PHIL 1300: Philosophy of Mathematics Josh Schechter MWF 11:00–11:50 AM Fall 2013
Course Information
Instructor Josh Schechter Office: Corliss-Bracket, 45 Prospect St., Room 307 Office Hours: 1–3 PM Mondays (and by appointment) Email: [email protected] Phone: 863-3225 (office)
Course Description This course provides an introduction to the philosophy of mathematics. We will discuss the nature of mathematical objects: Are they mental constructions, do they inhabit some Platonic realm, or are there no mathematical objects at all? We will also discuss the status of our knowledge of mathematics: How is it that we know mathematical truths? How is that we are justified in reasoning as we do in mathematics?
The course will be divided into two main parts. The first part will be devoted to discussing the history of the philosophy of mathematics. We will read works by Plato, Kant, Mill, Frege, Russell, Heyting, and Brouwer, among others. This part of the course will culminate in a discussion of the important developments in the late nineteenth and early twentieth centuries, with the emergence of Formalism, Intuitionism, and Logicism. In the second part of the course, we will turn to contemporary debates in the philosophy of mathematics. We will discuss Fictionalism, neo-Fregeanism, Plenituninous Platonism, and Structuralism, among other views.
Several sessions of the course will involve discussion of technical issues in logic and mathematics. The relevant results will be explained without presupposing any specific background knowledge. However, it is recommended that students have taken a course in logic, mathematics, or a related subject prior to taking this course.
Course Requirements There will be two 4-6 page papers, one 9-12 page paper, and a final exam.
Each short paper will be worth 15% of your grade. The final paper will be worth 30% of your grade. The final exam will be worth 30%. Class participation will be worth the remaining 10% of your grade. Required Texts The following required books are available at the bookstore: • (Shapiro) Stewart Shapiro (2000) Thinking about Mathematics: The Philosophy of Mathematics (Oxford, Oxford University Press); • (B&P) Paul Benacerraf and Hilary Putnam, eds. (1983) Philosophy of Mathematics: Selected Readings , 2nd edition (Cambridge, Cambridge University Press); nd • (Frege) Gottlob Frege (1884) The Foundations of Arithmetic . Translated by J. L. Austin, 2 revised edition (Northwestern University Press, 1980).
Additional readings will be available on OCRA or distributed in class. The OCRA password is “math”.
Texts on Reserve The following books are on reserve at the Rock: • Paul Benacerraf and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings , 2nd edition • Laurence BonJour, In Defense of Pure Reason • John Burgess and Gideon Rosen, A Subject with No Object • Hartry Field, Science without Numbers • Gottlob Frege, The Foundations of Arithmetic • Immanuel Kant, Prolegomena to Any Future Metaphysics • Penelope Maddy, Realism in Mathematics • A.W. Moore, The Infinite • Plato, Republic • Stewart Shapiro, Thinking about Mathematics: The Philosophy of Mathematics
Tentative Due Dates Paper 1: October 9th Paper 2: November 13th Final Paper: December 20 th
Final Exam: Friday, December 20 th at 2 PM PHIL 1300: Philosophy of Mathematics Josh Schechter MWF 11:00–11:50 AM Fall 2013
(Tentative) Schedule of Readings
Introduction Wednesday, September 4th Introduction to the Philosophy of Mathematics
Friday, September 6 th No Class
Monday, September 9 th Philosophical Puzzles about Mathematics (Shapiro) chapters 1-2 Optional: (OCRA) Mark Balaguer, “A Guide for the Perplexed”
Plato, Kant, and Mill Plato Wednesday, September 11 th Plato (OCRA) Plato, Meno [focus on pp. 71 through 78]
Friday, September 13th Plato on Mathematics (Shapiro) sections 3.1–3.2 (OCRA) Plato, Republic , book 6 [focus on the bottom of 199 through 207]
Monday, September 16th Plato on Mathematics II (Shapiro) section 3.3 (OCRA) Plato, Republic , book 7
Wednesday, September 18th Aristotle on Mathematics (Shapiro) section 3.4 (OCRA) Aristotle, Metaphysics M, parts 1-3 Optional: (OCRA) David Bostock, “Aristotle’s Philosophy of Mathematics”
Kant Friday, September 20 th Kant’s Challenge: The Synthetic A Priori (Shapiro) chapter 4.1–2 (OCRA) Immanuel Kant, Prolegomena to Any Future Metaphysics (excerpts)
Monday, September 23rd Kant on Geometry and Arithmetic and Geometry (OCRA) Lisa Shabel, “Kant's Philosophy of Mathematics” Optional: (OCRA) Philip Kitcher, “Kant and the Foundations of Mathematics” Optional: (OCRA) Michael Friedman, “Kant’s Theory of Geometry” Mill Wednesday, September 25th Mill’s Empiricism (Shapiro) chapter 4.3 (OCRA) J. S. Mill, A System of Logic Ratiocinative and Inductive , Book II: V–VI
Friday, September 27 th No Class
Monday, September 30 th Challenges for Mill (Frege) sections 1–17, 23–25.
The Big Three: Logicism, Formalism, and Intuitionism Logicism – Frege and Russell Wednesday, October 2 nd Frege’s Logicism (Shapiro) section 5.1 (Frege) sections 55–67
Friday, October 4th Frege’s Logicism II
Monday, October 7th The Julius Caesar Problem (Frege) sections 68–83
Wednesday, October 9 th Set Theory and Russell’s Paradox Notes on Set Theory
Friday, October 11th Russell’s Logicism (Shapiro) section 5.2 (B&P) Russell, Introduction to Mathematical Philosophy (excerpts) (OCRA) Soames, Philosophical Analysis in the Twentieth Century , vol. 1, ch. 6
Monday, October 14 th No Class
Wednesday, October 16th Logicism Assessed
Formalism and Hilbert’s Program Friday, October 18th Term Formalism and Game Formalism (Shapiro) sections 6.1–6.2 Frege, Grundgesetze der Arithmetik , 86-137
Monday, October 21 st Hilbert’s Programme (Shapiro) section 6.3 (B&P) John von Neumann, “The Formalist Foundations of Mathematics” (B&P) David Hilbert, “On the Infinite”
Wednesday, October 23rd Formalism and Gödel’s Incompleteness Theorems (Shapiro) section 6.4 Notes on the Incompleteness Theorems Optional: (OCRA) Richard Zach, “Hilbert’s Program”
Friday, October 25th Hilbert’s Programme II
Monday, October 28th Formalism Assessed
Intuitionism – Brouwer, Heyting, and Dummett Wednesday, October 30 th Introduction to Intuitionism (Shapiro) sections 7.1–7.2
Friday, November 1st Brouwer’s Intuitionism (B&P) L. E. J. Brouwer, “Intuitionism and Formalism” (B&P) L. E. J. Brouwer, “Consciousness, Philosophy, and Mathematics” Optional: (OCRA) Michael Detlefsen, “Brouwerian Intuitionism”
Monday, November 4th Heyting’s Intuitionism (Shapiro) section 7.3 (B&P) Arend Heyting, “The Intuitionist Foundations of Mathematics” Notes on Intuitionist Real Analysis
Wednesday, November 6th Dummett’s Intuitionism (Shapiro) section 7.4 (B&P) Michael Dummett, “The Philosophical Basis of Intuitionistic Logic”
Friday, November 8th Intuitionism Assessed
Logical Positivism Monday, November 11th Verificationism and Conventionalism (Shapiro) section 5.3 (B&P) A. J. Ayer, “The Apriori” (B&P) Rudolf Carnap, “Empiricism, Semantics, and Ontology”
Wednesday, November 13th Quine against Positivism (B&P) W. V. Quine, “Truth by Convention” Optional: Paul Horwich, “Conventionalism”
Recent Developments Friday, November 15th The Benacerraf-Field Problem (B&P) Paul Benacerraf, “Mathematical Truth” (OCRA) Hartry Field, Introduction to Realism, Mathematics and Modality
Monday, November 18 th Responses to the Benacerraf-Field Problem (OCRA) John Burgess and Gideon Rosen, A Subject with No Object (excerpts)
Wednesday, November 20th Rational Insight (Shapiro) section 8.1 (B&P) Kurt Gödel, “What is Cantor’s Continuum Hypothesis?”, pp. 482–5 Optional: (OCRA) Charles Parsons, “Reason and Intuition”
Friday, November 22 nd Rational Insight II (OCRA) Lawrence BonJour, In Defense of Pure Reason, chapters 4 and 6
Monday, November 25 th The Indispensability Argument; Quine’s Naturalism (Shapiro) section 8.2 (OCRA) Mark Colyvan, “Indispensability Arguments” W.V. Quine, “Two Dogmas of Empiricism” Optional: W. V. Quine, Word and Object (excerpts) Optional: Hilary Putnam, “Philosophy of Logic”
Wednesday, November 27 th Maddy’s Naturalism (Shapiro) section 8.3 Penelope Maddy, Realism in Mathematics (excerpts)
Friday, November 29 th No Class
Monday, December 2 nd Fictionalism (Shapiro) section 9.1 Hartry Field, Science without Numbers (excerpts) OCRA: Stephen Yablo, “Go Figure: A Path through Fictionalism”
Wednesday, December 4th Neo-Logicism (Shapiro) section 5.4 Bob Hale and Crispin Wright, The Reason’s Proper Study (excerpts) Optional: (OCRA) Richard Heck, “The Julius Caesar Objection”
Friday, December 9th Structuralism (Shapiro) chapter 10 (B&P) Paul Benacerraf, “What Numbers Could Not Be” (OCRA) Charles Parsons, “The Structuralist View of Mathematical Objects”
Monday, December 11 th Wrap Up