The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field. Read more...
1 Introduction
Part I Algorithm, Consistency, and Epistemic Randomness
2 Algorithms and the Mathematical Foundations of Computer Science
3 The Second Incompleteness Theorem: Reflections and Ruminations
4 Iterated Definability, Lawless Sequences, and Brouwer’s Continuum
5 A Semantics for In-Principle Provability
Part II Mind and Machines
6 Collapsing Knowledge and Epistemic Church’s Thesis
7 Gödel’s Disjunction
8 Idealization, Mechanism, and Knowability
Part III Absolute Undecidability
9 Provability, Mechanism, and the Diagonal Problem
10 Absolute Provability and Safe Knowledge of Axioms
11 Epistemic Church’s Thesis and Absolute Undecidability
