"When Stephen Wolfram's groundbreaking A New Kind of Science was published in 2002, its exploration and analysis of the computational universe of simple programs launched a scientific revolution. Twenty years later, the ideas and results of the book have found countless applications across science, technology and elsewhere-including the recent Wolfram Physics Project and its breakthrough in fundamental physics-and the book has indeed spawned what can only be described as a new kind of science. Here Wolfram reflects on the first two decades of A New Kind of Science, discussing some of the major implications that have emerged so far, as well as his far-reaching new thinking building on the conceptual framework developed in A New Kind of Science. Written in Wolfram's popular and accessible style, the book provides a window into one of the most vibrant intellectual developments of our time. Recognizing A New Kind of Science's significance not only in science but also in the arts, the book includes a gallery of pieces created over the past 20 years by artists inspired by the book"--

Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments. In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and 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, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding.Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics.ContributorsScott 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