Please note that the information on these pages here is tentative and will be updated frequently during the semester.
Schedule
Part 1: Introduction
Week 1 (Introduction and philosophical approaches)
(1) Tue, Sep 4 |
- Introduction to the course
- Readings:
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(2) Thu, Sep 6 |
- Approaches, questions, and positions
- Readings:
- Dutilh Novaes, "Different methodologies in philosophical investigation" (2012)
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Part 2: From things to numbers (Arithmetic)
Week 2–3 (Cognition and learning)
(3) Tue, Sep 12 |
- Homework 1 due
- The cognitive basis of mathematics
- Readings:
- De Cruz, Schlimm, and Neth, "The cognitive basis of arithmetic" (2010)
- Gaber and Schlimm, "Mathematical cogntion" (2014)
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(4) Thu, Sep 13 |
- Learning arithmetic
- Readings:
- Sarnecka, "Learning to represent exact numbers" (2015)
- Schlimm, "Numbers through numerals" (2018)
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(5) Tue, Sep 18 |
- Homework 2 due
- Mathematical notation: Numerals
- Readings:
- Schlimm and Neth, "Modeling Ancient and Modern Arithmetic Practices: Addition and Multiplication with Arabic and Roman Numerals" (2008)
- Dutilh-Novaes, "Mathematical reasoning and external symbolic systems" (2013)
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Part 3: Historical positions (Geometry)
Week 3–4 (Ancient mathematics and philosophy)
(6) Thu, Sep 20 |
- Euclidean geometry
- Readings:
- Dunham, "Euclid’s proof of the Pythagorean theorem" (1990)
- Manders, "The Euclidean diagram" (1995)
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(7) Tue, Sep 25 |
- Homework 3 due
- Plato and Aristotle
- Readings:
- Excerpt from Plato's "Meno"
- Shapiro, Ch. 3, "Plato's rationalism and Aristotle"
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Week 4 (Kant)
(8) Thu, Sep 27 |
- Critical Summary (1): Topic announcement due (see "Guidelines")
- Kant
- Readings:
- Excerpts from the "Prolegomena"
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Part 4: Developments in 19th century geometry and some philosophical reactions
Week 5 (Developments in geometry)
(9) Tue, Oct 2 |
- Homework 4 due
- Quiz 1 (in class)
- Analytic and projective geometry
- Readings:
- Gray, "Projective geometry and the axiomatization of mathematics"
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(10) Thu, Oct 4 |
- Critical summary (2): Outline due
- Non-Euclidean geometry
- Readings:
- Eichholtz, "The discovery of non-Euclidean geometry"
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Week 6 (19th century empiricism)
(11) Tue, Oct 9 |
- Homework 5 due
- Pasch and empiricism
- Readings:
- Schlimm, "Pasch's philosophy of mathematics" (2010)
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(12) Thu, Oct 11 |
- Critical summary (3): Full draft to peer reviewers
- Frege and Hilbert on geometry
- Readings:
- Frege, "On concept and object" (1892)
- Hilbert, "Foundations of Geometry" (1899)
- Frege-Hilbert correspondence (1895–1903)
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Part 5: Foundational positions in the 20th century
Week 7 (Logicism)
(13) Tue, Oct 16 |
- Homework 6 due
- Frege's logicism
- Readings:
- Frege, "Foundations of Arithmetic" (1884)
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(14) Thu, Oct 18 |
- Critical summary (4): Return peer review to authors
- Russell's theory of types
- Readings:
- Shapiro, Ch. 5, "Logicism: Is mathematics (just) logic?"
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Week 8 (Formalism and its limitations)
(15) Tue, Oct 23 |
- Homework 7 due
- Formalism and Hilbert's Program
- Readings:
- Sieg, "Hilbert's programs: 1917-1922" (1999)
- Zach, "Hilbert's program then and now" (2006)
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(16) Thu, Oct 25 |
- Critical Summaries due
- Gödel's incompleteness theorems
- Readings:
- Nagel and Newman, "Gödel's proof" (1956)
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Week 9 (Intuitionism and Logical positivism)
(17) Tue, Oct 30 |
- Homework 8 due
- Quiz 2 (in class)
- Intuitionism
- Readings:
- Brouwer, "Intuitionism and formalism" (1912)
- Heyting, "Disputation" (1956)
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(18) Thu, Nov 1 |
- Term paper (1): Topic announcement
- Carnap and Quine
- Readings:
- Quine, "On what there is" (1948)
- Carnap, "Empiricism, semantics, ontology" (1950)
- Quine, "Two dogmas of empiricism" (1951)
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Part 6: Current positions
Week 10-11 (Current positions: Realism and structuralism)
(19) Tue, Nov 6 |
- Homework 9 due
- Set theory and Realism
- Readings:
- Gödel, "What is Cantor's Continuum Hypothesis?" (1947)
- Parsons, "Mathematical intuition" (1979)
- Maddy, "Perception and intuition" (1990)
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(20) Thu, Nov 8 |
- Term paper (2): Abstract
- Developments in algebra: Group theory
- Readings:
- Rothman, "The short life of Évariste Galois" (1982)
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(21) Tue, Nov 13 |
- Homework 10 due
- Structuralism
- Readings:
- Benacerraf, "What numbers could not be" (1965)
- Shapiro, Ch. 10, "Structuralism"
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Part 7: Mathematical practice and methodology
(22) Thu, Nov 15 |
- Term paper (3): Outline
- Proofs and refutations
- Readings:
- Lakatos, "Proofs and refutations I–IV" (1963-64)
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(23) Tue, Nov 20 |
- Quiz 3 (in class)
- Homework 11 due
- Challenging foundationalism
- Readings:
- Lakatos, "A renaissance of empiricism in recent philosophy of mathematics" (1978)
- Rav, "Why do we prove theorems?" (1999)
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(24) Thu, Nov 22 |
- Term paper (4): Full draft to reviewers
- Axioms
- Readings:
- Schlimm, "Axioms in mathematical practice" (2013)
- Rodin, "On the constructive axiomatic method" (2018)
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(25) Tue, Nov 27 |
- Term paper (5): Return peer reviews
- Homework 12 due
- Mathematical notation: Logic
- Readings:
- Schlimm, "On Frege's Begriffsschrift notation for propositional logic: Design principles and trade-offs" (2017)
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(26) Thu, Nov 29 |
- Term paper due
- Review & outlook
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