24.801: Philosophy of Mathematics
Fall 2000
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.
Instructor
Stephen Yablo, Philosophy
Office: E39314
Phone: 2580740
Office Hours: by appointment
Email: yablo@mit.edu
Meetings
Thursdays 2:005:00 in 39335
Readings
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.
Requirements

one 25page term paper, due at the end of term

one 7orso page paper, due in the seventh week

classroom participation
Notes

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.
TENTATIVE SCHEDULE:
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 TruthValues?"
September 14 Platonism & Realism: Modal Strategies

*Putnam, "Mathematics Without Foundations"

*Kessler, "Mathematics and Modality"

*Field, "Realism, Mathematics, and Modality"
September 21 Fregean Platonism: Bad Company

*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"
September 28 Fregean Platonism: Julius Caesar

*Wright, excerpts from Frege and the Conception of Numbers as Objects

*Rosen, "Refutation of Nominalism (?)"

Heck, "The Julius Caesar Objection"
October 5 Fregean Platonism: Logical Issues

*Rayo and Yablo, "Nominalism thru DeNominalization"

*Wright, "Frege's Paradox"

Shapiro & Weir, "NeoLogicist Logic"
October 12 Fictionalism

*Field, introduction to Realism, Math, and Modality

*Yablo, "Go Figure: A Path Through Fictionalism"
October 19 Figuralism

*Balaguer, "An Account of Indispensable Applications of Mathematics"

*Yablo, "Myth of the Seven"
October 26 Are Numbers Necessary?

*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"
November 9 Mathematical Practice: Explanation

*Friedman, "Explanation and Scientific Understanding"

*Kitcher, "Explanatory Unification"

*Steiner, "Mathematical Explanation"

Detlefsen, "Fregean Hierarchies and Mathematical Explanation"
November 16 Mathematical Practice: New Objects

*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

Shapiro & Weir, "New V, ZF, and Abstraction"

Shapiro & Weir, "NeoLogicist Logic"

Rosen, "A Study in Modal Deviance"

Rosen, "Refutation of Nominalism(?)"

Detlefsen, "Fregean Hierarchies and Mathematical Explanation"

Field, "The Conceptual Contingency of Abstract Objects"

Hale/Wright, "Nominalism and the Contingency of Abstract Objects"

Hale, "Is Platonism Epistemologically Bankrupt?"

Goodman, "Mathematics as a Natural Science"

Kessler, "Mathematics and Modality"

Putnam, "Mathematics Without Foundations"

Gemes, "Explanation, Unification, and Content"

Manders, "Domain Extension and the Philosophy of Mathematics"

Manders, "Logical and Conceptual Relations in Mathematics"

Kitcher, "Innovation and Understanding in Mathematics"

Detlefsen, "Fregean Hierarchies and Mathematical Explanation"

Maddy, "The Roots of Contemporary Platonism"

Burgess, "Why I Am Not A Nominalist"

Linsky & Zalta, "Naturalized Platonism versus Platonized Naturalism"

Linsky & Zalta, review of Balaguer, Platonism and AntiPlatonism
in Mathematics

Kitcher, "Frege's Epistemology"

Steiner, "Mathematics, Explanation, and Scientific Knowledge"

McCarthy, "Platonism and Possibility"

Friedman, "Explanation and Scientific Understanding"

Chihara, "Mathematical Discovery and Concept Formation"

Wright, "On the Philosophical Significance of Frege's Theorem"

Wright, "On the Harmless Impredicativity of N= (Hume's Principle)"

Boolos, "Is Hume's Principle Analytic?"

Parsons, "Wright on Abstraction and Set Theory"

Heck, "The Julius Caesar Objection"

Sandborg, "Mathematical Explanation and the Theory of WhyQuestions"

Detlefsen, "Fregean Hierarchies & Math. Explanation"

Steiner, "Mathematical Explanation"

Kitcher, "Explanatory Unification"

Manders, "Logic and Conceptual Relns in Math"

Beall, "From FullBlooded Plat to Really..."

Tappenden, "Extending Knowledge and Fruitful Concepts"

Cheyne, "Problems with Profligate Platonism"

Tennant, "On the Necessity of Mathematical Objects"