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Philosophy 344M—Philosophy of Mathematics Jon Erling Litland

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Philosophy 344M—Philosophy of Mathematics Jon Erling Litland Philosophy 344M—Philosophy of Mathematics Jon Erling Litland Course Description Professor Jon Erling Litland e-mail: [email protected] Meeting time MWF: 10-11, WAG 308 Mathematics raises a host of philosophical questions: What is mathematics Office hours WF: 1-2, WAG 221 about? Do mathematical objects exist? If so, how can we have knowledge of them? More generally: how can we know mathematical facts? What is the relationship between truth and proof? Are there unknowable mathematical facts? Why is mathematics so useful for describing the physical world? Is there a unique foundation of mathematics? Can we discover new axioms for mathematics? We will study these questions by reading classic texts of the philosophy of mathematics, including Frege’s seminal Foundations of Arithmetic. Through- out the class we will take particular care to understand how various positions in the philosophy of mathematics can be characterized precisely and what changes in mathematical practice might be required if we opt for one position over another. The course will not require any specific mathematical background knowl- edge, but some familiarity with symbolic logic will be very helpful.1 1 Some familiarity with mathematical induction will also be helpful, though not required. Textbook The two textbooks for the course are: • Øystein Linnebo, Philosophy of Mathematics. • Frege, Foundations of Arithmetic, Translation by J.L. Austin.2 2 It is important that you get this translation. Copies have been ordered to the UT bookstore, but feel free to order your copies from Amazon.com or anywhere else. Apart from that we will read a range of articles. These—in addition to supplementary materials and lectures notes—will be available on the Canvas page for the course. Grading Assignments and grading Grades will be determined as follows. Participation counts for 10 % of the final grade.3 There will be 4 group assignments where you, working 3 The participation grade is determined by attendance and contribution to class room discussion. philosophy 344m—philosophy of mathematics 2 in groups, will work through how various positions in the philosophy of mathematics will deal with various problems. These assignments count for 4 Assignments will be given numerical 30% of the grade. There will be two papers: a shorter paper (4-5 pages) due grades 0–100. At the end of the course letters grades will be determined as in the just before spring break, and a longer paper (8-10 pages) due after the end of table below class. The shorter paper counts for 25 % of the grade; the longer paper counts 4 93-100% = A 90-92% = A- for 35% of the grade. 87-89 % = B+ 83-86% = B 80-82 % = B- 77-79 % = C+ 73-76% = C 70-72 % = C- 67-69 % = D+ 63-66% = D 60-62 % =D- Miscellaneous Policies 59 = F Table 1: Grade Conversion Attendance: Attendance is mandatory.5 5 You can have 5 unexcused absences without any e↵ect on your attendance grade. After 5 absences, each unexcused absence Late homework: Handing in papers late will lead to 10 percentage points lowers your attendance grade by 20 points. being deducted from the paper. Then another 10 percentage points will be After 10 unexcused absences you fail the class. deducted for each additional 24 hours.6 Electronic Devices: Laptops and tablets are allowed in class for note-taking. 6 To avoid a lateness penalty bring documen- Cell-phones are to be turned o↵ in class. tation of illness. Do not ask for an extension or reprieve without bringing documentation. No exceptions will be made. Disabilities: “Students with disabilities may request appropriate academic ac- commodations from the Division of Diversity and Community Engagement, Services for Students with Disabilities, 471-6259.” Collaboration and Plagiarism: If you are caught plagiarizing on an assignment you will automatically fail the assignment and be reported to the dean of students. If you are caught plagiarizing again you will automatically fail the class and be reported, again, to the dean of students. Religious holidays: By UT Austin policy, you must notify me of your pending absence at least fourteen days prior to the date of observance of a religious holy day. If you must miss a class, an examination, a work assign- ment, or a project in order to observe a religious holy day, you will be given an opportunity to complete the missed work within a reasonable time after the absence. philosophy 344m—philosophy of mathematics 3 Tentative Schedule with readings All references to Linnebo, are, unless otherwise specified to the textbook for the course.7 7 For an up to date list of readings check the Canvas site for the course. • Week 1, Jan 22, Jan 24: The problems of the philosophy of mathematics. – Linnebo, Chapter 1 • Week 2, Jan 27, 29, & 31: Frege’s Logicism. – Linnebo, chapter 2. – Frege, Foundations of Arithmetic • Week 3, Feb 3, 5, & 7: Formalism and Deductivism; – Linnebo Chapter 3. – Hilary Putnam. “The thesis that mathematics is logic”. In: Mathe- matics, Matter and Method: Philosophical Papers. 2nd ed. Vol. 1. Cambridge University Press, 1979, pp. 12–42 – Frege, Foundations of Arithmetic §86-125 • Week 4, Feb 10, 12, & 14: Finitism, and Hilbert’s program – Linnebo, Chapter 4. – W. W. Tait. “Finitism”. In: Journal of Philosophy 78 (1981), pp. 524– 546; – David Hilbert. “On the infinite”. In: Philosophy of Mathematics: Selected Readings. Ed. by Paul Benacerraf and HilaryEditors Putnam. 2nd ed. Cambridge University Press, 1984, pp. 183–201 • Week 5, Feb 17, 19, & 21 – 13, 14. Intuitionism. Linnebo chapter 5. + Notes • Week 6, Feb 24, 26 & 28: Empiricism and Indispensability – Linnebo, Chapter 6 – W. V. Quine. “Two Dogmas of Empiricism”. In: Philosophical Review 60 (1951), pp. 20–43 – Penelope Maddy. “Indispensability and practice”. In: Journal of Philosophy 89 (1992), pp. 275–289 • Week 7, Mar 2, Mar 4, & Mar 6. Nominalism. Epistemological motiva- tions. – Linnebo Chapter 7. philosophy 344m—philosophy of mathematics 4 – Paul Benacerraf. “Mathematical truth”. In: Journal of Philosophy 70 (1973), pp. 661–679. – Excerpts from Hartry Field. Realism, mathematics, and modalty. Oxford: Blackwell, 1989 • Week 8, Mar 9, 11, & 13: Nominalism carried out. – Hartry Field. “Realism and anti-realism about mathematics”. In: Philosophical Topics 13 (1982), pp. 45–69 – Stephen Yablo. “The Myth of Seven”. In: Fictionalism in Metaphysics. Ed. by Mark Eli Kalderon. Clarendon Press, 2005, pp. 88–115 – Melia J. “On What There’s Not”. In: Analysis 55 (1995), pp. 223–29 • Short paper due: Friday Mar 13. • Week 9, No classes: Spring Break • Week 10, Mar 23, 25 & 27: Mathematical intuition. – Linnebo, Chapter 8 – Penelope Maddy. “Perception and Mathematical Intuition”. In: Philo- sophical Review 89.2 (1980), pp. 163–196 – Charles Parsons. “Mathematical Intuition”. In: Proceedings of the Aristotelian Society 80 (1979) • Week 11, Mar 30, Apr 1, 3; Neo-logicist abstraction. – Linnebo, Chapter 9. – Bob Hale and Crispin Wright. “The Metaontology of Abstraction”. In: Metametaphysics: New Essays on the Foundations of Ontology. Ed. by David John Chalmers, David Manley, and Ryan Wasserman. Oxford University Press, 2009, pp. 178–212 – Richard Heck. “Frege’s Theorem”. In: The Harvard Review of Philoso- phy 7.1 (1999), pp. 56–73 • Week 12, (Apr 8, 10. TBD. Litland away at conference): Set theory: the iterative conception – Linnebo, Chapter 10 • Week 13: Apr 13, 15, 17. More set theory – Linnebo, Chapter 10 G. Boolos. “The Iterative Conception of Set”. In: Logic, Logic, Logic. Cambridge, MA: Harvard University Press, 1998, pp. 13–29 • Week 14 , Apr 20, 22, & 24: indefinite extensibility and whence the contradiction philosophy 344m—philosophy of mathematics 5 – Michael Dummett. Frege: Philosophy of Mathematics. London: Duckworth, 1991, ch 17 George Boolos. “Whence the contradiction?” In: Proceedings of the Aristotelian Society Supplementary Volume 67 (1993), pp. 213–233 – Kit Fine. “4. Our Knowledge of Mathematical Objects”. In: Oxford Studies in Epistemology (2006) • Week 15, Apr 27, 29, & May 1; Reduction and structuralism. – Linnebo Chapter 11. – Paul Benacerraf. “What numbers could not be”. In: Philosophical Review 74 (1965), pp. 47–73 – Michael D. Resnik. “Mathematics as a Science of Patterns: Ontology and Reference”. In: Noûs 15.4 (1981), pp. 529–550 • Week 16, May 4, 6, & 8; Structuralism and Indiscernibles. Review. – Øystein Linnebo. “Structuralism and the Notion of Dependence”. In: The Philosophical Quarterly 58.230 (2008), pp. 59–79. eprint: http: //pq.oxfordjournals.org/content/58/230/59.full.pdf+html – Leitgeb and Ladyman8 8 Hannes Leitgeb and James Ladyman. “Criteria of Identity and Structuralist • Final paper due: May 11. Ontology”. In: Philosophia Mathematica 16.3 (2008), pp. 388–396. philosophy 344m—philosophy of mathematics 6 References Benacerraf, Paul. “Mathematical truth”. In: Journal of Philosophy 70 (1973), pp. 661–679. – “What numbers could not be”. In: Philosophical Review 74 (1965), pp. 47– 73. Boolos, G. “The Iterative Conception of Set”. In: Logic, Logic, Logic. Cam- bridge, MA: Harvard University Press, 1998, pp. 13–29. Boolos, George. “Whence the contradiction?” In: Proceedings of the Aris- totelian Society Supplementary Volume 67 (1993), pp. 213–233. Dummett, Michael. Frege: Philosophy of Mathematics. London: Duckworth, 1991. Field, Hartry. “Realism and anti-realism about mathematics”. In: Philosophi- cal Topics 13 (1982), pp. 45–69. – Realism, mathematics, and modalty. Oxford: Blackwell, 1989. Fine, Kit. “4. Our Knowledge of Mathematical Objects”. In: Oxford Studies in Epistemology (2006). Hale, Bob and Crispin Wright. “The Metaontology of Abstraction”. In: Metametaphysics: New Essays on the Foundations of Ontology. Ed. by David John Chalmers, David Manley, and Ryan Wasserman. Oxford University Press, 2009, pp. 178–212. Heck, Richard. “Frege’s Theorem”. In: The Harvard Review of Philosophy 7.1 (1999), pp. 56–73. Hilbert, David. “On the infinite”. In: Philosophy of Mathematics: Selected Readings. Ed. by Paul Benacerraf and HilaryEditors Putnam. 2nd ed. Cambridge University Press, 1984, pp.
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