Abstract
According to what one might label the traditional view of proof in mathematics, proofs have the following characteristics. They are knowable a priori, the knowledge they provide is certain, rather than merely probable, they are surveyable, and, because of these other features, a mathematical proof is convincing to one who understands it. Opponents of this view typically drew their motivation not from the study of mathematics, but rather from a more general antipathy to apriority in epistemology and necessity in metaphysics (Mill, Putnam and Quine all spring to mind here). Tymoczko (1979) provides a different sort of challenge, taking as its cue a development within mathematics itself, rather than being based on philosophical scruples over apriority or necessity.
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References
Appel K (1984) The use of the computer in the proof of the four color theorem. Proc Am Philos Soc 128(1):35–39
Appel K, Haken W, Koch J (1977) Every planar map is four colorable. Ill J Math 21:429–567
Arkoudas K, Bringsjord S (2007) Computers, justification, and mathematical knowledge. Mind Mach 17(2):185–202
Azzouni J (1994) Metaphysical myths, mathematical practice. Cambridge University Press, New York
Baker A (2008) Experimental mathematics. Erkenntnis 68(3):331–344
Barberousse A, Vorms M (2014) About the warrants of computer-based empirical knowledge. Synthese 191(15):3595–3620
Barendregt H, Wiedijk F (2005) The challenge of computer mathematics. Philos Trans 363(1835):2351–2375. The Nature of Mathematical Proof
Barwise J (1989) Mathematical proofs of computer system correctness. Not Am Math Soc 36:844–851
Bassler OB (2006) The surveyability of mathematical proof: a historical perspective. Synthese 148(1):99–133
Brown GS (1972) Laws of form. Br J Philos Sci 23(3):291–292
Burge T (1993) Content preservation. Philos Rev 102:457–488
Burge T (1998) Computer proof, apriori knowledge, and other minds. Philos Perspect 12:1–37
Corfield D (2003) Towards a philosophy of real mathematics. Cambridge University Press, Cambridge
Detlefsen M, Luker M (1980) The four-color theorem and mathematical proof. J Philos 77(12):803–820
Gödel K (1983) What is cantor’s continuum problem? In: Paul B, Hilary P (eds). Cambridge University Press, Cambridge, pp 470–485. (Reprinted in Philosophy of Mathematics, 2nd ed.)
Gonthier G (2005) A computer-checked proof of the four colour theorem. http://www2.tcs.ifi.lmu.de/~abel/lehre/WS07-08/CAFR/4colproof.pdf
Gonthier G (2008) Formal proof – the four-color theorem. Notices AMS 55(11):1382–1393
Hales T (2008) Formal proof. Notices AMS 55(11):1370–1380
Hales T et al (2011) The flyspeck project. http://code.google.com/p/flyspeck/wiki/FlyspeckFactSheet
Harrison J (2008) Formal proof – theory and practice. Notices AMS 55(11):1395–1406
Hirsch M et al (1994) Responses to “theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics”. In: Jaffe A, Quinn F (eds) Bulletin of the American Mathematical Society Volume 30, Number 2, pp 178–207
Jaffe A, Quinn F (1993) ‘Theoretical mathematics’: toward a cultural synthesis of mathematics and theoretical physics. Bulletin (New Series) AMS 29(1):1–13
Johansen MW, Misfeldt M (2016) Computers as a source of a posteriori knowledge in mathematics. Int Stud Philos Sci 30(2):111–127
Katz JJ (1998) Realistic rationalism, Bradford
Kauffman L (2000) The Robbins Problem – computer proofs and human proofs. http://homepages.math.uic.edu/~kauffman/Robbins.htm
Krakowski I (1980) The four color problem reconsidered. Philos Stud 38(1):91–96
Kripke S (1972) Naming and necessity. Harvard University Press, Cambridge, MA
Levin M (1981) On Tymoczko’s argument for mathematical empiricism. Philos Stud 39:79–86
MacKenzie D (1999) Slaying the kraken: the sociohistory of a mathematical proof. Soc Stud Sci 29(1):7–60
MacKenzie D (2001) Mechanizing proof: computing, risk and trust. MIT Press, Mass
MacKenzie D (2005) Computing and the cultures of proving. Philos Trans A Math Phys Eng Sci 363(1835):2335–2347; discussion 2347–50. PMID: 16188609. https://doi.org/10.1098/rsta.2005.1649
Marcus R, McEvoy M (eds) (2016) An historical introduction to the philosophy of mathematics. Bloomsbury Academic
McEvoy M (2008) The epistemological status of computer proofs. Philos Math 16:374–387
McEvoy M (2013) Experimental mathematics, computers and the a priori. Synthese 190(3):397–412
Pollack R (1997) How to believe a machine-checked proof. Basic Res Comput Sci 97(Report Series):1–18
Prawitz D (2008) Proofs verifying programs and programs producing proofs. In: Lupacchini R, Corsi G (eds) Deduction, computation, experiment: exploring the effectiveness of proof. Springer, Berlin/Helderberg/New York, pp 81–94
Rav Y (1999) Why do we prove theorems? Philos Math 7(1):5–41
Resnik M (1997) Mathematics as a science of patterns. Oxford University Press, New York
Rota G (1997) The phenomenology of mathematical proof. Synthese 111(2):183–196
Secco and Pereira GD (2017) Proofs versus experiments: wittgensteinian themes surrounding the four-color theorem. In: Silva M (ed) How Colours matter to philosophy. Springer, Cham, pp 289–307. https://www.semanticscholar.org/paper/Proofs-Versus-Experiments%3A-Wittgensteinian-Themes-Secco-Pereira/15f28edc77c7f5c54f0dea0db67e8fea8816710f
Shanker SG (1986) The Appel-Haken solution of the four-color problem. In: Shanker (ed) Ludwig Wittgenstein: critical assessments, vol 3. Croom Helm, London, pp 395–412
Swart ER (1980) The philosophical implications of the four-color problem. Am Math Mon 87(9):697–707
Symons J, Alvarado R (2019) Epistemic entitlements and the practice of computer simulation. Mind Mach 29(1):37–60
Teller P (1980) Computer proof. J Philos 77(12):797–803
Tymoczko T (1979) The four-color problem and its philosophical significance. J Philos 76:57–82
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McEvoy, M. (2022). Revisiting ‘The New 4CT Problem’. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_38-1
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