Syllabus

Class Notes

Readings

Handouts

Links

 

 

 

 

 

Texts

Assignments

Office Hours

Schedule

Deadline Summary

pdf syllabus

 

 

 

Syllabus

 

Course Description and Overview:

Mathematics has long had a 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.

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.” Kripke implored, “There is no mathematical substitute for philosophy.”

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 into the twentieth century. We will survey a range of views about the status of mathematics and our intellectual capacities.

The second half of the course focuses generally on a debate among mathematical platonists, those who believe that mathematical objects exist or that mathematical statements are true. On one side, the indispensabilists believe that all knowledge must be grounded in sense experience. Since we have no direct sense experience of mathematical objects, our beliefs about them are justified indirectly, by their uses in scientific theories which are constructed to explain or account for our observations.

Opposing the indispensabilists, autonomy platonists believe that the robustness of our mathematical beliefs shows that we have cognitive capacities beyond mere perception, capacities which ground our mathematical beliefs. Toward the end of the course, we will look at recent work on one of those purported capacities, mathematical intuition, including some of my own work on the debate.

 

 

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Texts

Most readings are available in my forthcoming An Historical Introduction to the Philosophy of Mathematics (HIPM).
Some will come from my monograph, Autonomy Platonism and the Indispensability Argument (AP and the IA).
You will have access to those books.
Other readings will appear on the course website. Two excellent secondary sources are :

James Robert Brown. Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures, 2nd edition. New York: Routledge, 2008.

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:

The readings listed below.
Twenty reading précis (10%)
Two or three seminar presentations/papers (30%; 10% each)
Term paper (8-12 pages) (40%)

 Final exam (20%)

             Readings: are to be completed before the class indicated. Each chapter of An Historical Introduction to the Philosophy of Mathematics includes both primary sources and introductory commentary by the editors. The primary sources are required reading.

             Reading précis: are distillations of some argument in an assigned reading, 100–150 words. 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 précis 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. Writing précis should not be a burden. Just do a little writing as you read.
The first ten précis are due on Friday, March 11, at 4pm. The last ten précis are due on Friday, May 6, 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.

             Two or three times during the term, you will lead the class in a presentation of a seminar paper. The paper (750-1200 words) is due to be submitted for a grade by 4pm three days after your presentation. (So, papers presented on a Tuesday are due on Friday; papers presented on Thursday are due on Sunday.) Each chapter of An Historical Introduction to the Philosophy of Mathematics contains Themes to Explore, which may be useful topics for a seminar paper. You may be creative with your presentation. Your grade for the seminar paper will depend on both the paper and your presentation of it.

             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 3. A précis of your argument with an annotated bibliography is due on Thursday, March 31. A full draft of your term paper is due on Tuesday, April 19. The final draft is due on Tuesday, May 3. I hope that you will be able to use one or more of your seminar papers as part of your term paper. 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 11, from 7pm to 10pm. Preparatory questions will be distributed and posted on the course website.

             Accessibility and Diversity of Learning Styles: Your well-being and success in this course are important to me. Different people learn best in different ways. Please come talk with me about how best to balance your individual needs and learning style with my expectations for the course. If you are eligible for testing accommodations, please also see Allen Harrison, Associate Dean of Students for Multicultural Affairs and Accessibility Services in the Office of the Dean of Students, Elihu Root House.

             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 enforced

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Office Hours

My office hours for the Spring 2016 term are 10:30am - noon, Tuesdays and Thursdays.

My office is 202 College Hill Road, Room 210.


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Schedule:

Class Date Topic Primary Readings, to be completed before class
1 Tuesday, January 19 Mathematics and the Philosophy of Mathematics Brown, Chapter 1
2 Thursday, January 21 Pythagoras and the Pythagoreans HIPM 1: Pythagoras, Parmenides, and Zeno’s Paradoxes
3 Tuesday, January 26 Plato’s Platonism HIPM 2: Plato
from HIPM 3: Criticisms of Plato’s Theory of Forms
4 Thursday, January 28 Aristotle’s Qua HIPM 3: Aristotle
*Lear, “Aristotle’s Philosophy of Mathematics
5 Tuesday, February 2 Mathematics, Clarity, and Distinctness HIPM 4: The Rationalists, through p 131
*Kline, “Coordinate Geometry
*Kline, “The Mathematization of Science"
6 Thursday, February 4 Innate Ideas HIPM 4: The Rationalists
*Kline, “The Creation of the Calculus
7 Tuesday, February 9 Lockean Conceptualism HIPM 5: The Empiricists, to p 194
8
Thursday, February 11 Hume’s Distinction HIPM 5: The Empiricists, pp 202–end
 9– 10 Tuesday, February 16– Thursday, February 18 The Synthetic A Priori HIPM 6: Kant
*Kitcher, “Kant and the Foundations of Mathematics
11 Tuesday, February 23 Cantor’s Paradise HIPM 8: Cantor’s Transfinites
*Tiles, “Numbering the Continuum
 12– 13 Thursday, February 25– Tuesday, March 1 Logicism HIPM 9: Logicism
*Russell,“On Our Knowledge of General Principles
*Russell, “How A Priori Knowledge is Possible
14 Thursday, March 3
Term Paper Abstracts Due
Formalism and Incompleteness HIPM 10: Formalism
15 Tuesday, March 8 Intuitionism HIPM 11: Intuitionism
16 Thursday, March 10 Conventionalism HIPM 12: Conventionalism
Friday March 11 First 10 Reading Précis are due
Spring Break
17 Tuesday, March 29 The Problem HIPM 15: The Benacerraf Problem
18 Thursday, March 31
Term Paper Précis due
Two Dogmas of Empiricism Quine, “Two Dogmas of Empiricism”
*Grice and Strawson, “In Defense of a Dogma
19 Tuesday, April 5 The Indispensability Argument HIPM 16: Quine, Quine, and Putnam
Quine on Recreation
*Marcus, AP and the IA, Chapter 2: The Quinean Indispensability Argument
20 Thursday, April 7 Dispensabilism HIPM 16: Field, from Science without Numbers
*MacBride, “Listening to Fictions: A Study of Fieldian Nominalism
21 Tuesday, April 12 The Weasel Leng, “What's Wrong with Indispensability? (Or, the Case for Recreational Mathematics)
*Marcus, AP and the IA,
Chapter Four: The Weasel
22 Thursday, April 14 Autonomy Platonism I Marcus, AP and the IA, Chapter One: Platonism: An Overview
*HIPM 20: Contemporary Apriorism
23 Tuesday, April 19
Term Paper Draft Due
The Unfortunate Consequences Marcus, AP and the IA, Chapter Five: The Unfortunate Consequences
24 Thursday, April 21 Intuition in Mathematics I Chudnoff, “Awareness of Abstract Objects
25 Tuesday, April 26 Intuition in Mathematics II Chudnoff, “Intuition in Mathematics
26 Thursday, April 28 Autonomy Platonism II Marcus, “The Indispensabilist and the Autonomist
27 Tuesday, May 3
Term Paper Due
Reflective Equilibrium Goodman, “The New Riddle of Induction,” §1–§2 (pp 59–65)
Rawls, A Theory of Justice, §§3–4, 9
Russell, “Logical Atomism,” to top of p 326
28 Thursday, May 5 Autonomy Platonism III Marcus, AP and the IA, Chapter 11: Circles and Justification

Friday, May 6, 4pm: Final ten précis due

Wednesday, May 11, 7-10pm: Final Exam

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Deadline Summary:

Thursday, March 3: Term paper abstracts and bibliography
Friday, March 11, 4pm: First ten reading précis
Thursday, March 31: Term paper précis and annotated bibliography
Tuesday, April 19: First draft of term paper

Tuesday, May 3: Final draft of term paper
Friday, May 6, 4pm: Final ten Précis due
Wednesday, May 1 7-10pm: Final Exam


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