Godel’s Disjunction The scope and limits of mathematical knowledge 1st Edition by Leon Horsten, Philip Welch 0198759592 9780198759591

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Product details:

ISBN 10: 0198759592 

ISBN 13: 9780198759591

Author: Leon Horsten; Philip Welch

The logician Kurt Gödel 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 Gödel’s disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

Table of contents:

1. Introduction

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

6. Collapsing Knowledge and Epistemic Church’s Thesis

7. Gödel’s Disjunction

8. Idealization, Mechanism, and Knowability

9. Provability, Mechanism, and the Diagonal Problem

10. Absolute Provability and Safe Knowledge of Axioms

11. Epistemic Church’s Thesis and Absolute Undecidability

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Tags: Leon Horsten, Philip Welch, Godel’s, Disjunction