Computers, justification, and mathematical knowledge
Minds and Machines 17 (2) (2007)
| 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 | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | No categories specified (fix it) | |||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,664 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesMark McEvoy (2008). The Epistemological Status of Computer-Assisted Proofs. Philosophia Mathematica 16 (3):374-387.
N. Shankar (1994). Metamathematics, Machines, and Gödel's Proof. Cambridge University Press.
Andrew Aberdein (2006). Proofs and Rebuttals: Applying Stephen Toulmin's Layout of Arguments to Mathematical Proof. In Marta Bílková & Ondřej Tomala (eds.), The Logica Yearbook 2005. Filosofia.
Carlo Cellucci (2008). Why Proof? What is a Proof? In Giovanna Corsi & Rossella Lupacchini (eds.), Deduction, Computation, Experiment. Exploring the Effectiveness of Proof, pp. 1-27. Springer.
Jeremy Avigad (2006). Mathematical Method and Proof. Synthese 153 (1):105 - 159.
Casey Rufener (2011). The Four-Color Theorem Solved, Again: Extending the Extended Mind to Philosophy of Mathematics. Res Cogitans 2 (1):215-228.
Edwin Coleman (2009). The Surveyability of Long Proofs. Foundations of Science 14 (1-2):27-43.
Analytics
Monthly downloads |
Added to index2009-01-28Total downloads11 ( #99,458 of 549,007 )Recent downloads (6 months)1 ( #63,261 of 549,007 )How can I increase my downloads? |
My notes
Discussion
