Paper Info
Title
Computers, Justification, and Mathematical Knowledge
Abstract
The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system.
Year
DOI
Venue
2007
10.1007/s11023-007-9063-5
Minds and Machines
Keywords
Field
DocType
A priori,Justification,Proofs,Certificates,Four-color theorem,Mathematical knowledge
Analytic proof,Probabilistically checkable proof,Computer-assisted proof,Computer science,Structural proof theory,Proof theory,Mathematical proof,Artificial intelligence,Proof complexity,Proof assistant
Journal
Volume
Issue
ISSN
17
2
0924-6495
Citations 
PageRank 
References 
12
0.84
6
Authors
2
Name
Order
Citations
PageRank
Konstantine Arkoudas118619.63
Selmer Bringsjord228753.30