These are some works relevant to the topic of the seminar. I have been lazy about bibliographical details, but Google will supply these. To avoid copyright violations, some of the links are to copies kept in a password-protected directory.
The following collections contains a number of related essays:
The following provide some high-level reflection on mathematical understanding:
- Paolo Mancosu, editor, The philosophy of mathematical practice
- Henri Poincaré, Science and method.
- Ludwig Wittgenstein, Philosophical investigations.
- Jacques Hadamard, Psychology of invention in the mathematical field.
- Jeremy Avigad, "Understanding proofs," pdf.
- Jeremy Avigad, "Understanding, formal verification, and the philosophy of mathematics," pdf.
- Arthur Jaffe and Frank Quinn, "'Theoretical mathematics': toward a cultural synthesis of mathematics and theoretical physics," html.
- William Thurston, "On proof and progress in mathematics," html.
- Other responses to Jaffe and Quinn: html.
Traditional philosophical views:
- Immanuel Kant, Critique of pure reason. B introduction: pdf.
Transcendental doctrine of method, discipline of pure reason in dogmatic use:
On the schematism of the pure concepts of understanding:
- Michael Friedman, Kant and the exact sciences; in particular, chapter 2, "Concepts and intuitions in mathematics," pdf.
- Lisa Shabel, "Kant's philosophy of mathematics,"
- Lisa Shabel, "Kant on the 'Symbolic Construction' of Mathematical Concepts," pdf.
- Charles Parsons, "Kant's philosophy of arithmetic,"
- Edmund Husserl, The philosophy of arithmetic, excerpt: pdf.
- Gottlob Frege, The foundations of arithmetic (the Grundlagen)
- Gottlob Frege, "Sense and reference," pdf, and "Comments on sense and reference," pdf.
- Gottlob Frege, "Letter to Husserl," pdf.
- Gottlob Frege, "Function and concept"
- Gottlob Frege, "Concept and object"
- Gottlob Frege, The basic laws of arithmetic (the Grundgesetze)
- Gottlob Frege, "Thought," pdf.
- Patricia Blanchette, "Frege on shared belief and total functions," pdf.
- Mark Wilson, "The royal road to geometry"
- Mark Wilson, "Frege's mathematical setting," pdf.
- Mark Wilson, "Ghost world: a context for Frege's context principle"
- Richard Dedekind, "On the introduction of new functions in mathematics"
- Richard Heck and Robert May, "The composition of thoughts"
There is a vast literature on concepts more generally. These may be helpful:
- Kenneth Manders, "Domain extension and the philosophy of mathematics"
- Mark Wilson, "Can we trust logical form?" pdf.
- Mark Wilson, "Enlarging one's stall, or how did all these sets get in here?" pdf.
- Sheldon Smith, "Incomplete understanding of concepts: the case of the derivative," pdf.
- Jamie Tappenden, "Mathematical concepts and definitions," pdf.
- Jamie Tappenden, "Mathematical concepts: fruitfulness and naturalness"
- Jeremy Avigad, "Number theory and elementary arithmetic" (section 4, in particular)
Historians of mathematics often use the term:
- Frank Keil, "Spiders in the web of belief: the tangled relations between concepts and theories," pdf.
- Ray Jackendoff, "What is a concept, that a person might grasp it?" pdf.
- Ray Jackendoff, "Conceptual semantics," pdf.
- Ray Jackendoff, Semantics and cognition
- James Pustejovsky, "The generative lexicon," pdf.
- Bob Carpenter, The logic of typed feature structures
- Bob Carpenter, Type-logical semantics
- Jerry Fodor, Concepts: where cognitive science went wrong, Chapters 1 and 2, pdf.
- Ruth Millikan, "Biosemantics," pdf.
- Mark Wilson, Wandering significance
- Israel Kleiner, "Evolution of the function concept: a brief survey," jstor.
- Hans Wussing, The genesis of the abstract group concept.
- The Mactutor "Overview of the history of mathematics," html.
Mathematical explanation and proof
- Ludwig Wittgenstein, Remarks on the foundations of mathematics, Part VI:
Part VII: pdf.
- Imre Lakatos, Proofs and refutations
- Solomon Feferman, "The logic of mathematical discovery vs. the logical structure of mathematics,"
- Mark Steiner, "Mathematical explanation," pdf.
- David Sandborg, Explanation in mathematical practice
Chapter 4: (and title page) pdf,
Chapter 5: pdf,
Chapter 6: pdf,
- John Harrison, "A formal proof of Pick's theorem," pdf.
- Jamie Tappenden, "Proof style and understanding in mathematics I: visualization, unification and axiom choice"
- Jeremy Avigad, "Mathematical method and proof"
- Johannes Hafner and Paolo Mancosu, "The varieties of mathematical explanation"
- Paolo Mancosu, "Mathematical explanation: why it matters," pdf.
- Paolo Mancosu, "Mathematical style," SEP.
- Paolo Mancosu, "Explanation in mathematics," SEP.
- Andrew Arana, "Logical and semantic purity," pdf.
- Michael Detlefsen, "Purity as an ideal of proof," pdf.
- Andrew Arana and Michael Detlefsen, "Purity of methods"
Mathematical problem solving
In addition, there are very many problem collections and solving guides in mathematics, of which the following is a small sample.
- George Polya, How to solve it, Part I: pdf.
- George Polya, Mathematics and plausible reasoning, volume I: induction and analogy in mathematics, excerpts: pdf.
- George Polya, Mathematics and plausible reasoning, volume II: patterns of plausible inference, excerpts: pdf.
- George Polya, Mathematical discovery: on understanding, learning, and teaching problem solving.
- Allen Newell, "The heuristic of George Polya and its relation to artificial intelligence," pdf.
- Allen Newell and Herbert Simon, Human problem solving
- Herbert Simon, Models of thought, volumes 1 and 2.
- Herbert Simon, "Explaining the ineffable: AI on the topics of intuition, insight, and inspiration," pdf
- Alan Schoenfeld, Mathematical problem solving, pdf.
- Wayne Wickelgren, How to solve problems, excerpts: pdf,
- John Hayes, The complete problem solver
- Craig Kaplan and Herbert Simon, "In search of insight," pdf.
The following may be helpful in characterizing mathematical expertise more generally:
- Paolo ney de Souza and Jorge-Nuno Silva, Berkeley problems in mathematics.
- George Polya and Jeremy Kilpatrick, Stanford mathematics problem book.
- Goerge Polya and Gabor Szego, Problems and theorems in analysis, volumes 1 and 2.
- Arthur Engel, Problem-solving strategies.
- Terence Tao, Solving mathematical problems.
- Valentin Boju and Louis Funar, The math problems notebook.
- Masayoshi Hata, Problems and solutions in real analysis.
- K. A. Ericsson, N. Charness, P. J. Feltovich, and R. R. Hoffman, The Cambridge handbook of expertise and expert performance
- K. Anders Ericsson and Neil Charness, "Expert performance: its structure and acquisition," pdf.
- William Chase and Herbert Simon, "Perception in chess," pdf.
- Michelene T . H. Chi, Paul J. Feltovich, and Robert Glaser, "Categorization and representation of physics problems by experts and novices," pdf.
Cognitive science and mathematics education
Some of the general cognitive literature on analogy may be helpful:
- John Anderson, Rules of the mind
- Mark Singley and John Anderson, The transfer of cognitive skill, excerpt: pdf.
- John Anderson and Kenneth Koedinger, "Reifying implicit planning in geometry: guidelines for
model-based intelligent tutoring system design"
- Vincent Aleven, Kenneth Koedinger, and Karen Cross, "Tutoring answer explanation fosters learning with understanding"
- Robert Siegler and Zhe Chen, "Development of rules and strategies: balancing the old and the new"
- Kenneth Koedinger, "Cognitive tutors: technology bringing learning sciences to the classroom"
- Bethany Rittle-Johnson, Robert Siegler, and Martha Alibali, "Developing conceptual understanding and procedural skill in mathematics: an iterative process," pdf.
- Bethany Rittle-Johnson and Kenneth Koedinger, "Iterating between lessons on concepts and procedures can improve mathematics knowledge"
- John Stamper and Kenneth Koedinger, "Human-machine student model discovery and improvement
- Alan Schoenfeld, editor, Cognitive science and mathematics education.
- Keith Devlin, "What is conceptual understanding?" html.
- Martina Rau et al., "Intelligent tutoring systems with multiple representations and self-explanation prompts support learning of fractions," pdf.
- Martina Rau et al., "Sense making alone doesn't do it: fluency matters too! ITS support for robust learning with multiple representations," pdf.
- Jo Boaler, "The role of contexts in the mathematics classroom: do they make mathematics more 'real'?", pdf.
- Jennifer Kaminski et al, "The advantage of abstract examples in learning math," pdf.
- Guershon Harel and James Kaput, "The role of conceptual entities and their symbols in building advanced mathematical concepts"
- Guershon Harel, Evan Fuller, and Jeffrey M. Rabin, "Attention to meaning by algebra teachers"
David Marr's book, Vision, describes and important and influential approach to thinking about cognitive representations and processes.
- D. Gentner, K. J. Holyoak, and B. N. Kokinov, The analogical mind: Perspectives from cognitive science.
- D. Gentner and C. Toupin, "Systematicity and surface similarity in the development of analogy."
- K. J. Holyoak, and P. Thagard, "Analogical mapping by constraint satisfaction."
- David Marr, Vision (excerpts), pdf.
- Tomaso Poggio, "The levels of understanding framework, revisited," pdf.
- David Danks, "Computational realism, levels, and constraints" (a draft of a book chapter), pdf.
Visualization and diagrammatic reasoning
- David Marr, Vision.
- Gerard Allwein and Jon Barwise, Logical reasoning with diagrams
- Janice Glasgow, N. Hari Narayanan, and B. Chandrasekaran, Diagrammatic reasoning: cognitive and computational perspectives
- Jill Larkin and Herbert Simon, "Why a diagram is (sometimes) worth ten thousand words," pdf.
- Marcus Giaquinto, Visual thinking in mathematics: an epistemological study, Chapter 4, Chapter 5, Chapter 6, Chapter 8, Chapter 12.
- Jeremy Avigad, review of Giaquinto's book
- Marcus Giaquinto, "Visualizing in mathematics"
- Marcus Giaquinto, "Cognition of struture"
- Paolo Mancosu, "Visualization in logic and mathematics," pdf
- Ken Manders, "The Euclidean diagram"
- John Mumma, "Proofs, pictures and Euclid," pdf.
- John Mumma, "The role of geometric content in Euclid's diagrammatic reasoning", pdf.
- Jeremy Avigad, Edward Dean, and John Mumma, "A formal system for Euclid's Elements," pdf.
- John Anderson and Ken Koedinger, "Abstract planning and perceptual chunks: Elements of expertise in geometry." pdf.
Insights from the history of mathematics
- Philip Kitcher, The nature of mathematical knowledge
- Ken Manders, "Euclid or Descartes: representation and responsiveness"
- Jeremy Avigad, "Methodology and metaphysics in the development of Dedekind's theory of ideals,"
- Andrew Arana and Paolo Mancosu, "On the relationship between plane and solid geometry," pdf.
- Jeremy Avigad and Rebecca Morris, "Character and object," pdf.
- Mohan Ganesalingam, "The language of mathematics," html.
- Andrei Paskevich, "The syntax and semantics of the ForTheL language," pdf.
- The Naproche project, html.
- Jeremy Avigad, "Mathematical type inference"
- Mark Wilson, "The unreasonable uncooperativeness of mathematics in the natural sciences"
- Chris Pincock, "Towards a philosophy of applied mathematics"
- Robert Batterman, The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence.
Considering automated reasoning in mathematics would take as too far afield, but here are two general references:
The field used to be divided into "logic-based" methods and "human" or "heuristic" methods. The latter not as popular these days, but some logic-based methods are tending back in that direction. The following provide a sense of early work along those lines.
- John Harrison, Handbook of practical logic and automated reasoning.
- J. Alan Robinson and Andrei Voronkov, editors, Handbook of automated reasoning, volumes 1 and 2.
- Donald Loveland, "Automated theorem-proving: a quarter century," pdf.
- Woody Bledsoe, "Non-resolution theorem proving," pdf.
- Douglas Lenat, "Automated theory formation in mathematics," pdf.