Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T02:54:16.766Z Has data issue: false hasContentIssue false
  • English
  • Français

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

Published online by Cambridge University Press:  31 August 2021

Joshua Habgood-Coote Open the ORCID record for Joshua Habgood-Coote [Opens in a new window]  and
Fenner Stanley Tanswell
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

Keywords

Social epistemologymathematical practiceproofgroup theoryclassification of finite simple groupscollective knowledgemathematical knowledge
Type
Article
Information
Episteme , Volume 20 , Issue 2 , June 2023 , pp. 281 - 307
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allo, P., Van Bendegem, J.P. and Van Kerkhove, B. (2013). ‘Mathematical Arguments and Distributed Knowledge.’ In Aberdein, A. and Dove, I.J (eds), The Argument of Mathematics, pp. 339–60. New York, NY: Springer.CrossRefGoogle Scholar
Alperin, J.L., Brauer, R. and Gorenstein, D. (1970). ‘Finite Groups with Quasi-dihedral and Wreathed Sylow 2-subgroups.’ Transactions of the American Mathematical Society 151, 1–261.Google Scholar
Andersen, L.E. (2017 a). ‘On the Nature and Role of Peer Review in Mathematics.’ Accountability in Research 24, 177–92.CrossRefGoogle ScholarPubMed
Andersen, L.E. (2017 b). ‘Outsiders Enabling Scientific Change: Learning from the Sociohistory of a Mathematical Proof.’ Social Epistemology 31, 184–91.CrossRefGoogle Scholar
Andersen, L.E. (2020). ‘Acceptable Gaps in Mathematical Proofs.’ Synthese 197, 233–47.CrossRefGoogle Scholar
Andersen, L.E., Andersen, H. and Sørensen, H.K. (Forthcoming). ‘The Role of Testimony in Mathematics.’ Synthese.Google Scholar
Antonutti Marfori, M. (2010). ‘Informal Proofs and Mathematical Rigour.’ Studia Logica 96, 261–72.CrossRefGoogle Scholar
Appel, K. and Haken, W. (1977). ‘Every Planar Map is Four Colorable. Part I: Discharging.’ Illinois Journal of Mathematics 21, 429–90.Google Scholar
Appel, K., Haken, W. and Koch, J. (1977). ‘Every Planar Map is Four Colorable. Part II: Reducibility.’ Illinois Journal of Mathematics 21, 491–567.Google Scholar
Aschbacher, M. (1977). ‘A Characterization of Chevalley Groups over Fields of Odd Characteristic.’ Annals of Mathematics 106, 353–468.CrossRefGoogle Scholar
Aschbacher, M. (1980). ‘The Classification of Finite Simple Groups.’ Mathematical Intelligencer 3, 59–65.CrossRefGoogle Scholar
Aschbacher, M. (2004). ‘The Status of the Classification of Finite Simple Groups.’ Notices of the American Mathematical Society 51, 736–40.Google Scholar
Aschbacher, M. and Smith, S.D. (2004 a). The Classification of Quasithin Groups: I. Structure of Strongly Quasithin K-groups. Providence, RI: American Mathematical Society.Google Scholar
Aschbacher, M. and Smith, S.D. (2004 b). The Classification of Quasithin Groups: II Main Theorems: The Classification of Simple QTKE-groups. Providence, RI: American Mathematical Society.Google Scholar
Atiyah, M., Borel, A., Chaitin, G.J., Friedan, D., Glimm, J., Gray, J.J., Hirsch, M.W., Mac Lane, S., Mandelbrot, B.B., Ruelle, D., Schwarz, A., Uhlenbeck, K., Thom, R., Witten, E. and Zeeman, C. (1994). ‘Responses to: A. Jaffe and F. Quinn.’ Bulletin of the American Mathematical Society (New Series) 30, 178–207.Google Scholar
Avigad, J. (2006). ‘Mathematical Method and Proof.’ Synthese 153, 105–59.CrossRefGoogle Scholar
Avigad, J. (2008). ‘Understanding Proofs.’ In Mancosu, P. (ed.), The Philosophy of Mathematical Practice, pp. 317–53. Oxford: Oxford University Press.CrossRefGoogle Scholar
Azzouni, J. (1994). Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Barany, M. (2010). ‘‘[B]ut this is Blog Maths and we're Free to Make up Conventions as we go Along': Polymath1 and the Modalities of ‘Massively Collaborative Mathematics’.’ Proceedings of the 6th International Symposium on Wikis and Open Collaboration. New York, NY: ACM. http://doi.acm.org/10.1145/1832772.1832786.Google Scholar
Barany, M. (2016). ‘Fellow Travelers and Traveling Fellows: The Intercontinental Shaping of Modern Mathematics in Mid-twentieth Century Latin America.’ Historical Studies in the Natural Sciences 46, 669–709.CrossRefGoogle Scholar
Barany, M. (2018). ‘Integration by Parts: Wordplay, Abuses of Language, and Modern Mathematical Theory on the Move.’ Historical Studies in the Natural Sciences 48, 259–99.CrossRefGoogle Scholar
Barany, M. (2020). ‘Abstract Relations: Bibliography and the Infra-structures of Modern Mathematics.’ Synthese. https://doi.org/10.1007/s11229-020-02683-3.CrossRefGoogle Scholar
Bassler, O.B. (2006). ‘The Surveyability of Mathematical Proof: A Historical Perspective.’ Synthese 148, 99–133.CrossRefGoogle Scholar
Bird, A. (2010). ‘Social Knowing: The Social Sense of ‘Scientific Knowledge’.’ Philosophical Perspectives 24, 23–56.CrossRefGoogle Scholar
Brauer, R., Suzuki, M. and Wall, G.E. (1958). ‘A Characterization of the One-dimensional Unimodular Projective Groups Over Finite Fields.’ Illinois Journal of Mathematics 2, 718–45.CrossRefGoogle Scholar
Buzzard, K. (n.d.). ‘What is the Xena Project?’ https://web.archive.org/web/20200811184328/https://xenaproject.wordpress.com/what-is-the-xena-project/.Google Scholar
Buzzard, K. (2019). ‘Does Anyone “Know” a Proof of Fermat's Last Theorem?’ https://xenaproject.wordpress.com/2019/09/27/does-anyone-know-a-proof-of-fermats-last-theorem/.Google Scholar
Carter, J. (2019). ‘Philosophy of Mathematical Practice – Motivations, Themes and Prospects.’ Philosophia Mathematica (III) 27, 1–32.CrossRefGoogle Scholar
Coleman, E. (2009). ‘The Surveyability of Long Proofs.’ Foundations of Science 14, 27–43.CrossRefGoogle Scholar
Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R. (1985). The ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press.Google Scholar
Corneli, J., Martin, U., Murray-Rust, D., Pease, A., Puzio, R. and Nesin, G.R. (2017). ‘Modelling the Way Mathematics is Actually Done.’ FARM 2017: Proceedings of the 5th ACM SIGPLAN International Workshop on Functional Art, Music, Modelling, and Design, pp. 10–19. New York, NY: Association for Computing Machinery.CrossRefGoogle Scholar
Davies, B. (2005). ‘Wither Mathematics.’ Notices of the AMS 52, 1350–6.Google Scholar
Dawson, J.W. Jr. (2006). ‘Why do Mathematicians Re-prove Theorems?’ Philosophia Mathematica (III) 14, 269–86.CrossRefGoogle Scholar
Daston, L. (2019). ‘The Coup D'Oeil: On a Mode of Understanding.’ Critical Inquiry 45(2), 307–31.CrossRefGoogle Scholar
Dutilh Novaes, C. (Forthcoming). The Dialogical Roots of Deduction. Cambridge: Cambridge University Press.Google Scholar
Easwaran, K. (2009). ‘Probabilistic Proofs and Transferability.’ Philosophia Mathematica (III) 17, 341–62.CrossRefGoogle Scholar
Easwaran, K. (2015). ‘Rebutting and Undercutting in Mathematics.’ Philosophical Perspectives 29, 146–62.CrossRefGoogle Scholar
Ernest, P. (1998). Social Constructivism as a Philosophy of Mathematics, Albany, NY: State University of New York Press.Google Scholar
Fallis, D. (1997). ‘The Epistemic Status of Probabilistic Proof.’ Journal of Philosophy 94, 165–86.CrossRefGoogle Scholar
Fallis, D. (2003). ‘Intentional Gaps in Mathematical Proofs.’ Synthese 134, 45–69.CrossRefGoogle Scholar
Feit, W. (1983). ‘Review: Daniel Gorenstein, Finite Simple Groups, an Introduction to their Classification.’ Bulletin of the American Mathematical Society 8(1), 120–4.CrossRefGoogle Scholar
Fong, P. (n.d.). MR0284499. ‘MathSciNet Review of Alperin et al. (1970).’ https://mathscinet.ams.org/mathscinet-getitem?mr=284499.Google Scholar
Geist, C., Löwe, B. and Kerkhove, B. (2010). ‘Peer Review and Knowledge by Testimony in Mathematics.’ In Löwe, B. and Müller, T. (eds), PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pp. 155–78. London: College Publications.Google Scholar
Gilbert, M. (1987). ‘Modelling Collective Belief.’ Synthese 73, 185–204.CrossRefGoogle Scholar
Goldberg, S. (2010). ‘If That Were True I Would Have Heard About It By Now.’ In Relying on Others: An Essay in Epistemology, pp. 155–84. Oxford: Oxford University Press.CrossRefGoogle Scholar
Goldman, A.I. (1999). Knowledge in a Social World. Oxford: Oxford University Press.CrossRefGoogle Scholar
Gorenstein, D. (1979). ‘The Classification of Finite Simple Groups 1. Simple Groups and Local Analysis.’ Bulletin of the American Mathematical Society 1, 43–200.CrossRefGoogle Scholar
Gorenstein, D. (n.d.). MR0249504. ‘MathSciNet Review of Walter (1969).’ https://mathscinet.ams.org/mathscinet-getitem?mr=249504.Google Scholar
Gorenstein, D (1980 a). ‘An Outline of the Classification of Finite Simple Groups.’ In Santa Cruz Conference on Finite Groups. Vol. 37, Proceedings of Symposia in Pure Mathematics, pp. 3–28. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D (1980 b). The Classification of Finite Simple Groups. The University Series in Mathematics. New York, NY: Springer.Google Scholar
Gorenstein, D. (1982). Finite Simple Groups: An Introduction to their Classification. New York, NY: Springer.CrossRefGoogle Scholar
Gorenstein, D. (1983). The Classification of Finite Simple Groups. Vol. 1. Groups of Noncharacteristic 2 Type. The University Series in Mathematics. New York, NY: Plenum Press.CrossRefGoogle Scholar
Gorenstein, D. and Harada, K. (1974). ‘Finite Groups Whose 2-Subgroups are Generated by at Most 4 Elements.’ Memoirs of the American Mathematical Society 147, 1–464.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (1994). The Classification of the Finite Simple Groups, Number 1. Mathematical Surveys and Monographs 40.1. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Gorenstein, D., Lyons, R., and Solomon, R. (1995). The Classification of the Finite Simple Groups, Number 2. Mathematical Surveys and Monographs, Vol. 40.2. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (1997). The Classification of the Finite Simple Groups, Number 3. Mathematical Surveys and Monographs, Vol. 40.3. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (1999). The Classification of the Finite Simple Groups, Number 4: Part II, Chapters 1–4: Uniqueness Theorems. Mathematical Surveys and Monographs, Vol. 40.4. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (2002). The Classification of the Finite Simple Groups, Number 5. Mathematical Surveys and Monographs, Vol. 40.5. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Gorenstein, D., Lyons, R., and Solomon, R. (2004). The Classification of the Finite Simple Groups, Number 6: Part IV: The Special Odd Case. Mathematical Surveys and Monographs, Vol. 40.6. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (2018 a). The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a. Mathematical Surveys and Monographs, Vol. 40.7. Providence, RI: American Mathematical Society.Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R. (2018 b). The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed. Mathematical Surveys and Monographs, Vol. 40.8. Providence, RI: American Mathematical Society.Google Scholar
Gowers, T. and Nielsen, M. (2009). ‘Massively Collaborative Mathematics.’ Nature 461, 879–81.CrossRefGoogle ScholarPubMed
Greiffenhagen, C. (2014). ‘The Materiality of Mathematics: Presenting Mathematics at the Blackboard.’ British Journal of Sociology 65, 502–28.CrossRefGoogle ScholarPubMed
Grier, D. (2005). When Computers Were Human. Princeton, NJ: Princeton University Press.Google Scholar
Habgood-Coote, J. (2019 a). ‘Group Knowledge, Questions, and the Division of Epistemic Labour.’ Ergo: An Open Access Journal of Philosophy 6, art. 33 2019–20.Google Scholar
Habgood-Coote, J. (2019 b). ‘Knowledge-how, Abilities, and Questions.’ Australasian Journal of Philosophy 97, 86–104.CrossRefGoogle Scholar
Habgood-Coote, J. (2020). ‘Group Inquiry.’ Erkenntnis. https://doi.org/10.1007/s10670-020-00232-5.CrossRefGoogle Scholar
Hamami, Y. and Morris, R.L. (2020). ‘Philosophy of Mathematical Practice: A Primer for Mathematics Educators.’ ZDM. https://doi.org/10.1007/s11858-020-01159-5.CrossRefGoogle Scholar
Hersh, R. (1991). ‘Mathematics has a Front and Back.’ Synthese 88, 127–33.CrossRefGoogle Scholar
Hersh, R. (1993). ‘Proving is Convincing and Explaining.’ Educational Studies in Mathematics 24, 389–99.CrossRefGoogle Scholar
Jaffe, A. and Quinn, F. (1993). ‘Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics.’ Bulletin of the American Mathematical Society (New Series) 29, 1–13.CrossRefGoogle Scholar
Johansen, M.W. and Misfeldt, M. (2016). ‘Computers as a Source of A Posteriori Knowledge in Mathematics.’ International Studies in the Philosophy of Science 30, 111–27.CrossRefGoogle Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. Oxford: Oxford University Press.Google Scholar
Kneebone, G.T. (1957). ‘The Philosophical Basis of Mathematical Rigour.’ Philosophical Quarterly 7, 204–23.CrossRefGoogle Scholar
Lane, L.D. (2017). Bridge Between Worlds: Relating Position and Disposition in the Mathematical Field. PhD thesis, University of Edinburgh.Google Scholar
Larvor, B. (2012). ‘How to Think About Informal Proofs.’ Synthese 187, 715–30.CrossRefGoogle Scholar
Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Löwe, B. (2016). ‘Philosophy or Not? The Study of Cultures snd Practices of Mathematics.’ In Ju, S., Löwe, B., Müller, T. and Xie, Y. (eds), Cultures of Mathematics and Logic. Selected Papers from the Conference in Guangzhou, China, 9–12 November 2012, pp. 23–42. Basel: Birkhäuser.CrossRefGoogle Scholar
Mackenzie, D. (2001). Mechanizing Proof: Computing, Risk and Trust. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
MacKenzie, D. and Spinardi, G. (1995). ‘Tacit Knowledge, Weapons Design, and the Uninvention of Nuclear Weapons.’ American Journal of Sociology 101(1), 44–99.CrossRefGoogle Scholar
Martin, U. (2015). ‘Stumbling Around in the Dark: Lessons from Everyday Mathematics.’ In Felty, A.P. and Middeldorp, A. (eds), Proceedings of CADE (Conference on Automatic Deduction) 25. Lecture Notes in Artificial Intelligence 9195, pp. 29–51. Cham: Springer International.Google Scholar
Martin, U. and Pease, A. (2013). ‘Mathematical Practice, Crowdsourcing and Social Machines.’ International Conference on Intelligent Computer Mathematics, pp. 98–119. Berlin: Springer.CrossRefGoogle Scholar
Merton, R.K. (1957). ‘Priorities in Scientific Discovery: A Chapter in the Sociology of Science.’ American Sociological Review 22, 635–59.CrossRefGoogle Scholar
Ornes, S. (2015). ‘The Whole Universe Catalog.’ Scientific American 313, 68–75.CrossRefGoogle ScholarPubMed
Pease, A., Aberdein, A. and Martin, U. (2019). ‘Explanation in Mathematical Conversations: An Empirical Investigation.’ Philosophical Transactions of the Royal Society A 377. https://doi.org/10.1098/rsta.2018.0159.Google ScholarPubMed
Pease, A., Martin, U., Tanswell, F. and Aberdein, A. (2020). ‘Using Crowdsourced Mathematics to Understand Mathematical Practice.’ ZDM. https://doi.org/10.1007/s11858-020-01181-7.CrossRefGoogle Scholar
Polymath, D.H.J. (2012). ‘A New Proof of the Density Hales-Jewett Theorem.’ Annals of Mathematics 175, 1283–327.Google Scholar
Polymath, D.H.J. (2014 a). ‘Variants of the Selberg sieve, and bounded intervals containing many primes.’ Research in the Mathematical Sciences 1. https://doi.org/10.1186/s40687-014-0012-7.CrossRefGoogle Scholar
Polymath, D.H.J. (2014 b). ‘The ‘bounded Gaps Between Primes’ Polymath Project – A Retrospective.’ https://arxiv.org/abs/1409.8361.Google Scholar
Rav, Y. (1999). ‘Why Do We Prove Theorems?’ Philosophia Mathematica (III) 7, 5–41.CrossRefGoogle Scholar
Rittberg, C., Tanswell, F. and Van Bendegem, J.P. (2018). ‘Epistemic Injustice in Mathematics.’ Synthese. https://doi.org/10.1007/s11229-018-01981-1.CrossRefGoogle Scholar
Solomon, R. (2001). ‘A Brief History of the Classification of the Finite Simple Groups.’ Bulletin of the American Mathematical Society 38(3), 315–52.CrossRefGoogle Scholar
Sosa, E. (2007). A Virtue Epistemology: Apt Belief and Reflective Knowledge, Volume 1. Oxford: Oxford University Press.CrossRefGoogle Scholar
Steingart, A. (2012). ‘A Group Theory of Group Theory: Collaborative Mathematics and the ‘Uninvention’ of a 1000-page Proof.’ Social Studies of Science 42(2), 185–213.CrossRefGoogle Scholar
Tanswell, F. (2015). ‘A Problem with the Dependence of Informal Proofs on Formal Proofs.’ Philosophia Mathematica (III) 23(3), 295–310.CrossRefGoogle Scholar
Tanswell, F. (2016). Proof, Rigour and Informality: A Virtue Account of Mathematical Knowledge. PhD Thesis, University of St Andrews.Google Scholar
Tanswell, F. (Forthcoming). ‘Go Forth and Multiply: On Actions, Instructions and Imperatives in Mathematical Proofs.’ In Brown, J. and Beuno, O. (eds), Essays on the Philosophy of Jody Azzouni. Cham: Springer.Google Scholar
Tanswell, F. and Kidd, I.J. (2020). ‘Mathematical Practice and Epistemic Virtue and Vice.’ Synthese. https://doi.org/10.1007/s11229-020-02664-6.CrossRefGoogle Scholar
Thompson, J. (1968, 1970, 1971, 1973, 1974a, 1974b). ‘Nonsolvable Finite Groups All of Whose Local Subgroups are Solvable: I–VI.’ Bulletin of the American Mathematical Society 74 (1968), 383–437; Pacific Journal of Mathematics 33 (1970), 451–536; 39 (1971), 483–534; 48 (1973), 511–92; 50 (1974), 215–97; 51 (1974), 573–630.CrossRefGoogle Scholar
Thurston, W.P. (1994). ‘On Proof and Progress in Mathematics.’ Bulletin of the American Mathematical Society 30(2), art. MR1249357.CrossRefGoogle Scholar
Tymoczko, T. (1979). ‘The Four-Color Problem and Its Philosophical Significance.’ Journal of Philosophy 76, 57–83.CrossRefGoogle Scholar
Van Kerkhove, B. and Van Bendegem, J.P. (2008). ‘Pi on Earth, or Mathematics in the Real World.’ Erkenntnis 68, 421–35.CrossRefGoogle Scholar
Wagenknecht, S. (2014). ‘Opaque and Translucent Epistemic Dependence in Collaborative Scientific Practice.’ Episteme 11, 475–92.CrossRefGoogle Scholar
Walter, H. (1967). ‘Finite Groups with Abelian Sylow 2-subgroups of Order 8.’ Inventiones Mathematicae 2, 332–76.CrossRefGoogle Scholar
Walter, J.H. (1969). ‘The Characterization of Finite Groups with Abelian Sylow 2-Subgroups.’ Annals of Mathematics 89, 405–514.CrossRefGoogle Scholar
Weiner, M. (2007). ‘Norms of Assertion.’ Philosophy Compass 2, 187–95.CrossRefGoogle Scholar
Wittgenstein, L. (1956). Remarks on the Foundations of Mathematics, 1st edition. von Wright, G.H., Rhees, R. and Anscombe, G.E.M. (eds), translated by Anscombe, G.E.M. Cambridge, MA: MIT Press.Google Scholar
Zagzebski, L.T. (1996). Virtues of the Mind: An Inquiry into the Nature of Virtue and the Ethical Foundations of Knowledge. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Zollman, K.J.S. (2018). ‘The Credit Economy and the Economic Rationality of Science.’ Journal of Philosophy 115, 5–33.CrossRefGoogle Scholar