Bibliography
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.
General references
The following collections contains a number of related essays:
- Paolo Mancosu, editor, The philosophy of mathematical practice
The following provide some high-level reflection on mathematical understanding:
- 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.
Mathematical concepts
Traditional philosophical views:
- Immanuel Kant, Critique of pure reason. B introduction: pdf.
Transcendental doctrine of method, discipline of pure reason in dogmatic use:
pdf.
On the schematism of the pure concepts of understanding:
pdf.
- Michael Friedman, Kant and the exact sciences; in particular, chapter 2, "Concepts and intuitions in mathematics," pdf.
- Lisa Shabel, "Kant's philosophy of mathematics,"
pdf.
- Lisa Shabel, "Kant on the 'Symbolic Construction' of Mathematical Concepts," pdf.
- Charles Parsons, "Kant's philosophy of arithmetic,"
pdf.
- 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"
Contemporary views:
- 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)
There is a vast literature on concepts more generally. These may be helpful:
- 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
Historians of mathematics often use the term:
- 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:
pdf.
Part VII: pdf.
- Imre Lakatos, Proofs and refutations
- Solomon Feferman, "The logic of mathematical discovery vs. the logical structure of mathematics,"
pdf.
- Mark Steiner, "Mathematical explanation," pdf.
- David Sandborg, Explanation in mathematical practice
Chapter 4: (and title page) pdf,
Chapter 5: pdf,
Chapter 6: pdf,
references: 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"
doi,
pdf.
- 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
- 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,
pdf,
pdf
- John Hayes, The complete problem solver
- Craig Kaplan and Herbert Simon, "In search of insight," pdf.
In addition, there are very many problem collections and solving guides in mathematics, of which the following is a small sample.
- 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.
The following may be helpful in characterizing mathematical expertise more generally:
- 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
- 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
using data"
- 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"
Some of the general cognitive literature on analogy may be helpful:
- 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's book, Vision, describes and important and influential approach to thinking about cognitive representations and processes.
- 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,"
pdf.
- Andrew Arana and Paolo Mancosu, "On the relationship between plane and solid geometry," pdf.
- Jeremy Avigad and Rebecca Morris, "Character and object," pdf.
Mathematical language
- 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"
Applied mathematics
- 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.
Automated reasoning
Considering automated reasoning in mathematics would take as too far afield, but here are two general references:
- John Harrison, Handbook of practical logic and automated reasoning.
- J. Alan Robinson and Andrei Voronkov, editors, Handbook of automated reasoning, volumes 1 and 2.
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.
- Donald Loveland, "Automated theorem-proving: a quarter century," pdf.
- Woody Bledsoe, "Non-resolution theorem proving," pdf.
- Douglas Lenat, "Automated theory formation in mathematics," pdf.