Philosophy of Mathematics
Readings marked with a * are particularly important. Those marked with a ! are ones that I find interesting.
- Collections and Introductions
Topics:
- Logicism: a) Frege b) Russell c) Neo-Fregeans
- Intuitionism
- Formalism: Hilbert's Programme (including Godel's Incompleteness Theorems)
- The Rise of Set Theory
- Empiricism
- Structuralism
Ontology and Epistemology
- Platonism
- Indispensibility
- Fictionalism
- Mathematics and Knowledge
- Lakatos
Historical Figures
- Plato and Aristotle
- Kant
- Mill
- Wittgenstein
i. Collections and Introductions
The following are very useful collections of papers on the philosophy of mathematics that I will sometimes pick articles from:
Van Heijenoort ed. From Frege to Gödel, a source book in Mathematical Logic 1879-1931 (Harvard UP, 1967) [Referred to below as "van heijnoort"]
*Benacerraf, P. & Putnam H. eds Philosophy of Mathematics, selected readings (2nd edition, Cambridge UP, 1983) [There are two editions of this book, with slightly different essays in each one. I will usually refer to the 2nd edition as "B&P".]
Hart, W.D. ed The Philosophy of Mathematics (OUP, 1996) ["Hart"]
!!!Boolos, George., Logic, Logic and Logic (Harvard UP, 1998) ["LL&L". Personally one of my favourite books in philosophy (and I don't even like logic really!) Can be more technical than others but also more insightful]
This next book is a introductory book that divides people who teach the philosophy of mathematics. Personally I like it since it gives brief, lucid and understandable overviews of topics that are often quite difficult to understand from the source material. The worry is that students will rely on it too much: try as hard as you can to avoid doing this, I know the book pretty well so I'll be able to tell! I'll refer to it as "Shapiro" in what follows.
Shapiro, Stewart. Thinking about mathematics: the philosophy of mathematics (OUP, 1991) ["Shapiro"]
Also, the Stanford Encyclopedia has some fantastic articles on some of these topics (in particular those by Edward Zalta on Frege).
*Shapiro, Chapter 5
Essay Titles: Is mathematics a branch of logic? Did either of Russell or Frege establish logicism? ‘Arithmetic may not be logic, but it is analytic.’ Discuss. Explain the view that mathematics is really logic in light of the fact that mathematics, unlike logic, is about specific entities such as numbers and functions. Why is Russell's paradox a threat to this view?
*Frege, G. The Foundations of Arithmetic. trans. J. L. Austin (Oxford: Blackwell, 1950). [auf Deutsch: Die Grundlagen der Arithmetik] esp. secs. 55-91, 106-9. These sections reprinted as Frege, G, "The Concept of Number" in B&P.
Parsons, Charles. 1965, ‘Frege's Theory of Number’, in M. Black (ed.), Philosophy in America, Ithaca: Cornell, 180-203. Reprinted in Parsons, Charles. Mathematics in Philosophy: Selected Essays (Cornell, 1983)
*Russell, B. "Letter to Frege" and Frege's reply in Van Heijenoort
!Boolos, G., 'The Consistency of Frege's Foundations of Arithmetic', in J. J. Thomson (ed.), On Being and Saying, Cambridge, MA: MIT Press, pp. 3-20 Reprinted in Hart and LL&L
Essay Titles: How Successful was Frege's attempt to reduce mathematics to logic? Did Frege present good grounds for his view of numbers in The Foundations of Arithmetic?
*Russell, B. Introduction to Mathematical Philosophy (George, Allen & Unwin, 1919) esp. chs 1-3, 12-13, Reprinted in B&P
Russell, B. "Mathematical Logic as Based on the Theory of Types," American Journal of Mathematics, 30, 222-262. Reprinted in Russell, B. Logic and Knowledge (London: Allen and Unwin) 1956, 59-102, Reprinted in van Heijenoort.
!Ramsey, F.P. ‘The Foundations of Mathematics’, in The Foundations of Mathematics and Other Logical Essays, ed. R.B. Braithwaite, (London: Routledge & Kegan Paul, 1931)
Chihara, C.S. Ontology and the Vicious-Circle Principle (Cornell UP, 1973), Chapter 1: "Russell's solution to the paradoxes".
Essay titles: What problems led Russell to develop type theory? Does Russell deal with these problems satisfactorily? Explain and evaluate Russell's simple theory of types and his justification for it.
*Hale, B., and C. Wright, "Introduction", in The Reason's Proper Study (Oxford: Oxford University Press, 2001).
Boolos, G., ‘Saving Frege From Contradiction’, in Proceedings of the Aristotelian Society, 87 (1986/1987): 137 - 151; Reprinted in LL&L
!Boolos, G., 1997, ‘Is Hume's Principle Analytic?’, in Heck, R., (ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett, (Oxford: Oxford University Press, 1997) 245-262; Reprinted in LL&L
Demopoulos, W., (ed.), 1995, Frege's Philosophy of Mathematics, Cambridge: Harvard University Press. [You don't need to read all of the essays in here! Just anything that strikes you as interesting]
Wright, C., 1997, ‘On the Philosophical Significance of Frege's Theorem’, in in Heck, R., (ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett, (Oxford: Oxford University Press, 1997) 245-262;
Essay Titles: ‘Arithmetic may not be logic, but it is analytic.’ Discuss.
Initial reading
*Brouwer, LEJ. 1913, "Intuitionism and Formalism", Bulletin of AMS 20; reprinted in B&P
Shapiro, S., "Intuitionism: Is something wrong with our logic?", Thinking about mathematics: The philosophy of mathematics, Chapter 7, (OUP:1998)
Heyting, A. Intuitionism: an introduction, (Amsterdam: North Holland, 1956), Chapter 1, "Dispution"; reprinted in B&P
Dummett, M. 'The Philosophical Basis of Intuitionistic Logic', in Truth and other Enigmas, (London: Duckworth 1978)
Essay Title: Do the intuitionists provide us with any good reason to reject the law of the excluded middle?
Further Material
Brouwer, LEJ. 1928, "Mathematics, science and language", Monatshefte fur Mathematik und Physik 36; English Translation by Walter P. van Sigt in Paolo Mancosu (ed.) From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s (OUP: Oxford, 1998)
Heyting, A. 1931 "Die intuitionistische Grundlegung der Mathematik", Erkenntnis 2; English translation by Erna Putnam and Gerald J. Masey, "The intuitionistic foundations of mathematics" in B&P
!Dummett: Elements of Intuitionism, 2nd edn., (Oxford: OUP 2000) Introductory Remarks, Chapter 1
Parsons, C. 1980, "Mathematical Intuition", Proceedings of the Aristotelian Society 80, pp.145-168
van Sigt, WP. "Brouwer's intuitionist programme" in Paolo Mancosu (ed.) From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s (OUP: Oxford, 1998)
iv. Formalism: Hilbert's Programme
Initial Reading:
Hilbert, D., "On the infinite", English Translation by Stedan Bauer-Mengelberg, Jean van Heijenoort (ed.) From Frege to Godel: A source book in mathematical logic 1878-1931 (HUP: Cambridge, Mass., 1967) 367-384 (Omit the last part of the paper where Hilbert sketches a proof of Cantor's continuum hypothesis, which is unsuccesful)
Bernays, P., "The philosophy of mathematics and Hilbert's proof theory" in Paolo Mancosu (ed.) From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s (OUP: Oxford, 1998)
*Macousu, P., "Hilbert and Bernays on metamathematics", in Paolo Mancosu (ed.) From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s (OUP: Oxford, 1998)
Shapiro, S., "Formalism: do mathematical statements mean anything?", Thinking about mathematics: The philosophy of mathematics, Chapter 6, (OUP:1998)
Extra reading:
Giaquinto, M., "Hilbert's philosophy of mathematics", British Journal for the Philosophy of Science 34 (1983), pp.119-32 Kreisel, G., "Hilbert's Programme", Dialectica 12 (1958), pp.346-372; revised version in B&P
Smorynski, C., "The incompleteness theorems", Sections 1 and 2, Jon Barwise (ed.) Handbook of mathematical logic, (North-Holland Publishing Co: Amsterdam, 1977) pp.821-9
*Craig Smorynski, "Hilbert's Programme", CWI Quarterly, Vol. 1, (1988) pp3-59 (Ask me for this)
Sieg, W., "Hilbert's program sixty years later", The Journal of Symbolic Logic 53 (1988), pp.338-48
Essay Title: Explain Hilbert's Programme. Was it successful?
George Boolos "The Iterative Conception of Set" The Journal of Philosophy, Vol. 68, No. 8, Philosophy of Logic and Mathematics. (Apr. 22, 1971), pp. 215-231. Here (Jstor)
*Benacerraf, P., 'What Numbers Could Not Be', Philosophical Review, 74 (1965), 47-73. Reprinted in B&P
Shapiro, S., 1997. Philosophy of Mathematics: Structure and Ontology, Oxford: Oxford University Press Chapter 3 Online
Hellman, G., 1989. Mathematics without Numbers, Oxford: Clarendon Press Chapter 1 Online
Isaacson, D. “Mathematical intuition and objectivity”, Alexander George (ed.), Mathematica and Mind, Oxford University Press, 1994, pp. 118-140 Also in Hart
Resnik, M. 1982. ‘Mathematics as a Science of Patterns: Epistemology’. Noûs 16: 95-105.
——. 1981. ‘Mathematics as a Science of Patterns: Ontology and Reference’. Noûs 15: 529-566.
Essay Question: What is a "structure"? Assess the view that mathematics is the science of structures?
Ontology and Epistemology
Quine, W. V., ‘Carnap and Logical Truth’ (1954) in Ways of Paradox and other Essays (New York: Random House, 1966) Reprinted in B&P
*Parsons, C., ‘Quine’s Philosophy of Mathematics’ in L. E. Hahn, and P. A. Schilpp, eds., The Philosophy of W. V. Quine (La Salle, Ill.: Open Court, 1985)
*Quine, W. V., ‘Reply to Charles Parsons’ in L. E. Hahn, and P. A. Schilpp, eds., The Philosophy of W. V. Quine (La Salle, Ill.: Open Court, 1985)
Maddy, P., 1992, ‘Indispensability and Practice’ Journal of Philosophy 89: 275-289.
Sober, E., 1993, ‘Mathematics and Indispensability’. Philosophical Review 102: 35-57.
Essay Titles: Is mathematics indispensible to science?
*Field, H., Science without Numbers (Oxford: Blackwell, 1980)
*Yablo, Stephen, 'Go Figure: A path through fictionalism', to appear in Midwest Studies of Philosophy, available here, Sections 1-6
Feferman, S. ‘Why a Little Bit goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics’ in In the Light of Logic (Oxford: Oxford University Press, 1998) Ch.14.
Papineau, D., Philosophical Naturalism (Oxford: Blackwell, 1993) Ch.5.
Shapiro, S. 1983. ‘Conservativeness and Incompleteness’. Journal of Philosophy 80: 521-530; reprinted in Hart
Essay Titles: EITHER (a) Does the evidence that supports a testable part of our best total scientific
view of the world also support the mathematics needed in that part?
OR (b) If it turned out that empirical science could in principle dispense with the
natural numbers in favour of a system of spatial points, should we then take
it that numbers do not exist?
xi. Mathematical and Knowledge
*Benacerraf, P., 'Mathematical Truth' (1983) in B&P
!Maddy, P., ‘Perception and Mathematical Intuition’. Philosophical Review (1980) 84: 163-96. Reprinted in Hart
Maddy, P. Realism in Mathematics (Oxford: Clarendon Press, 1990)
Field, H., ‘Realism, Mathematics & Modality’. In Realism, Mathematics & Modality (Oxford: Blackwell, 1989) Sec. 2.
Azzouni, J., Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences, (Cambridge: CUP, 1994), Sections 1.7-1.9
Essay Titles: Is Benacerraf right in claiming that mathematical truth and mathematical knowledge exclude each other or can you think of a way out of Benacerraf’s Dilemma? Assess the view that our knowledge of basic arithmetic is acquired by inductive generalization from the evidence of the senses.
*Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery, John Worrall and Elie Zahar (eds), Cambridge University Press, (1976)
*Larvor, Brendan. Lakatos: An Introduction, Routledge (1998), Chapters 1 and 2
Lakatos, Imre. “What does a mathematical proof prove?”, published posthumously in John Worrall and Gregory Currie (eds) Imre Lakatos Philosophical Papers volume 2: Mathematics, science and epistemology, Cambridge University Press, (1978)
Larvor, Brendan. “Lakatos as historian of mathematics”, Philosophia Mathematica 5 (1997)