, Volume 17, Issue 2, pp 185202
First online:
Computers, Justification, and Mathematical Knowledge
 Konstantine ArkoudasAffiliated withCognitive Science Department, Computer Science Department, RPI Email author
 , Selmer BringsjordAffiliated withCognitive Science Department, Computer Science Department, RPI
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The original proof of the fourcolor 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 computationintensive custombuilt 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 customwritten 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 fourcolor theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system.
Keywords
A priori Justification Proofs Certificates Fourcolor theorem Mathematical knowledge Title
 Computers, Justification, and Mathematical Knowledge
 Journal

Minds and Machines
Volume 17, Issue 2 , pp 185202
 Cover Date
 200707
 DOI
 10.1007/s1102300790635
 Print ISSN
 09246495
 Online ISSN
 15728641
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 A priori
 Justification
 Proofs
 Certificates
 Fourcolor theorem
 Mathematical knowledge
 Industry Sectors
 Authors

 Konstantine Arkoudas ^{(1)}
 Selmer Bringsjord ^{(1)}
 Author Affiliations

 1. Cognitive Science Department, Computer Science Department, RPI, Troy, NY, USA, 12180