Summer 2024
43190350 Philosophy of Mathematics

Course Information

Instructor Dr. Bruno Bentzen

TA Huayu Guo

Office Location Chengjun Complex 4, 306

Office Hours 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:

1. understand the relationship between mathematics and philosophy

2. distinguish the main positions in the philosophy of mathematics

3. have a basic understanding of foundational crisis in mathematics

4. acquire a better understanding of some contemporary views

Assessment and grades

• Homework 60%

• Final paper 40%

Homework

There will be three homework assignments throughout the course. They will be assigned every two weeks on Fridays and will be due in two weeks time in class. I wish to evaluate your performance, so your homework should reflect your own efforts. Each assignment must be completed and turned in on time. If you missed the deadline because you were ill or for some other valid reason, please send me an email. Your work must be written legibly. Your homework will be graded and returned to you on the Wednesday before the next due date. Please remember to name and date your assignments.

Final 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) and must be written in English. Your English level will not be subject to evaluation, only the content of your paper. However, you should communicate your ideas clearly and effectively. Again, do not forget to name and date your paper — please also add your student number. Your paper must also contain a clear title, introduction, exposition of your problem and ideas, and conclusion. You will be graded according to the following three criteria below:

• You demonstrate that you understand well the issues you are writing about;

• Your central idea is clear and related to the theme;

• Your arguments given to support your main claims are sound.

Detailed information about the content of the final paper will be provided as it approaches.

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: Frege's logicism (cont.) (Shapiro ch. 5.4, Linnebo ch.9)

Lecture 6: Russell's doctrine of types (Shapiro ch. 5, Linnebo ch.9.2–3)

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's intuitionistic mathematics (Shapiro ch. 7.1–3, Linnebo ch.5)

Lecture 11: Heyting's meaning explanations (Shapiro ch. 7.4, Linnebo ch.5)

Lecture 12: Dummett's meaning-theoretic turn (Shapiro ch. 7.4, Linnebo ch.5)

Lecture 13: Gödel's views on intuition, mathematical naturalism (Shapiro ch.8.1-3, Linnebo chs.6 and 7)

Lecture 14: Nominalism in mathematics (Shapiro ch.9, Linnebo ch.7)

Lecture 15: The iterative conception of sets (Linnebo ch.10)

Lecture 16: Structuralism (Shapiro ch.10)

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