Syllabus
Course Description and Overview:
This course is divided into two parts. The first part, covering roughly the first nine weeks of the term, is an historical survey of the philosophical questions which arise from considering how to explain our knowledge of mathematics. Do we have a priori knowledge of necessary truths? Is our knowledge of mathematics empirical? Do we really have mathematical knowledge at all? The readings in the first part of the course, covered mainly chronologically, range from ancient philosophy through the twentieth century, with special attention paid to the fruitful period between Frege, in the late nineteenth century, and Gödel. We will devote the second part of the course, the last five weeks of the term, to recent work, including my own, on the indispensability argument.
Mathematics has a long and prominent place in philosophy. Plato’s students were implored to excel in mathematics; a sign over the door to his Academy said, “Let no one enter who is ignorant of geometry.” Aristotle wrote, “Mathematics has come to be the whole of philosophy for modern thinkers” (Metaphysics I.9: 992a32).
Some prominent philosophers in the early modern period were mathematicians, including Descartes, who developed analytic geometry, and Leibniz, who developed the calculus. In the late nineteenth and early twentieth centuries, philosophers including Frege and Russell made advances in the foundations of mathematics proper. In recent years, many philosophers have made contributions to set theory and mathematical logic, independently of their philosophical work.
In the other direction, mathematicians from Euclid forward have contributed to philosophy. Cantor’s work on transfinite numbers transformed the philosopher’s concept of infinity, which had played a central role in philosophical debate about God and the origins of the universe for millennia. Other philosophical topics like necessity and contingency have received mathematical treatment which has changed the way philosophers argue about these concepts. Indeed some mathematicians, like Hilbert, Gödel, von Neumann, and Tarski, are central philosophical figures.
Even philosophers who have not contributed to mathematics have made mathematical insights central to their work. Berkeley tried to debunk the calculus on philosophical grounds. Kant’s transcendental idealism begins with the question of what the structure of our reasoning must be in order to yield mathematical certainty. Wittgenstein’s Remarks on the Foundations of Mathematics contain core elements of his philosophical positions.
Still, even philosophers who spend time with mathematics deny that the relationship of mathematics to philosophy is particularly close. Wittgenstein wrote that philosophy, “Leaves mathematics as it is, and no mathematical discovery can advance it.” (Philosophical Investigations, §124) Kripke implored that, “There is no mathematical substitute for philosophy.”
In this course, in addition to examining the philosophical questions which arise from considerations of our knowledge of mathematics, we will try to see what makes mathematics so interesting to philosophers, and also what contributions mathematics can make to philosophy.
Texts
James Robert Brown, Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures, New York: Routledge, 2000.
Stewart Shapiro, Thinking About Mathematics: The Philosophy of Mathematics, New York: Oxford, 2000.
Assignments and Grading:
Your responsibilities this course include the following, with their contributions to your grade calculation in parentheses:
All the primary readings listed below, including seminar papers
Twenty reading prècises (10%)
Two seminar papers/presentations (2-4 pages; 5-10 minutes) (40%; 20% each)
Term paper (8-12 pages) (30%)
Final exam (20%)
Readings: Readings are to be completed before the class indicated. The Primary Readings are required; the secondary readings are optional. Some secondary readings, notably the readings from the Brown and Shapiro texts, are introductory elucidations of the primary readings. Some secondary readings are further scholarly articles on a given topic, critical commentaries on the primary readings, or extended studies of a point we will study only briefly. All of the readings on the syllabus that are not from either the Brown or Shapiro texts will be accessible from the course website. The course bibliography includes further readings, many of which are also accessible from the course website.
Reading prècises: Reading prècises are 100- to 150- word summaries, or distillations, of some portion of an assigned reading. In preparing for most classes, you should write one prècis before class. You may choose to write about an entire reading, or to focus on a small portion of one reading. If there is more than one reading, you may choose one reading on which to focus. You need not complete prècises for the two classes in which you are presenting a seminar paper. In lieu of up to five prècises, you can write a list of 6-8 detailed questions on the reading. Your twenty prècises are due on Friday, December 10, at 4pm. You will mainly be graded on the completion of twenty prècises, rather than their quality. I expect that the prècises will be useful to you in preparing both for classes and for the final exam.
Seminar papers: Many classes will run as extended discussions of a 750- to 1500-word seminar paper. Seminar papers should assimilate the assigned readings and summarize the main arguments. I also encourage you to include some critical analysis. You are instigating class discussion, focusing our thoughts on the central theses, and raising questions. It is good practice to end a seminar paper with a few questions you believe will be useful for the class to consider. Each seminar paper is due at noon by email to all seminar participants the day before the class in which it will be discussed (i.e. Sunday or Tuesday). This deadline is necessary for all participants in the seminar to be able to read the paper and prepare comments and questions for class.
You will lead the class on the day we discuss your seminar paper. You may be creative with your presentation. You may focus on the content of your paper. You may also discuss any particular difficulties in the material, or topics that you were unable to cover in the paper. Your grade for the seminar paper will depend on both the paper and your presentation of it. Each student in the course will write and present two seminar papers.Term Papers: Your term papers will be completed in three stages. A one-paragraph abstract of you paper is due on Wednesday, October 13. A full draft of your term paper is due on Monday, November 15. The final draft is due on Monday, December 6. See the Paper Assignment handout for various options for paper topics. I will be happy to meet with you to discuss your topic, in advance. Failure to hand in a draft, or handing in an insufficient draft, will reduce your final paper grade by two steps (e.g. from B+ to B-).
Final Exam: The final exam will be on Wednesday, December 15, from 9am to noon. Preparatory questions will be posted on the course website.
The Hamilton College Honor Code will be strictly enforced
Office Hours
My office hours for the Fall 2010 term are 10:30am - noon, Monday through Friday. My office is in room 201 of 210 College Hill Road, which is at the northwest corner of CHR and Griffin Road.
Date | Topic | Primary Readings | Secondary Readings | Seminar Paper | |
1 | Monday, August 30 | What is Mathematics? What is Philosophy of Mathematics? | Brown, Chapter 1 Shapiro, pp 21-29 |
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2 | Wednesday, September 1 | Pythagoras and the Pythagoreans | Kline, "The Creation of Classical Greek Mathematics" Kline, "The Greek Rationalization of Nature" |
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3 | Monday, September 6 | Plato's Platonism | Selections from Plato on Mathematics Aristotle, Metaphysics I.9 |
Shapiro, pp 49-63 Brown, Chapter 2 |
Noah |
4 | Wednesday, September 8 | Aristotle | Aristotle, Metaphysics XIII.1-3 Aristotle, Physics II.2 Lear, "Aristotle's Philosophy of Mathematics" |
Shapiro, pp 63-71 Aristotle, Metaphysics XIII-XIV |
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5 | Monday, September 13 | Modern Rationalism I | Descartes, Third and Fifth Meditations Descartes, Synthetic Presentation from Second Replies Leibniz, "Meditations on Knowledge, Truth, and Ideas" |
Kline, "Coordinate Geometry" Kline, "The Mathematization of Science" |
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6 | Wednesday, September 15 | Modern Rationalism II | Locke, Essay, Bk 1, Ch. 2 Leibniz, Selections from New Essays |
Kline, "The Creation of the Calculus" | |
7 | Monday, September 20 | Modern Empiricism | Locke, Selections on Mathematics Selections from Berkeley's Principles Selections from Hume on Mathematics |
Sam | |
8 | Wednesday, September 22 | The Synthetic A Priori I | Kant, Prolegomena, §§1-2 Selections from Kant's Critique |
Shapiro, pp 76-91 | Dan |
9 | Monday, September 27 | The Synthetic A Priori II | |||
10 | Wednesday, September 29 | Radical Empiricism | Mill, System of Logic, Book II, §V and §VI Frege, from The Foundations of Arithmetic, I |
Shapiro, pp 91-102 | Nick |
11 | Monday, October 4 | Cantor's Paradise | Tiles, "Cantor's Transfinite Paradise" | Dauben, "Cantor's Philosophy of the Infinite" Tiles, "Numbering the Continuum" |
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12 | Wednesday, October 6 | Logicism | Frege, from The Foundations of Arithmetic, II Russell, "Letter to Frege" Frege, "Letter to Russell" |
Shapiro, pp 107-115 Russell, "On Our Knowledge of General Principles" Russell, "How A Priori Knowledge is Possible" |
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13 | Monday, October 11 | Formalism and Incompleteness | Hilbert, "On the Infinite" Johann von Neumann, "The Formalist Foundations of Mathematics" |
Brown, Chapter 5 Shapiro, pp 140-168 Smullyan, "The General Idea Behind Gödel's Proof" |
Hime |
14 | Wednesday, October 13 Abstracts due | Gödel Platonism | Gödel, "What is Cantor's Continuum Problem? (1964)" | Shapiro, pp 201-212 Brown, Chapter 11 Feferman, et al., "Introductory Note..." Gödel, "What is Cantor's Continuum Problem? (1947)" |
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15 | Monday, October 18 | Gödel Catch-Up Day | |||
16 | Wednesday, October 20 | Intuitionism | Brouwer, "Intuitionism and Formalism" Heyting, "Disputation" |
Brown, Chapter 8 Shapiro, pp 172-189 Brouwer, "Consciousness, Philosophy, and Mathematics" |
Noah |
17 | Monday, October 25 | Conventionalism | Carnap, "Empiricism, Semantics and Ontology" Ayer, "The A Priori" |
Shapiro, pp 124-133 |
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18 | Wednesday, October 27 | The Problem | Benacerraf, "Mathematical Truth" | Field, "Knowledge of Mathematical Entities" Shapiro, pp 29-39 |
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19 | Monday, November 1 | Two Dogmas of Empiricism | Quine, "Two Dogmas of Empiricism" | Shapiro, pp 212-220 Grice and Strawson, "In Defense of a Dogma" |
Hime |
20 | Wednesday, November 3 | The Indispensability Argument | Quine, "On What There Is" Quine, "Success and the Limits of Mathematization" Quine on Recreation |
Marcus, "Quine's Indispensability Argument" | |
21 | Monday, November 8 | Dispensabilism I | Field, from Science without Numbers | Shapiro, pp 226-237 Brown, Chapter 4 |
Dan |
22 | Wednesday, November 10 | Dispensabilism II | Field, from Science without Numbers | Field, "Introduction: Fictionalism, Epistemology, and Modality" MacBride, "Listening to Fictions: A Study of Fieldian Nominalism |
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23 | Monday, November 15 Draft due |
The Weasel | Melia, "Weaseling Away the Indispensability Argument" |
Colyvan, "Mathematics and Aesthetic Considerations in Science?" Melia, "Response to Colyvan" |
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24 | Wednesday, November 17 | The Eleatic and the Indispensabilist | Colyvan, "The Eleatic Principle" | Colyvan, "The Quinean Backdrop" Marcus, "The Eleatic and the Indispensabilist" |
Nick |
25 | Monday, November 29 | Mathematical Recreation | Leng, "What's Wrong with Indispensability? (Or, the Case for Recreational Mathematics)" | Marcus, "Why the Indispensability Argument Does Not Justify Belief in Mathematical Objects" | Sam |
26 | Wednesday, December 1 | The Explanatory Argument | Baker, "Are There Genuine Mathematical Explanations of Physical Phenomena?" Mancosu, "Mathematical Explanation: Problems and Prospects," §1-§3 |
Lyon and Colyvan, "The Explanatory Power of Phase Spaces" | |
27 | Monday, December 6 Paper due |
The Nominalist Against the Explanatory Argument | Bangu, "Inference to the Best Explanation and Mathematical Realism" | ||
28 | Wednesday, December 8 | The Platonist Against the Explanatory Argument | Marcus, "Explanation and Indispensability (short version)" | Brown, Chapter 3 Marcus, "Explanation and Indispensability (long version)" Marcus, "Toward Autonomy Realism" |