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.
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Notes
- 1.
(Milton, 1953b, 495, book I, lines 1–7)
- 2.
(Parsons 2008, 166)
- 3.
(LI, pp. 765 and 787, texts combined)
- 4.
(Wittgenstein, 1981, prop. 5.4731, p. 129)
- 5.
In (Hanna, 2006a Chapters 6 and 7), I work out Kant’s idea that mathematical knowledge is grounded on reflective self-consciousness together with the imagination.
- 6.
One way of doing this would be via “plenitudinous platonism”: For every consistently imaginable mathematical statement, there is a corresponding mathematical object. (See, e.g., Balaguer, 1998.) This construes imaginability as conceivability. But there are other ways of thinking about the imagination, e.g., Kant’s conception of the productive imagination as a “schematizing” (i.e., mental modeling) capacity (Kant 1997, A84–147/B116–187, and esp. A120 n.). In (Hanna, 2006b, Chapter 6), I extend BD to logical knowledge, and then develop a strategy for solving the extended BD that starts with the thesis that a reasonable epistemology should be modeled on the imagination, not on perception. So by the classification scheme described here, strictly speaking, that earlier solution counts as a pre-emptive negative or skeptical solution. But to the extent that the present solution postulates the innate presence of mental modeling abilities in sense perception, it also postulates the innate presence of the capacity for imagination within the capacity for sense perception. So in that sense, the present positive or anti-skeptical solution is really only an extension and refinement of the earlier solution.
- 7.
- 8.
See (and hear) (Numminen, 2009).
- 9.
(Milton, 1953a, p. 487, book XII, lines 641–649).
- 10.
I am very grateful to the organizers (especially Mirja Hartimo, Leila Haaparanta, Juliette Kennedy, and Sara Heinämaa) of and also the participants in (especially William Tait), the Phenomenology and Mathematics conference at the University of Tampere, Finland in May 07, where I presented an earlier version of this paper, for all their help—critical, editorial, philosophical, and otherwise.
References
Balaguer, M. 1998. Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press.
Benacerraf, P. 1965. What Numbers Could Not Be. Philosophical Review 74: 47–73.
Benacerraf, P. 1973. Mathematical Truth. Journal of Philosophy 70: 661–680.
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.
Block, N. 1980a. Troubles with Functionalism. In Readings in the Philosophy of Psychology, ed. N. Block, Vol 1, 268–305. Cambridge: Harvard Univ. Press.
Block, N. 1980b. What is Functionalism? In Readings in the Philosophy of Psychology, ed. N. Block, Vol. 1, 171–184. Cambridge: Harvard Univ. Press.
Byrne, A. and Logue, H. (eds.) 2009. Disjunctivism: Contemporary Readings. Cambridge, MA: MIT Press.
Chomsky, N. 1986. Knowledge of Language. Westport, CN: Praeger.
Field, H. 1980. Science without Numbers: A Defense of Nominalism. Princeton, NJ: Princeton University Press.
Field, H. 1989. Realism, Mathematics, and Modality. Oxford: Blackwell.
Frege, G. 1953. Foundations of Arithmetic. 2nd edition, Trans. J.L. Austin. Evanston, IL: Northwestern University Press.
Giaquinto, M. 2007. Visual Thinking in Mathematics. Oxford: Oxford Univ. Press.
Haddock, A. and McPherson, F. (eds.) 2008. Disjunctivism: Perception, Action, Knowledge. Oxford: Oxford Univ. Press.
Hale, B. 1987. Abstract Objects. Oxford: Blackwell.
Hale, B. and Wright, C. 2001. The Reason’s Proper Study. Oxford: Clarendon/Oxford University Press.
Hale, B. and Wright, C. 2002. Benacerraf’s Dilemma Revisited. European Journal of Philosophy 10: 101–129.
Hanna, R. 2001. Kant and the Foundations of Analytic Philosophy. Oxford: Oxford University Press.
Hanna, R. 2002. Mathematics for Humans: Kant’s Philosophy of Arithmetic Revisited. European Journal of Philosophy 10: 328–353.
Hanna, R. 2006a. Kant, Science, and Human Nature. Oxford: Oxford Univ. Press.
Hanna, R. 2006b. Rationality and the Ethics of Logic. Journal of Philosophy 103: 67–100.
Hanna, R. 2006c. Rationality and Logic. MA: MIT Press: Cambridge.
Hanna, R. and Maiese, M. 2009. Embodied Minds in Action. Oxford: Oxford University Press.
Jackson, F. 1996. Mental Causation. Mind 105: 377–413.
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.
Katz, J. 1995. What Mathematical Knowledge Could Be. Mind 104: 491–522.
Kim, J. 2006. Philosophy of Mind. 2nd edition. Boulder: Westview.
Kripke, S. 1982. Wittgenstein on Rules and Private Language. Cambridge, MA: Harvard University Press.
Martin, M. G. F. 2006. On Being Alienated. In Perceptual Experience, eds. T. Gendler and J. Hawthorne, 354–410. Oxford: Clarendon/Oxford University Press.
Milton, J. 1953a. Paradise Lost. In The Poems of John Milton. 2nd edition, ed. J. Milton, 204–487. New York: Ronald Press.
Milton, J. 1953b. Paradise Regained. In The Poems of John Milton. 2nd edition, ed. J. Milton, 495–544. New York: Ronald Press.
Numminen, M. A. 2009. Wovon Man Nicht Sprechen Kann, Darüber Muss Man Schweigen. At ULR = http://www.youtube.com/watch?v=57PWqFowq-4.
Parsons, C. 1983. Kant’s Philosophy of Arithmetic. In Mathematics in Philosophy, C. Parsons, 119–149. New York: Cornell Univ. Press.
Parsons, C. 2008. Mathematical Thought and its Objects. Cambridge: Cambridge University Press.
Potter, M. 1990. Sets: An Introduction. Oxford: Clarendon/Oxford Univ Press.
Potter, M.. 2000. Reason’s Nearest Kin. Oxford: Oxford University Press.
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.
Searle, J. 1984. Minds, Brains, and Science. Cambridge: Harvard University Press.
Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. New York: Oxford Univ Press.
Shapiro, S. 1998. Induction and Indefinite Extensibility: The Gödel Sentence is True, But Did Someone Change the Subject? Mind 107: 597–624.
Shapiro, S. 2000. Thinking about Mathematics. Oxford: Oxford University Press.
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.
Struik, D. J. 1967. A Concise History of Mathematics. New York: Dover.
Troelstra, A. S. and Dalen, D. V. 1998. Constructivism in Mathematics: An Introduction, vol. 1. Amsterdam: North Holland.
Wittgenstein, L. 1981. Tractatus Logico-Philosophicus, Trans. C.K. Ogden. London: Routledge and Kegan Paul.
Wittgenstein, L. 1983. Remarks on the Foundations of Mathematics. 2nd edition, Trans. G.E.M. Anscombe. Cambridge, MA: MIT Press.
Wright, C. 1983. Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen Univ. Press.
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Hanna, R. (2010). Mathematical Truth Regained. In: Hartimo, M. (eds) Phenomenology and Mathematics. Phaenomenologica, vol 195. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3729-9_8
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