Outline

Meeting 14: December 9

Simon and Larkin

• characterization of the project
• role of diagrammatic representations
• pulley example
• geometry example

Giaqiunto, Chapter 12

• examples
• diagrammatic vs. symbolic

Summary and recap

• framing the project
• central questions
• what we've seen
• conclusions

Meeting 13: November 30

Wickelgren

• 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

Schoenfeld

• heuristics, and specificity
• examples of "special cases"
• examples of "setting subgoals"
• forms of control
• expert vs. novice behavior
• belief

Meeting 11: November 9

Polya

• notions of "plausible inference"
• supports: instances, extreme cases
• case study: a limit inequality
• proof that motivates the discovery

Schoenfeld

• resources, heuristics, control, belief
• discussion of resources

Meeting 10: November 2

Polya

• 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

Polya

• 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

Tappendenden

• concepts, explanation, natural kinds, and essences
• case studies: the Legendre symbol, and the definition of "prime"

Mancosu on explanation

Steiner

• Examples: summing 1 to n, irrationality of sqrt(2), Pythagorean theorem
• Steiner's analysis
• critique

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

Husserl

• 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

• 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

Jackendoff

• I-semantics vs. E-semantics
• Sketch of the theory: categories, attributes, constructors

Frege

• Sense and reference, means of representation
• The telescope analogy

Meeting 2: September 7

Kant

• Understanding vs. sensibility (concepts vs. intuitions)
• Cognition of mathematical objects, example from Euclid
• Schemata
• 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