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:
    • Syllabus
(2) Thu, Sep 6
  • Approaches, questions, and positions
  • Readings:
    • Dutilh Novaes, "Different methodologies in philosophical investigation" (2012)

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)
(4) Thu, Sep 13
  • Learning arithmetic
  • Readings:
    • Sarnecka, "Learning to represent exact numbers" (2015)
    • Schlimm, "Numbers through numerals" (2018)
(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)

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)
(7) Tue, Sep 25
  • Homework 3 due
  • Plato and Aristotle
  • Readings:
    • Excerpt from Plato's "Meno"
    • Shapiro, Ch. 3, "Plato's rationalism and Aristotle"

Week 4 (Kant)

(8) Thu, Sep 27
  • Critical Summary (1): Topic announcement due (see "Guidelines")
  • Kant
  • Readings:
    • Excerpts from the "Prolegomena"

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"
(10) Thu, Oct 4
  • Critical summary (2): Outline due
  • Non-Euclidean geometry
  • Readings:
    • Eichholtz, "The discovery of non-Euclidean geometry"

Week 6 (19th century empiricism)

(11) Tue, Oct 9
  • Homework 5 due
  • Pasch and empiricism
  • Readings:
    • Schlimm, "Pasch's philosophy of mathematics" (2010)
(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)

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)
(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?"

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)
(16) Thu, Oct 25
  • Critical Summaries due
  • Gödel's incompleteness theorems
  • Readings:
    • Nagel and Newman, "Gödel's proof" (1956)

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)
(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)

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)
(20) Thu, Nov 8
  • Term paper (2): Abstract
  • Developments in algebra: Group theory
  • Readings:
    • Rothman, "The short life of Évariste Galois" (1982)
(21) Tue, Nov 13
  • Homework 10 due
  • Structuralism
  • Readings:
    • Benacerraf, "What numbers could not be" (1965)
    • Shapiro, Ch. 10, "Structuralism"

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)
(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)
(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)
(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)
(26) Thu, Nov 29
  • Term paper due
  • Review & outlook

Instructor:
Dirk Schlimm
Department of Philosophy
Leacock 916

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