Kurt Gödel and the Foundations of Mathematics

Gödel's legacy is still very much in evidence. His legacy is overwhelming, particularly in the arena of general mathematical and philosophical inquiry.

The extent of Gödel's impact in the more restricted domain of mathematical practice, however, is more open to question. In fact, Macintyre provides an in-depth assessment of this impact in Chapter 1 of this volume. However, even in this comparatively specialized domain, Gödel's impact is seen to be substantial. As discussed in this chapter, particularly in Section 18.11, I believe that the potential impact of Gödel's work on mathematical practice is also extensive. Although the full realization of this potential impact will have to wait for some new breakthroughs, I have every confidence that these breakthroughs will materialize.

Generally speaking, current mathematical practice has now become very far removed from general mathematical and philosophical inquiry, where Gödel's legacy is most obviously decisive. However, there are signs that some of our most distinguished mathematicians recognize the need for some sort of reconciliation: “Mathematicians took the role of philosophers, but I want to bring the philosophers back in. I hope someday we will be able to explain mathematics in a philosophical way using philosophical methods” (Atiyah, 2008a, 2008b).

I will not attempt to exhaustively discuss the full impact of Gödel's work and all of the ongoing important research programs that it suggests, as this would require a book-length manuscript.