Meta-math reference list

By “meta-mathematics” I mean discussions on: the culture, the psychology/cognitive science, the history, or the (humanistic) philosophy of mathematics. I also include the category “reflection,” which means personal stories and reflections on the mathematical process, written by practicing mathematicians (some of them quite well-known, e.g. Poincare, Thurston).

This list is in no way comprehensive.  It’s just a small slice of what’s out there, but the slice that I’ve read and enjoyed.   If you know of something that should be here, let me know! (leave a comment).  The Journal of Humanistic Mathematics is also a great source.

First is a table; below that you’ll find very brief descriptions.

Culture Psychology History Philosophy Reflection
BOOKS
Byers X X
Corfield X
Davis & Hersh X X X X X
Hadamard X X X
Hersh, ed. X X X X
Hersh and J-S. X X
Lakatos X X X
Lakoff & Nuñez X X X
Wilder X X
Wolcott X X X
ARTICLES
Atiyah, et. al. X X
Davis & Hersh X X
Dyson X X
Enzensberger X
Jaffe and Quinn X
Thurston X X X
Tymoczko X X
White X

Books

W. Byers, How Mathematicians Think, Princeton Univ. Press, 2010. 424 pgs.

William Byers, a mathematician and Zen Buddhist, shows the role that ambiguity, paradox, and contradiction play in how we do mathematics. He proposes recentering mathematics on the ideas, and perhaps even injecting more subjectivity into mathematics.

D. CorfieldTowards a Philosophy of Real Mathematics, Cambridge Univ. Press, 2003. 300 pgs.

From Spencer:  “My interpretation is that he is arguing for a Philosophy of maths which is actually more about how mathematics is really done, rather than what mathematics is, whether it is `truth’ and the potentially tiresome preoccupation with introspective foundational issues that one sometimes associates with the phrase “Philosophy of Mathematics”. He seems to take a lot of inspiration from Jaffe and Quinn, Lakatos, Davis and Hersh etc.”

P. Davis and R. Hersh, The Mathematical Experience, Mariner Books, 1999. 464 pgs.

The ground-breaking book that simultaneously showed non-mathematicians what it’s like to be a mathematician, and challenged mathematicians to think about the cultural and experiential context of their work.

J. Hadamard, The Psychology of Invention in the Mathematical Field, Dover, 1954. 145 pgs.

Inspired by Poincaré’s description of his mathematical process, this accomplished French mathematician surveyed many mathematicians of his day, and compiled and analyzed their responses.  The book includes the questionnaire used, and Albert Einstein’s response verbatim.

R. Hersh, ed., 18 Unconventional Essays on the Nature of Mathematics, Springer, 2005. 347 pgs.

Includes many interesting articles: critiques of the conventional math philosophies, the Thurston and White articles described below, a case-study of the “mangle of practice” in constructing the quaternions, and others.

R.Hersh and V. John-Steiner, Loving and Hating Mathematics, Princeton Univ. Press, 2010. 428 pgs.

Full of stories from the (mostly recent) history of math, this fascinating and thorough book gives concrete examples of the emotional and social side of mathematics.

I. Lakatos, Proofs and Refutations, Cambridge Univ. Press, 1976. 188 pgs.

Lakatos demonstrates the interplay between definition, conjecture, and proof, and shows that informal reasoning plays a key role in the doing of mathematics.

G. Lakoff and R. Nuñez, Where Mathematics Comes From, Basic Books, 2001. 512 pgs.

These cognitive scientist begin a study of the cognitive science of mathematics, applying findings in that science to explain how our brains process and do mathematics.

R. Wilder, Mathematics as a Cultural System, Pergamon Press, 1981. 188 pgs.

Conjectures cultural patterns within the evolution of mathematics.

L. Wolcott, My Name is Not Susan: A Love Story Between Mathematics and Non-Mathematics, Daikon Pickle Productions, 2009. 142 pgs.

A collection of essays about math and non-math (e.g. math and music, math and travel, math and meditation), stories about doing math, and analysis of the mathematical culture.  Available for free download here.


Articles

M. Atiyah et al., Responses to “Theoretical Mathematics”’: Toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. vol. 30, no.2, April 1994, p. 178-207.

A collection of interesting responses to the article by Jaffe and Quinn (see  below), from a collection of well-known mathematicians.

P. Davis and R. Hersh, The Ideal Mathematician, from “The Mathematical Experience”, Mariner Books, 1999.

A funny six-page caricature of the “ideal mathematician”, presenting many of the cultural quirks and philosophical contradictions most of us carry.

F. Dyson, Birds and Frogs, Notices Amer. Math. Soc. 56 (2009), no. 2, 212–223.

Freeman Dyson explains that “mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details… The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it.” He gives historical examples of both birds and frogs.

H. M. Enzensberger, Drawbridge Up: Mathematics – A Cultural Anathema, A. K. Peters, 2001. 48 pgs.

This excellent essay by the poet and essayist Hans Enzensberger asks why it is that mathematicians are isolated and the public is mathematically illiterate. He offers several possible causes and remedies.

A. Jaffe and F. Quinn, “Theoretical Mathematics”’: Toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. vol. 29, no.1, July 1993, p. 1-13.

Jaffe and Quinn propose changing the publishing conventions to allow for more speculative and conjectural mathematics to be presented and debated.

W. P. Thurston, On proof and progress in mathematics, Bull. Amer. Math. Soc. vol. 30 no. 2, April 1994, p. 161-177.

William Thurston, a brilliant mathematician, gives a particularly thoughtful response to Jaffe and Quinn’s article. He discusses the different modes of communication in math, the incredible variety of ways of understanding math, the nature of math and proof, and the motivations of mathematicians.

T. Tymoczko, Value judgements in mathematics: Can we treat mathematics as an art?, in “Essays in Humanistic Mathematics”, ed. Alvin M. White, MAA Notes #32, 1993.

Tymoczko makes the case that the development of mathematics continually requires value judgements, and proposes that aesthetic criticism can provide these judgements. He makes the case for treating mathematics more like an art, and even analyzes the proof of the Fundamental Theorem of Algebra as an art critic would.

L. A. White, The locus of mathematical reality: an anthropological footnote. In “18 Unconventional Essays on the Nature of Mathematics.” ed. Reuben Hersh. Springer, 2006.

Leslie White, an anthropologist, confidently explains that mathematics is an element of human culture, and therefore objective yet culturally contingent.

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One Response to “Meta-math reference list”

  1. Spencer Says:

    David Corfield’s Towards a Philosophy of Real Mathematics is also in this vein (it is a book). My interpretation is that he is arguing for a Philosophy of maths which is actually more about how mathematics is really done, rather than what mathematics is, whether it is `truth’ and the potentially tiresome preoccupation with introspective foundational issues that one sometimes associates with the phrase “Philosophy of Mathematics”. He seems to take a lot of inspiration from Jaffe and Quinn, Lakatos, Davis and Hersh etc.

    Nice List!

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