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 socalled "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 highschool algebra and geometry.
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.
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 hardtounderstand 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 hardworking 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.
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.
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 153155) and
proofs
or propositions 1, 2, 4, 7, 16, and 47.
You may ignore the extensive commentary.
Sections 1 and 22.1.2 of Netz's The Shaping of Deduction. 
Reading Assignment 1 
1/9  Aristotelian Logic
Lecture 3 Slides 
Sections 15.5 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.

Reading Assignment 3 
1/14  Criticisms of Abstract Ideas
Handout 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 6768 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 
None. 
1/19  No Class  None  None 
1/21 
Eudoxian Theory of Proportion and Greek Number Theory
Lecture 7 Slides 
Euclid's Elements:

Reading Assignment 5 
1/23 
Plato's Theory of Recollection
Lecture 8 Slides 
Meno. Translator's Introduction (pages 870871) and Lines 79e86a.
Phaedo Translator's introduction (pages 4950) and Lines 72e  77a. 
Reading Assignment 6 
1/26 
Plato's Philosophy of Mathematics
Lecture 9 Slides 
Republic. Translator's Introduction (pages 971972) 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 3741; Sections 3.43.5, 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 Rubric 

2/2  Descartes' Early Philosophy of Mathematics I
Discussion Questions 
Rules 24, 10, and 1215 of Descartes' Rules for the Direction of the Mind.  Reading Assignment 10 
2/4  Descartes' Early Philosophy of Mathematics II

Rules 1622 of Descartes' Rules for the Direction of the Mind.
Descartes. Letter to Beeckman from 1619. 
Reading Assignment 11 
2/6  Descartes' Geometry

Descartes. Geometrie.

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 518 of
G.W.
Leibniz: Philosophical Essays. Ed. Ariew and Garber.
"A Study in the Early Logical Calculus." Pages 371373 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 14, 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 15, 7, and Part III. Section 1.
Hume. Enquiry Concerning Human Understanding. Section 4. Lines 13. 
None. 
2/23  Kant

Kant. Prolegomena to Any Future Metaphysics.
Preamble (pp. 1522) and Sections 613 (pp. 3238). 
Reading Assignment 16 
2/25  Aristotle on the infinite

Euclid. Elements. Book IX: Proposition 20.
Aristotle. Physics.

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 325329 (Sections α and β) The Works of Archimedes.

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 115.  Reading Assignment 20 
3/6  Infinite Divisibility

Hume. A Treatise of Human Nature. Book I. Part II. Sections 14.  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 426434, 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 15.  Reading Assignment 23 
3/13  Class WrapUp  None  None 
3/20  Third Paper Due at Midnight 