Advertisement

Mathematical Truth Regained

  • Robert Hanna
Chapter
  • 735 Downloads
Part of the Phaenomenologica book series (PHAE, volume 195)

Abstract

Benacerraf’s Dilemma (BD), as formulated by Paul Benacerraf in “Mathematical Truth,” is about the apparent impossibility of reconciling a “standard” (i.e., classical Platonic) semantics of mathematics with a “reasonable” (i.e., causal, spatiotemporal) epistemology of cognizing true statements. In this paper I spell out a new solution to BD. I call this new solution a positive Kantian phenomenological solution for three reasons: (1) It accepts Benacerraf’s preliminary philosophical assumptions about the nature of semantics and knowledge, as well as all the basic steps of BD, and then shows how we can, consistently with those very assumptions and premises, still reject the skeptical conclusion of BD and also adequately explain mathematical knowledge. (2) The standard semantics of mathematically necessary truth that I offer is based on Kant’s philosophy of arithmetic, as interpreted by Charles Parsons and by me. (3) The reasonable epistemology of mathematical knowledge that I offer is based on the phenomenology of logical and mathematical self-evidence developed by early Husserl in Logical Investigations and by early Wittgenstein in Tractatus Logico-Philosophicus.

Keywords

Actual World Mathematical Knowledge Sense Perception Mathematical Truth Peano Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Balaguer, M. 1998. Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press.Google Scholar
  2. Benacerraf, P. 1965. What Numbers Could Not Be. Philosophical Review 74: 47–73.CrossRefGoogle Scholar
  3. Benacerraf, P. 1973. Mathematical Truth. Journal of Philosophy 70: 661–680.CrossRefGoogle Scholar
  4. Benacerraf, P. 1996. What Mathematical Truth Could Not Be – I. In Benacerraf and his Critics, eds. A. Morton and S. P. Stich, 9–59. Oxford: Blackwell.Google Scholar
  5. Block, N. 1980a. Troubles with Functionalism. In Readings in the Philosophy of Psychology, ed. N. Block, Vol 1, 268–305. Cambridge: Harvard Univ. Press.Google Scholar
  6. Block, N. 1980b. What is Functionalism? In Readings in the Philosophy of Psychology, ed. N. Block, Vol. 1, 171–184. Cambridge: Harvard Univ. Press.Google Scholar
  7. Byrne, A. and Logue, H. (eds.) 2009. Disjunctivism: Contemporary Readings. Cambridge, MA: MIT Press.Google Scholar
  8. Chomsky, N. 1986. Knowledge of Language. Westport, CN: Praeger.Google Scholar
  9. Field, H. 1980. Science without Numbers: A Defense of Nominalism. Princeton, NJ: Princeton University Press.Google Scholar
  10. Field, H. 1989. Realism, Mathematics, and Modality. Oxford: Blackwell.Google Scholar
  11. Frege, G. 1953. Foundations of Arithmetic. 2nd edition, Trans. J.L. Austin. Evanston, IL: Northwestern University Press.Google Scholar
  12. Giaquinto, M. 2007. Visual Thinking in Mathematics. Oxford: Oxford Univ. Press.CrossRefGoogle Scholar
  13. Haddock, A. and McPherson, F. (eds.) 2008. Disjunctivism: Perception, Action, Knowledge. Oxford: Oxford Univ. Press.Google Scholar
  14. Hale, B. 1987. Abstract Objects. Oxford: Blackwell.Google Scholar
  15. Hale, B. and Wright, C. 2001. The Reason’s Proper Study. Oxford: Clarendon/Oxford University Press.CrossRefGoogle Scholar
  16. Hale, B. and Wright, C. 2002. Benacerraf’s Dilemma Revisited. European Journal of Philosophy 10: 101–129.CrossRefGoogle Scholar
  17. Hanna, R. 2001. Kant and the Foundations of Analytic Philosophy. Oxford: Oxford University Press.Google Scholar
  18. Hanna, R. 2002. Mathematics for Humans: Kant’s Philosophy of Arithmetic Revisited. European Journal of Philosophy 10: 328–353.CrossRefGoogle Scholar
  19. Hanna, R. 2006a. Kant, Science, and Human Nature. Oxford: Oxford Univ. Press.CrossRefGoogle Scholar
  20. Hanna, R. 2006b. Rationality and the Ethics of Logic. Journal of Philosophy 103: 67–100.Google Scholar
  21. Hanna, R. 2006c. Rationality and Logic. MA: MIT Press: Cambridge.Google Scholar
  22. Hanna, R. and Maiese, M. 2009. Embodied Minds in Action. Oxford: Oxford University Press.Google Scholar
  23. Jackson, F. 1996. Mental Causation. Mind 105: 377–413.CrossRefGoogle Scholar
  24. Kant, I. 1997. Critique of Pure Reason. Trans. P. Guyer and A. Wood. Cambridge: Cambridge Univ. Press. When citing the first Critique, I follow the common practice of giving page numbers from the A (1781) and B (1787) German editions only.Google Scholar
  25. Katz, J. 1995. What Mathematical Knowledge Could Be. Mind 104: 491–522.CrossRefGoogle Scholar
  26. Kim, J. 2006. Philosophy of Mind. 2nd edition. Boulder: Westview.Google Scholar
  27. Kripke, S. 1982. Wittgenstein on Rules and Private Language. Cambridge, MA: Harvard University Press.Google Scholar
  28. Martin, M. G. F. 2006. On Being Alienated. In Perceptual Experience, eds. T. Gendler and J. Hawthorne, 354–410. Oxford: Clarendon/Oxford University Press.CrossRefGoogle Scholar
  29. Milton, J. 1953a. Paradise Lost. In The Poems of John Milton. 2nd edition, ed. J. Milton, 204–487. New York: Ronald Press.Google Scholar
  30. Milton, J. 1953b. Paradise Regained. In The Poems of John Milton. 2nd edition, ed. J. Milton, 495–544. New York: Ronald Press.Google Scholar
  31. Numminen, M. A. 2009. Wovon Man Nicht Sprechen Kann, Darüber Muss Man Schweigen. At ULR = http://www.youtube.com/watch?v=57PWqFowq-4.
  32. Parsons, C. 1983. Kant’s Philosophy of Arithmetic. In Mathematics in Philosophy, C. Parsons, 119–149. New York: Cornell Univ. Press.Google Scholar
  33. Parsons, C. 2008. Mathematical Thought and its Objects. Cambridge: Cambridge University Press.Google Scholar
  34. Potter, M. 1990. Sets: An Introduction. Oxford: Clarendon/Oxford Univ Press.Google Scholar
  35. Potter, M.. 2000. Reason’s Nearest Kin. Oxford: Oxford University Press.Google Scholar
  36. Schacter, D. L. 1990. Perceptual Representation Systems and Implicit Memory: Towards a Resolution of the Multiple Memory Systems Debate. Annals of the New York Academy of Science 608: 543–571.CrossRefGoogle Scholar
  37. Searle, J. 1984. Minds, Brains, and Science. Cambridge: Harvard University Press.Google Scholar
  38. Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. New York: Oxford Univ Press.Google Scholar
  39. Shapiro, S. 1998. Induction and Indefinite Extensibility: The Gödel Sentence is True, But Did Someone Change the Subject? Mind 107: 597–624.CrossRefGoogle Scholar
  40. Shapiro, S. 2000. Thinking about Mathematics. Oxford: Oxford University Press.Google Scholar
  41. Skolem, T. 1967. The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains. In From Frege to Gödel, ed. J. van Heijenoort. Cambridge, MA: Harvard University Press.Google Scholar
  42. Struik, D. J. 1967. A Concise History of Mathematics. New York: Dover.Google Scholar
  43. Troelstra, A. S. and Dalen, D. V. 1998. Constructivism in Mathematics: An Introduction, vol. 1. Amsterdam: North Holland.Google Scholar
  44. Wittgenstein, L. 1981. Tractatus Logico-Philosophicus, Trans. C.K. Ogden. London: Routledge and Kegan Paul.Google Scholar
  45. Wittgenstein, L. 1983. Remarks on the Foundations of Mathematics. 2nd edition, Trans. G.E.M. Anscombe. Cambridge, MA: MIT Press.Google Scholar
  46. Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen Univ. Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Robert Hanna
    • 1
  1. 1.University of ColoradoBoulderUSA

Personalised recommendations