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 19thcentury 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 
 NonEuclidean geometry
 Readings:
 Eichholtz, "The discovery of nonEuclidean 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)
 FregeHilbert 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: 19171922" (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 89 (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 CantorZermelo 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 910 (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 1011 (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 1213 (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 tradeoffs" (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)
