Online research in philosophy

This paper uses a proof of Gödels theorem, implemented on a computer, to explore how a person or a computer can examine such a proof, understand it, and evaluate its validity. It is argued that, in order to recognize it (1) as Gödel's theorem, and (2) as a proof that there is an undecidable statement in the language of PM, a person must possess a suitable semantics. As our analysis reveals no differences between the processes required by people and machines to understand Gödel's theorem and manipulate it symbolically, an effective way to characterize this semantics is to model the human cognitive system as a Turing Machine with sensory inputs. La logistique n'est plus stérile: elle engendre la contradicion! – Henri Poincaré ‘Les mathematiques et la logique’.

The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including those of Gödel and Church-Rosser. The computer verification using the Boyer-Moore theorem prover yields precise and rigorous proofs of these difficult theorems. It also demonstrates the range and power of automated proof checking technology. The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction.

This paper is concerned with real proofs as opposed to formal proofs, and specifically with the ultimate reason of real proofs (`Why Proof?') and with the notion of real proof (`What is a Proof?').

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.

Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, and so there is no reason to believe that computer-assisted proofs are not a priori. Thanks are due to Michael Levin, David Corfield, and an anonymous referee for Philosophia Mathematica for their helpful comments. Earlier versions of this paper were presented at the Hofstra University Department of Mathematics colloquium series, and at the 2005 New Jersey Regional Philosophical Association; I am grateful to both audiences for their comments. CiteULike    Connotea    Del.icio.us    What's this?

Bertrand Russell’s 1906 article ‘The Theory of Implication’ contains an algebraic weak completeness proof for classical propositional logic. Russell did not present it as such. We give an exposition of the proof and investigate Russell’s view of what he was about, whether he could have appreciated the proof for what it is, and why there is no parallel of the proof in Principia Mathematica.

The computer played an essential role in the proof given by Kenneth Appel and Kenneth Henken of the Four-Color Theorem (4CT).1 First proposed in 1852 by Francis Guthrie, the four color problem is to determine whether four colors are sufficient to color any map on a plane so that no adjacent regions have the same color. Appel and Heken’s proof involves a lemma that a certain ‘avoidable’ set U of configurations is reducible. The proof of this critical lemma requires certain combinatorial checks which are too long to do by hand. The job was done by an IBM 370/168, using over 1200 hours of computer time. In 1977, Appel and Heken, assisted by John Koch, published the proof, and the 4CT has since been considered an established result. No one has seen the entire proof of the reducibility lemma. It was too long to print out; even if it had been, no one would be able to run through it step by step.

We give a new proof of Friedman's conjecture that every uncountable Δ 1 1 set of reals has a member of each hyperdegree greater than or equal to the hyperjump.

Preface -- Introduction -- There is only one reality -- The ultimate perspective and the ultimate drama -- Proof #1: Science -- Proof #2: History -- Proof #3: Prophecy -- Proof #4: Supernatural -- Proof #5: Psychology -- Proof #6: Sociology -- Proof #7: Inerrancy -- Proof #8: Micro-science -- Proof #9: Logic -- Proof #10: The only provably -- Inerrant, complete system -- Why proof is important -- Personal iplications of proof.