Gödel's Disjunction: The scope and limits of mathematical knowledge
Leon Horsten and Philip Welch
Abstract
The logician Kurt Gödel 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 mathematicalmind ismore powerful than any computer. These arguments, and counte ... More
The logician Kurt Gödel 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 mathematicalmind ismore 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 of 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.
Keywords:
Gödel’s disjunction,
absolute provability,
absolute undecidability,
mathematical proof,
incompleteness,
Church’s Thesis
Bibliographic Information
| Print publication date: 2016 |
Print ISBN-13: 9780198759591 |
| Published to Oxford Scholarship Online: November 2016 |
DOI:10.1093/acprof:oso/9780198759591.001.0001 |
Authors
Affiliations are at time of print publication.
Leon Horsten, editor
Professor of Philosophy, University of Bristol
Philip Welch, editor
Professor of Mathematical Logic, University of Bristol
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