Abstract
We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics is correct. This is not to say that arguments for platonism — that is, arguments for the existence of abstract objects such as numbers, sets, Hilbert spaces, and so on that comprise the subject matter of mathematics — are either uninteresting or unneeded. Nevertheless, I am not personally interested in such arguments. The reason for this is simple: I have never2 doubted that there are abstract objects, and I have never doubted that mathematics is about (some of) them. While arguments for the existence of abstract objects are philosophically important, such debates hold little interest for me when my intuitions already fall so strongly on one side.
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References
Benacerraf, P. (1973), “Mathematical Truth”, Journal of Philosophy 70: 661–680.
Boghossian, P. (1997), “Analyticity”, in Hale and Wright [1997]: 331–367.
Boolos, G. (1997), “Is Hume’s Principle Analytic?”, in Boolos [1998]: 301–314, reprinted in Cook [2007]: 3–15.
Boolos, G. (1998), Logic, Logic, and Logic, Cambridge, MA, Harvard University Press.
Boolos, G. and Heck, R. (1998). “Die Grundlagen der Arithmetik §82 — 83”, in Boolos [1998]: 315–338.
Cook, R. (ed.) (2007), The Arché Papers on the Mathematics of Abstraction, Dordrecht, Springer.
Field, H. (1984), “Is Mathematical Knowledge Just Logical Knowledge?” The Philosophical Review XCIII: 509–552.
Fine, K. (2002), The Limits of Abstraction, Oxford, Oxford University Press.
Frege, G. (1893, 1903), Grundgezetze der Arithmetik I & II, Hildesheim, Olms.
Frege, G. (1979a), “Sources of Knowledge of Mathematics and the Mathematical Natural Sciences”, in Frege [1979b]: 269–270.
Frege, G. (1979b), Posthumous Writings, Chicago, University of Chicago Press.
Frege, G. (1980), Die Grundlagen der Arithmetic, J. Austin (trans.), Evanston, IL, Northwestern University Press.
Frege, G. (1997), The Frege Reader, M. Beaney (ed.), Oxford, Blackwell.
Hale, R. (2000), “Reals by Abstraction”, Philosophia Mathematica 8: 100–123, reprinted in Cook [2007]: 175–196.
Hale, R. and Wright, C. (1997), A Companion to the Philosophy of Language, Oxford, Blackwell.
Hale, R. and Wright, C. (2001), The Reason’s Proper Study, Oxford, Oxford University Press.
Heck, R. (ed.) (1997), Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford, New York: Oxford University Press.
Heck, R. (1993), “The Development of Arithmetic in Frege’s Grundgesetze der Arithmetik”, Journal of Symbolic Logic 10: 153–17
Heck, R. (2005), “Julius Caesar and Basic Law V”, Dialectica 59: 161–178.
Kant, I. (1965), The Critique of Pure Reason, Norman Smith (trans.), New York, St Martins Press.
MacBride, F. (2003), “Speaking with Shadows: A Study of Neo-Fregeanism”, British Journal of the Philosophy of Science 54: 103–163.
MacFarlane, J. (2002), “Frege, Kant, and the Logic in Logicism”, The Philosophical Review 111: 25–65.
Quine, W. (1960), “Two Dogmas of Empiricism”, The Philosophical Review 60: 20–43.
Russell, B. (1902), “Letter to Frege”, in van Heijenoort [1967]: 124–125.
Shapiro, S. (1991), Foundations Without Foundationalism: The Case for Second-order Logic, Oxford, Oxford University Press.
van Heijenoort, J. (1967), From Frege to Gödel: A Sourcebook in Mathematical Logic, Cambridge MA, Harvard University Press.
Weir, A. (2004), “Neo-Fregeanism: An Embarassment of Riches”, Notre Dame Journal of Formal Logic 44: 13–48, reprinted in Cook [2007]: 383–420.
Wright, C. (1997), “On the Philosophical Significance of Frege’s Theorem”, in Heck [1997]: 201–244, reprinted in Hale and Wright [2001]: 272–306.
Wright, C. (1999), “Is Hume’s Principle Analytic?”, Notre Dame Journal of Formal Logic 40: 6–30, reprinted in Hale and Wright [2001]: 307–333 and in Cook [2007]: 17–43.
Wright, C. (2000), “Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint”, Notre Dame Journal of Formal Logic 41: 317–334, reprinted in Cook [2007]: 253–272.
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© 2009 Roy T. Cook
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Cook, R. (2009). New Waves on an Old Beach: Fregean Philosophy of Mathematics Today. In: Bueno, O., Linnebo, Ø. (eds) New Waves in Philosophy of Mathematics. New Waves in Philosophy. Palgrave Macmillan, London. https://doi.org/10.1057/9780230245198_2
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DOI: https://doi.org/10.1057/9780230245198_2
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