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.