How can or should the recent developments in mathematics influence the philosophy of mathematics? (Q2702589)

The author diagnoses a stagnation in the philosophy of mathematics which is still dominated by topics debated in early 20th century, especially concerning demarcations between foundational positions like Platonism, formalism, logicism, intuitionism and constructivism. ``In the second half of our century,'' the author claims, ``there have been no really important debates or new innovations in the philosophy of mathematics'' (p. 259).NEWLINENEWLINENEWLINEHere to change this situation the author suggests to reconsider the meaning of mathematical sentences and theories. He supports this suggestion by discussing five fields of investigation for a future philosophy of mathematics: (1) Independence results of set theory; (2) the enormous range of applications of mathematics; (3) chaos theory, especially conceptions of causality connected with this theory; (4) experimental mathematics with its effects on the notion of proof (e.g. computer proofs); (5) algorithmic information theory as developed and described by G. Chaitin.NEWLINENEWLINENEWLINEAs a precondition (``absolute fixed point'', p. 266) for any future philosophy of mathematics the author emphasizes: ``the beginning and indeed the center of any philosophy of mathematics must lie in `mathematical practice', the work of the `working mathematician', mathematical experience'' (p. 266).NEWLINENEWLINEFor the entire collection see [Zbl 0948.00029].