Компьютер как новая реальность математики. I. Personal account
Аннотация
В последние десятилетия много говорится о компьютерных доказательствах (comuputer proofs, computer aided proofs, computer verified proofs и т. д.). Совершенно очевидно, что в еще большей степени появление и распространение компьютеров изменило приложения математики. О чем однако говорится гораздо меньше, это о том, как компьютеры изменили саму математику, отношение математиков к математической реальности, как возможности ее непосредственно наблюдать, так и понимание того, что вообще мы можем надеяться доказать. Я рассказываю о своем собственном опыте использования компьютеров как инструмента и об опыте использования компьютеров в работах моих коллег, которые я наблюдал с близкого расстояния. Этот опыт радикально изменил мои взгляды на многие аспекты функционирования математики, в частности на ее преподавание. Первая часть носит общий мемуарно-философский характер, следующие посвящены нескольким важным конкретным продвижениям, полученным с помощью компьютеров в алгебре и теории чисел.
Литература
M. O. Asanov and V. G.Parfenov, “Final’nye sorevnovaniya chempionata mira po programmirovaniyu potryasayushchii uspekh peterburgskikh komand” [The final competitions of the World Programming Championship an amazing success of the St. Petersburg teams], Computer Tools in Education, no. 2, pp. 11–18, 2001 (in Russian).
N. A. Vavilov and A. Yu. Luzgarev, “Chevalley group of type E7 in the 56-dimensional representation,” Zap. Nauchn. Semin. POMI, vol. 386, pp. 5–99, 2011 (in Russian).
N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “Chevalley group of type E6 in the 27-dimensional representation,” Zap. Nauchn. Semin. POMI, vol. 338, pp. 5–68, 2006 (in Russian).
N. A. Vavilov, O. A. Ivanov, G. A. Lushnikova, and V. G. Khalin, Uroki matematiki pri pomoshchi Mathematica [Math lessons with Mathematica], Saint Petersburg, Russia: OTsEiM, 2008 (in Russian).
N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica dlya nematematika [Mathematica for nonmathematician], [In Print], Saint Petersburg, Russia, 2020 (in Russian).
O. A. Ivanov, Easy as Pi?: An Introduction to Higher Mathematics, Saint Petersburg, Russia: Izd-vo SPbGU, 1995.
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 1. Introduction,” Computer tools in school, no. 1, pp. 5–14, 2010 (in Russian).
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 2. Computer experiments with Maxima,” Computer tools in school, no. 2, pp. 3–13, 2010 (in Russian).
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 3. Basics of programming in Maxima,” Computer tools in school, no. 4, pp. 3–12, 2010 (in Russian).
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 4. Algebra with Maxima,” Computer tools in school, no. 4, pp. 3–13, 2010 (in Russian).
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 5. Calculus with Maxima,” Computer tools in school, no. 5, pp. 3–15, 2010 (in Russian).
O. A. Ivanov, “CAS Maxima in school mathematics education. Lesson 6. Conclusion: ideas, principles, approaches,” Computer tools in school, no. 6, pp. 3–13, 2010 (in Russian).
O. A. Ivanov, Elementarnaya matematika dlya shkol’nikov, studentov i prepodavatelei [Elementary mathematics for schoolchildren, students and teachers], Moscow: MTsNMO, 2019 (in Russian).
O. A. Ivanov and G. M. Fridman, Diskretnaya matematika i programmirovanie v Wolfram Mathematica [Discrete Mathematics and Programming in Wolfram Mathematica], Saint Petersburg, Russia: Piter, 2019 (in Russian).
K. F. Klein, Elementary mathematics from a higher standpoint, Moscow: Nauka, 1987 (in Russian).
D. E. Knuth, The TeXbook, Protvino, Russia: AO RDTeX, 1993 (in Russian).
D. E. Knuth, The Art of Computer Programming. 1: Fundamental Algorithms, Moscow: Vil’yams, 2000 (in Russian).
D. E. Knuth, The Art of Computer Programming. 2: Seminumerical Algorithms, Moscow: Vil’yams, 2000 (in Russian).
D. E. Knuth, The Art of Computer Programming. 3: Sorting and Searching, Moscow: Vil’yams, 2000 (in Russian).
Yu. I. Manin, Dokazuemoe i nedokazuemoe [Provable and non-provable], Moscow: Sovetskoe radio, 1979 (in Russian).
Yu. I. Manin, Vychislimoe i nevychislimoe [Computable and non-computable], Moscow: Sovetskoe radio, 1980 (in Russian).
V. A Petrov and S. N. Pozdnyakov, “Zaochnaya shkola sovremennogo programmirovaniya. Zanyatie 1. algoritmy nad tselymi chislami” [Correspondence school of modern programming. Lesson 1. Algorithms on Integers], Computer Tools in Education, no. 1, pp. 39–49, 1999 (in Russian).
G. P´olya, Mathematics and Plausible Reasoning, Moscow: Nauka, 1975 (in Russian).
A. A. Semenov, Razlozhenie Bruhat dlinnykh kornevykh torov v gruppakh Chevalley [Bruhat decomposition of long root tori in Chevalley groups], PhD dissertation, SPbSU, Saint Petersburg, Russia, 1991 (in Russian).
M. D. Spivak, The joy of TeX: A gourmet guide to typesetting with the AMS-TeX macro package, Moscow: Mir, 1993 (in Russian).
G. Steinhaus, Zadachi i razmyshleniya [Challenges and Reflections], Moscow: Mir, 1974 (in Russian).
D. H. Bailey and J. M. Borwein, “Experimental mathematics: examples, methods and implications,” Notices Amer. Math. Soc., vol. 52, no. 5, pp. 502–514, 2005.
D. H. Bailey and J. M. Borwein, “Exploratory experimentation and computation,” Notices Amer. Math. Soc., vol. 58, no. 10, pp. 1410–1419, 2011.
G. Carlin, Brain droppings, New York, NY, USA: Hyperion, 1998.
A. Chenciner, “Le vrai, le faux, l’insignifiant,” L’Institut de m´ecanique c´eleste et de calcul des ´eph´em´erides (IMCCE), 2015. [Online] (in French). Available: https://perso.imcce.fr/alain-chenciner/Vrai_{}faux_{}insignifiant.pdf
J. H. Conway et al., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, New York, NY, USA: Oxford University Press, 1985.
D. van Dantzig, “Is 101010 a finite number?” Dialectica, vol. 9, no. 3/4, pp. 273–277, 1955; doi: 10.2307/42964599
L. Di Martino and M. C. Tamburini, “2-generation of finite simple groups and some related topics,” in enerators and relations in groups and geometries, A. Barlotti et al., eds., NATO ASI Series, vol. 333, Dordrecht, Netherlands: Springer, pp. 195–233, 1991; doi: 10.1007/978-94-011-3382-1_8
A. Halevy, P. Norvig, and F. Pereira, “The unreasonable effectiveness of data,” Intelligent Systems, IEEE, vol. 24, no. 2, pp. 8–12, 2009; doi: 10.1109/MIS.2009.36
H. A Helfgott, “Minor arcs for Goldbach’s problem,” arXiv:1205.5252v4, [math.NT], 30 Dec 2013.
H. A Helfgott, “Major arcs for Goldbach’s problem,” arXiv:1305.2897v4, [math.NT], 14 April 2014.
H. A Helfgott, “The Ternary Goldbach Conjecture is true,” arXiv:1312.7748v2, [math.NT], 17 Jan 2014.
H. A Helfgott, “The ternary Goldbach problem,” arXiv:1501.05438v2, [math.NT], 27 Jan 2015.
H. A Helfgott and D. J. Platt, “Numerical verification of the ternary Goldbach conjecture up to 8.875 · 1030,” Exp. Math., vol. 22, no. 4, pp. 406–409, 2003; doi: 10.1080/10586458.2013.831742
A. Jaffe and F. Quinn, “Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics,” Bull. Amer. Math. Soc. (N.S.), vol. 29, no. 1, pp. 1–13, 1993; doi: 10.1090/S0273-0979-1993-00413-0
W. Jockusch, J. Propp, and P. Shor, “Random domino tilings and the arctic circle theorem,” arXiv:math/9801068v1 [math.CO], 13 Jan 1998.
D. E. Knuth, “Mathematics and computer science: coping with finiteness,” Science, vol. 194, no. 4271, pp. 1235–1242, 1976; doi: 10.1126/science.194.4271.1235
A. S. Kulikov et al., “Competitive programmer’s core skills,” in Coursera. [Online Course]. Available: https://www.coursera.org/learn/competitive-programming-core-skills
E. Plotkin, A. Semenov, and N. Vavilov, “Visual basic representations: an atlas,” Internat. J. Algebra Comput, vol. 8, no. 1, pp. 61–95, 1998; doi: 10.1142/S0218196798000053
F. Quinn, “A revolution in mathematics? What really happened a century ago and why it matters today,” Notices Amer. Math. Soc., vol. 59, no. 1, pp. 31–37, 2012; doi: 10.1090/noti787
O. Ramar´e, “Etat des lieux ´ ”, in Luminy Institute of Mathematics, 2002. [Online] (in French). Available: http://iml.univ-mrs.fr/~ramare/Maths/ExplicitJNTB.pdf
H. Steinhaus, Mi¸edzy duchem a materi ¸a po´sreniczy matematyka, Warsaw: PWN, 2000 [in Polish].
M. C. Tamburini, “Generation of certain simple groups by elements of small order,” Istit. Lombardo Accad. Sci. Lett. Rend. A, vol. 121, pp. 21–27, 1987.
N. Vavilov, “Structure of Chevalley groups over commutative rings,” in Proc. Conference Non-associative Algebras and Related Topics (Hiroshima, 1990), River Edge, NJ, USA: World Sci. Publ., 1991, pp. 219–335.
N. Vavilov, “Do it yourself structure constants for Lie algebras of types El,” Zap. Nauchn. Semin. POMI, vol. 281, pp. 60–104, 2001.
N. Vavilov, “Reshaping the metaphor of proof,” Philos Trans. Roy. Soc. A, vol. 377, no. 2140, pp. 1–18, 2019; doi: 10.1098/rsta.2018.0279
B. L. van der Waerden, Moderne Algebra, Berlin: Springer-Verlag, 1931.
H. Weber, Lehrbuch der Algebra, Braunschweig, Germany: F. Vieweg und Sohn, 1896 [in German].
S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Boston, MA, USA: Addison-Wesley Publishing Company, 2003.
S. Wolfram, “A new kind of science,” in Wolfram Media, Inc., 2002. [Online]. Available: urlhttps://www.wolframscience.com/nks/
S. Wolfram, “Finally we may have a path to the fundamental theory of physics and it’s beautiful,” [News], in Stephen Wolfram site, Apr. 29, 2020. [Online]. Available: https://writings.stephenwolfram.com/2020/04
D. Zeilberger, “Theorems for a price: tomorrow’s semi-rigorous mathematical culture,” Notices Amer. Math. Soc., no. 40, pp. 978–981, 1993.
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