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PHIL √-1 Introduction to Philosophy of Mathematics

An "imaginary" and (so far) unapproved course in the philosophy of mathematics.

Lecturer: Josh Smith email: [email protected] Office hours: TBD.

Overview There seems to be something remarkable about mathematics that sets it apart from the natural sciences. While the natural sciences are concerned with real entities, mathematics appears to be concerned with abstract concepts. The methods of mathematics differ, too. We seem to acquire mathematical knowledge through deductive reasoning, quite differently than the inductive methods used in the natural sciences. We are left to wonder how and why mathematics is effective, what mathematical entities are, and how we come to mathematical knowledge at all. Our course will take us on a whirlwind tour of the philosophy of mathematics, beginning with a brief history of some traditional views. We will work through a variety of schools and debates: logicism, formalism, intuitionism, platonism, nominalism, and structuralism. You need not know these words now; you should by semester's end. We will tie everything together by looking at a puzzling current mathematical-cosmological "Theory of Everything," its philosophical roots and its philosophical objectors.

Prerequisites Option 1: MATH 70 (Algebra II) and MATH 109 (Precalculus) with a grade of C or better. Option 2: Passing high school Calculus, or Precalculus or a trigonometry based course elsewhere, with a grade of B or better. (NB: These prerequisites are the same as in MATH 226. The course prerequisites should be flexible enough to be open to philosophy students. This may require some revision. Also, MATH 226, for comparison, is open to all graduate students.)

Texts [Core] Stewart Shapiro, Thinking about Mathematics (Oxford University Press, 2000). [Anthology] Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics: Selected Readings 2nd ed. (Cambridge University Pres, 1984).

Readings This course includes core and anthology readings. Both are required. Seldom will more than two readings be assigned for a given class. It is expected that you will come to class prepared and ready to discuss. Grading and Assignments This course assumes a certain level of intellectual maturity. You will therefore not be tested on your ability to recall information. Grades will be based on your ability to engage with the ideas of the course and with your fellow students. Short writing assignments will ensure you are making progress. These assignments start early, as our early topics are foundational for our later topics.

Final Paper: Worth 30% of your grade. Max 10 pages. Details will be given later in the semester.

Short Writing Assignments: Worth 50% of your grade. Ten short (1-page) assignments in response to prompt questions, separately worth 5% of your grade.

Participation: Worth 20% of your grade. Participation in on-line discussions will be due each week. Suggested minimum: at least one insightful response to two different threads.

Attendance Attendance is expected and required. A sign-in sheet will be circulated at random.

Meetings and Reading Assignments Much of this course will follow the layout of Shapiro, with some exceptions. Mill, for example, is presented early in Shapiro, but we address Mill in the context of structuralism. Additional readings from the anthology and outside sources will further elaborate upon the foundations laid in our core text. NB: "B+P" refers to our anthology by Benacerraf and Putnam.

1. Introduction to our Introduction Shapiro Ch. 1+2

2. One tradition in the philosophy of mathematics Shapiro 51-63 [13] Plato, excerpt from Meno (on iLearn)

3. An alternative tradition in the philosophy of mathematics Shapiro 73-91 [19] Selections from Kant's Critique of Pure Reason (on iLearn)

4. Logicism: Frege Shapiro 107-115 [9] Frege, 1884, Selections from "The concept of number," B+P 130-159 [29]

5. Logicism: Russell Shapiro 115-124 [10] Russell, 1919, Selections from Introduction to Mathematical Philosophy, B+P 160-182 [22]

6. Logicism: Carnap Shapiro 124-133 [10] Carnap 1950

7. (The End of) Logicism: Quine Shapiro 124-133 [10] A.J. Ayer "The A Priori" Quine "Two Dogmas"

8. Formalism: Hilbert Shapiro 140-165 [26] Hilbert, "On the Infinite"

9. Formalism: Gödel and Incompleteness Shapiro 165-170 [6] Excerpt(s) from Godel's Proof

10. Formalism: Curry Shapiro 165-170 [16] Curry 1954 or Curry 1958

11. Intuitionism: Brouwer Shapiro 172-185 [14] Brouwer 1948

12. Intuitionism: Heyting Shapiro 185-189 [5] Heyting 1931 and/or 1956

13. Intuitionism: Dummett Shapiro 190-197 [8] Dummett 1973 or 1991

14. Platonism: Godel Shapiro 201-11 [11] Godel, "What is Cantor's Continuum Problem?"

15. Platonism: Maddy Shapiro 220-4 [5] Maddy, Realism in mathematics

16. Platonism: A Challenge (Benacerraf) Shapiro 24-33 [10] Benacerraf, "Mathematical Truth"

17. Platonism: (Empirically) Quine and Putnam Shapiro 91-102 [12] Quine, "Two Dogmas" SEP "Indispensability Arguments"

18. Platonism: A Review Shapiro 211-220 [10] SEP "Platonism in Mathematics"

19. Nominalism: Field Shapiro 226-237 [12] Field, "Realism and Anti-Realism about Mathematics"

20. Nominalism: Chihara Shapiro 237-256 [20] Chihara 1990

21. Structuralism: Benacerraf

22. Structuralism: Resnik

23. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" Putnam + Quine

24. Unreasonably Effective: Tegmark's Mathematical Universe Hypothesis Tegmark

25. Reasonably Uneffective: Tegmark's Detractors Hut, Alford, Vilenkin

26. Reasonably Uneffective: Tegmark's Detractors Page, Ward

27. Still Unreasonably Effective: The External Reality Hypothesis