We'll continue discussing visualization and diagrammatic reasoning:
And then we'll decompress and reflect on the semester.
- Giaquinto, Visual thinking in mathematics: an epistemological study, Chapter 8, Chapter 12.
- Larkin and Simon, "Why a diagram is (sometimes) worth ten thousand words," pdf.
- Mancosu, "Visualization in logic and mathematics," pdf
This week, we'll have a brief discussion of Wickelgren, and then move on to talk about Euclidean diagrammatic reasoning. Read any of the following:
- Mumma, "Proofs, pictures and Euclid," pdf.
- Mumma, "The role of geometric content in Euclid's diagrammatic reasoning", pdf.
- Avigad, Dean, and Mumma, "A formal system for Euclid's Elements," pdf.
This week, we'll finish the discussion of Schoenfeld, and start discussing Wickelgren's take on problem solving: pdf,
This week we will finish up the discussion of Polya; specifically, we'll talk about the excerpt from Patterns of plausible inference (volume II of Mathematics and plausible reasoning).
After that, we will start discussing Schoenfeld, Mathematical problem solving, pdf.
This week will be devoted to the Polya, and the readings below.
This week will discuss Arana's paper, and then move on to Polya.
- Polya, How to solve it, Part I: pdf.
- Polya, Mathematics and plausible reasoning, excerpts from volume I: pdf.
- Polya, Mathematics and plausible reasoning, excerpts from volume II: pdf.
This week is something of an interlude: we will discuss the draft of
We will focus on sections 4-8.
- Avigad and Morris, "Character and object," arxiv.
The week after, we will finish up the discussion of explanation (including Arana's paper), and move on to problem solving. The latter will eventually include Polya, Schoenfeld, and Wickelgren, in particular.
We will continue with mathematical explanation and proof:
- Sandborg, Explanation in mathematical practice
Chapter 4: (and title page) pdf,
Chapter 5: pdf.
- Avigad, "Mathematical method and proof," pdf.
- Arana, "Logical and semantic purity," pdf.
If you are interested in Pick's theorem, see also John Harrison's "A formal proof of Pick's theorem," on the bibliography.
Solomon Feferman's "The logic of mathematical discovery vs. the logical structure of mathematics," a critique of Lakatos' Proofs and Refutations, is also worth reading.
This week, we will discuss and the paper by Tappenden, and then move on to explanation:
- Tappenden, "Mathematical concepts and definitions," pdf.
- Mancosu, "Mathematical explanation: why it matters," pdf.
- Steiner, "Mathematical explanation," pdf.
This week, we will finish the discussion of Wilson, and discuss Husserl.
- Husserl, The philosophy of arithmetic, excerpt: pdf.
- In case it may be helpful, here is my review of a book by Marcus Giaquinto, which tries to tease
apart normative and psychological issues: pdf.
Read at least one of these papers by Mark Wilson:
- "Frege's mathematical setting," pdf.
- "Enlarging one's stall, or how did all these sets get in here?" pdf.
Finally, look at the following study in mathematics education:
- Bethany Rittle-Johnson, Robert Siegler, and Martha Alibali, "Developing conceptual understanding and procedural skill in mathematics: an iterative process," pdf.
Concepts in general:
- Read Jackendoff's "What is a concept, that a person might grasp it?" pdf.
Frege on sense and reference (choose readings from among these as you wish):
- "Sense and reference," pdf, and "Comments on sense and reference," pdf.
- "Letter to Husserl," pdf.
- "Thought," pdf.
Kant's account of mathematics:
There are additional links to material on Kant, and concepts, in the bibliography.
- Read the introdution to the second edition of the Critique of pure reason for an overview of Kant's project. pdf.
- Read the characterization of mathematical cognition in "The discipline of pure reason in dogmatic use," focusing on pages 630-636. pdf.
- Take a look at his account of the schemata, focusing on pages 273-274. pdf.
- Read chapter 2 of Friedman's Kant and the exact sciences (chapter 1 is optional). pdf.
- Take a look at Avigad, "Understanding, formal verification, and the philosophy of mathematics," pdf, for a framing of the issues.
- Let me know if you would like to be on the seminar e-mail list.