Instructors: Michael Harris (Mathematics) and Justin Clarke-Doane (Philosophy)
Location and time: Tuesdays 4:10-6:30 PM, location TBA
Office hours: TBA
Schedule: The class meets once weekly, probably on Wednesday afternoon for 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.
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:
Week 1: Interplay between Philosophy and Mathematics (Clarke-Doane & Harris)
Plato, Meno (excerpts)
Berkeley, The Analyst
Mazur, Imagining Numbers (Especially √−15) (selections)
Olsson, The Weil Conjectures (selections)
Russell, ‘On Some Difficulties with the Theory of Transfinite Numbers and Order Types’
Harris, ‘Notes from the 2022 Fields Medal Symposium’
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’
Optional: Putnam, Philosophy of Mathematics: Why Nothing Works
Week 3: The Legacy of Logicism (Harris)
Azzouni, ‘The Derivation-Indicator View of Mathematical Practice’
Doxiadis & Papadimitriou, Logicomix
Turing, ‘Intelligent Machinery’
Harris, ‘The Central Dogma of Mathematical Formalism’
Ording, 99 Variations on a Proof, selections
Optional: Goldfarb, 'Poincaré against the Logicists’
Week 4: Formal Limitations (Clarke-Doane)
Paul Cohen, ‘Independence Results in Set Theory’
Gödel, ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’
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 (selections)
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).
Optional, Kennedy, 'Gödel, Turing and the Iconic/Performative Axis,' Philosophies 2022, 7(6), 141; https://doi.org/10.3390/philosophies7060141
Week 6: Minds and Machines (Harris)
Stephanie Dick, After Math: (Re)configuring Minds, Proof, and Computing in the Postwar United States (selections)
Donald MacKenzie, Mechanizing Proof: Computing, Risk, and Trust (selections)
Harris, ‘Do Androids Prove Theorems in their Sleep?’
Optional: Avigad, ‘The Mechanization of Mathematics’
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: Harris, ‘Ce qu'il faut comprendre afin de distinguer lestravaux d'Andrew Wiles des “déchets”’
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’
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)