Godel’s Disjunction: The scope and limits of mathematical knowledge

✍ Scribed by Horsten, Leon; Welch, Philip

• Publisher: Oxford University Press
• Year: 2016
• Tongue: English
• Leaves: 288
• Edition: 1
• Category: Library

✦ Synopsis

The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not 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.

✦ Table of Contents

Content: Part I. Algorithm, consistency and epistemic randomness. Algorithms and the mathematical foundations computer science / Walter Dean
The second incompleteness theorem: reflectons and ruminations / Albert Visser
Iterated definability, lawless sequences, and Brouwer’s continuum / Joan Rand Moschovakis
A semantics for in-principle provability / T. Achourioti – Part II. Mind and machines. Collapsing Knowledge and Epistemic Church's Thesis / Timothy J. Carlson
Godel's Disjunction / Peter Koellner
Idealization, mechanism, and knowability / Stewart Shapiro – Part III. Absolute Undecidability. Provability, mechanism, and the diagonal problem / Graham Leach-Krouse
Absolute Provability and Safe Knowledge of Axioms / Timothy Williamson
Epistemic Church's Thesis and Absolute Undecidability / Marianna Antonutti Marfori and Leon Horsten.

✦ Subjects

Gödel, Kurt.;Mathematics -- Philosophy.