Philosophy : Seminar in Epistemology: Knowledge, Truth, and Mathematics

Russell Marcus, Instructor. Email me.

Hamilton College, Spring 2008

 

Syllabus

Note: this version of the syllabus will be kept up-to-date during the term, if changes are necessary. You can see the pdf version of the syllabus here.


Meeting Times and Place:

Texts:

Course Description and Overview:

This course is a survey of the philosophical questions which arise from considering historical and contemporary approaches to explaining 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 divide into three periods. The first part of the course surveys historical positions through the mid-nineteenth century. The second part of the course focuses on the fruitful period between Frege and Gödel. The last part covers contemporary approaches.

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.” Some prominent philosophers in the early modern period were mathematicians, including Descartes, who helped to found 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 often think about mathematics, and 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.

Assignments and Grading

Most class discussions will be initiated by short (2-4 page) papers written by the members of the seminar. Each student will be expected to complete three seminar papers during the term. Students will also be expected to complete a longer paper, in two drafts (preliminary and final). You have a choice of either a participation and preparation grade, or a final exam grade. By the penultimate week of classes, I will be able to tell you what your participation and preparation grade will be. At that time, you may choose to take a final exam, in its place. Assignments will be weighted as follows:

1. All the primary readings listed below, including seminar papers.
2. Seminar papers (45%; 15% each)
3. Term paper (30%)
4. Participation and preparation / final exam (25%)

The preliminary draft of your term paper is due on April 14. If you choose to expand one of your first two seminar papers, the April 14 draft must show evidence of further research. 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-).

Some notes about assignments:

  • You are expected to complete the primary readings for each class.
  • The secondary readings are suggested. For some classes, you may not complete the secondary readings. If you are writing a seminar paper, you should certainly include the secondary readings in your preparation.
  • Readings are to be completed before the class indicated below.
  • For further work, especially for the topic you choose for your term paper, see the course bibliography.
  • More detailed information on assignments is available here.
  • The Hamilton College Honor Code will be strictly enforced.
I will prepare reading guides for some primary sources, lists of questions corresponding to each reading. You can use the reading guides to help you determine your comprehension of the assignments. In addition, the final exam will be based directly on the reading guides.

Some General Notes on Class Discussion and Seminar Papers:

Classes will generally run as discussions of one or two seminar papers. Seminar papers should assimilate the assigned readings and summarize the main arguments. They must demonstrate attempts to grapple with the primary sources. You should also consider the secondary readings. Critical discussion is encouraged, and need not be fully developed. 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 discuss.

Each seminar paper is due at noon 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. Classes will begin with an opportunity to present your paper, at which time you may discuss any particular difficulties in the material, or topics that you were unable to cover in the paper.

Students may be allowed to work on one seminar paper, either their second or third, in pairs, with a consequent increase in expectations of length and breadth.

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

Schedule:

Note: Full bibliographic references for each of the entries below is available in the course bibliography.

Class 1: What is mathematics? What is philosophy of mathematics? (1/21)

  • Readings:
    Brown, Chapter 1
    Shapiro, pp 21-29

Class 2: Pythagoras and the Pythagoreans (1/23)

Class 3: Plato’s Platonism (1/28)

Class 4: Aristotle (1/30)

Class 5: Modern Rationalism I (2/4)

Class 6: Modern Rationalism II (2/6)

Class 7: Modern Empiricism (2/11)

Class 8: The Synthetic A Priori, I (2/13)

Class 9: The Synthetic A Priori, II (2/18)

  • Seminar Paper: Heather
  • Primary Readings
    See Readings for Class 8

Class 10: Radical Empiricism (2/20)

Class 11-12: Cantor’s Paradise (2/25-27)

Class 13: Logicism (3/3)

Class 14: Formalism (3/5)

  • Seminar Paper: Matt
  • Primary Reading
    Hilbert, “On the Infinite”
  • Secondary Readings

    Brown, pp 62-71
    Shapiro, pp 140-165

Class 15: Incompleteness (3/10)

Class 16: Intuitionism (3/12)

Class 17: Carnap (3/31)

Class 18: Wittgenstein’s Conventionalism (4/2)

Class 19: Wittgenstein’s Conventionalism (4/7)

  • Seminar Paper: Jazmine
  • Primary Readings

    See Class 17, plus...
    Ayer, “The A Priori”

(Note: we will not have class on 4/9, as I will be away for a conference. We will make up that class on Friday, 4/25, at 1:15pm. Room TBA.)

Class 20: Gödel Platonism (4/14)

Class 21: The Problem (4/16)

Class 22: Quine, I (4/21)

Class 23: Quine, II (4/23)

Class 24: Structuralism, I (Friday, 4/25)

Class 25: Structuralism, II (4/28)

  • Seminar Paper: James
  • Primary Reading
    Shapiro, “Structure”
  • Secondary Reading

    • Shapiro, pp 275-289

Class 26: Fictionalism (4/30)

Class 27: Contemporary Platonism (5/5)

Class 28: Computer Proofs (5/7)

Epitaph

 

 

Excised assignments

Modalism

Contemporary Platonism