Intentional Gaps In Mathematical Proofs

Albers, Donald J., Gerald L. Alexanderson, and Constance Reid: 1990, More Mathematical People, Harcourt Brace Jovanovich, Boston.Google Scholar

Anglin, W. S.: 1991, ‘Mathematics and Value’, Philosophia Mathematica (2) 6(2), 145–173.Google Scholar

Azzouni, Jody: 1994, Metaphysical Myths, Mathematical Practice, Cambridge University Press, New York.Google Scholar

Babai, Laszlo, Lance Fortnow, Leonid A. Levin, and Mario Szegedy: 1991, ‘Checking Computations in Polylogarithmic Time’, Proceedings of the 23rd ACM Symposium on Theory of Computing, pp. 21–31.Google Scholar

Ball, W. W. R.: 1960, A Short Account of the History of Mathematics, Dover Publications, New York.Google Scholar

Blum, Manuel: 1987, ‘How to Prove a Theorem So No One Else Can Claim It’, Proceedings of the International Congress of Mathematicians 1986, American Mathematical Society, Providence, RI, pp. 1444–1451.Google Scholar

Corcoran, John: 1973, ‘Gaps Between Logical Theory and Mathematical Practice’, in Mario Bunge (ed.), The Methodological Unity of Science, D. Reidel Publishing Co., Boston, pp. 23–50.Google Scholar

Dauben, Joseph W.: 1994, ‘Topology: Invariance of Dimension’, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness, Routledge, pp. 939–946.Google Scholar

Davis, Philip J.: 1972, ‘Fidelity in Mathematical Discourse: Is One and One Really Two?’, American Mathematical Monthly 79(3), 252–263.Google Scholar

Dawson, John W.: 1985, ‘The Reception of Godel's Incompleteness Theorems’, PSA 1984, 2, 253–271.Google Scholar

Descartes, René: 1952, The Geometry of René Descartes, trans. David E. Smith and Marcia L. Latham, Open Court Publishing Company, La Salle, IL.Google Scholar

Descartes, René: 1927, Selections, trans. Ralph M. Eaton, Charles Scribner's Sons, New York.Google Scholar

Dieudonné, Jean: 1992, ‘The Concept of “Rigorous Proof”’, Mathematics-The Music of Reason, Springer-Verlag, Berlin, pp. 235–238.Google Scholar

Enderton, Herbert B.: 1972, A Mathematical Introduction to Logic, Academic Press, San Diego.Google Scholar

Fallis, Don: 1997, ‘The Epistemic Status of Probabilistic Proof’, Journal of Philosophy 94(4), 165–186.Google Scholar

Faltings, Gerd: 1995, ‘The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles’, Notices of the American Mathematical Society 42(7), 743–746.Google Scholar

Frege, Gottlob: 1980, The Foundations of Arithmetic, 2nd edn., trans. J. L. Austin, Northwestern University Press, Evanston, IL.Google Scholar

Gödel, Kurt: 1992, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, trans. B. Meltzer, Dover Publications, New York.Google Scholar

Hadamard, Jacques: 1945, The Psychology of Invention in the Mathematical Field, Princeton University Press, Prentice.Google Scholar

Hardwig, John: 1985, ‘Epistemic Dependence’, Journal of Philosophy 82(7), 335–349.Google Scholar

Harel, David: 1989, The Science of Computing, Addison-Wesley, Reading, MA.Google Scholar

Hersh, Reuben: 1993, ‘Proving Is Convincing and Explaining’, Educational Studies in Mathematics 24(4), 389–399.Google Scholar

Hume, David: 1967, in L. A. Selby-Bigge (ed.), A Treatise of Human Nature, Clarendon Press, Oxford.Google Scholar

Jackson, Allyn: 1994, ‘Update on Proof of Fermat's Last Theorem’, Notices of the American Mathematical Society 41(3), 185–186.Google Scholar

Kershner, R. B.: 1968, ‘On Paving the Plane’, American Mathematical Monthly 75(8), 839–844.Google Scholar

Kitcher, Philip: 1984, The Nature of Mathematical Knowledge, Oxford University Press, New York.Google Scholar

Kitcher, Philip and William Aspray: 1988, ‘An Opinionated Introduction’, in Philip Kitcher and William Aspray (eds.), History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, pp. 3–57.Google Scholar

Lakatos, Imre: 1976, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge University Press, New York.Google Scholar

Lehman, Hugh: 1980, ‘An Examination of Imre Lakatos, Philosophy of Mathematics’, The Philosophical Forum 12(1), 33–48.Google Scholar

Mac Lane, Saunders: 1996, ‘Structure in Mathematics’, Philosophia Mathematica (3) 4(2), 174–183.Google Scholar

Maddy, Penelope: 1988, ‘Believing the Axioms’, Journal of Symbolic Logic 53, 481–511, 736-764.Google Scholar

Plato: 1948, Euthyphro, Apology, Crito, trans. F. J. Church and Robert D. Cumming, Liberal Arts Press, New York.Google Scholar

Poincaré, Henri: 1952, Science and Method, trans. Francis Maitland, Dover Publications, New York.Google Scholar

Pólya, George: 1957, How To Solve It, 2nd edn., Princeton University Press, Princeton.Google Scholar

Pólya, George: 1954, Mathematics and Plausible Reasoning, Vol. 1, Princeton University Press, Princeton.Google Scholar

Rabin, M. O.: 1980, ‘Probabilistic Algorithm for Testing Primality’, Journal of Number Theory 12, 128–138.Google Scholar

Reid, Constance: 1986, Hilbert-Courant, Springer-Verlag, New York.Google Scholar

Ribet, Kenneth A.: 1993, ‘Wiles Proves Taniyama's Conjecture; Fermat's Last Theorem Follows’, Notices of the American Mathematical Society 40(6), 575–576.Google Scholar

Schattschneider, Doris: 1981, ‘In Praise of Amateurs’, in Klarner and David A. Belmont (eds.), The Mathematical Gardner, Wadsworth International, Belmont, CA, pp. 140–166.Google Scholar

Shakespeare, William: 1963, The Tragedy of Julius Caesar, Signet Classic. New York.Google Scholar

Shoenfield, Joseph R.: 1967, Mathematical Logic, Addison-Wesley, Reading, MA.Google Scholar

Smale, Steve: 1981, ‘The Fundamental Theorem of Algebra and Complexity Theory’, Bulletin of the American Mathematical Society 4, 1–36.Google Scholar

Steiner, Mark: 1975, Mathematical Knowledge, Cornell University Press, Ithaca, NY.Google Scholar

Stewart, Ian: 1996, From Here to Infinity, Oxford University Press, New York.Google Scholar

Struik, D. J.: 1969, A Source Book in Mathematics, 1200-1800, Harvard University Press, Cambridge, MA.Google Scholar

Tarski, Alfred: 1956, ‘The Concept of Truth in Formalized Languages’, Logic, Semantics, Metamathematics: Papers From 1923 to 1938, trans. J. H. Woodger, Clarendon Press, Oxford.Google Scholar