24.801: Philosophy of Mathematics

Outline

This seminar takes up four or five related topics. First, the relations if any between realism (the view that mathematics is "objective") and platonism (which postulates the existence of mathematical objects).   Second, a recently influential version of platonism whose authors mean their approach to be consistent with antirealism. This is "Fregean Platonism" as proposed originally by Crispin Wright.   Third, the relations if any between fictionalism as maintained by Hartry Field and figuralism as maintained by yours truly.  Fourth, the alleged necessity of mathematical objects and mathematical truth.  Fifth,  if there's time,  the bearing of all this on mathematical practice.

Stephen Yablo, Philosophy
Office: E39-314
Phone: 258-0740
Office Hours: by appointment
E-mail: yablo@mit.edu

Thursdays 2:00-5:00 in 39-335

Papers will be available for copying in the rack by the graduate student mailboxes. See below for a list of likely readings. There may be the odd classroom handout as well.

• one 25-page term paper, due at the end of term
• one 7-or-so page  paper, due in the seventh week
• classroom participation

• Starred readings will be explicitly discussed.  Unstarred quite likely won't.
• Not all the unstarred readings listed for a given week will be equally important; I'll try to indicate the week before what the focus will be.
• Depending on class size and interest, presentations may be folded into the mix somewhere.

September 7   Platonism & Realism: Actualist Strategies

• *Field, "Mathematical Objectivity and Mathematical Objects"
• *Balaguer, "A Theory of Mathematical Correctness and Mathematical Truth"
• Field, "Which Undecidable Sentences Have Determinate Truth-Values?"

• *Putnam, "Mathematics Without Foundations"
• *Kessler, "Mathematics and Modality"
• *Field, "Realism, Mathematics, and Modality"

• *Wright, excerpts from Frege and the Conception of Numbers as Objects
• *Boolos, "The Standard of Equality for Numbers"
• *Wright, "The Philosophical Significance of Frege's Theorem"
• Wright, "On the Harmless Impredicativity of N= (Hume's Principle)"
• Boolos, "Is Hume's Principle Analytic?"
• Shapiro & Weir, "New V, ZF, and Abstraction"

• *Wright, excerpts from Frege and the Conception of Numbers as Objects
• *Rosen, "Refutation of Nominalism (?)"
• Heck, "The Julius Caesar Objection"

• *Rayo and Yablo, "Nominalism thru De-Nominalization"
• *Wright, "Frege's Paradox"
• Shapiro & Weir, "Neo-Logicist Logic"

• *Field, introduction to  Realism, Math, and Modality
• *Yablo, "Go Figure: A Path Through Fictionalism"

• *Balaguer, "An Account of Indispensable Applications of Mathematics"
• *Yablo, "Myth of the Seven"

• *Hale and Wright,  "Nominalism and the Contingency of Abstract Objects"
• *Field, "The Conceptual Contingency of Abstract Objects"
• Tennant, "On the Necessity of Mathematical Objects"

November 2  Or Do They Just Seem That Way?

• *Rosen, "A Study in Modal Deviance"
• *Yablo, "Essence and Existence for Abstract Objects"

• *Friedman, "Explanation and Scientific Understanding"
• *Kitcher, "Explanatory Unification"
• *Steiner, "Mathematical Explanation"
• Detlefsen, "Fregean Hierarchies and Mathematical Explanation"

• *Manders, "Domain Extension and the Philosophy of Mathematics"
• *Maddy, excerpts from Naturalism in Mathematics

November 23 Thanksgiving Holiday
 

November 30 Mathematical Practice: New Axioms

• *Maddy, "Believing the Axioms"
• *Maddy, excerpts from Naturalism in Mathematics

December 7 Mathematical Practice: New Concepts

• *Tappenden, "Extending Knowledge and Fruitful Concepts"
• *Chihara, "Mathematical Discovery and Concept Formation"

Readings

1. Shapiro & Weir, "New V, ZF, and Abstraction"
2. Shapiro & Weir, "Neo-Logicist Logic"
3. Rosen, "A Study in Modal Deviance"
4. Rosen, "Refutation of Nominalism(?)"
5. Detlefsen, "Fregean Hierarchies and Mathematical Explanation"
6. Field, "The Conceptual Contingency of Abstract Objects"
7. Hale/Wright, "Nominalism and the Contingency of Abstract Objects"
8. Hale, "Is Platonism Epistemologically Bankrupt?"
9. Goodman, "Mathematics as a Natural Science"
10. Kessler, "Mathematics and Modality"
11. Putnam, "Mathematics Without Foundations"
12. Gemes, "Explanation, Unification, and Content"
13. Manders, "Domain Extension and the Philosophy of Mathematics"
14. Manders, "Logical and Conceptual Relations in Mathematics"
15. Kitcher, "Innovation and Understanding in Mathematics"
16. Detlefsen, "Fregean Hierarchies and Mathematical Explanation"
17. Maddy, "The Roots of Contemporary Platonism"
18. Burgess, "Why I Am Not A Nominalist"
19. Linsky & Zalta, "Naturalized Platonism versus Platonized Naturalism"
20. Linsky & Zalta, review of Balaguer, Platonism and Anti-Platonism in Mathematics
21. Kitcher, "Frege's Epistemology"
22. Steiner, "Mathematics, Explanation, and Scientific Knowledge"
23. McCarthy, "Platonism and Possibility"
24. Friedman, "Explanation and Scientific Understanding"
25. Chihara, "Mathematical Discovery and Concept Formation"
26. Wright, "On the Philosophical Significance of Frege's Theorem"
27. Wright, "On the Harmless Impredicativity of N= (Hume's Principle)"
28. Boolos, "Is Hume's Principle Analytic?"
29. Parsons, "Wright on Abstraction and Set Theory"
30. Heck, "The Julius Caesar Objection"
31. Sandborg, "Mathematical Explanation and the Theory of Why-Questions"
32. Detlefsen, "Fregean Hierarchies & Math. Explanation"
33. Steiner, "Mathematical Explanation"
34. Kitcher, "Explanatory Unification"
35. Manders, "Logic and Conceptual Relns in Math"
36. Beall, "From Full-Blooded Plat to Really..."
37. Tappenden, "Extending Knowledge and Fruitful Concepts"
38. Cheyne, "Problems with Profligate Platonism"
39. Tennant, "On the Necessity of Mathematical Objects"