# Summer 2024

43190350 Philosophy of Mathematics

Wed 4:15–5:50 PM / Fri 8:00–9:35 AM at 3 North Hall, 217

Course Information

Instructor Dr. Bruno Bentzen

TA Huayu Guo

Ofﬁce Location Chengjun Complex 4, 306

Ofﬁce Hours Wed 3:00–5:00 PM (or by appointment)

Email bbentzen at zju.edu.cn

Credits 32 hours (8 weeks)

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, Russell's response to the paradox with this doctrine of types, Carnap's logical positivism and neologicism, 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, Dummett's meaning-theoretic argument for the adoption of intuitionistic logic over classical logic, Gödel's views on mathematical intuition and Maddy's naturalism, nominalism in mathematics, the iterative conception of sets, Benacerraf's identification problem and structuralism, and 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.

Some lectures draw on notes on additional material which will be made available in advance. 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:

understand the relationship between mathematics and philosophy

distinguish the main positions in the philosophy of mathematics

have a basic understanding of foundational crisis in mathematics

acquire a better understanding of some contemporary views

Assessment and grades

Final paper 40%

Homework 50%

Participation 10%

Final paper

There will be one major writing assignment at the end of the course. Your paper must have one page (around 500 words) and must be written in English. You should name and date your paper — it must also contain a title, introduction, exposition of your problem and ideas, and conclusion. You will be graded based on the clarity and structure of your exposition:

How well you can understand the issues you are writing about

How good the arguments you offer are

Your English level will NOT be subject to evaluation, only the content of your paper. But you should at least be able to communicate your ideas clearly and effectively. Detailed information about the content of the ﬁnal paper will be provided as it approaches.

Homework

Each homework assignment must be completed and turned in on time. I wish to evaluate your performance, so your homework should reﬂect your own efforts. Homework will be assigned weekly every Thursday and will be due on the following Thursday in class. No late homework will be accepted. If you missed the deadline because you were ill or for some other valid reason, please send me an email. Your solutions must be written legibly. The corrected homework will be returned to you on the Tuesday that follows the due date.

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.

Feedback

I 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, neither positively nor negatively. But it will help me to see my lectures from different angles and develop new ways of improving them.

Schedule (tentative)

Lecture 1: Some philosophical challenges imposed by mathematics (Shapiro ch. 1–2)

Lecture 2: Overview of some positions, Plato and Aristotle (Shapiro ch. 3, Linnebo ch.1.7)

Lecture 3: Kant's views on arithmetic (Shapiro ch. 4.1–2, Linnebo ch.1.6)

Lecture 4: Frege's logicism (Shapiro ch. 5.1, Linnebo ch.2)

Lecture 5: Russell's doctrine of types (Shapiro ch. 5, Linnebo ch.9.2–3)

Lecture 6: Carnap's logical positivism and neologicism (Shapiro ch. 5.3, Linnebo ch.9)

Lecture 7: Term and game formalism (Shapiro ch. 6.1–2, Linnebo ch.3)

Lecture 8: Deductivism and Hilbert's views on geometry (Shapiro ch. 6.2, Linnebo ch.3)

Lecture 9: Hilbert's program, infinity, and Gödel's incompleteness (Shapiro ch. 6.3–4, Linnebo ch.4)

Lecture 10: Brouwer and Heyting on intuitionism (Shapiro ch. 7.1–3, Linnebo ch.5)

Lecture 11: Dummett's meaning-theoretic turn (Shapiro ch. 7.4, Linnebo ch.5)

Lecture 12: Gödel's views on intuition and mathematical naturalism (Shapiro ch.8.1-3, Linnebo chs.6 and 7)

Lecture 13: Nominalism in mathematics (Shapiro ch.9, Linnebo ch.7)

Lecture 14: The iterative conception of sets (Linnebo ch.10)

Lecture 14: Structuralism (Shapiro ch.10)

Lecture 16: Univalent foundations and automated proof checking (private notes)

Please let me know if you find any broken links.