Varieties of mathematical understanding

Avigad, Jeremy

doi:10.1090/bull/1726

Varieties of mathematical understanding
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Bull. Amer. Math. Soc. 59 (2022), 99-117 Request permission

Abstract:

This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a possible resolution.

References

Jeremy Avigad, Methodology and metaphysics in the development of Dedekind’s theory of ideals, The architecture of modern mathematics, Oxford Univ. Press, Oxford, 2006, pp. 159–186. MR 2258019
Jeremy Avigad, The mechanization of mathematics, Notices Amer. Math. Soc. 65 (2018), no. 6, 681–690. MR 3792862
Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh (eds.), Handbook of satisfiability, IOS Press, 2009.
Jonathan Borwein and David Bailey, Mathematics by experiment, 2nd ed., A K Peters, Ltd., Wellesley, MA, 2008. Plausible reasoning in the 21st Century. MR 2473161
Jonathan Borwein and Keith Devlin, The computer as crucible, A K Peters, Ltd., Wellesley, MA, 2009. An introduction to experimental mathematics. MR 2464847
Joshua Brakensiek, Marijn Heule, John Mackey, and David Narváez, The resolution of Keller’s conjecture, International Joint Conference on Automated Reasoning (IJCAR) 2020 (Nicolas Peltier and Viorica Sofronie-Stokkermans, eds.), Springer, 2020, pp. 48–65.
Elizabeth R. Chen, Michael Engel, and Sharon C. Glotzer, Dense crystalline dimer packings of regular tetrahedra, Discrete Comput. Geom. 44 (2010), no. 2, 253–280. MR 2671012, DOI 10.1007/s00454-010-9273-0
Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc. 64 (2017), no. 2, 102–115. MR 3587715, DOI 10.1090/noti1474
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. MR 3664817, DOI 10.4007/annals.2017.185.3.8
J. H. Conway and S. Torquato, Packing, tiling, and covering with tetrahedra, Proc. Natl. Acad. Sci. USA 103 (2006), no. 28, 10612–10617. MR 2242647, DOI 10.1073/pnas.0601389103
Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR 0197380
W. T. Gowers, The two cultures of mathematics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 65–78. MR 1754768
W. T. Gowers, Decompositions, approximate structure, transference, and the Hahn-Banach theorem, Bull. Lond. Math. Soc. 42 (2010), no. 4, 573–606. MR 2669681, DOI 10.1112/blms/bdq018
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379, DOI 10.4007/annals.2008.167.481
Niccolò Guicciardini, Reading the Principia, Cambridge University Press, Cambridge, 1999. The debate on Newton’s mathematical methods for natural philosophy from 1687 to 1736. MR 1725356, DOI 10.1017/CBO9780511524752
Niccolò Guicciardini, Isaac Newton on mathematical certainty and method, Transformations: Studies in the History of Science and Technology, MIT Press, Cambridge, MA, 2009. MR 2562084, DOI 10.7551/mitpress/9780262013178.001.0001
Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray, and Sharon C. Glotzer, Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra, Nature 462 (2009), 773–777.
Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat Dang, John Harrison, Le Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat Thang Nguyen, Quang Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai An Ta, Nam Trung Tran, Thi Diep Trieu, Josef Urban, Ky Vu, and Roland Zumkeller, A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017), e2, 29. MR 3659768, DOI 10.1017/fmp.2017.1
Thomas C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. MR 2179728, DOI 10.4007/annals.2005.162.1065
Yacin Hamami and Rebecca L. Morris, Philosophy of mathematical practice: a primer for mathematics educators, ZDM Mathematics Education 52 (2020), 1113–1126.
Ott-Heinrich Keller, Über die lückenlose Erfüllung des Raumes mit Würfeln, J. Reine Angew. Math. 163 (1930), 231–248 (German). MR 1581241, DOI 10.1515/crll.1930.163.231
Andrzej P. Kisielewicz, Rigid polyboxes and Keller’s conjecture, Adv. Geom. 17 (2017), no. 2, 203–230. MR 3652241, DOI 10.1515/advgeom-2017-0004
Jeffrey C. Lagarias and Peter W. Shor, Keller’s cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 279–283. MR 1155280, DOI 10.1090/S0273-0979-1992-00318-X
John Mackey, A cube tiling of dimension eight with no facesharing, Discrete Comput. Geom. 28 (2002), no. 2, 275–279. MR 1920144, DOI 10.1007/s00454-002-2801-9
Oskar Perron, Über lückenlose Ausfüllung des $n$-dimensionalen Raumes durch kongruente Würfel, Math. Z. 46 (1940), 1–26 (German). MR 3041, DOI 10.1007/BF01181421
Oskar Perron, Über lückenlose Ausfüllung des $n$-dimensionalen Raumes durch kongruente Würfel. II, Math. Z. 46 (1940), 161–180 (German). MR 2185, DOI 10.1007/BF01181436
S. Szabó, A reduction of Keller’s conjecture, Period. Math. Hungar. 17 (1986), no. 4, 265–277. MR 866636, DOI 10.1007/BF01848388
Yuri Tschinkel, About the cover: on the distribution of primes—Gauss’ tables, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 1, 89–91. MR 2201552, DOI 10.1090/S0273-0979-05-01096-7
Maryna S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991–1015. MR 3664816, DOI 10.4007/annals.2017.185.3.7
Richard S. Westfall, Never at rest, Cambridge University Press, Cambridge, 1980. A biography of Isaac Newton. MR 741027
D. T. Whiteside (ed.), The mathematical papers of Isaac Newton, Cambridge University Press, Cambridge, 1976 xlvii+706 pp 5 plates. Vol. VII: 1691–1695; With the assistance in publication of A. Prag. MR 0505130