7AAN2056: Philosophy of Mathematics Syllabus Academic year 2016/17

School of Arts & Humanities Department of Philosophy 7AAN2056: Philosophy of Mathematics Syllabus Academic year 2016/17 Basic information Credits: 20 Module Tutor: Dr Tamsin de Waal Office: Rm 702 Consultation time: TBC Semester: 1 Lecture time and venue: Tuesday 16:00-17:00, St David s Rm, King s Building, Strand Campus Class time and venue: Tuesday 17:00-18:00, Rm 306, Philosophy Building, Strand Campus Module description What is the subject matter of mathematics? Is it abstract mathematical objects, or can apparent facts about mathematical objects be reduced to facts about something else? Assuming we have knowledge of mathematical facts, how is this knowledge acquired? Despite being essential to the sciences (and often thought of as one of the sciences), the non- empirical nature of mathematics raises epistemological and metaphysical questions quite distinct from those that arise in, say, physics. This course will examine approaches to answering these questions, including varieties of Platonism, and various forms of nominalism. We ll also take a close look at the role of mathematics in the sciences, with the aim of evaluating one of the key arguments in the debate between the Platonist and the nominalist: the indispensability argument. In the last two weeks of the course, we ll focus on the role of representations, such as geometric diagrams, in mathematical reasoning. All students are required to prepare the required reading in advance of both the lecture and seminar each week. See the outline below for details. Key texts books to consider purchasing: Shapiro, S., Thinking About Mathematics (OUP 2000). Brown, J.R., Philosophy of Mathematics (Routledge 2008). Colyvan, M., An Introduction to the Philosophy of Mathematics (CUP 2012). Benacerraf, P., and Putnam, H., Philosophy of Mathematics: Selected Readings, 2 nd ed. (CUP 1983). S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP 2005). 1

Assessment Methods and Deadlines Formative assessment: one 2,000-3,000- word essay due Friday December 9 th 2016, by 16:00. Summative assessment: one 4,000- word essay due Wednesday January 18 th 2017, by 16:00. (This may answer the same question as the formative essay. Summative essays should be submitted via KEATS). Lecture Outline (plus suggested readings) Week One (Sept 27 th ), Introduction: Philosophy and Mathematics Brown, J.R., Philosophy of Mathematics (Routledge 2008), ch.1. Shapiro, S., Thinking about Mathematics (OUP 2000), ch.s.1-2. Week Two (Oct 4 th ), Plato S. Shapiro, Thinking about Mathematics (OUP, 2000), ch.3. Plato s Meno, 82b- 85d. Plato s Republic, Bk VI, 509d- 511e. Aristotle s Metaphysics, Bks M and N, trans. J. Annas (OUP 1988) (and cf. Annas intro). Annas, J., An Introduction to Plato s Republic (OUP 1981), ch.11. Mueller, I., Mathematical method and philosophical truth, in R. Kraut, ed., The Cambridge Companion to Plato (CUP 1992), 170-99. Denyer, N., Sun and Line: The Role of the Good, in G.R.F Ferrari, ed., The Cambridge Companion to Plato s Republic (CUP 2007), 284-309. Burnyeat, M.F., Plato on why mathematics is good for the soul, in Proceedings of the British Academy 103, 1-82. Reprinted in T. Smiley, ed., Mathematics and Necessity (OUP 2001). Pritchard, P., Plato s Philosophy of Mathematics (Academia- Verlag 1995). Wedberg, A. Plato s Philosophy of Mathematics (Almqvist & Wiksell 1955). Week Three (Oct 11 th ), Modern Platonism: Gödel, Maddy, and the Epistemological Challenge Shapiro, S., Thinking about Mathematics (OUP 2000), ch.8. Cf. also Brown, J.R., Philosophy of Mathematics (Routledge 2008), ch.2. Maddy, P., Perception and Intuition in Mathematics, Philosophical Review (1980). Benacerraf, P., Mathematical Truth, Journal of Philosophy, 70 (19) (1973), 661 80. (Also in BP). Gödel, K., Russell's Mathematical Logic (1944), and What is Cantor's Continuum Problem? (1964), in Benacerraf and Putnam (1983), 447-485. See also Some basic theorems on the foundations of mathematics and their implications. Maddy, P., Realism in Mathematics (OUP 1990). Parsons, C., Mathematical Intuition, Proceedings of the Aristotelian Society, 80 (1979), 142-168. Benacerraf, P., What mathematical truth could not be, in A. Morton and S. Stich, eds., Benacerraf and His Critics (Blackwell 1996). Page 2

Linnebo, Ø., Epistemological Challenges to Mathematical Platonism, Philosophical Studies, vol. 129, no. 3 (2006), 545-74. Balaguer, M., Platonism and Anti- Platonism in Mathematics (OUP 1998). Restall, G., Just what is full- blooded platonism?, Philosophia Mathematica 11, no. 1 (2003), 82-91. Week Four (Oct 18 th ), Modern Platonism: The Indispensibility Argument Colyvan, M., An Introduction to the Philosophy of Mathematics (CUP 2012), ch 3. Putnam, H., Philosophy of Logic (Harper & Row 1971), VIII. Maddy, P., Indispensability and Practice, Journal of Philosophy, vol. 89, no. 6 (1992), 275-289. Quine, W.V.O., On What There Is, and Two Dogmas of Empiricism, in From a Logical Point of View, 2nd edition (Harvard University Press 1980). See also Carnap and Logical Truth in BP, 355-76. Colyvan, M., The Indispensability of Mathematics (OUP 2001). Field, H., Realism, Mathematics and Modality (Blackwell 1989), esp. introduction. Maddy, P., Naturalism in Mathematics (OUP 1997). See also Naturalism and Ontology, Philosophia Mathematica 3 (3) (1995), 248 270. Also Three Forms of Naturalism, in S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP, 2005), 437-459. Sober, E., Mathematics and Indispensability, Philosophical Review, vol. 102, no.1 (1993), 35-57. Resnik, M.D., Scientific vs. Mathematical Realism: The Indispensibility Argument, Philosophia Mathematica, 3 (2) (1995), 166-74. Cf. also Quine and the Web of Belief, in S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP, 2005), 412-436. Week Five (Oct 25 th ), Logicism Shapiro, S., Thinking about Mathematics (OUP 2000), ch.5. Frege, G., The Foundations of Arithmetic, trans. J.L. Austin (Blackwell 1950), 55-91, 106-109 (or see Benacerraf, P., and Putnam, H., eds., Philosophy of Mathematics (CUP 1983), 130-159). Russell, B., Introduction to Mathematical Philosophy (Allen and Unwin 1919), 1-19, 194-206 (or see Benacerraf, P., and Putnam, H., eds., Philosophy of Mathematics (CUP 1983), 160-182). Russell, B., Letter to Frege (1902) and reply, in J. van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic (Harvard University Press 1967), 124-128. George, A., and Velleman, D.J., Philosophies of Mathematics (Blackwell 2002), ch.s.2-3. Giaquinto, M., The Search for Certainty: A Philosophical Account of Foundations of Mathematics (OUP 2002), ch.s.2-3. Potter, M., Reason s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap (OUP 2000), chs.2-5. Urquhart, A., The Theory of Types, in N. Griffin, ed., The Cambridge Companion to Bertrand Russell (CUP 2003), 286-309. See also Beaney, M., Russell and Frege, 128-70. Dummett, M., Frege: Philosophy of Mathematics (Duckworth 1991). Demopoulos, W., and Clark, P., The Logicism of Frege, Dedekind, and Russell, in S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP 2005), 129-165. Wright, C., and Hale, R., `Logicism in the twenty- first century' in S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP, 2005), 166-202. Page 3

Wright, C., and Hale, R., The Reason s Proper Study: Essays towards a Neo- Fregean Philosophy of Mathematics (OUP 2001). READING WEEK (OCT 31 ST NOV 4 TH ) Week Six (Nov 8 th ), Nominalism (Guest lecturer: John Heron) Colyvan, M., An Introduction to the Philosophy of Mathematics (CUP 2012), ch.4. Leng, M., Mathematics and Reality (OUP 2010), ch.3, 45-75. Colyvan, M., There is no easy road to nominalism, Mind, vol. 119, no. 474 (2010), 285-386. Shapiro, S., Thinking about Mathematics (OUP 2000), ch.9. Field, H., Science without Numbers (Princeton University Press 1980) (2 nd ed. OUP). Burgess, J.P., and Rosen, G.A., A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics (OUP 1997). Cf. also Nominalism Reconsidered, in S. Shapiro, ed., The Oxford Handbook of Mathematics and Logic (OUP 2005), 515-535. Burgess, J.P., 'Mathematics and Bleak House', Philosophia Mathematica 12 (1) (2004), 18-36. Yablo, S., 'The Myth of the Seven', in M. Kalderon, ed., Fictionalism in Metaphysics (OUP 2005), 88-115. Melia, J., 'Weaseling Away the Indispensability Argument', Mind 109 (2000), 453-79. Chihara, C.S., Constructibility and Mathematical Existence (OUP 1990). Cf. also Nominalism, in S. Shapiro, ed., The Oxford Handbook of Mathematics and Logic (OUP 2005), 483-514. Week Seven (Nov 15 th ), Structuralism. Shapiro, S., Thinking about Mathematics (OUP 2000), ch.10. Benacerraf, P., What numbers could not be, Philosophical Review, 74 (1965), 47 73. Resnik, M.D., Mathematics as a Science of Patterns: Ontology and Reference, Nous 15 (1981), 529-550. See also Mathematics as a Science of Patterns: Epistemology, Nous 16 (1982), 95-105. Resnik, M.D., Mathematics as a Science of Patterns (OUP 1997). Shapiro, S., Philosophy of Mathematics: Structure and Ontology (OUP 1997). See also Mathematics and Reality, Philosophy of Science, 50 (1983), 523-448. Hellman, G., Mathematics without numbers: Towards a Modal- Structural Interpretation (OUP 1989). See also Structuralism, in S. Shapiro, ed., The Oxford Handbook of Mathematics and Logic (OUP 2005), 536-62. Parsons, C., The Structuralist View of Mathematical Objects, Synthese 84 (1990), 303-346. McLarty, C., What Structuralism Achieves, in P. Mancosu, ed., The Philosophy of Mathematical Practice (OUP 2008), 354-69. Hale, R., Structuralism s unpaid epistemological debts, Philosophia Mathematica, 4 (2) (1996), 124-147. MacBride, F., Can Structuralism solve the Access Problem?, Analysis 64.4 (2004), 309-317. Cf. also Structuralism Reconsidered, in S. Shapiro, ed., The Oxford Handbook of Mathematics and Logic (OUP 2005), 563-89. Page 4

Week Eight (Nov 22 nd ), The Applicability of Mathematics. Colyvan, M., An Introduction to the Philosophy of Mathematics (CUP 2012), chs.5-6. Baker, A., Are there genuine mathematical explanations of physical phenomena?, Mind vol.114 (2005), 223-38. Wigner, E.P., The unreasonable effectiveness of mathematics in the natural sciences. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959. Communications on Pure and Applied Mathematics 13, no. 1 (1960), 1-14. Grattan- Guinness, I., Solving Wigner s Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences, Mathematical Intelligencer, vol.30 (2008), 7-17. Steiner, M., The Application of Mathematics to Natural Science, Journal of Philosophy, vol.86 (1989), 449-80. See also The Applicabilities of Mathematics, Philosophia Mathematica, vol.3 (1995), 129-56. Pincock, C., A new perspective on the problem of applying mathematics, Philosophia Mathematica (3) vol. 12 (2004), 135 161. Batterman, R.W., On the explanatory role of mathematics in empirical science, British journal for the Philosophy of Science vol. 61, no.1 (2010), 1-25. Mancosu, P., Mathematical explanation: Why it matters, in P. Mancosu, ed., The Philosophy of Mathematical Practice (OUP 2008), 134-49. Week Nine (Nov 29 th ), The Role of Representations in Mathematics: Plato and Kant Shapiro, S., Thinking about Mathematics (OUP 2000), ch.4. Kant, I., Critique of Pure Reason, A1-16/B1-30, A162-6/B202-7, A708-38/B736-66, trans. and eds. P. Guyer & A. Wood (CUP 1998). For Plato, the reading for wk 2 is relevant again here. Cf. also Republic, Bk VII, 528e- 531d (and cf. esp. the papers in J.P. Anton, ed., Science and the Sciences in Plato (Caravan Books 1980) on this passage). Brittan, G., Kant s Philosophy of Mathematics, in G. Bird, ed., A Companion to Kant (Blackwell 2006), ch.15. Posy, C.J., ed., Kant s Philosophy of Mathematics (Kluwer 1992). Shabel, L., Apriority and Application: Philosophy of Mathematics in The Modern Period, in S. Shapiro, ed., The Oxford Handbook of Mathematics and Logic (OUP 2005), 29-50. See also Kant on the symbolic construction of mathematical concepts, in Studies in History and Philosophy of Science, Part A 29 (4) (1998), 589-621. Shabel, L., Mathematics in Kant's Critical Philosophy: Reflections on Mathematical Practice (Routledge 2003). Friedman, M., Kant and the Exact Sciences (Harvard University Press 1992), Part I. See also Geometry, Construction, and Intuition in Kant and His Successors, in G. Scher and R. Tieszen, eds., Between Logic and Intuition: Essays in Honor of Charles Parsons (CUP 2000), 186 218. Also Kant on geometry and spatial intuition, Synthese vol. 186, no.1 (2012), 231-255. Carson, E., Kant on Intuition in Geometry, Canadian Journal of Philosophy, 27 (4) (1997), 489 512. Page 5

Week Ten (Dec 6 th ), The Role of Representations in Mathematics: Contemporary views Brown, J.R., Philosophy of Mathematics (Routledge 2008), esp. chs.3 and 6. Giaquinto, M., Visualizing in mathematics, in P. Mancosu, ed., The philosophy of mathematical practice (OUP 2008), 23 33. Colyvan, M., An Introduction to the Philosophy of Mathematics (CUP 2012), ch.8. Giaquinto, M., Visual Thinking in Mathematics (OUP 2007). Mancosu, P., Jørgensen, K.F., Pedersen, S.A., eds., Visualization, Explanation and Reasoning Styles in Mathematics (Springer- Verlag 2005). Barwise, J. and Etchemendy, J., Visual information and valid reasoning, in G. Allwein & J. Barwise, eds., Logical Reasoning with Diagrams (OUP 1996), 3-26. De Cruz, H., and De Smedt, J., Mathematical symbols as epistemic actions, Synthese, vol.190 (1) (2013), 3-19. Greaves, M., The Philosophical Status of Diagrams (CSLI Publications 2002). Manders, K., The Euclidean Diagram (1995), in P. Mancosu, ed., The philosophy of mathematical practice (OUP 2008), 80-133. Lakatos, I., Proofs and Refutations: The Logic of Mathematical Discovery (CUP 1976). Detlefsen, M., Formalism, in S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP 2005), 237-317. On aspects of the Philosophy of Mathematics that we won t be covering in detail on this course, in particular, finitism and intuitionism, see esp. George, A., and Velleman, D.J., Philosophies of Mathematics (Blackwell 2002). Also S. Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and Logic (OUP 2005), and Shapiro, S., Thinking about Mathematics (OUP 2000). The Stanford Encyclopedia of Philosophy is a good online resource plato.stanford.edu Page 6