Official, printable course outlines in PDF format can be found at the end of this page.

## Contents

## Course Description

The philosophy of mathematics deals with, as its name suggests, philosophical issues that are raised by mathematics. Perhaps the most important difference between mathematics and the other sciences is that mathematics deals with entirely abstract concepts (such as number, set, function). Mathematicians do not conduct experiments to confirm their hypotheses, they find proofs. One important issue then, for instance, is clarifying the role of proofs in mathematics, and with it, the nature of mathematical knowledge. Another issue is the question of the status of mathematical objects—do they inhabit some Platonic realm of mathematical objects, are they constructions of the mind, or are there perhaps, strictly speaking, no mathematical objects, are they fictitious? In the course of discussing these issues, we will first focus on three important schools of thought: logicism, intuitionism, and formalism. These schools are associated with, respectively, Gottlob Frege and Bertrand Russell, Immanuel Kant and L. E. J. Brouwer, and with David Hilbert. In the second half of the course, we will study some of the more recent developments in the philosophy of mathematics (after 1950). This includes the debate about the metaphysics of mathematical objects, in particular, whether there are any (realism) or not (nominalism), and the status of mathematical truths; structuralism in mathematics; mathematical explanation; and empiricist and naturalist approaches to mathematics.

## Prerequisites and Preparation

Two previous courses in Philosophy, one of which must be PHIL 367 or 467, and one of which must be a 400 or higher level course; **or consent of the Department. **Due to the nature of the course, I will be very open to waiving prerequisites, especially if you have advanced training in logic (such as PHIL 379 or 479) or in mathematics. Please email rzach@ucalgary.ca if you would like to take the course but do not have the listed prerequisites.

Although Phil 279 (Logic I) and 379 (Logic II) are not formally prerequisites, you will have a hard time in this course if you have not at least taken Logic I. Obtaining a background in logic (at least Phil 279, preferably 379) is therefore highly recommended.

## Required Text

Alexander George and Daniel J. Velleman,

Philosophies of Mathematics(Blackwell, 2002)Gottlob Frege,

The Foundations of Arithmetic(Blackwell/Northwestern, 1980)A package of readings.

All will be available at the University of Calgary Bookstore.

## Requirements

*Undergraduate students:* Six short writing assignments (250 words max., 5% each for a total of 30%, graded on pass/fail basis), a final paper (2500–4000 words max., 40%), and an in-class presentation on the topic of your paper (10–15 minutes, 10%). Class participation counts for 20% of your grade.

*Graduate students:* Six short writing assignments (250 words max., 5% each for a total of 30%, graded on pass/fail basis), a final paper (5000 words max., 40%), and preparing and leading a one-hour lecture/seminar (10%). Class participation counts for 20% of your grade. There will be a separate discussion section reserved for graduate students.

*Due Dates:* The final paper will be due the last week of class (i.e., April 13).

The aim of the short writing assignments is to get you engaged with the readings before they are discussed in class. A short writing assignment consists in a one-page writeup of something you think of while you do the reading. It can be a question together with an attempted answer, a worry, a short note on a connection between the reading for that week and some other topic (from this or another class). Ideally, it would be something that can form the basis of a discussion in class.

You have to turn in six such short essays. You can choose when you submit them, and on what you want to write. They are due on Tuesdays at 6pm and you may submit them by email. However, you have to complete at least two of these assignments in weeks 1–6, and at least two in weeks 7–12.

## Evaluation

Your final paper, your presentation, and your course participation will receive a letter grade reflecting the level of mastery of the material shown by the work you submit. The meanings of letter grades are defined in the *Calendar*, for written work, they amount roughly to the following criteria:

- A
- Excellent—superior performance, showing comprehensive understanding of subject matter. (Your writing is clear and concise; your assignments make obvious that your understanding of the issues and arguments is correct and complete; you show superior ability in representing and assessing others’ philosophical arguments; you show significant ability for original philosophical thinking).
- B
- Good—clearly above average performance with knowledge of subject matter generally com-plete. (You show a good grasp of the assigned reading; but either your writing is not perfectly clear or your assignments are largely only expository and don’t show the critical ability required for an A).
- C
- Satisfactory—basic understanding of the subject matter. (Your work shows that you’ve worked through the reading and attended class, but your assignments misrepresent the arguments or views we’re discussing, or your criticisms are off the mark.)
- D
- Minimal pass—marginal performance. (Your work is unclear or confused; or you grossly misunderstood or misrepresent the arguments or views we’re discussing.)
- F
- Fail—Unsatisfactory performance. (Your work fails to show that you’ve made a serious attempt at coming to grips with the material; or your writing borders on the incomprehensible.)

In computing your final grade, your marks will be converted to grade points and averaged according to the weights given above. The correspondence of letter grades with grade points is defined in the *Calendar* (A = 4, B = 3, C = 2, D = 1, F = 0). “Slash” grades are possible with grade point values 0.5 below the higher grade (e.g., A/B = 3.5). For short writing assignments, a “pass” will receive 4 grade points, a fail, 0.

The final grade will be the letter grade corresponding to the weighted average of your assignments, paper, presentation, and participation plus a margin of 0.1. For the final grade, +’s and -‘s are possible, too; as defined in the *Calendar*, +/- adds/subtracts 0.3 grade points. In other words, a course average of 3.9 or higher receives an A; between 3.6 and 3.9, an A–; between 3.2 and 3.6, a B+; between 2.9 and 3.2, a B; and so on. There is no D- grade; to earn a D you require a course average of at least 0.9. The A+ grade is reserved for “truly outstanding” performance.

## Assignments and Policies

### Late work and extensions

A final paper handed in late will be penalized by the equivalent of one grade point per calendar day. A short writing assignment handed in late will be penalized by one half grade point for every two hours (or part thereof) it is late (i.e., it will receive a 3.5 if handed in after 6pm but before 8pm, a 3.0 if handed in before 10pm, etc.).

### Plagiarism

You will find the University policy on plagiarism at the end of the printed version of this outline. Plagiarism is a very serious academic offense. It is not limited to copying papers wholesale from the Internet; close paraphrase of the texts, of the lectures, or of anyone (other than you) without attribution constitutes plagiarism. Your assignments should only contain your own formulations. You should use direct quotes from the texts sparingly, and clearly mark them as such by using quotation marks and giving a source reference. When in doubt, consult with the instructor. **Plagiarism will result in a failing grade in the course and a report to the Dean’s office.**

### Checking your grades and reappraisals of work

University policies for reappraisal of term work and final grades apply (see the section “Reappraisal of Grades and Academic Appeals” of the Academic Regulations in the *Calendar*). In particular, term work will only be reappraised within 15 days of the date you are advised of your marks. Please keep track of your assignments (make sure to pick them up in lecture or in office hours) and your marks (check them on the website) and compare them with the graded work returned to you.

## Course Website

A course website on U of C’s BlackBoard server has been set up. You will be automatically registered if you’re registered in the class. To access the BlackBoard site, you can either go directly to blackboard.ucalgary.ca and log in with your UCIT account name and password, or you can access it through the myUofC portal (my.ucalgary.ca; log in with your eID). If you don’t have an eID or UCIT account, see elearn.ucalgary.ca/help.html.

## Tentative Syllabus and Due Dates

This is a tentative syllabus to give you a rough idea what topics we will cover when.

- [
**Week 1: Kant and Frege**] January 12.Kant,

*Prolegomena*

Frege,*Foundations*, Sections 1–28 - [
**Week 2: Frege’s Logicism**] January 19.Frege,

*Foundations*, Sections 29–69

George and Velleman,*Philosophies of Mathematics*, Ch. 2Benacerraf, “Frege: The last logicist”

MacFarlane, “Frege, Kant, and the logic in logicism” - [
**Week 3: Frege and Dedekind**] January 26Frege,

*Foundations*, Sections 70–109

Dedekind, “The nature and meaning of numbers”Tait, “Frege versus Cantor and Dedekind: On the concept of number”

Reck, Dedekind’s Structuralism: An Interpretation and Partial Defense” - [
**Week 4: Russell’s Paradox, The Theory of Types, Set Theory**] February 2Russell, Letter to Frege

Russell, “Mathematical logic as based on the theory of types”

George and Velleman,*Philosophies of Mathematics*, Ch. 3Gödel, “Russell’s mathematical logic”

Urquhart, “The theory of types” - [
**Week 5: Intuitionism**] February 9Weyl, “On the new foundational crisis of mathematics”

Brouwer, “Mathematics, science, and language”

Heyting, “The intutionist foundation of mathematics”

George and Velleman,*Philosophies of Mathematics*, Ch. 4 and 5.Brouwer, “Intuitionistic reflections on formalism”

- [
**Week 6: Hilbert’s Program**] February 16George and Velleman,

*Philosophies of Mathematics*, Ch. 6

Hilbert, “On the infinite”

Bernays, “The philosophy of mathematics and Hilbert’s proof theory”Hilbert, “The foundations of mathematics”

Mancosu, “Hilbert and Bernays on metamathematics”

von Neumann, “The formalist foundation of mathematics”

Zach, “Hilbert’s Program” - [
**Week 7: Hilbert’s Program, Logicism, and Gödel’s Theorems**] March 2Detlefsen, “On interpreting Gödel’s second theorem”

George and Velleman,*Philosophies of Mathematics*, Ch. 7Zach, “Gödel’s first incompleteness theorem and mathematical instrumentalism”

- [
**Week 8: Mathematical Objects and Mathematical Objectivity I**] March 9Putnam, “What is mathematical truth?”

Putnam,*Philosophy of Logic,*Sections IV–IX.

Benacerraf, “Mathematical truth”

Tait, “Truth and proof: The platonism of mathematics” - [
**Week 9: Mathematical Objects and Mathematical Objectivity II**] March 16Burgess, “Why I am not a nominalist”

Maddy, “Indispensability and practice”

Putnam, “Mathematics without foundations”

Field, “Is mathematical knowledge just logical knowledge?” - [
**Week 10: Structuralism in Mathematics**] March 23Benacerraf, “What numbers could not be”

Parsons, “The structuralist view of mathematical objects”

Field, “Mathematical objectivity and mathematical objects”Reck and Price, “Structures and structuralism in contemporary philosophy of mathematics”

- [
**Week 11: Explanation in Mathematics**] March 30Steiner, “Mathematical explanation”

Resnik and Kushner, “Explanation, independence, and realism in mathematics”

Mancosu, “Mathematical explanation: problems and prospects”Sandborg, “Mathematical explanation and the theory of why-questions”

- [
**Week 12: Empiricism and Naturalism in Mathematics**] April 6Lakatos, “A renaissance of empiricism in the recent philosophy of mathematics?”

Kitcher, “Mathematical naturalism”

Maddy, “Naturalism” - [
**Week 13: Student Presentations**] April 13

## Bibliography

### Assigned Readings (included in reader)

- Immanuel Kant. 1783. Selections from
*Polegomena to Any Future Metaphysics*(Sections 1–6). Reprinted from: Immanuel Kant,*Prolegomena*(Paul Carus, trans.), Open Court, 1902, 13–49. - Richard Dedekind. 1888. Was sind und was sollen die Zahlen? Reprinted from: William Bragg Ewald,
*From Kant to Hilbert: A Source Book in the Foundations of Mathematics*, vol. 2. Oxford: Oxford University Press, 1996, pp. 787–833. - Bertrand Russell. 1902. Letter to Frege. Reprinted from Jean van Heijenoort,
*From Frege to Göodel: A Source Book in Mathematical Logic*. Cambridge, MA: Harvard University Press, 1967, 124–125. - Bertrand Russell. 1908. Mathematical logic as based on the theory of types.
*American Journal of Mathematics*30, 222–262. Reprinted from Jean van Heijenoort,*From Frege to Gödel*. Cambridge, MA: Harvard University Press, 1967, 150–182. - Hermann Weyl. 1921. On the new foundational crisis of mathematics. Reprinted from: Paolo Mancosu, ed.,
*From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, New York and Oxford: Oxford University Press, 1998, 86–118. - L. E. J. Brouwer. 1929. Mathematics, science, and language. Reprinted from: Paolo Mancosu, ed.,
*From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, New York and Oxford: Oxford University Press, 1998, 45–53. - David Hilbert. 1925. On the infinite. Reprinted from: Paul Benacerraf and Hilary Putnam, eds.,
*Philosophy of Mathematics*, 2nd ed., Cambridge, MA: Harvard University Press, 1983, 183–201. - Paul Bernays. 1930. The philosophy of mathematics and Hilbert’s proof theory. Reprinted from: Paolo Mancosu, ed.,
*From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, New York and Oxford: Oxford University Press, 1998, 234–265. - Rudolf Carnap, Arend Heyting, and John von Neumann. 1931. Symposium on the foundations of mathematics. Reprinted from: Paul Benacerraf and Hilary Putnam, eds.,
*Philosophy of Mathematics*, 2nd ed., Cambridge, MA: Harvard University Press, 1983, 41–65. - Michael Detlefsen. 1979. On interpreting Gödel’s second theorem.
*Journal of Philosophical Logic*8, 279–313. - Hilary Putnam. 1975. What is mathematical truth? In: Hilary Putnam,
*Mathematics, Matter, and Method*, 2nd, ed., Cambridge University Press, 1979, 60–78. - Hilary Putnam. 1971. Selections from
*Philosophical Logic*(sections 4–9), Harper and Row. Reprinted from: Hilary Putnam,*Mathematics, Matter, and Method*, 2nd, ed., Cambridge University Press, 1979, 337–357. - Paul Benacerraf. 1973. Mathematical truth.
*The Journal of Philosophy*70, 661–679. - William W. Tait, 1986. Truth and proof: The platonism of mathematics.
*Synthèse*69, 341–370. - Penelope Maddy. 1992. Indispensability and practice.
*The Journal of Philosophy*89, 275–289. - John P. Burgess. 1983. Why I am not a nominalist.
*Notre Dame Journal of Formal Logic*24, 93–105. - Hilary Putnam. 1967. Mathematics without foundations.
*The Journal of Philosophy*64, 5–22. Reprinted in: Hilary Putnam,*Mathematics, Matter, and Method*, 2nd, ed., Cambridge University Press, 1979, 43–59. - Hartry Field. 1984. Is mathematical knowledge just logical knowledge?
*The Philosophical Review*93, 509–552. - Paul Benacerraf. 1965. What numbers could not be.
*The Philosophical Review*74, 47–73. - Charles Parsons. 1990. The structuralist view of mathematical objects.
*Synthèse*84, 303–346. - Hartry Field. 1998. Mathematical objectivity and mathematical objects. In: Stephen Laurence and Cynthia Macdonald, eds.,
*Contemporary Readings in the Foundations of Metaphysics*, Blackwell, 387–403. Reprinted from: Hartry Field,*Truth and the Absence of Fact*, Oxford University Press, 2001, 315–331. - Mark Steiner. 1978. Mathematical explanation.
*Philosophical Studies*34, 135–151. - Michael D. Resnik and David Kushner. 1987. Explanation, independence, and realism in mathematics.
*British Journal for the Philosophy of Science*38, 141–158. - Paolo Mancosu. 2001. Mathematical explanation: problems and prospects.
*Topoi*20, 97–117. - Imre Lakatos. 1976. A renaissance of empiricism in the recent philosophy of mathematics?
*British Journal for the Philosophy of Science*27, 201–23. - Philip Kitcher. 1988. Mathematical naturalism. In: William Aspray and Philip Kitcher, eds.,
*History and Philosophy of Modern Mathematics*, Minnesota Studies in the Philosophy of Science 11, University of Minnesota Press, 293–325. - Penelope Maddy. 1997. Mathematical naturalism. Ch. III.4 of Penelope Maddy,
*Naturalism in Mathematics*. New York and Oxford: Oxford University Press, pp. 181–205.

### Additional Suggested Readings (Not in Reader)

- Paul Benacerraf. 1981. Frege: The last logicist. In: Peter French, et al., (eds),
*Midwest Studies in Philosophy VI*, University of Minnesota Press, pp. 17–35. - John MacFarlane. 2002. Frege, Kant, and the logic in logicism.
*The Philosophical Review*111, 25–65. - Erich H. Reck. 2003. Dedekind’s structuralism: An interpretation and partial defense.
*Synthèse*137, 369–419 - Kurt Gödel. 1944. Russell’s mathematical logic. In: Arthur Schilpp, ed.,
*The Philosophy of Bertrand Russell*, La Salle, IL: Open Court. - Alasdair Urquhart. 2003. The theory of types. In Nicholas Griffin, ed.,
*The Cambridge Companion to Bertrand Russell*, Cambridge: Cambridge University Press, 286–309. - L. E. J. Brouwer. 1928. Intuitionistic reflections on formalism. Reprinted from: Paolo Mancosu, ed.,
*From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, New York and Oxford: Oxford University Press, 1998, 40–44. - David Hilbert. 1928. The foundations of mathematics. Reprinted from: Jean van Heijenoort,
*From Frege to Gödel*. Harvard University Press, 1967, 464–479. - Paolo Mancosu. 1998. Hilbert and Bernays on metamathematics. In: Paolo Mancosu, ed.,
*From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s*, New York and Oxford: Oxford University Press, 149-188. - Richard Zach. 2003. Hilbert’s Program.
*Stanford Encyclopedia of Philosophy*, http://plato.stanford.edu/entries/hilbert-program/ - Richard Zach. 2004. Gödel’s first incompleteness theorem and mathematical instrumentalism. Mancuscript.
- Erich H. Reck, Michael P. Price. 2000. Structures and structuralism in contemporary philosophy of mathematics.
*Synthèse*125, 341–383. - David Sandborg. 1998. Mathematical explanation and the theory of why-questions.
*British Journal for the Philosophy of Science*49, 603–624.