Summer 2025
43190350 Philosophy of Mathematics

Course Information

Instructor Dr. Bruno Bentzen

TA Jilin Wang

Office Location Chengjun Complex 4, 306

Office Hours by appointment

Email bbentzen at zju.edu.cn

Credits 32 hours (8 weeks)

Language English

Course Description

This course is a general overview of the philosophy of mathematics, focusing on the particular challenges that mathematics poses for philosophy, some historical developments since Kant that led to the foundational crisis at the beginning of the twentieth century with Russell's paradox, and some contemporary views. The main topics discussed include the rise and fall of Frege's logicism (and its revival with neologicism), Russell's response to the paradox with his doctrine of types, term and game formalism, deductivism and Hilbert's axiomatization of geometry, Hilbert's program and the problem of infinity, the impact of Gödel's's incompleteness theorems, Brouwer's proposal of intuitionism and Heyting's further development of his views, and Dummett's meaning-theoretic argument for the adoption of intuitionistic logic over classical logic. If time allows, we will also cover either Gödel's views on mathematical intuition and Maddy's naturalism, nominalism in mathematics, the iterative conception of sets, Benacerraf's identification problem and structuralism, or Voevodsky's univalent axiom and univalent foundations.

Course Materials and Resources

This course is roughly based on the following textbooks:

• Shapiro, Stewart. Thinking about mathematics: The philosophy of mathematics. Oxford Univerity Press, 2020.

• Linnebo, Øystein. Philosophy of mathematics. Princeton University Press, 2017.

In addition, lecture slides will be made available to you online before class each week. They can always be found here along with all my teaching materials. Please note that old slides from past classes might occasionally be updated for corrections or minor revisions. Students interested in complementary readings are also encouraged to check the list of primary and secondary literature that will be suggested in class.

Course objectives

Upon the successful completion of this course, you will:

1. understand the relationship between mathematics and philosophy;

2. distinguish the main positions in the philosophy of mathematics;

3. have a basic understanding of the foundational crisis in mathematics;

4. acquire a better understanding of some contemporary views.

Assessment and grades

• Term paper 40%

• Presentation 50%

• Participation 10%

Grades are awarded on a scale from 0 to 100, where 100 is the best grade and 60 is the minimum passing grade.

Term paper: There will be one major writing assignment at the end of the course. Your paper must have one page (no more than 500 words), contain your name, date, and student number, and must be written in English. Your English level will not be subject to evaluation, but your prose has to be clear. Please include a title, introduction, exposition of your problem and ideas, and conclusion. Your paper should be submitted digitally to our TA in PDF format. Detailed information about possible topics and the deadline will be provided in due time. Excellent papers will show me that you went beyond the material covered in class and studied the subject in depth. More specifically, you will be graded according to the following three criteria:

• Clarity: your central claims are stated precisely and your paper has a clear structure;

• Strength: your arguments provide plausible support for their conclusions;

• Command: you demonstrate a good command of the concepts under discussion.

Presentation: You will be required to give oral presentations on our last day of classes. Presentations must be given in English, should make use of slides, and should last for about 5-7 minutes. Your oral English skills will not be evaluated, only the content of your presentation. There will be a short QA session of about 2-5 minutes after each presentation. Students in the audience are especially encouraged to ask questions to the presenting student. You should send your slides to our TA at least one day in advance. The content of the presentations will be selected from topics covered in class and related subjects. Presenting on the same topic as your term paper is especially encouraged. Additional information about the presentations will be given in due time.

Participation: I will not require you to speak up in class, but any form of engagement with the lectures is highly encouraged. The use of mobile phones, computers, and other portable devices is permitted for taking notes and class-related activities only. 

Attendance policy

You are expected to attend every lecture and be on time. If you cannot come to class due to an emergency, please let me know as soon as possible. If you miss a class it is your responsibility to make up the material missed and catch up with your classmates. 

Plagiarism and AI policies

I wish to evaluate your performance, so your work should reflect your own efforts. You can discuss your paper or presentation with others, but do not copy their work and submit them as your own. Any form of cheating and plagiarism is prohibited and will be taken as a serious offense by the university. The use of AI editing tools such as Grammarly or Hemingway Editor as language aids is permitted. However, submitting AI-generated papers (in part or in whole) such as those generated by ChatGPT or DeepSeek prompts without permission is considered cheating. Submitting AI-generated text as your own is no different from plagiarism and I will reserve the right to run AI writing detectors and request an impromptu oral explanation of your ideas whenever the suspicion arises.

Feedback

I always welcome feedback, be it positive or negative. If you wish, you can do this by speaking to me directly after class, sending me an email, or, if you prefer, sending me an anonymous note. Giving feedback will not have any effect on your grade, either positively or negatively. But it will help me to see my lectures from different angles and develop new ways of improving them.

Special Accommodations

Please contact me if you have a disability or other circumstances that require special accommodations.

Schedule

The following schedule is tentative and subject to change with fair notice:

Lecture 1: 

• Overview of some basic positions

• Philosophy and mathematical practice

• Reading recommendations: Shapiro ch. 1–2

Lecture 2:

• Prehistory of the philosophy of mathematics

• Plato's views of mathematics

• Aristotle's criticism of Plato

• Reading recommendations: Shapiro ch. 3, Linnebo ch.1.7

Lecture 3: 

• Aristotle's logic

• Euclid's Elements 

• Reading recommendations: Shapiro ch. 1–2

Lecture 4: 

• Kant's philosophy of mathematics

• Reading recommendations: Shapiro ch. 1–2

Lecture 5: 

• Kant's philosophy of mathematics

• Mill's epistemology for mathematics and logic 

• Reading recommendations: Shapiro ch. 4.1–3, Linnebo ch.1.6

Lecture 6: 

• The foundational crisis

• Frege's logic in Begriffsschrift

• Frege's project in Grundlagen

• Reading recommendations: Shapiro ch. 5.4, Linnebo ch.9

Lecture 7: 

• Paper announcement

• Frege's logic in Grundgesetze

• Russell's paradox and neologicism today

• Russell's logicism in Principles of Mathematics

• Reading recommendations: Shapiro ch. 5, Linnebo ch.9.2–3

Lecture 8: 

• From Principles of Mathematics to the doctrine of types

• The doctrine of types in Principia Mathematica

• Reading recommendations: Shapiro ch. 5, Linnebo ch.9.2–3

Lecture 9: 

• The doctrine of types in Principia Mathematica

• Syntax and semantics of formal systems

• Reading recommendations: Shapiro ch. 5, Linnebo ch.9.2–3

Lecture 10: 

• Term and game formalism, and deductivism

• Hilbert's program and Gödel's incompleteness

• Reading recommendations: Shapiro ch. 6.1–4, Linnebo ch.3–4

Lecture 11: 

• Hilbert's finitism

• Gödel's incompleteness theorems

• Reading recommendations: Shapiro ch. 7.1–3, Linnebo ch.5

Lecture 12: 

• The emergence of non-Euclidean geometry

• Brouwer's first act of intuitionism

• Reading recommendations: Shapiro ch. 7.1–4, Linnebo ch.5

Lecture 13: 

• Primer on intuitionistic logic

• Brouwer's second act of intuitionism

• Reading recommendations: Shapiro ch.8.1-3, Linnebo chs.6 and 7

Lecture 14: 

• Heyting's meaning explanations

• Dummett's meaning-theoretic turn

Lecture 15: 

• Holiday

Lecture 16: 

• Dummett's meaning-theoretic turn

• Group presentations

Past instances

• Summer 2024

Please let me know if you find any broken links.