Instructors: Michael Harris (Mathematics) and Justin Clarke-Doane (Philosophy)
Location and time: Tuesdays 4:10-6:00 PM, 301M Fayerweather; those who wish can continue in room 528 Mathematics until 6:40.
Office hours: Tuesday 3-4 PM (11-12 AM January 28), Thursday 10-11 (starting January 30), and by appointment.
Schedule: The class meets once weekly on Tuesday afternoon for (up to) 150 minutes. A provisional list of topics and readings for each session can be found below.
Mathematics and philosophy have an ancient association. Euclid was a member of Plato's Academy, which is supposed to have read 'Let no one ignorant of geometry enter' above the door. Since then, philosophical questions have motivated mathematical research, and mathematical developments have informed philosophical activity. In this seminar, we discuss problems at the intersection of philosophy and mathematics, including: What is a mathematical proof? How does mathematical proof relate to formal proof in the sense of philosophers? What are the limitations of formal proof, and what bearing do they have on the philosophy of mathematics? Is the philosophy of mathematical practice different from the philosophy of mathematics? How do we select mathematical axioms? Is there ‘one true mathematics’ or could mathematics have been very different? What is mathematical understanding; is it only accessible to professional mathematicians, or to humans more generally? Do we understand infinity? What is the role of mathematics in culture broadly? How does its philosophy affect this?
Note: The Philosophy Department will be hosting a conference entitled Ultrafinitism: Physics, Mathematics, & Philosophy during the semester. All are welcome to attend.
Structure of the Course
Each student will write a term paper (of about 20 pages) or a shorter midterm paper (of about 10 pages) and a final paper (of about 10 pages) each on a topic of their choice. Students should clear their topics with us well ahead of the end of the semester. Mathematical, historical, and philosophical topics are all encouraged. Use of generative AI is not allowed.
This is a discussion-based seminar. Class participation will count for 25% of one' grade. One’s paper will constitute the remaining 75%.
Tentative Course Schedule:
(When not indicated otherwise, texts should be available in the Courseworks Files.)
Week 1: Interplay between Philosophy and Mathematics (Clarke-Doane & Harris)
Plato, Meno (excerpts)
Berkeley, The Analyst
Mazur, Imagining Numbers (Especially √−15) (selections, on reserve)
Russell, ‘On Some Difficulties with the Theory of Transfinite Numbers and Order Types’
Harris, ‘Notes from the 2022 Fields Medal Symposium’
Articles on FrontierMath:
• FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI (read the introduction and look over the sample problems)
• "'Brutal' math test stumps AI but not human experts"
• Can AI do maths yet? Thoughts from a mathematician
Optional: Olsson, The Weil Conjectures (selections, on reserve)
Week 2: The 20th Century 'Crisis' (Clarke-Doane)
Burgess, Rigor and Structure (selections)
Frege, Conceptual Notation, Preface and Ch. 1
Heyting, ‘The Intuitionist Foundation of Mathematics’
Hilbert, ‘On the Infinite’ (in van Heijenoort, From Frege to Gödel)
Optional: Putnam, Philosophy of Mathematics: Why Nothing Works
S. M. Srivastava, 'How did Cantor Discover Set Theory and Topology?'
Week 3: The Legacy of Logicism (Harris)
Doxiadis & Papadimitriou, Logicomix
Floyd, Wittgenstein's Philosophy of Mathematics
Wittgenstein, Tractatus Logico-Philosophicus, propositions 6.01 to 6.241, and 7
Turing, ‘Intelligent Machinery’
Harris, ‘The Central Dogma of Mathematical Formalism'
Ording, 99 Variations on a Proof, Proofs 3, 6, 7, 26, 48, 52, 62
Optional: Goldfarb, 'Poincaré against the Logicists’,
Post, 'Introduction to a general theory of elementary propositions,' (in Heijenoort, pp. 265-9), for those unfamiliar with symbolic logic.
Chemla, ed, The History of Mathematical Proof in Ancient Traditions, Chapter 14: 'Dispelling mathematical doubts: assessing mathematical correctness of algorithms in Bhāskara’s commentary on the mathematical chapter of the Āryabhat.īya' (A. Keller); Chapter 15: 'Argumentation for state examinations: demonstration in traditional Chinese and Vietnamese mathematics' (A. Volkov)
Week 4: Formal Limitations (Clarke-Doane)
Paul Cohen, ‘Independence Results in Set Theory’
Gödel, ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’ (in van Heijenoort, From Frege to Gödel)
Skolem, ‘Some Remarks on Axiomatized Set Theory’
Optional: Hartry Field, ‘Are Our Logical and Mathematical Concepts Highly Indeterminate?’
Optional: Nelson, ‘Hilbert’s Mistake’
Week 5: Informal Proof and other mathematical activities (Harris)
Harris, Mathematics without Apologies, Chapters 3 and 8.
Thurston, ‘On Proof and Progress in Mathematics’
Rav, ‘Why do we Prove Theorems?’
De Toffoli, Groundwork for a fallibilist account of mathematics, The Philosophical Quarterly 71 (2021), no. 4, 823–844.
Mazur, 'Conjecture,' Synthese 111 (2):197-210 (1997).
Yves André, “Réflexions sur l’écriture et le style en mathématique”
Optional, Kennedy, 'Gödel, Turing and the Iconic/Performative Axis,' Philosophies 2022, 7(6), 141; https://doi.org/10.3390/philosophies7060141
McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology,' (introduction and sections 3 and 4) (see also Week 12)
Week 6: Minds and Machines (Harris)
Stephanie Dick, After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States (selections)
Alan Turing, 'Intelligent machinery'
Donald MacKenzie, 'Computing and the cultures of proving'
Harris, ‘Do Androids Prove Theorems in their Sleep?’
Kevin Buzzard, 'The rise of formalism in mathematics' (slides from his talk at the 2022 International Congress of Mathematicians)
Optional: Jeremy Avigad, ‘The Mechanization of Mathematics’
Donald MacKenzie, Mechanizing Proof: Computing, Risk, and Trust (chapters 1-4)
Week 7: Justification (Clarke-Doane)
Ash & Clarke-Doane, ‘Intuition and Observation’
Gödel, ‘Russell’s Mathematical Logic’
Quine, ‘Two Dogmas of Empiricism’
Russell, ‘The Regressive Method for Discovering the Premises of Mathematics’
Week 8: Contingency (Clarke-Doane & Harris)
Clarke-Doane, Mathematics and Metaphilosophy, Ch. 3
Hamkins, ‘How the Continuum Hypothesis Could have been a Fundamental Axiom’
Lakatos, Proofs and Refutations (selections)
Optional: De Freitas and Sinclair, What Is a Mathematical Concept?, particularly the articles of Netz, Corfield, and De Freitas-Sinclair.
Asok, Constructing projective modules
Week 9: Pluralism (Clarke-Doane)
Blue, ‘What is it to be a Solution to Cantor’s Continuum Problem?’
Clarke-Doane, ‘What is Logical Monism?’
Gödel, ‘What is Cantor’s Continuum Hypothesis?’
Hamkins, ‘The Set-theoretic Multiverse’
Optional: Priest, Mathematical Pluralism
Week 10: Mathematics and Culture (Harris)
Broch, The Sleepwalkers (selections)
Cormac McCarthy, Stella Maris (selections)
Engelhardt, Modernism, Fiction and Mathematics (selections)
Musil, ‘The Mathematical Man’
Pynchon, Gravity’s Rainbow or Against the Day (selections)
Labatut, When We Cease to Understand the World and/or The Maniac
Zamyatin, We (selections)
Week 11: Philosophy of Mathematical Practice (Clarke-Doane & Harris)
Burgess & De Toffoli, ‘What is Mathematical Rigor?’
Confield, Toward a Philosophy of Real Mathematics (selections)
Mancosu, Philosophy of Mathematical Practice (Introduction)
Wittgenstein, Lectures on the Foundations of Mathematics, Cambridge, 1939 (selections)
Optional: Davis & Hersh, The Mathematical Experience
Week 12: Grothendieck and structuralist mathematics (Harris)
McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology,' (introduction and sections 3 and 4)
Bourbaki, ‘The Architecture of Mathematics’
Grothendieck, Harvest and Sowing (selections)
Grothendieck, ‘The Cohomology Theory of Abstract Algebraic Varieties’
Mazur, 'When is one thing equal to some other thing?'
McLarty, ‘The Rising Sea: Grothendieck on Simplicity and Generality’
Harris, Mathematics without Apologies, Chapter 7.
Optional: France Culture 5-part podcast on Grothendieck
Fonseca, Colonel Lágrimas (selections, in Spanish)
Week 13: Mathematics & Metaphysics (Clarke-Doane & Harris)
Carnap, The Logical Structure of the World (selections)
Clarke-Doane, Morality and Mathematics (selections)
Penrose, The Road to Reality (selections)
Rota, ‘The Pernicious Influence of Mathematics upon Philosophy’
Lautman, Mathematics, Ideas and the Physically Real (selections)