In the 1930s a series of seminal works published by Alan Turing Kurt Godel Alonzo Church & others established the theoretical basis for computability. This work advancing precise characterizations of effective algorithmic computability was the culmination of intensive investigations into the foundations of mathematics. In the decades since the theory of computability has moved to the center of discussions in philosophy computer science & cognitive science. In this volume distinguished computer scientists mathematicians logicians & philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing Godel & Church including the Church-Turing thesis & Godel's response to Church's & Turing's proposals. Other chapters cover more recent technical developments including computability over the reals Godel's influence on mathematical logic & on recursion theory & the impact of work by Turing & Emil Post on our theoretical understanding of online & interactive computing; & others relate computability & complexity to issues in the philosophy of mind the philosophy of science & the philosophy of mathematics. Contributors: Scott Aaronson Dorit Aharonov B. Jack Copeland Martin Davis Solomon Feferman Saul Kripke Carl J. Posy Hilary Putnam Oron Shagrir Stewart Shapiro Wilfried Sieg Robert I. Soare Umesh V. Vazirani