence of structures and, therefore, at bottom, can be treated as logical claims.)
	Putnam, H. (1971) Philosophy of Logic, New York: Harper. (Presentation of a basically empiricist, pragmatist view of logic and mathematics.)
	Putnam, H. (1975) ‘What is Mathematical Truth?’, in Mathematics, Matter and Method: Philosophical Papers, vol. 1, Cambridge: Cambridge University Press. (Defends a realist, though not immaterialist, view of mathematics. Once again stresses the basic motif that mathematics and the natural sciences are not qualitatively different.)
	Putnam, H. (1980) ‘Models and Reality’, Journal of Symbolic Logic 45 (3): 464–482; repr. in Benacerraf and Putnam (1964), 2nd edn, 421–444. (Argues against even moderate forms of metaphysical realism in mathematics and defends a verificationist view of meaning in mathematics.)
	Quine, W.V. (1948) ‘On What There Is’, Review of Metaphysics 2: 21–38. (Argument for mathematical realism.)
	Quine, W.V. (1951) ‘Two Dogmas of Empiricism’, repr. in From a Logical Point of View, Cambridge, MA: Harvard University Press, 1953; repr. New York: Harper & Row, 1963. (Influential attack on the analytic/synthetic distinction and empiricist reductionism of the logical empiricists.)
	Quine, W.V. (1954) ‘Carnap and Logical Truth’, repr. in Benacerraf and Putnam (1964). (Criticism of Carnap’s conception of logical truth.)
	Resnik, M. (1981) ‘Mathematics as a Science of Patterns: Ontology and Reference’, Noûs 15: 529–550. (Formulates and defends a structuralist conception of meaning in mathematics.)
	Resnik, M. (1982) ‘Mathematics as a Science of Patterns: Epistemology’, Noûs 16: 95–105. (Formulates a structuralist view of mathematical knowledge.)
	Robinson, A. (1979) Selected Papers of Abraham Robinson, vol. 1, Model Theory and Algebra, ed. H.J. Keisler et al., Amsterdam: North Holland. (Collection of basic papers of the Robinson school applying model theory to algebra. See especially ‘On the Application of Symbolic Logic to Algebra’, the paper in which Robinson presented his ideas for applying logic to the resolution of problems in algebra.)
	Russell, B.A.W. (1903) The Principles of Mathematics, Cambridge: Cambridge University Press; 2nd edn, London: Allen & Unwin, 1937; repr. London: Routledge, 1992. (Russell’s first full-length development of his logicist views.)
	Russell, B.A.W. (1906) ‘Les paradoxes de la logique’, Revue de métaphysique et de morale 14: 627–650; trans. ‘On ‘‘Insolubilia’’ and their Solution by Symbolic Logic’, in Essays in Analysis, ed. D. Lackey, London: Allen & Unwin, 1973. (Reply to Poincaré’s views concerning the relationship between logic and mathematics. Also includes an early statement of the hypothetico-deductive conception of the justification of the basic laws of mathematics.)
	Russell, B.A.W. (1907) ‘The Regressive Method of Discovering the Premises of Mathematics’, in Essays in Analysis, ed. D. Lackey, London: Allen & Unwin, 1973. (Useful statement of the methodological view underlying Russell’s logicism.)
	Russell, B.A.W. (1919) Introduction to Mathematical Philosophy, London: Allen & Unwin; repr. London: Routledge. (Useful statement of Russell’s post-Principia views on logic and mathematics. Discusses some of the weaknesses of his logicist position.)
	Russell, B.A.W. (1973) Essays in Analysis, ed. D. Lackey, London: Allen & Unwin. (Useful collection of Russell’s writings, some not previously published.)
	Shapiro, S. (1983) ‘Mathematics and Reality’, Philosophy of Science 50: 523–548. (Statement and defence of a structuralist view of mathematics.)
	Sieg, W. (1988) ‘Hilbert’s Program Sixty Years Later’, Journal of Symbolic Logic 53: 338–348. (Discussion of the progress and changes in Hilbert’s programme since its formulation in the 1920s.)
	Simpson, S. (1988) ‘Partial Realization of Hilbert’s Program’, Journal of Symbolic Logic 53: 349–363. (Useful survey and basic exposition of the work of the so-called ‘reverse mathematics’ programme begun by H. Friedman. Argues that though Hilbert’s programme in its original form is refuted by G2, a significant portion of it can still be carried out.)
	Troelstra, A. (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Berlin, Heidelberg and New York: Springer. (Metamathematical study of various intuitionist systems.)
	Troelstra, A. (1977) ‘Proof Theory and Constructive Mathematics’, in J. Barwise (ed.) Handbook of Mathematical Logic, Amsterdam: North Holland. (Survey of proof-theoretic work on intuitionist systems.)
	Weyl, H. (1927) ‘Philosophie der Mathematik und Naturwissenschaft’, in Handbuch der Philosophie, Munich: Oldenbourg; revised and expanded trans. Philosophy of Mathematics and Natural Science, Princeton, NJ: Princeton University Press, 1949. (Valuable collection of essays in the philosophy of mathematics by one of the twentieth century’s pre-eminent scientific thinkers.)
	Whitehead, A.N. and Russell, B.A.W. (1910) Principia Mathematica, vol. 1, Cambridge: Cambridge University Press, 2nd edn, 1925. (The classic symbolic formalization of logic and mathematics; a basis of much of the greatest work in mathematical logic and the foundations of mathematics in the twentieth century.)
	Wittgenstein, L.J.J. (1939) Wittgenstein’s Lectures on the Foundations of Mathematics: Cambridge 1939, ed. C. Diamond, Chicago, IL: University of Chicago Press, 1976. (A very different view from the logicism of the Tractatus.)
	Wittgenstein, L.J.J. (1956) Remarks on the Foundations of Mathematics, ed. G.H. von Wright, R. Rhees and G.E.M. Anscombe, trans. G.E.M. Anscombe, Oxford: Blackwell, and Cambridge, MA: Harvard University Press, 2nd edn, 1967, part 1, appendix 2, 54–63; 3rd edn, 1978. (Presents statements of the later Wittgenstein’s view that a proof of a mathematical theorem functions to logically exclude doubt through its establishment of a theorem as a norm governing a language game.)