What are the limits of mathematical explanation? Interview with Charles McCarty by Piotr Urbańczyk
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Aug 22, 2016
Keywords:
mathematics logic mathematical explanation limits of explanation mathematical proof proof-core intuitionism constructivsim Gödel’s Incompleteness Theorems intuitionistics mathematics classical mathematics Axiom of Choice
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Abstract
An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation.
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McCarty, D. C., & Urbańczyk, P. (2016). What are the limits of mathematical explanation? Interview with Charles McCarty by Piotr Urbańczyk. Philosophical Problems in Science (Zagadnienia Filozoficzne W Nauce), (60), 119–137. Retrieved from https://zfn.edu.pl/index.php/zfn/article/view/369
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References
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Beeson, M.J., 1985. Foundations of constructive mathematics. Metamathematical Studies. Berlin: Springer-Verlag, pp. XXIII+466.
Bishop, E., 1967. Foundations of constructive analysis. New York: McGraw-Hill, pp. XIII+370.
Chomsky, N., 1968. Language and mind. New York: Harcourt Brace Jovanovich, pp. VII+88.
Collingwood, R.G., 1938. Principles of art. Oxford: Clarendon Press, pp. xi+347.
Dedekind, R., 1888. Was sind und was sollen die Zahlen? Braunschweig: F. Vieweg. pp. xv+58.
Du Bois-Reymond, P., 1882. Allgemeine Functionentheorie. Erster Theil. Tübingen: Verlag der H. Laupp’schen Buchhandlung, pp. XIV+292.
Gödel, K., 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by J. van Heijenoort from the original German and reprinted in: S. Feferman et al. (Eds.), Collected works. Volume I. Publications 1929–1936. New York, NY: Oxford University Press 1986, pp. 144–195.
Gödel, K., 1933. On the interpretation of the intuitionistic propositional calculus. Translated from the original German and reprinted in: S. Feferman et al. (Eds.), Collected works. Volume I. Publications 1929–1936. New York, NY: Oxford University Press 1986, pp. 301–303.
Gödel, K., 1961. The modern development of the foundations of mathematics in light of philosophy. Translated by Eckehart Koehler and William Craig, reprinted in: S. Feferman et al. (Eds.), Collected works. Volume III. Unpublished essays and lectures. New York, NY: Oxford University Press 1995, pp. 375–387.
Hardy, G.H., 1940. A mathematician’s apology. London, UK: Cambridge University Press, pp. vii+93.
Hilbert, D., 1902. Mathematical problems. Translated by M.W. Newson from the original German and reprinted in Bulletin of the American Mathematical Society (Series 2), 8, pp. 437–439.
Hilbert, D., 1930. Logic and the knowledge of nature. Translated by W. Ewald from the original German and reprinted in: W. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics. Volume 2. Oxford: Clarendon Press 1996, pp. 1157–1165.
Kleene, S.C., 1952. Recursive functions and intuitionistic mathematics. In: Proceedings of the International Congress of Mathematicians. Cambridge, Mass., USA. Aug. 30 – Sep. 6, 1950. Volume 1. Providence: American Mathematical Society, pp. 679–685.
Klein, F., 1911. The Evanston Colloquium Lectures on Mathematics. Reprinted in part in: W. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics. Volume 2. Oxford: Clarendon Press 1996, pp. 958.
Russell, B., Whitehead, A.N., 1910–1913. Principia mathematica. Volumes I–III. Cambridge: Cambridge University Press, pp. xiii+666, xxxiv+772, x+491.
Simpson, S., 1999. Subsystems of second-order arithmetic. Berlin: Springer-Verlag, pp. xiv+444.
Skolem, T., 1923. The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. Translated by Stefan Bauer-Mengelberg, reprinted in: J. Van Heijenoort (Ed.), From Frege to Gödel. A source book in mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press, pp. 302–333.
Tarski, A., Mostowski, A., Robinson, R., 1953. Undecidable theories. Amsterdam: North-Holland Publishing Company, pp. xi+98.
Beeson, M.J., 1985. Foundations of constructive mathematics. Metamathematical Studies. Berlin: Springer-Verlag, pp. XXIII+466.
Bishop, E., 1967. Foundations of constructive analysis. New York: McGraw-Hill, pp. XIII+370.
Chomsky, N., 1968. Language and mind. New York: Harcourt Brace Jovanovich, pp. VII+88.
Collingwood, R.G., 1938. Principles of art. Oxford: Clarendon Press, pp. xi+347.
Dedekind, R., 1888. Was sind und was sollen die Zahlen? Braunschweig: F. Vieweg. pp. xv+58.
Du Bois-Reymond, P., 1882. Allgemeine Functionentheorie. Erster Theil. Tübingen: Verlag der H. Laupp’schen Buchhandlung, pp. XIV+292.
Gödel, K., 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by J. van Heijenoort from the original German and reprinted in: S. Feferman et al. (Eds.), Collected works. Volume I. Publications 1929–1936. New York, NY: Oxford University Press 1986, pp. 144–195.
Gödel, K., 1933. On the interpretation of the intuitionistic propositional calculus. Translated from the original German and reprinted in: S. Feferman et al. (Eds.), Collected works. Volume I. Publications 1929–1936. New York, NY: Oxford University Press 1986, pp. 301–303.
Gödel, K., 1961. The modern development of the foundations of mathematics in light of philosophy. Translated by Eckehart Koehler and William Craig, reprinted in: S. Feferman et al. (Eds.), Collected works. Volume III. Unpublished essays and lectures. New York, NY: Oxford University Press 1995, pp. 375–387.
Hardy, G.H., 1940. A mathematician’s apology. London, UK: Cambridge University Press, pp. vii+93.
Hilbert, D., 1902. Mathematical problems. Translated by M.W. Newson from the original German and reprinted in Bulletin of the American Mathematical Society (Series 2), 8, pp. 437–439.
Hilbert, D., 1930. Logic and the knowledge of nature. Translated by W. Ewald from the original German and reprinted in: W. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics. Volume 2. Oxford: Clarendon Press 1996, pp. 1157–1165.
Kleene, S.C., 1952. Recursive functions and intuitionistic mathematics. In: Proceedings of the International Congress of Mathematicians. Cambridge, Mass., USA. Aug. 30 – Sep. 6, 1950. Volume 1. Providence: American Mathematical Society, pp. 679–685.
Klein, F., 1911. The Evanston Colloquium Lectures on Mathematics. Reprinted in part in: W. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics. Volume 2. Oxford: Clarendon Press 1996, pp. 958.
Russell, B., Whitehead, A.N., 1910–1913. Principia mathematica. Volumes I–III. Cambridge: Cambridge University Press, pp. xiii+666, xxxiv+772, x+491.
Simpson, S., 1999. Subsystems of second-order arithmetic. Berlin: Springer-Verlag, pp. xiv+444.
Skolem, T., 1923. The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. Translated by Stefan Bauer-Mengelberg, reprinted in: J. Van Heijenoort (Ed.), From Frege to Gödel. A source book in mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press, pp. 302–333.
Tarski, A., Mostowski, A., Robinson, R., 1953. Undecidable theories. Amsterdam: North-Holland Publishing Company, pp. xi+98.