Syllabus

Class Notes

Readings

Handouts

Links

 

 

 

Texts

Assignments

Preparing for Class

Office Hours

Schedule

Deadline Summary

pdf syllabus

 

 

 

 

 

Syllabus

 

Course Description and Overview:

This course is divided into two parts. The first part, covering roughly the first eight weeks of the term, is an historical survey of philosophical questions about mathematics. Do we have a priori knowledge of necessary truths? Is our knowledge of mathematics empirical? Do we 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. We will devote the second part of the course, the last six weeks of the term, to recent work 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.”

The second half of the course focuses on a topic of central importance in contemporary philosophy of mathematics: the indispensability argument. The central problem in the philosophy of mathematics is to explain how we can have knowledge of the abstract objects of mathematics. Rationalists claim that we have a non-sensory capacity for understanding mathematical truths, but that claim appears incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies. Logicists argue that mathematical truths are just complex logical truths, but we can not reduce mathematics to logic without adding substantial portions of set theory to our logic. Fictionalists deny that there are any mathematical objects.

The indispensability argument attempts to justify our mathematical beliefs while avoiding any appeal to rational insight, to provide a platonist ontology with an empiricist epistemology. We will read a variety of recent work, including excerpts from my manuscript in progress, Autonomy Platonism and the Indispensability Argument.

 

 

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Texts

Most primary readings are available on this page. In addition, I am assigning optional but vaulable secondary readings from two excellent sources:

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.

 

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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 précis (10%)
                          Two seminar papers/presentations (2-4 pages) (30%; 15% each)
                          Term paper (8-12 pages) (40%)

                          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 précis: Reading précis are 100- to 150- word distillations of some argument in an assigned reading. A précis is a skeletal version of an argument, the argument in its most efficient form. 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 portion of the reading. If there is more than one reading, choose one on which to focus. You need not complete précis for the two classes in which you are presenting a seminar paper. In lieu of up to five précis, you can write a list of 6-8 detailed questions on the reading. The first ten précis are due on Friday, March 14, at 4pm. The last ten précis are due on Friday, May 9, at 4pm. You will mainly be graded on the completion of twenty précis, rather than their quality. I expect that the précis 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. 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. Monday or Wendesday). 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 four stages. A one–to-two-paragraph abstract of your paper with a proposed bibliography is due on Thursday, March 6. A précis of your argument with an annotated bibliography is due on Thursday, April 3. A full draft of your term paper is due on Tuesday, April 22. The final draft is due on Tuesday, May 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, May 14, from 7pm to 10pm. Preparatory questions will be posted on the course website.

             On Grades: Grades on assignments will be posted on Blackboard, along with a running total, which I call your grade calculation. Your grade calculation is a guide for me to use in assigning you a final grade. There are no rules binding how I translate your grade calculation into a letter grade. The Hamilton College key for converting letter grades into percentages is not a tool for calculating your final grade. I welcome discussion of the purposes and methods of grading, as well as my own grading policies.

             Both the Writing Center and the Oral Communications Center have an astoundingly wonderful set of resources to help you write and speak more effectively.

             The Hamilton College Honor Code will be strictly enforced

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Preparing for Class:


Preparation for most classes will consist of the following tasks

1. Completing the primary readings (preferably two-to-three times)
2. Writing a reading précis
3. Reading a seminar paper (if there is one) and preparing comments and questions
4. Completing as much of the secondary readings as you can


For the two classes in which you will be presenting, your preparation will of course differ. I encourage you to find time to meet with me in advance.

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Office Hours

My office hours for the Spring 2014 term are 10:30am - noon, Tuesdays and Thursdays.

My office is 202 College Hill Road, Room 210.


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Schedule:



Date Topic Primary Readings Secondary Readings Seminar Paper
1 Tuesday, January 21 Mathematics and the Philosophy of Mathematics? Brown, Chapter 1
Shapiro, pp 21-29
 
2 Thursday, January 23 Pythagoras and the Pythagoreans Kline, "The Creation of Classical Greek Mathematics"
Kline, "The Greek Rationalization of Nature"
 
3 Tuesday, January 28 Plato's Platonism Selections from Plato on Mathematics
Aristotle, Metaphysics I.9
Shapiro, pp 49-63
Brown, Chapter 2
 
4 Thursday, January 30 Aristotle Aristotle, Metaphysics XIII.1-3
Aristotle, Physics II.2
Lear, "Aristotle's Philosophy of Mathematics"
Shapiro, pp 63-71
Aristotle, Metaphysics XIII-XIV
 
5 Tuesday, February 4 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"
 
6 Thursday, February 6 Modern Rationalism II Locke, Essay, Bk 1, Ch. 2
Leibniz, Selections from New Essays
Kline, "The Creation of the Calculus"  
7 Tuesday, February 11 Modern Empiricism Locke, Selections on Mathematics
Selections from Berkeley's Principles
Selections from Hume on Mathematics
8 Thursday, February 13 The Synthetic A Priori I Kant, Prolegomena, §§1-2
Selections from Kant's Critique
Shapiro, pp 76-91  
9 Tuesday, February 18 The Synthetic A Priori II Kitcher, "Kant and the Foundations of Mathemtics"  
10 Thursday, February 20 Radical Empiricism Mill, System of Logic, Book II, §V and §VI
Frege, from The Foundations of Arithmetic, I
Shapiro, pp 91-102 Genesis
11 Tuesday, February 25 Cantor's Paradise Tiles, "Cantor's Transfinite Paradise" Tiles, "Numbering the Continuum" Austin
12 Thursday, February 27 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"
Jason
13 Tuesday, March 4 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"
Jack
14 Thursday, March 6
Term Paper Abstracts due
Intuitionism Brouwer, "Intuitionism and Formalism"
Heyting, "Disputation"
Brown, Chapter 8
Shapiro, pp 172-189
Brouwer, "Consciousness, Philosophy, and Mathematics"
Shaq
15 Tuesday, March 11 Conventionalism Carnap, "Empiricism, Semantics and Ontology"
Ayer, "The A Priori"

Shapiro, pp 124-133
Brown, Chapter 9

 
16 Thursday, March 13 Two Dogmas of Empiricism Quine, "Two Dogmas of Empiricism" Shapiro, pp 212-220
Grice and Strawson, "In Defense of a Dogma"
 
  Friday March 14 First 10 Reading Précis are due      
17 Tuesday, April 1 The Problem Benacerraf, "Mathematical Truth"
Field, "Knowledge of Mathematical Entities"
Shapiro, pp 29-39 Jason
18 Thursday, April 3
Term Paper Précis due
The Indispensability Argument Quine, "On What There Is"
Quine, "Existence and Quantification"

Quine on Recreation
Azzouni, "On 'On What There Is'"
Marcus, "Chapter 2: The Quinean Indispensability Argument"
Austin
19 Tuesday, April 8 Dispensabilism I Field, from Science without Numbers Shapiro, pp 226-237
Brown, Chapter 4
Jack
20 Thursday, April 10 Dispensabilism II Field, "Introduction: Fictionalism, Epistemology, and Modality"
MacBride, "Listening to Fictions: A Study of Fieldian Nominalism
Melia, "Field's Programme: Some Interference"
 
21 Tuesday, April 15 The Weasel Melia, "Weaseling Away the Indispensability Argument"
Colyvan, "Mathematics and Aesthetic Considerations in Science?"
Melia, "Response to Colyvan"
Genesis
22 Thursday, April 17 The Eleatic and the Indispensabilist Colyvan, "The Quinean Backdrop"
Colyvan, "The Eleatic Principle"
Marcus, "The Eleatic and the Indispensabilist" Shaq
23 Tuesday, April 22
Term Paper Draft due
Mathematical Recreation Leng, "What's Wrong with Indispensability? (Or, the Case for Recreational Mathematics)" Maddy, "Indispensability and Practice"
Sober, "Mathematics and Indispensability"
 
24 Thursday, April 24 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"  
25 Tuesday, April 29 Bangu Against the Explanatory Argument Bangu, "Inference to the Best Explanation and Mathematical Realism"
Bangu, "Indispensability and Explanation"
   
26 Thursday, May 1 The Platonist Against the Explanatory Argument Marcus, "How Not to Enhance the Indispensability Argument" Marcus, "Chapter 5: The Unfortunate Consequences"
Brown, Chapter 3
 
27 Tuesday, May 6
Term Paper due
Autonomy Platonism I: Plenitudinous Platonism Balaguer, "On Plenitudinous Platonism"
Balaguer, "A New Platonist Epistemology"
Marcus, "Chapter 9: Two Versions of Autonomy Platonism, §1-§2"  
28 Thursday, May 8 Autonomy Platonism II Marcus, "Chapter 9: Two Versions of Autonomy Platonism, §3-§7"
Marcus, "Chapter 10: Circles"
Shapiro, pp 201-212
 

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Deadline Summary:

Thursday, March 6: Term paper abstracts and bibliography
Friday, March 14, 4pm: First ten reading précis
Thursday, April 3: Term paper précis and annotated bibliography
Tuesday, April 22: First draft of term paper

Tuesday, May 6: Final draft of term paper
Friday, May 9, 4pm: Final ten Précis due
Wednesday, May 14, 7-10pm: Final Exam


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