nLab development and current state of mathematics

This is not part of philosophy in a narrow sense, but the opinions on what is good mathematics, how mathematics could develop better, and how it should be stimulated to develop, as well as overviews of the current state of mathematics and its important points, is an important part of reflection, but also of the practical action of a mathematician.

On the relation between conjectural and proved mathematics styles

This is a topic of a well-known discussion series in 1990s.

• Arthur Jaffe, Frank Quinn, “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics , Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1–13; MR94h:00007,math.HO/9307227, doi

• Michael Atiyah, Armand Borel, G. J. Chaitin, Daniel Friedan, James Glimm, Jeremy J. Gray, Morris W. Hirsch, Saunders MacLane, Benoit B. Mandelbrot, David Ruelle, Albert Schwarz, Karen Uhlenbeck, René Thom, Edward Witten, Christopher Zeeman, Responses to “Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics”, by A. Jaffe and F. Quinn, Bull. Am. Math. Soc. 30:178-207, 1994, math.HO/9404229

• Michael Atiyah, Reflections on geometry and physics, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 1–6, Int. Press, Cambridge, MA, 1995.

• William P. Thurston, On proof and progress in mathematics, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 161–177, MR94m:00006, doi, arxiv/math.HO

• David Corfield at el. Wittgenstein and Thurston on understanding, nCafe

Somewhat later and with different scope, but related to this series, is also a long article

• Frank Quinn, Contributions to a science of contemporary mathematics, pdf

Some other issue

• The big picture? in mathematics.