several historical points. 1. GENERAL REMARKS Gödel's legacy is still very much in evidence. His legacy is overwhelmingly decisive, particularly in the arena of general mathematical and2 philosophical inquiry. The extent of Gödel's impact in the more restricted domain of mathematical practice is more open to question. In fact, there is an in depth assessment of this impact in Macintyre 2009. But even in this comparatively specialized domain, Gödel's impact is seen to be substantial. As indicated here, particularly in section 12, we believe that the potential impact of Gödel's work on mathematical practice is also overwhelming. However, the full realization of this potential impact will have to wait for some new breakthroughs. We 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 decisively overwhelming. However, there are some signs that some of our most distinguished mathematicians recognize the need for some sort of reconciliation. Here is a quote from Atiyah M. 2008b: "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".3 We will not attempt to properly discuss the full impact of Gödel's work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, (Wang 1987, 1996), (Dawson 2005), and the historically comprehensive five volume set (Gödel

We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([15], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (Zermelo-Fraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian

Abstract not found

a total function which overtakes all total recursive functions. ∗

We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([14], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (Zermelo-Fraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian

Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axioms. 1.