Crowdsourcing a list of general resources on the purpose, value, and nature of mathematics

Parallel to the other crowdsourced resource drive on this blog, I would like to collect books, writings and other resources that discuss or explain the purpose(s), value(s), and nature of mathematics. As seen in reactions to recent events, one can see many oversimplified or inaccurate perceptions of mathematics, such as

• “Mathematical research is about solving open problems.”
• “The main reason for posing an open problem is to obtain its solution.”
• “We should only work on problems that are of immediate benefit to society.”
• “We should prioritize maximizing the number of problems solved.”

The question of what mathematics actually is is far more complex than these assertions might suggest. Here are some classic books and articles on these topics:

• Vladimir Arnold, “On teaching mathematics“, 1997 (somewhat provocative in tone, but still interesting)
• Philip Davis, Reuben Hersh, “The Mathematical Experience“, 1981
• Timothy Gowers, “The two cultures of mathematics“, 2000
• Timothy Gowers, “Mathematics: a very short introduction“, 2002
• G. H. Hardy, “A Mathematician’s Apology“, 1940 (somewhat dated at this point, but still interesting)
• Rueben Hersh, “What is mathematics, really?“, 1997
• Imre Lakatos, “Proofs and refutations“, 1976
• Bill Thurston, “On proof and progress in mathematics“, 1994
• Cedric Villani, “Birth of a Theorem“, 2015

Here are some more informal links:

• Matt Might, The illustrated guide to a Ph.D., 2010.
• A collaboration between myself and Zach Wienersmith on sphere packing, as a five part comic strip (Apr 9, 2026)
• Illustrating the impact of the mathematical sciences – a series of posters by the National Academy of Sciences. (I was on the committee to create these posters.) (2023)

I can also list a few writings, posts, and interviews I have done on these topics:

• There is more to mathematics than rigour and proofs (2007)
• What is good mathematics? (2007)
• The value in “Dumb” questions (2022)
• What makes for ‘Good’ Mathematics (2024)
• What does it mean to think like a mathematician? (Mar 14, 2026)
• Interview with “Big Think” (relating to my forthcoming book) (Aug 28, 2026)
• On remembering what it is like to be a child: curiosity-driven play, the “largest number” game, and what premature optimization for the nominal goal extinguishes (Sep 9, 2026)

Please contribute more links in the comments below!