Please note that the information on these pages here is tentative and will be updated frequently during the semester.

Schedule

Part I: Historical positions

Week 1 (Introduction and philosophical approaches)

(1) Tue, Sep 4
  • Introduction to the course
  • Readings:
    • Handout 0: Syllabus and two geometric problems
(2) Thu, Sep 7
  • Approaches, questions, and positions
  • Readings:
    • Dutilh Novaes, "Different methodologies in philosophical investigation"

Week 2 (Plato, Aristotle, and Kant)

(3) Tue, Sep 12
  • Homework 1 due
  • Plato and Aristotle
  • Readings:
    • Excerpt from Plato's "Meno"
    • Shapiro, Ch. 3, "Plato's rationalism and Aristotle"
    • Dunham, "Euclid's proof of the Pythagorean Theorem" (pp. 27–40)
(4) Thu, Sep 14
  • Kant
  • Readings:
    • Excerpts from the "Prolegomena"

Part II: Developments in 19th century geometry and some philosophical reactions

Week 3 (Developments in 19th-century geometry)

(5) Tue, Sep 19
  • Homework 2 due
  • Analytic and projective geometry
  • Readings:
    • Gray, "Projective geometry and the axiomatization of mathematics"
(6) Thu, Sep 21
  • Non-Euclidean geometry
  • Readings:
    • Eichholtz, "The discovery of non-Euclidean geometry"

Week 4 (Pasch, Hilbert, and Frege on geometry)

(7) Tue, Sep 26
  • Homework 3 due
  • Pasch and empiricism
  • Readings:
    • Schlimm, "Pasch's philosophy of mathematics" (2010)
(8) Thu, Sep 28
  • Topic announcement for Critical Summary due (see "Guidelines")
  • Hilbert and Frege on geometry
  • Readings:
    • Frege, "On concept and object" (1892)
    • Hilbert, "Foundations of Geometry" (1899)
    • Frege-Hilbert correspondence (1895–1903)

Part III: Traditional positions in the 20th century

Week 5 (Logicism)

(9) Tue, Oct 3
  • Homework 4 due
  • Quiz 1 (in class)
  • Frege's logicism
  • Readings:
    • Frege, "Foundations of Arithmetic" (1884)
(10) Thu, Oct 5
  • Russell's theory of types
  • Readings:
    • Shapiro, Ch. 5, "Logicism: Is mathematics (just) logic?"

Week 6 (Formalism and its limitations)

(11) Tue, Oct 10
  • Homework 5 due
  • Hilbert's Programme
  • Readings:
    • Sieg, "Hilbert's programs: 1917-1922" (1999)
    • Zach, "Hilbert's program then and now" (2006)
(12) Thu, Oct 12
  • Full draft of Critical Summary to peer reviewers
  • Gödel's incompleteness theorems
  • Readings:
    • Nagel and Newman, "Gödel's proof" (1956)

Week 7 (Intuitionism and Logical positivism)

(13) Tue, Oct 17
  • Homework 6 due
  • Intuitionism
  • Readings:
    • Brouwer, "Intuitionism and formalism" (1912)
    • Heyting, "Disputation" (1956)
(14) Thu, Oct 19
  • Feedback on Critical Summaries to authors
  • Carnap and Quine
  • Readings:
    • Quine, "On what there is" (1948)
    • Carnap, "Empiricism, semantics, ontology" (1950)
    • Quine, "Two dogmas of empiricism" (1951)

Week 8-9 (Set theory: Axiomatizations and realism)

(15) Tue, Oct 24
  • Homework 7 due
  • Axiomatizing set theory
  • (Guest lecturer: Moritz Bodner)
  • Readings:
    • Zermelo, "Investigation into the Foundations of Set Theory I" (1908)
    • Fraenkel, "On the Foundations of Cantor-Zermelo Set Theory" (1922)
    • von Neumann, "The Axiomatisation of Set Theory" (1928)
(16) Thu, Oct 26
  • Critical Summaries due
  • Gödel's theorems and set theory
  • (Guest lecturer: Moritz Bodner)
  • Readings:
    • Tarski, "A General Method in Proofs of Undecidability" (1953)
(17) Tue, Oct 31
  • Homework 8 due
  • Quiz 2 (in class)
  • Set theory and Realism
  • Readings:
    • Gödel, "What is Cantor's Continuum Hypothesis?" (1947)
    • Parsons, "Mathematical intuition" (1979)
    • Maddy, "Perception and intuition" (1990)

Week 9-10 (Algebra and structuralism)

(18) Thu, Nov 2
  • Topic announcement of term paper topic due (6pm)
  • Developments in algebra: Group theory
  • Readings:
    • Rothman, "The short life of Évariste Galois" (1982)
(19) Tue, Nov 7
  • Homework 9 due
  • Structuralism
  • Readings:
    • Benacerraf, "What numbers could not be" (1965)
    • Shapiro, Ch. 10, "Structuralism"

Part IV: Philosophy of mathematical practice and mathematical cognition

Week 10-11 (Lakatos)

(20) Thu, Nov 9
  • Milestone 1 (abstract, bibliography)
  • Challenging foundationalism
  • Readings:
    • Lakatos, "A renaissance of empiricism in recent philosophy of mathematics" (1978)
    • Lakatos, "Proofs and refutations I" (1963)
    • Schlimm, "Axioms in mathematical practice" (2013)
(21) Tue, Nov 14
  • Homework 10 due
  • Proofs and refutations
  • Readings:
    • Lakatos, "Proofs and refutations II–IV" (1963)
    • Schlimm, "Mathematical concepts and investigative practice" (2012)

Week 11 (Mathematical cognition)

(22) Thu, Nov 16
  • Milstone 2 (outline of term paper)
  • The cognitive basis of mathematics
  • Readings:
    • De Cruz, Schlimm, and Neth, "The cognitive basis of arithmetic" (2010)
    • Gaber and Schlimm, "Mathematical cogntion" (2014)

Week 12-13 (Mathematical notations)

(23) Tue, Nov 21
  • Homework 11 due
  • Numerals
  • Readings:
    • Schlimm and Neth, "Modeling ancient and modern arithmetic practices" (2008)
    • Dutilh Novaes, "Mathematical reasoning and external symbol systems" (2013)
(24) Thu, Nov 23
  • Quiz 3 (in class)
  • (Full draft to reviewers)
  • Logical notations
  • Readings:
    • Schlimm, "On Frege's Begriffsschrift notation for propositional logic: Design principles and trade-offs" (2017)
(25) Tue, Nov 28
  • Homework 12 due
  • Diagrams
  • (Comments back to authors)
  • Readings:
    • Brown, "Knots and notation" (2008)
    • De Toffoli and Giardino, "Envisioning Transformations—The Practice of Topology" (2014)
    • Giaquinto, The Epistemology of Visual Thinking in Mathematics (2015); in particular Ch. 3 on "Visual thinking and proof"
(26) Thu, Nov 30
  • Final paper due
  • Review
  • Readings:
    • "Math and aftermath" (1997)

Instructor:
Dirk Schlimm
Department of Philosophy
Leacock 916

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