Meeting 14: December 9
Simon and Larkin
- characterization of the project
- role of diagrammatic representations
- pulley example
- geometry example
Giaqiunto, Chapter 12
- diagrammatic vs. symbolic
Summary and recap
- framing the project
- central questions
- what we've seen
Meeting 13: November 30
- comparison with Schoenfeld; focus on problems rather than cognition
- comparison with Polya: emphasis on AI, algorithms
- overview of the book
- problem "similarity"
Avigad, Dean, and Mumma on Euclidean geometry
Meeting 12: November 16
- heuristics, and specificity
- examples of "special cases"
- examples of "setting subgoals"
- forms of control
- expert vs. novice behavior
Meeting 11: November 9
- notions of "plausible inference"
- supports: instances, extreme cases
- case study: a limit inequality
- proof that motivates the discovery
- resources, heuristics, control, belief
- discussion of resources
Meeting 10: November 2
- preface to How to Solve It
- Polya's list
- rectangular prism example
- preface to Induction and analogy
- generalization, specialization, and analogy (and the Pythagorean theorem)
- Euler and the Basel problem
Meeting 9: October 26
Method and proof
- three proofs of the sum of squares theorem
- constraints on a model of proof to support the evaluations
Arana on purity
- logical vs. semantical purity
- case studies: casus irreducibilus, Desargue's theorem, induction, Gödel's theorem
- Normative aspects, rationality
- Models of the mind
Meeting 8: October 19
Becky Morris' presentation of "Character and object," and discussion.
Meeting 7: October 12
Discussion of Chapters 4 and 5 of Sandbord's thesis:
- various proofs of Pick's theorem
- two (among possibly more) axes of explanation: reduction of arbitrariness, and directness (in this case, geometric interpretation)
Concern: how to ground the analysis and make it more precise?
Method and proof:
- interactive theorem proving
- the corresponding view of proof
Meeting 6: October 5
Discussion: "explanation" vs. "epistemic virtue"
- concepts, explanation, natural kinds, and essences
- case studies: the Legendre symbol, and the definition of "prime"
Mancosu on explanation
- Examples: summing 1 to n, irrationality of sqrt(2), Pythagorean theorem
- Steiner's analysis
Meeting 5: September 28
Comparing the understandings of "concept" in Kant, Jackendoff, Rittle-Johnson et al., Frege, Wilson, Husserl, Tappenden
Back to Wilson
- "Imaginary points" revisited
- Ideals in number theory
- Historical background
- Husserl's project (from the preface)
- The difference between "primary" and "psychical" relations
- A concept map: the pen, the key, the watch, something, one, collective combination, multiplicity, cardinal number
Meeting 4: September 21
Rittle-Johnson et al.
- conceptual knowledge vs. procedural knowledge
- problem representations
- testing procedures
- discussion of the model
Wilson on Frege's mathematical background
See here for more on involutions, including instructions on running a neat demo by Rob Lewis.
- Relative logicism
- "Imaginary points"
Meeting 3: September 14
Back to "senses" or "meanings." Some dichotomies:
- internal / private / subjective vs. external / public / objective
- descriptive / behavioral vs. normative / theoretical (propositional attitudes)
- cognitive significance / how we grasp things vs. informational significance / how the world is
- appropriateness / "methodology" / "pragmatics" vs. justification, correctness, truth
- knowing how or why vs. knowing that
- I-semantics vs. E-semantics
- Sketch of the theory: categories, attributes, constructors
- Sense and reference, means of representation
- The telescope analogy
Meeting 2: September 7
- Understanding vs. sensibility (concepts vs. intuitions)
- Cognition of mathematical objects, example from Euclid
- Quanta, quantitatum, and time
- Kant's concerns, more general concerns
Meeting 1: August 31
Overview of seminar goals (roughly as described on the main page)
Discussion of some of the general difficulties in making sense of "concepts"
- "meanings," between syntax and reference
- do we really need a theory of concepts?
Background for reading Kant
- The project of the Critique
- Transcendental idealism
- Some attractive features of Kant's views