Computers, justification, and mathematical knowledge

Minds and Machines 17 (2):185-202 (2007)
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.
Keywords A priori   Certificates   Four-color theorem   Justification   Mathematical knowledge   Proofs
Categories (categorize this paper)
DOI 10.1007/s11023-007-9063-5
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 24,442
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA
Tyler Burge (1993). Content Preservation. Philosophical Review 102 (4):457-488.
Roderick M. Chisholm (1966). Theory of Knowledge. Englewood Cliffs, N.J., Prentice-Hall.
Hans Reichenbach (1939). Experience and Prediction. Philosophical Review 48 (5):536-538.

View all 10 references / Add more references

Citations of this work BETA
Selmer Bringsjord (2015). A Vindication of Program Verification. History and Philosophy of Logic 36 (3):262-277.

View all 7 citations / Add more citations

Similar books and articles

Monthly downloads

Added to index


Total downloads

57 ( #85,657 of 1,925,107 )

Recent downloads (6 months)

12 ( #67,378 of 1,925,107 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.