Phil 373: Introduction to Philosophy of Mathematics

Course Mechanics

Course Description

Galileo once claimed, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." Galileo's views about the importance of mathematics were/are not unique. Since Ancient Greece, philosophers have argued that mathematics is the paradigm of certain knowledge. Today, almost every science employs mathematics, as mathematical techniques are often thought to embody rigor and objectivity.

In light of these facts, it is perhaps surprising that, until the middle of the 20th century, the foundations of mathematics were hotly contested. Many eminent mathematicians of the nineteenth and twentieth centuries (e.g., Hilbert, Cauchy, Weierstraß, Dedekind, etc.) believed that accepted proofs of even elementary theorems from geometry and calculus lacked appropriate logical rigor. This perception led to the "search for rigor" in nineteenth century mathematics and the so-called "crisis" in the foundations of mathematics at the beginning of the twentieth century. This course is an historical introduction to the philosophical questions that prompted this search for rigor and the resulting foundational crisis.

We will focus on two central questions. First, to what degree are mathematical theorems are justified by rational insight, sensory experience, purely symbolic computations, or some combination of the three? We will trace answers to this question from Ancient Greece until the beginning of the nineteenth century, and we will pay close attention to how answers to this question are related to mathematical developments concerning the relationship between arithmetic and geometry. Second, what is the infinite, and how one can reason about infinite sets, spaces, and numbers without becoming entangled in contradictions?

The course presupposes no mathematical background, except familiarity with high-school algebra and geometry.

Course Goals

The course has three central goals. First, by the end of the semester, students should be able to state several key philosophical questions that prompted a search for a "foundation" for mathematics in the 19th and 20th centuries. Second, students should be able to explain how changes in mathematical practice changed philosophical theorizing (and vice versa). In particular, students should be able to identify features of Ancient Greek mathematical practice that led to central epistemological problems in modern philosophy about the justificatory roles of reason, sense perception, and symbolic computation. Finally, students should be able to reconstruct, critique, and develop epistemological arguments about the foundations of mathematics, and they should be able to communicate those arguments orally and in writing that is clear, brief, and precise.

Course Requirements

Philosophical thinking is a skill, not unlike playing the piano, riding a bike, or dancing. Learning a new skill requires practice, and the best way to practice philosophical thinking is to write and engage in spirited (but polite) debates with other philosophers. Thus, the central requirement of this course is to write three short papers in which you summarize an argument from the readings and defend it from potential objections. More details about the topics of the three papers can be found on the course website.

Additionally, before each class period, I will ask you to answer two or three short questions about the readings. Most of the questions require only one sentence to answer, and none require more than a paragraph. Please bring both the assigned readings and your answers with you to class. Some students view frequent assignments/assessments as "busy work" or as an instructor's attempt to gauge which students are working hardest. That is is not my intention at all. The philosophy of mathematics is a complex subject, and consequently, many of the assigned readings are somewhat difficult. When faced with hard-to-understand texts, it is easy to become discouraged and to give up. One of my central duties, as an instructor, is to ensure that you do not give up when concepts and/or arguments are initially difficult to understand. The purpose of the nightly questions is threefold: (i) to encourage you to read the assigned texts closely and actively, (ii) to prepare you for class discussions in which we will clarify and build upon the readings, and (iii) to provide me with feedback about which concepts are most difficult for students to understand.

Finally, you will not be able to understand the course material without attending class. I teach some courses (e.g., introductory logic) in which I recognize that, for some very bright, motivated and hard-working students, attending class is not always necessary. This is not such a course. The assigned readings are extremely difficult for a number of reasons, and without class discussion and lecture, you will likely learn very little. To encourage you to attend and to make sure you are active in class discussions, I will ask you a question at the end of every class about the material that has been covered that day, and I will give you fifteen minutes to write a short response to said question.

Submitting Assignments

Reading assignments are due at the beginning of class. You should bring a typed, hard copy of your assignment to class; your answers should not exceed the front side of one page. At the end of class, I will ask you to turn over your reading assignment and respond to a question about the material from class. You can write your response in pencil or pen.

All papers should be submitted electronically via Canvas. Please do not email me your papers unless you already have tried to upload them via Canvas. When a class exceeds even a small number of students (e.g., ten), it is difficult for an instructor to organize and maintain a record of students' work if it is submitted via email.


I use rubrics when assigning you grades on more substantial assignments (e.g., papers and presentations). Rubrics contain detailed descriptions of which skills you are performing well and which are in need of improvement. I encourage you to look at the rubrics before you write your papers so that you know exactly how you will be assessed. Even better, find a partner and grade each other's papers using the provided rubrics. Doing so gives you experience evaluating philosophical work and will improve your own writing.

I do not regrade assignments, but I would be happy to clarify why you received the grade that you did.

Your final grade will be calculated via a weighted average using the following weights:

It is necessary to write all three papers in order to pass the course. Your final grade will be converted to a four point scale using the following equation: Four Point Scale = Percentage/10 - 5.5.

For example, if your final percentage is 90%, then your final grade will be 3.5 = 9/10 - 5.5.

Course Files


Below is a table indicating readings and assignments that are due each class. If you are a registered student in the class, then you can download the readings from the link in the "Course Files" section above.

Date Topic Readings Assignment
1/5 Introduction to class themes

Lecture 1 Slides
None None
1/7 Greek geometry and the role of the diagram

Lecture 2 Slides
Book 1 of Euclid's Elements: Please read the Common Notions and Postulates (pages 153-155) and proofs or propositions 1, 2, 4, 7, 16, and 47. You may ignore the extensive commentary.

Sections 1 and 2-2.1.2 of Netz's The Shaping of Deduction.
Reading Assignment 1
1/9 Aristotelian Logic

Lecture 3 Slides
Sections 1-5.5 and 6 of Aristotle's Logic Reading Assignment 2
1/12 Locke on Abstract Ideas and Demonstration

Lecture 4 Slides
Locke's Essay Concerning Human Understanding.
  • Book II, Chapters 11 and 12.
  • Book IV Chapter 2: Sections 1-11.
Reading Assignment 3
1/14 Criticisms of Abstract Ideas

Group Assignment
Introduction to Berkeley's A Treatise Concerning the Principles of Human Knowledge.

Section 1.1.7 of Hume's A Treatise of Human Nature

Pages 67-68 of Shapiro's Thinking About Mathematics.
Reading Assignment 4
1/16 Criticisms of Abstract Ideas Review readings from previous class.
Groups' Reconstructions of Hume's Arguments
1/19 No Class None None
1/21 Eudoxian Theory of Proportion and Greek Number Theory

Lecture 7 Slides
Euclid's Elements:
  • Book V: Please read the definitions (pages 113-114) and proofs of propositions 1, 4, and 13.
  • Book VII: Please read the definitions (pages 277-279) and proofs of propositions 1 and 2.
  • Book X: Please read the commentary on pages 1-3, and the definitions on page 10.
Reading Assignment 5
1/23 Plato's Theory of Recollection

Lecture 8 Slides
Meno. Translator's Introduction (pages 870-871) and Lines 79e-86a.

Phaedo Translator's introduction (pages 49-50) and Lines 72e - 77a.
Reading Assignment 6
1/26 Plato's Philosophy of Mathematics

Lecture 9 Slides
Republic. Translator's Introduction (pages 971-972) and Book VII. Lines 514 - 530d. Reading Assignment 7
1/28 Aristotle's Philosophy of Mathematics

Discussion Questions
Chapter 3 of Shapiro's Thinking About Mathematics. Reading Assignment 8
1/30 Construction in Geometry Prior to Descartes

Bos. Redefining Geometrical Exactness.

Sections 1.1 -1.4, 1.6; Chapter 3 pages 37-41; Sections 3.4-3.5, 3.7; Section 5.1, 5.3; Chapter 6
Reading Assignment 9
1/31 First Paper Due by 8PM
Assignment Description
Suggestions for writing a philosophy paper
2/2 Descartes' Early Philosophy of Mathematics I

Discussion Questions
Rules 2-4, 10, and 12-15 of Descartes' Rules for the Direction of the Mind. Reading Assignment 10
2/4 Descartes' Early Philosophy of Mathematics II

Rules 16-22 of Descartes' Rules for the Direction of the Mind.

Descartes. Letter to Beeckman from 1619.
Reading Assignment 11
2/6 Descartes' Geometry

Descartes. Geometrie.
  • Book I. Pages 2-6.
  • Book II. Pages 40-48.
  • Book III. Pages 152-156.
Pages 317-324 of Kline's Mathematical Thought from Ancient to Modern Times. Volume 1.
Reading Assignment 12
2/9 Leibniz's Reaction to Descartes

"Letter to Queen Sophie Charlotte of Prussia" in G.W. Leibniz: Philosophical Essays. Ed. Ariew and Garber.

Section 42 "Critical Thoughts on the General Principles of Descartes." In Gottfried Wilhelm Leibniz: Philosophical papers and letters Read Leibniz's criticisms of Articles 1, 5, 26, 31/35, 43/45/46, and 75.
Reading Assignment 13
2/11 Class cancelled due to instructor illness. None. None.
2/13 Leibniz's Logic

"Preface to a universal characteristic" and "Samples of the Numerical Characteristic." Pages 5-18 of G.W. Leibniz: Philosophical Essays. Ed. Ariew and Garber.

"A Study in the Early Logical Calculus." Pages 371-373 of Gottfried Wilhelm Leibniz: Philosophical papers and letters. Ed. Loemker.
Reading Assignment 14
2/16 No class: President's day None. None.
2/18 Newton's Philosophy of Mathematics

Preface and Book I, Lemmas 1-4, Lemma 28 of Newton's Principia.

Domski. "The Constructible and the Intelligible in Newton's Philosophy of Geometry."
Reading Assignment 15
2/20 Hume

Hume. A Treatise of Human Nature. Book I. Part I. Sections 1-5, 7, and Part III. Section 1.

Hume. Enquiry Concerning Human Understanding. Section 4. Lines 1-3.
2/23 Kant

Kant. Prolegomena to Any Future Metaphysics.
Preamble (pp. 15-22) and Sections 6-13 (pp. 32-38).
Reading Assignment 16
2/25 Aristotle on the infinite

Euclid. Elements. Book IX: Proposition 20.

Aristotle. Physics.
  • Book III. Lines 202b30 - 208a25.
  • Book VI. Lines 231a18-233a31 and 239b5 - 240b8.
Reading Assignment 17
2/27 Infinity and Exhaustion

Reread the sections from Euclid on the Eudoxian theory of proportions.

Euclid. Elements. Book XII: Propositions 1 and 2.

Heath. A History of Greek Mathematics. Pages 325-329 (Sections α and β)

The Works of Archimedes.
  • "Measurement of the circle." Proposition 1 (pp. 91-93)
  • "Quadrature of the Parabola." Proposition 16 (pp. 244-245)
The Method of Archimedes.. Proposition 1.
Reading Assignment 18
3/2 The Invention of the Calculus

Kline. Mathematical Thought from Ancient to Modern Times. Chapter 17. Reading Assignment 19
3/4 Berkeley's Critique
Berkeley. The Analyst. Sections 1-15. Reading Assignment 20
3/6 Infinite Divisibility
Hume. A Treatise of Human Nature. Book I. Part II. Sections 1-4. Reading Assignment 21
3/7 Second Paper due by 8PM
3/9 The Search for Rigor
Kline. Mathematical Thought from Ancient to Modern Times. Pages 426-434, Chapter 18, and Chapter 40. Sections 1, 2, and 7. Reading Assignment 22
3/11 Construction of the reals and the intermediate value theorem
Dedekind. Continuity and Irrational Numbers. Introduction and Sections 1-5. Reading Assignment 23
3/13 Class Wrap-Up None None
3/20 Third Paper Due at Midnight