# (p.263) References

# (p.263) References

Bibliography references:

Aspray, W., and P. Kitcher [1988], *History and philosophy of modern mathematics*, *Minnesota studies in the philosophy of science* 11, Minneapolis, Minnesota University Press.

Azzouni, J. [1994], *Metaphysical myths, mathematical practice*, Cambridge, Cambridge University Press.

Balaguer, M. [1994], “Against (Maddian) naturalized epistemology,” *Philosophia Mathematica* (3) 2: 97–108.

Balaguer, M. [1995], “A Platonist epistemology,” *Synthese* 103: 303–325.

Barbut, M. [1970], “On the meaning of the word ‘structure’ in mathematics,” in *Introduction to structuralism*, ed. by M. Lane, New York, Basic Books, 367–388.

Barwise, J. [1985], “Model‐theoretic logics: Background and aims,” in *Model‐theoretic logics*, ed. by J. Barwise and S. Feferman, New York, Springer‐Verlag, 3–23.

Benacerraf, P. [1965], “What numbers could not be,” *Philosophical Review* 74: 47–73; reprinted in Benacerraf and Putnam [1983], 272–294.

Benacerraf, P. [1973], “Mathematical truth,” *Journal of Philosophy* 70: 661–679; reprinted in Benacerraf and Putnam [1983], 403–420.

Benacerraf, P., and H. Putnam [1983], *Philosophy of mathematics*, second edition, Cambridge, Cambridge University Press.

Bernays, P. [1935], “Sur le platonisme dans les mathématiques,” *L'enseignement mathématique* 34: 52–69; tr. as “Platonism in mathematics,” in Benacerraf and Putnam [1983], 258–271.

Bernays, P. [1961], “Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre,” in *Essays on the foundation of mathematics*, ed. by Y. Bar‐Hillel et al., Jerusalem, Magnes Press, 3–49.

Bernays, P. [1967], “Hilbert, David,” in *The encyclopedia of philosophy*, vol. 3. ed. By P. Edwards, New York, Macmillan and The Free Press, 496–504.

Beth, E. W., and J. Piaget [1966], *Mathematical epistemology and psychology*, Dordrecht, Holland, D. Reidel.

Bishop, E. [1967], *Foundations of constructive analysis*, New York, McGraw‐Hill.

Bishop, E. [1975], “The crises in contemporary mathematics,” *Historia Mathematica* 2: 505–517.

(p.264)
Blackburn, S. [1984], *Spreading the word*, Oxford, Clarendon Press.

Block, N. (ed.) [1980], *Readings in philosophy of psychology* 1, Cambridge, Harvard University Press.

Boolos, G. [1984], “To be is to be a value of a variable (or to be some values of some variables),” *Journal of Philosophy* 81: 430–449.

Boolos, G. [1985], “Nominalist Platonism,” *Philosophical Review* 94: 327–344.

Boolos, G. [1987], “The consistency of Frege's *Foundations of arithmetic*,” in *On being and saying: Essays for Richard Cartwright*, ed. by Judith Jarvis Thompson, Cambridge, MIT Press, 3–20.

Bourbaki, N. [1949], “Foundations of mathematics for the working mathematician,” *Journal of Symbolic Logic* 14: 1–8.

Bourbaki, N. [1950], “The architecture of mathematics,” *American Mathematical Monthly* 57: 221–232.

Bourbaki, N. [1968], *Theory of sets*, Paris, Hermann.

Brouwer, L. E. J. [1948], “Consciousness, philosophy and mathematics,” in Benacerraf and Putnam [1983], 90–96.

Buck, R. [1956], *Advanced calculus*, New York, McGraw‐Hill.

Burgess, J. [1983], “Why I am not a nominalist,” *Notre Dame Journal of Formal Logic* 24: 93–105.

Burgess, J. [1984], “Synthetic mechanics,” *Journal of Philosophical Logic* 13: 379–395.

Burgess, J. [1992], “Proofs about proofs: A defense of classical logic,” in *Proof, logic and formalization*, ed. by Michael Detlefsen, London, Routledge, 8–23.

Cantor, G. [1899], “Letter to Dedekind,” van Heijenoort [1967], 113–117.

Cantor, G. [1932], *Gesammelte Abhandlungen mathematischen und philosophischen Inhalts*, ed. by E. Zermelo, Berlin, Springer.

Carnap, R. [1931], “Die logizistische Grundlegung der Mathematik,” *Erkenntnis* 2: 91–105; tr. as “The logicist foundations of mathematics,” in Benacerraf and Putnam [1983], 41–52.

Carnap, R. [1934], *Logische Syntax der Sprache*, Vienna, Springer; tr. as *The logical syntax of language*, New York, Harcourt, 1937.

Carnap R. [1942], *Introduction to semantics*, Cambridge, Harvard University Press.

Carnap, R. [1950], “Empiricism, semantics, and ontology,” *Revue Internationale de Philosophie* 4: 20–40; reprinted in Benacerraf and Putnam [1983], 241–257.

Cartwright, N. [1983], *How the laws of physics lie*, Oxford, Oxford University Press.

Chihara, C. [1973], *Ontology and the vicious‐circle principle*, Ithaca, New York, Cornell University Press.

Chihara, C. [1984], “A simple type theory without Platonic domains,” *Journal of Philosophical Logic* 13: 249–283.

Chihara, C. [1990], *Constructibility and mathematical existence*, Oxford, Oxford University Press.

Chihara, C. [1993], “Modality without worlds,” in *Philosophy of mathematics: Proceedings of the fifteenth international Wittgenstein symposium* 1, ed. by J. Czermak, Vienna, Verlag‐ Hölder‐Pichler‐Tempsky, 253–268.

Coffa, A. [1986], “From geometry to tolerance: Sources of conventionalism in nineteenth‐ century geometry,” in *From quarks to quasars*: *Philosophical problems of modern physics*, University of Pittsburgh series, vol. 7, Pittsburgh, Pittsburgh University Press, 3–70.

Coffa, A. [1991], *The semantic tradition from Kant to Carnap*, Cambridge, Cambridge University Press.

Cooper, R. [1984], “Early number development: Discovering number space with addition and subtraction,” in *Origins of cognitive skills*, ed. by C. Sophian, Hillsdale, New Jersey, Erlbaum, 157–192.

(p.265)
Corry, L. [1992], “Nicolas Bourbaki and the concept of mathematical structure,” *Synthese* 92: 315–348.

Crossley, J., C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams [1972], *What is mathematical logic?* New York, Oxford University Press.

Crowell, R., and R. Fox [1963], *Introduction to knot theory*, Boston, Ginn.

Curry, H. [1950], *A theory of formal deducibility*, Notre Dame, Edwards Brothers.

Davidson, D. [1974], “On the very idea of a conceptual scheme,” *Proceedings and Addresses of the American Philosophical Association* 47: 5–20.

Dedekind, R. [1872], *Stetigkeit und irrationale Zahlen*, Brunswick, Vieweg;

tr. as *Continuity and irrational numbers*, in *Essays on the theory of numbers*, ed. by W. W. Beman, New York, Dover, 1963, 1–27.

Dedekind, R. [1888], *Was sind und was sollen die Zahlen?* Brunswick, Vieweg;

tr. as *The nature and meaning of numbers*, in *Essays on the theory of numbers*, ed. by W. W. Beman, New York, Dover, 1963, 31–115.

Dedekind, R. [1932], *Gesammelte mathematische Werke* 3, ed. by R. Fricke, E. Noether, and O. Ore, Brunswick, Vieweg.

Demopoulos, W. [1994], “Frege, Hilbert, and the conceptual structure of model theory,” *History and Philosophy of Logic* 15: 211–225.

Dieterle, J. [1994], *Structure and object*, Ph.D. dissertation, The Ohio State University.

Dummett, M. [1973], “The philosophical basis of intuitionistic logic,” in *Truth and other enigmas*, by M. Dummett, Cambridge, Harvard University Press, 1978, 215–247; reprinted in Benacerraf and Putnam [1983], 97–129.

Dummett, M. [1973a], *Frege*: *Philosophy of language*, New York, Harper and Row.

Dummett, M. [1977], *Elements of intuitionism*, Oxford, Oxford University Press.

Dummett, M. [1981], *The interpretation of Frege's philosophy*, Cambridge, Harvard University Press.

Dummett, M. [1991], *Frege: Philosophy of mathematics*, Cambridge, Harvard University Press.

Dummett, M. [1991a], *The logical basis of metaphysics*, Cambridge, Harvard University Press.

Edwards, H. [1988], “Kronecker's place in history,” in Aspray and Kitcher [1988], 139–144.

Etchemendy, J. [1988], “Tarski on truth and logical consequence,” *Journal of Symbolic Logic* 53: 51–79.

Feynman, R. [1967], *The character of physical law*, Cambridge, MIT Press.

Field, H. [1980], *Science without numbers*, Princeton, Princeton University Press.

Field, H. [1984], “Is mathematical knowledge just logical knowledge?” *Philosophical Review* 93: 509–552; reprinted (with added appendix) in Field [1989], 79–124.

Field, H. [1985], “On conservativeness and incompleteness,” *Journal of Philosophy* 82: 239–260; reprinted in Field [1989], 125–146.

Field, H. [1989], *Realism, mathematics and modality*, Oxford, Blackwell.

Field, H. [1991], “Metalogic and modality,” *Philosophical Studies* 62: 1–22.

Fine, A. [1986], *The shaky game: Einstein, realism and the quantum theory*, Chicago, University of Chicago Press.

Frege, G. [1879], *Begriffsschrift*, *eine der arithmetischen nachgebildete Formelsprache des reinen Denkens*, Halle, Louis Nebert; tr. in van Heijenoort [1967], 1–82.

Frege, G. [1884], *Die Grundlagen der Arithmetik*, Breslau, Koebner; *The foundations of arithmetic*, tr. by J. Austin, second edition, New York, Harper, 1960.

Frege, G. [1903], *Grundgesetze der Arithmetik* 2, Olms, Hildescheim.

Frege, G. [1903a], “Über die Grundlagen der Geometrie,” *Jahresbericht der Mathematiker‐ Vereinigung* 12: 319–324, 368–375.

(p.266)
Frege, G. [1906], “Über die Grundlagen der Geometrie,” *Jahresbericht der Mathematiker‐ Vereinigung* 15: 293–309, 377–403, 423–430.

Frege, G. [1967], *Kleine Schriften*, Darmstadt, Wissenschaftlicher Buchgesellschaft (with I. Angelelli).

Frege, G. [1971], *On the foundations of geometry and formal theories of arithmetic*, tr. by Eikee‐Henner W. Kluge, New Haven, Yale University Press.

Frege, G. [1976], *Wissenschaftlicher Briefwechsel*, ed. by G. Gabriel, H. Hermes, F. Kambartel, and C. Thiel, Hamburg, Felix Meiner.

Frege, G. [1980], *Philosophical and mathematical correspondence*, Oxford, Blackwell.

Freudenthal, H. [1962], “The main trends in the foundations of geometry in the 19th century,” in *Logic, methodology and philosophy of science*, Proceedings of the 1960 Congress, ed. by E. Nagel, P. Suppes, A. Tarski, Stanford, Stanford University Press, 613–621.

Friedman, M. [1983], *Foundations of space‐time theories: Relativistic physics and philosophy of science*, Princeton, Princeton University Press.

Friedman, M. [1988], “Logical truth and analyticity in Carnap's *Logical syntax of language*,” in Aspray and Kitcher [1988], 82–94.

Gandy, R. [1988], “The confluence of ideas in 1936,” in R. Herken, ed., *The universal Turing machine*, New York, Oxford University Press, 55–111.

Geach, P. [1967], “Identity,” *Review of Metaphysics* 21: 3–12.

Geach, P. [1968], *Reference and generality*, Ithaca, New York, Cornell University Press.

Geroch, R. [1985], *Mathematical physics*, Chicago, University of Chicago Press.

Gödel, K. [1944], “Russell's mathematical logic,” in Benacerraf and Putnam [1983], 447–469.

Gödel, K. [1951], “Some basic theorems on the foundations of mathematics and their implications,” in *Collected Works* 3, Oxford, Oxford University Press, 1995, 304–323.

Gödel, K. [1953], “Is mathematics syntax of language?” in *Collected Works* 3, Oxford, Oxford University Press, 1995, 334–362.

Gödel, K. [1964], “What is Cantor's continuum problem?,” in Benacerraf and Putnam [1983], 470–485.

Gödel, K. [1986], *Collected works* 1, Oxford, Oxford University Press.

Goldfarb, W. [1979], “Logic in the twenties: The nature of the quantifier,” *Journal of Symbolic Logic* 44: 351–368.

Goldfarb, W. [1988], “Poincaré against the logicists,” in Aspray and Kitcher [1988], 61–81.

Goldfarb, W. [1989], “Russell's reasons for ramification,” in *Rereading Russell*, *Minnesota studies in the philosophy of science* 12: 24–40.

Goldman, A. [1986], *Epistemology and cognition*, Cambridge, Harvard University Press.

Goodman, Nicolas [1979], “Mathematics as an objective science,” *American Mathematical Monthly* 86: 540–551.

Grassmann, H. [1972], *Gessammelte mathematische und physicalische Werke* 1, ed. by F. Engels, New York, Johnson Reprint Corporation.

Gupta, A., and N. Belnap [1993], *The revision theory of truth*, Cambridge, MIT Press.

Hale, Bob [1987], *Abstract objects*, Oxford, Blackwell.

Hallett, M. [1990], “Physicalism, reductionism and Hilbert,” in *Physicalism in mathematics*, ed. by A. D. Irvine, Dordrecht, Holland, Kluwer Academic Publishers, 183–257.

Hallett, M. [1994], “Hilbert's axiomatic method and the laws of thought,” in *Mathematics and mind*, ed. by Alexander George, Oxford, Oxford University Press, 158–200.

Hand, M. [1993], “Mathematical structuralism and the third man,” *Canadian Journal of Philosophy* 23: 179–192.

Hazen, A. [1983], “Predicative logics,” in *Handbook of philosophical logic* 1, ed. by D. Gabbay and F. Guenthner, Dordrecht, Holland, Reidel, 331–407.

(p.267)
Heath, T. [1921], *A history of Greek mathematics*, Oxford, Clarendon Press.

Hellman, G. [1983], “Realist principles,” *Philosophy of Science* 50: 227–249.

Hellman, G. [1989], *Mathematics without numbers*, Oxford, Oxford University Press.

Hellman, G. [1996], “Structuralism without structures,” *Philosophia Mathematica* (3) 4: 100–123.

Helmholtz, H. von [1921], *Schriften zur Erkenntnistheorie*, ed. by P. Hertz and M. Schlick, Berlin, Springer; tr. as *Epistemological writings*, Dordrecht, Holland Reidel, 1977.

Hersh, R. [1979], “Some proposals for reviving the philosophy of mathematics,” *Advances in mathematics* 31: 31–50.

Heyting, A. [1931], “The intuitionistic foundations of mathematics,” in Benacerraf and Putnam [1983], 52–61.

Heyting, A. [1956], *Intuitionism, An introduction*, Amsterdam, North Holland.

Hilbert, D. [1899], *Grundlagen der Geometrie*, Leipzig, Teubner; *Foundations of geometry*, tr. by E. Townsend, La Salle, Illinois, Open Court, 1959.

Hilbert, D. [1900], “Mathematische Probleme,” *Bulletin of the American Mathematical Society* 8 (1902): 437–479.

Hilbert, D. [1900a], “Über den Zahlbegriff,” *Jahresbericht der Deutschen Mathematiker‐ Vereinigung* 8: 180–194.

Hilbert, D. [1902], *Les principes fondamentaux de la géometrie*, Paris, Gauthier‐Villars; the French tr. of Hilbert [1899].

Hilbert, D. [1905], “Über der Grundlagen der Logik und der Arithmetik,” in *Verhandlungen des dritten internationalen Mathematiker‐Kongresses in Heidelberg vom 8 bis 13 August 1904*, Leipzig, Teubner, 174–185;

tr. as “On the foundations of logic and arithmetic,” in van Heijenoort [1967], 129–138.

Hilbert, D. [1925], “Über das Unendliche,” *Mathematische Annalen* 95: 161–190;

tr. as “On the infinite,” in van Heijenoort [1967], 369–392; Benacerraf and Putnam [1983], 83–201.

Hilbert, D. [1935], *Gesammelte Abhandlungen*, 3, Berlin, Springer.

Hilbert, D., and S. Cohn‐Vossen [1932], *Geometry and the imagination*, tr. by P. Nemenyi, New York, Chelsea Publishing Company, 1952.

Hodes, H. [1984], “Logicism and the ontological commitments of arithmetic,” *Journal of Philosophy* 81: 123–149.

Hodges, W. [1985], “Truth in a structure,” *Proceedings of the Aristotelian Society* 86 (1985–1986): 135–151.

Horwich, P. [1990], *Truth*, Oxford, Blackwell.

Huntington, E. [1902], “A complete set of postulates for the theory of absolute continuous magnitude,” *Transactions of the American Mathematical Society* 3: 264–279.

Kitcher, P. [1983], *The nature of mathematical knowledge*, New York, Oxford University Press.

Kitcher, P. [1986], “Frege, Dedekind, and the philosophy of mathematics,” in *Frege synthesized*, ed. by L. Haaparanta and J. Hintikka, Dordrecht, Holland, Reidel, 299–343.

Klein, F. [1921], *Gesammelte mathematische Abhandlungen* 1, Berlin, Springer.

Klein, J. [1968], *Greek mathematical thought and the origin of algebra*, Cambridge, MIT Press.

Kraut, R. [1980], “Indiscernibility and ontology,” *Synthese* 44: 113–135.

Kraut, R. [1993], “Robust deflationism,” *Philosophical Review* 102: 247–263.

Kreisel, G. [1967], “Informal rigour and completeness proofs,” *Problems in the philosophy of mathematics*, ed. by I. Lakatos, Amsterdam, North Holland, 138–186.

Kripke, S. [1965], “Semantical analysis of intuitionistic logic I,” in *Formal systems and recursive functions*, ed. by J. Crossley and M. Dummett, Amsterdam, North Holland, 92–130.

(p.268)
Kripke, S. [1975], “Outline of a theory of truth,” *Journal of Philosophy* 72: 690–716.

Kuhn, T. [1970], *The structure of scientific revolutions*, second edition, Chicago, University of Chicago Press.

Lakatos, I. [1976], *Proofs and refutations*, ed. by J. Worrall and E. Zahar, Cambridge, Cambridge University Press.

Lakatos, I. [1978], *Mathematics*, *science and epistemology*, ed. by J. Worrall and G. Currie, Cambridge, Cambridge University Press.

Landman, F. [1989], “Groups,” *Linguistics and Philosophy* 12: 559–605, 723–744.

Lawvere, W. [1966], “The category of categories as a foundation for mathematics,” in *Proceedings of the conference on categorical algebra in La Jolla, 1965*, ed. by S. Eilenberg et al., New York, Springer, 1–21.

Lebesgue, H. [1971], “A propos de quelques travaux mathematiques recents,” *Enseignement mathematique* (2) 17: 1–48.

Lévi‐Strauss, C. [1949], *Les Structures élémentaires de la parenté*, Paris, P.U.F.

Levy, A. [1960], “Principles of reflection in axiomatic set theory,” *Fundamenta Mathematicae* 49: 1–10.

Lewis, D. [1986], *On the plurality of worlds*, Oxford, Blackwell.

Lewis, D. [1991], *Parts of classes*, Oxford, Blackwell.

Lewis, D. [1993], “Mathematics is megethology,” *Philosophia Mathematica* (3) 1: 3–23.

Luce, L. [1988], “Frege on cardinality,” *Philosophy and Phenomenological Research* 48: 415–434.

Maddy, P. [1981], “Sets and numbers,” *Nous* 11: 495–511.

Maddy, P. [1988], “Mathematical realism,” *Midwest Studies in Philosophy* 12: 275–285.

Maddy, P. [1990], *Realism in mathematics*, Oxford, Oxford University Press.

Maddy, P. [1993], “Does V equal L?” *Journal of Symbolic Logic* 58: 15–41.

Maddy, P. [1998], “Naturalizing mathematical methodology,” in *Philosophy of mathematics today: Proceedings of an international conference in Munich*, ed. by M. Schirn, The Mind Association, Oxford, Oxford University Press, 75–93.

McCarty, C. [1987], “Variations on a thesis: Intuitionism and computability,” *Notre Dame Journal of Formal Logic* 28: 536–580.

McCarty, C. [1995], “The mysteries of Richard Dedekind,” in *From Dedekind to Gödel*, ed. by Jaakko Hintikka, *Synthese Library Series* 251, Dordrecht, The Netherlands, Kluwer Academic Publishers, 53–96.

McLarty, C. [1993], “Numbers can be just what they have to,” *Nous* 27: 487–498.

Moore, G. H. [1982], *Zermelo's axiom of choice*: *Its origins*, *development*, *and influence*, New York, Springer‐Verlag.

Myhill, J. [1960], “Some remarks on the notion of proof,” *Journal of Philosophy* 57: 461–471.

Nagel, E. [1939], “The formation of modern conceptions of formal logic in the development of geometry,” *Osiris* 7: 142–224.

Nagel, E. [1979], “Impossible numbers: A chapter in the history of modern logic,” in *Teleology revisited and other essays in the philosophy and history of science*, New York, Columbia University Press, 166–194.

Neurath, O. [1932], “Protokollsätze,” *Erkenntnis* 3: 204–214.

Nolt, J. [1978], *The language of set theory*, Ph.D. dissertation, The Ohio State University.

Padoa, A. [1900], “Essai d'une théorie algébrique des nombres entiers, précede d'une introduction logique à une théorie déductive qelconque,” in *Bibliothèque du Congrès international de philosophie*, Paris; tr. as “Logical introduction to any deductive theory,” in van Heijenoort [1967], 118–123.

(p.269)
Parsons, C. [1965], “Frege's theory of number,” in *Philosophy in America*, ed. by Max Black, Ithaca, New York, Cornell University Press, 180–203; reprinted in Parsons [1983], 150–175.

Parsons, C. [1977], “What is the iterative conception of set?” in *Logic*, *foundations of mathematics and computability theory*, ed. by R. Butts and J. Hintikka, Dordrecht, Holland, Reidel, 335–367; reprinted in Benacerraf and Putnam [1983], 503–529; and in Parsons [1983], 268–297.

Parsons, C. [1983], *Mathematics in philosophy*, Ithaca, New York, Cornell University Press.

Parsons, C. [1990], “The structuralist view of mathematical objects,” *Synthese* 84: 303–346.

Parsons, C. [1995], “Structuralism and the concept of set,” in *Modality, morality, and belief: Essays in honor of Ruth Barcan Marcus*, ed. by W. Sinnott‐Armstrong, D. Raffman, N. Ascher, Chicago, University of Chicago Press, 74–92.

Pasch, M. [1926], *Vorlesungen über neuere Geometrie*, Zweite Auflage, Berlin, Springer.

Plücker, J., [1846], *System der Geometrie des Raumes*, Dusseldorf.

Poincaré, H. [1899], “Des fondements de la géométrie,” *Revue de Métaphysique et de Morale* 7: 251–279.

Poincaré, H. [1900], “Sur les principes de la géométrie,” *Revue de Métaphysique et de Morale* 8: 72–86.

Poincaré, H. [1906], “Les mathématiques et la logique,” *Revue de Métaphysique et de Morale* 14: 294–317.

Poincaré, H. [1908], *Science et méthode*, Paris, Flammarion,

tr. as *Science and method, Foundation of* s*cience*, tr. by G. Halsted, New York, Science Press, 1921, 359–546.

Polya, G. [1954], *Mathematics and plausible reasoning*, Princeton, Princeton University Press.

Polya, G. [1977], *Mathematical methods in science*, Washington, D.C., Mathematical Association of America.

Posy, C. [1984], “Kant's mathematical realism,” *Monist* 67: 115–134.

Proclus [485], *Commentary on Euclid's elements I*, tr. by G. Morrow, Princeton, Princeton University Press, 1970.

Putnam, H. [1967], “Mathematics without foundations,” *Journal of Philosophy* 64: 5–22; reprinted in Benacerraf and Putnam [1983], 295–311.

Putnam, H. [1971], *Philosophy of logic*, New York, Harper Torchbooks.

Putnam, H. [1975], “What is mathematical truth?” in *Mathematics, matter and method*: *Philosophical papers*, vol. 1, by Hillary Putnam, Cambridge, Cambridge University Press, 60–78.

Putnam, H. [1980], “Models and reality,” *Journal of Symbolic Logic* 45: 464–482; reprinted in Benacerraf and Putnam [1983], 421–444.

Putnam, H. [1981], *Reason*, *truth and history*, Cambridge, Cambridge University Press.

Putnam, H. [1987], *The many faces of realism*, LaSalle, Illinois, Open Court.

Quine, W. V. O. [1937], “New foundations for mathematical logic,” *American Mathematical Monthly* 44: 70–80.

Quine, W. V. O. [1941], “Whitehead and the rise of modern logic,” in P. A. Schilpp, *The philosophy of Alfred North Whitehead*, New York, Tudor, 127–163.

Quine, W. V. O. [1951], “Two dogmas of empiricism,” *Philosophical Review* 60: 20–43.

Quine, W. V. O. [1960], *Word and object*, Cambridge, MIT Press.

Quine, W. V. O. [1969], *Ontological relativity and other essays*, New York, Columbia University Press.

Quine, W. V. O. [1981], *Theories and things*, Cambridge, Harvard University Press.

(p.270)
Quine, W. V. O. [1986], *Philosophy of logic*, second edition, Englewood Cliffs, New Jersey, Prentice‐Hall.

Quine, W. V. O. [1992], “Structure and nature,” *Journal of Philosophy* 89: 5–9.

Ramsey, F. [1925], “The foundations of mathematics,” *Proceedings of the London Mathematical Society* (2) 25: 338–384.

Resnik, M. [1966], “On Skolem's paradox,” *Journal of Philosophy* 63: 425–438.

Resnik, M. [1975], “Mathematical knowledge and pattern cognition,” *Canadian Journal of Philosophy* 5: 25–39.

Resnik, M. [1980], *Frege and the philosophy of mathematics*, Ithaca, New York, Cornell University Press.

Resnik, M. [1981], “Mathematics as a science of patterns: Ontology and reference,” *Nous* 15: 529–550.

Resnik, M. [1982], “Mathematics as a science of patterns: Epistemology,” *Nous* 16: 95–105.

Resnik, M. [1985], “How nominalist is Hartry Field's nominalism?,” *Philosophical Studies* 47: 163–181.

Resnik, M. [1988], “Mathematics from the structural point of view,” *Revue Internationale de Philosophie* 42: 400–424.

Resnik, M. [1988a], “Second‐order logic still wild,” *Journal of Philosophy* 85: 75–87.

Resnik, M. [1990] “Beliefs about mathematical objects,” in *Physicalism in mathematics*, ed. by A. D. Irvine, Dordrecht, Holland, Kluwer Academic Publishers, 41–71.

Resnik, M. [1992], “A structuralist's involvement with modality” (review of Hellman [1989]), *Mind* 101: 107–122.

Resnik, M. [1996], “Structural relativity,” *Philosophia Mathematics* (3) 4: 83–99.

Resnik, M. [1998], “Holistic mathematics,” in *Philosophy of mathematics today: Proceedings of an international conference in Munich*, ed. by M. Schirn, The Mind Association, Oxford, Oxford University Press, 227–246.

Rogers, H. [1967], *Theory of recursive functions and effective computability*, New York, McGraw‐Hill.

Russell, B. [1899] “Sur les axiomes de la géométrie,” *Revue de Métaphysique et de Morale* 7: 684–707.

Russell, B. [1903], *The principles of mathematics*, London, Allen and Unwin.

Russell, B. [1904], “Non‐euclidean geometry,” *Athenaeum* 4018: 592–593.

Russell, B. [1956], *An essay on the foundations of geometry*, New York, Dover; first published in 1897.

Russell, B. [1993], *Introduction to mathematical philosophy*, New York, Dover; first published in 1919.

Scanlan, M. [1888] “Beltrami's model and the independence of the parallel postulate,” *History and Philosophy of Logic* 9: 13–34.

Scanlan, M. [1991], “Who were the American postulate theorists?” *Journal of Symbolic Logic* 56: 981–1002.

Schoenfield, J. [1967], *Mathematical logic*, Reading, Massachusetts, Addison Wesley.

Shapiro, S. [1980], “On the notion of effectiveness,” *History and Philosophy of Logic* 1: 209–230.

Shapiro, S. [1983], “Conservativeness and incompleteness,” *Journal of Philosophy* 80: 521–531.

Shapiro, S. [1983a], “Mathematics and reality,” *Philosophy of Science* 50: 523–548.

Shapiro, S. [1983b], “Remarks on the development of computability,” *History and Philosophy of Logic* 4: 203–220.

(p.271)
Shapiro, S. [1985], “Epistemic and intuitionistic arithmetic,” in *Intensional mathematics*, ed. by S. Shapiro, Amsterdam, North Holland, 11–46.

Shapiro, S. [1987], “Principles of reflection and second‐order logic,” *Journal of Philosophical Logic* 16: 309–333.

Shapiro, S. [1989], “Logic, ontology, mathematical practice,” *Synthese* 79: 13–50.

Shapiro, S. [1991], *Foundations without foundationalism: A case for second‐order logic*, Oxford, Oxford University Press.

Shapiro, S. [1993], “Understanding Church's thesis, again,” *Acta Analytica* 11: 59–77.

Shapiro, S. [1993a], “Modality and ontology,” *Mind* 102: 455–481.

Shapiro, S. [1995], “Skolem paradox,” in *The Oxford companion to philosophy*, ed. by T. Honderich, Oxford, Oxford University Press, 827.

Shapiro, S. [1998], “Logical consequence: Models and modality,” *Philosophy of mathematics today: Proceedings of an international conference in Munich*, ed. by M. Schirn, The Mind Association, Oxford, Oxford University Press, 131–156.

Sher, G. [1991], *The bounds of logic*, Cambridge, MIT Press.

Sieg, W. [1990], “Physicalism, reductionism and Hilbert,” in *Physicalism in mathematics*, ed. by A. D. Irvine, Dordrecht, Holland, Kluwer Academic Publishers, 183–257.

Sieg, W. [1994], “Mechanical procedures and mathematical experience,” in *Mathematics and mind*, ed. by Alexander George, Oxford, Oxford University Press, 71–140.

Skolem, T. [1922], “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre,” *Matematikerkongressen i Helsingfors den 4–7 Juli 1922*, Helsinki, Akademiska Bokhandeln, 217–232;

tr. as “Some remarks on axiomatized set theory,” in van Heijenoort [1967], 291–301.

Stein, H. [1988], “*Logos*, logic, and *Logistiké*: Some philosophical remarks on the nineteenth century transformation of mathematics,” in Aspray and Kitcher [1988], 238–259.

Steiner, M. [1989], “The application of mathematics to natural science,” *Journal of Philosophy* 86: 449–480.

Tait, W. [1981], “Finitism,” *Journal of Philosophy* 78: 524–546.

Tait, W. [1986], “Truth and proof: The Platonism of mathematics,” *Synthese* 69: 341–370.

Tait, W. [1986a], “Critical notice: Charles Parsons' *Mathematics in philosophy*,” *Philosophy of Science* 53: 588–607.

Tait, W. [1997], “Frege versus Cantor and Dedekind on the concept of number,” in *Early analytic philosophy: Frege, Russell, Wittgenstein: Essays in honor of Leonard Linsky*, ed. by W. Tait, Chicago, Open Court, 213–248.

Tarski, A. [1933], “Der Warheitsbegriff in dem formalisierten Sprachen,” *Studia Philosophica* 1: 261–405;

tr. as “The concept of truth in formalized languages,” in Tarski [1956], 152–278.

Tarski, A. [1935], “On the concept of logical consequence,” in Tarski [1956], 417–429.

Tarski, A. [1944], “The semantic conception of truth and the foundations of semantics,” *Philosophy and Phenomenological Research* 4: 341–376.

Tarski, A. [1956], *Logic*, *semantics and metamathematics*, Oxford, Clarendon Press; second edition, ed. by John Corcoran, Indianapolis, Hackett, 1983.

Tarski, A. [1986], “What are logical notions?” (ed. by John Corcoran), *History and Philosophy of Logic* 7: 143–154.

Taylor, R. [1993], “Zermelo, reductionism, and the philosophy of mathematics,” *Notre Dame Journal of Formal Logic* 34: 539–563.

Tennant, N. [1987], *Anti‐realism and logic*, Oxford, Oxford University Press.

Tennant, N. [1997], “On the necessary existence of numbers,” Nous 31.

(p.272)
Turing, A. [1936], “On computable numbers, with an application to the *Entscheidungs‐ problem*,” *Proceedings of the London Mathematical Society* 42: 230–265; reprinted in *The Undecidable*, ed. by M. Davis, Hewlett, New York, The Raven Press, 1965, 116–153.

Turnbull, R. [1978], “Knowledge of the Forms in the later Platonic Dialogues,” *Proceedings and Addresses of the American Philosophical Association* 51: 735–758.

Vaihinger, H. [1913], *Die Philosophie des Als Ob*, Berlin, Verlag von Reuther & Reichard; tr. as *The philosophy of “as if* *,”* by C. K. Ogden, London, Routledge and Kegan Paul, 1935.

Van Fraassen, B. [1980], *The scientific image*, Oxford, Oxford University Press.

Van Heijenoort, J. [1967], *From Frege to Gödel*, Cambridge, Harvard University Press.

Van Heijenoort, J. [1967a], “Logic as calculus and logic as language,” *Synthese* 17: 324–330.

Veblen, O. [1904], “A system of axioms for geometry,” *Transactions of the American Mathematical Society* 5: 343–384.

Veblen, O. [1925], “Remarks on the foundations of geometry,” *Bulletin of the American Mathematical Society* 31: 121–141.

Wagner, S. [1987], “The rationalist conception of logic,” *Notre Dame Journal of Formal Logic* 28: 3–35.

Wang, H. [1974], *From mathematics to philosophy*, London, Routledge and Kegan Paul.

Wang, H. [1987], *Reflections on Kurt Gödel*, Cambridge, MIT Press.

Weinberg, S. [1986], “Lecture on the applicability of mathematics,” *Notices of the American Mathematical Society* 33.5.

Weyl, H. [1949], *Philosophy of mathematics and natural science*, Princeton, Princeton University Press; revised and augmented edition, New York, Athenaeum Press, 1963.

Wilson, M. [1981], “The double standard in ontology,” *Philosophical Studies* 39: 409–427.

Wilson, M. [1992], “Frege: The royal road from geometry,” *Nous* 26: 149–180.

Wilson, M. [1993], “There's a hole and a bucket, dear Leibniz,” *Midwest Studies in Philosophy* 18: 202–241.

Wilson, M. [1993a], “Honorable intensions,” in *Naturalism: A critical appraisal*, ed. by S. Wagner and R. Warner, Notre Dame, University of Notre Dame Press, 53–94.

Wittgenstein, L. [1953], *Philosophical investigations*, tr. by G. E. M. Anscombe, New York, Macmillan.

Wittgenstein, L. [1978], *Remarks on the foundations of mathematics*, tr. by G. E. M. Anscombe, Cambridge, MIT Press.

Wright, C. [1983], *Frege's conception of numbers as objects*, Aberdeen University Press.

Wright, C. [1992], *Truth and objectivity*, Cambridge, Harvard University Press.

Yaqub, A. [1993], *The liar speaks the truth: A defense of the revision theory of truth*, New York, Oxford University Press, 1993.

Zalta, E. [1983], *Abstract objects: An introduction to axiomatic metaphysics*, Dordrecht, Holland, Reidel.

Zermelo, E. [1904], “Beweis, dass jede Menge wohlgeordnet werden kann,” *Mathematische Annalen* 59: 514–516; tr. in van Heijenoort [1967], 139–141.