Przyczynek do historii „epsilonistyki”

Galina Sinkevich

Abstrakt


Przyjrzymy sie historii powstania jezyka „epsilon-delta” w pracach matematyków w XIX wieku. Pokazemy, iz mimo tego, ze oznaczenia zostały wprowadzone przez Cauchy’ego w roku 1823, w pełni definicja „epsilon-delta”pojawiła sie dopiero u Weierstrassa w roku 1861. Przytoczymy rózne interpretacje tej problematyki przez matematyków w czasach pózniejszych.

On the history of epsilontics

This is a review of the genesis of 'epsilon-delta' - language in works of mathematicians of the 19th century. It shows that although the symbols 'epsilon-delta' and were initially introduced in 1823 by Cauchy, no functional relationship for as a function of 'epsilon-delta' was ever specified by Cauchy. It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass’ definition of a limit. The article gives various interpretations of these issues later provided by mathematicians. This article presents the text of the same author [47] which is slightly redone and translated to Polish.

 

 


Bibliografia


A.–M. Ampére, Recherche sur quelques points de la théorie des fonctions dérivées qui conduisent á une nouvelle démonstration du théorême de Taylor, et à l’expression finie des termes qu’on néglige lorsqu’on arrête cette série à un terme quelconque, Journal de l’École Polytechnique, t. 6, n13 (1806), p. 148–181.

I. G. Bashmakova, On the role of interpretation in the history of mathematics, (Russian) Istor.–Mat. Issled., No. 30 (1986), 182–194, MR 0901194, Zbl 0615.01002.

B. Belhoste, Cauchy, 1789–1857 (French), Librairie Classique Eugène Belin, Paris, 1985, MR 0804232, Zbl 0593.01006.

B. Belhoste, Augustin Cauchy, Nauka, Moscow, 1997.

P. Błaszczyk, M. G. Katz, D. Sherry, Ten misconceptions from the history of analysis and their debunking, Found. Sci. 18, no. 1(2013), 43 74, MR 3031794, Zbl 1291.01018.

B. Bolzano, Paradoksy beskonechnogo (Paradoxes of the infinity), 1851, translation edited by I. Sleshinski, Mathesis, Odessa, 1911.

B. Bolzano, Rein analytischer beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, Prague, 1817, in: Bernard Bolzano (1781–1848). Bicentenary. Early mathematical works. With an introduction by Luboš Nový and Jaroslav Folta, Acta Historiae Rerum Naturalium necnon Technicarum Special Issue, 12. Lecture Notes in Pure and Applied Mathematics, 78. Ceskoslovenské Akademie Ved (CSAV), Prague, 1981, MR 666704, Zbl 0534.01024

F. Cajory, A history of the conception of limits and fluxions in Great Britain from Newton to Woodhouse, Open Court, Chicago and London, 1919, JFM 47.0035.12.

G. Cantor, Ueber die Ausdehnung eines Satzes aus der Theorie der

trigonometrischen Reihen (German), Math. Ann. 5, no. 1(1872), 123-132, MR 1509769, Zbl 04.0101.02

A. L. Cauchy, Algebricheskij analiz (Analyse Algébrique), translated by F. Ewald, V. Grigoriev, A. Iljin, Druck von Bär&Hermann, Leipzig, 1864.

A. L. Cauchy, Course d’Analyse de l’Ecole Royale Politechnique (1821). Analyse Algébrique w: Oeuvres complétes, Series 2, vol. 3., Cambridge Library Collection, Cambridge University Press, Cambridge, 2009, 1—471, MR 2856421, Zbl 1201.01042

A. L. Cauchy, Kratkoje izlozhenije urokov o differenzialnom i integralnom ischislenii (Résumé des leçons données sur le calcul infinitésimal), translation by Bunjakovski, St. Petersburg, 1831.

A. L. Cauchy, Note sur les séries convergentes dont les divers termes sont des fonctions continues d’une variable réelle ou imaginaire, entre des limites données, 1853, CR t. XXXVI, w: Oeuvres complétes, Series 1, vol. 12,Cambridge Library Collection, Cambridge University Press, Cambridge, 2009, p. 30–36, MR 2882785, Zbl 1201.01040

A. L. Cauchy, Résumé des leçons données sur le calcul infinitésimal, 1823, Oeuvres complétes, Series 2, vol. 4, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009, 9–261, MR 2865850, Zbl 1201.01051.

J. M. Child (ed.), The early mathematical manuscripts of Leibniz Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child, The Open Court Publishing Co., Chicago-London, 1920, JFM 47.0035.09.

J. D’Alembert, Limite (1765) w: Encyclopédie méthodique ou par ordre de matiéres, t. II, Padoue, 1789, p. 311–312.

S. Demidov, “Zakon nepreryvnosti” Leibniza i poniatie nepreryvnosti funkcii u Eulera (Leibniz’ law of continuity and the notion of continuous function in Euler), Istoriko-matematicheskie issledovania (Historical-mathematical researches), XXXII-XXXIII, 1990, p. 34–39.

U. Dini, Fondamenti per la teoria delle funzioni di variabili reali, Pisa, 1878.

A. Dorofeeva, Formirovanie poniatija nepreryvnoy funkcii (The formation of the notion of continuous function), Istoria i metodologija estestvennych nauk (History and methodology of natural sciences), Moscow State University, XI (Mathematics and mechanics), Moscow, 1971, p. 37–50.

P. Dugac, Elements d‘analyse de Karl Weierstrass, Paris, 1972.

P. Dugac, Poniatie predela i irrazionalnogo chisla, konzepzii Charles Meray i Karl Weierstrass (The notion of a limit and irrational number, concepts of Charles Méray and Karl Weierstrass), Istoriko matematicheskie issledovania (Historicalmathematical researches), XVIII, 1973, p. 176-180.

L. Euler, Institutiones calcului differentialis (1755) w: L. Euler, Differential calculus, vol. 2, Gostechizdat, Moscow–Leningrad, 1949.

L. Euler, Vvedenije v analis beskonechno malych (Introductio in analysin infinitorum), Moscow: Fismatgis, Vol. II, 1748.

J. B. Fourier, Théorie analitique de la chaleur, Oeuvres, Paris, v. 1, 1822, Zbl 1271.01048.

C. F. Gauss, Grundbegriffe der Lehre von der Reihen (1800), Werke, Leipzig: B. Bd. X/1, 1917, s. 390–394.

J. Grabiner, Who gave you the Epsilon? Cauchy and the Origin of Rigorous Calculus, Am. Math. Mon. 90 (1983), 185–194, MR 691368, Zbl 0517.26003.

I. Grattan–Guinness, Bolzano, Cauchy and the "new analysis” of the early nineteenth century, Arch. History Exact Sci. 6 (1970), no. 5, 372 400, MR 1554135, Zbl 0198.00601.

I. Grattan–Guinness, The mathematics of the past: distinguishing its history from our heritage, Historia Math. 31 (2004), no. 2, 163–185, MR 2055640, Zbl 1063.01023.

J. Gray, Plato‘s ghost. The modern transformation of mathematics, Princeton University Press, Princeton, NY, 2008, Zbl 1166.00005.

H. Hankel, Grenze, Allgemeine Enzyklopädie der Wissenschaften und Künste, Leipzig: Brockhaus, Vol. 90, 1870/71, 185—211.

E. Heine, Die Elemente der Functionenlehre, (German), J. Reine Angew. Math. 74 (1872), 172-–188, MR 1579539, JFM 04.0187.01.

S.-A.-J. L‘Huilier, Exposition Élémentaire des Principles des calcul supérieurs, 1786.

M. Katz, D. Sherry, Leibniz’s laws of continuity and homogeneity Notices Am. Math. Soc. 59, No. 11 (2012), 1550–1558, MR 3027109, Zbl 1284.03064.

T. Koetsier, Lakatos, Lakoff and Núnez: Towards a Satisfactory Definition of Continuity. In Explanation and Proof in Mathematics, Philosophical and Educational Perspectives, edited by G. Hanna, H. Jahnke, H. Pulte, Springer, 2009.

S. F. Lacroix, Traité du calcul differentiel et du calcul intégral, Paris, 1797, 1798, 1800.

S. F. Lacroix, Traité élementaire de calcul différentiel et de calcul intégral, Paris, 1806, 1828.

J. Lagrange, Mécanique analytique, Paris, 2–ed., v. 1.

J. Lagrange, Théorie des fonctions analytique, Oeuvres de Lagrange, v. 9, Paris, 1881.

J. G. Leathem, Volume and Surface Integrals Used in Physics, Cambridge University Press. VI u, 1905, s. 48, JFM 36.0358.01.

H. Lebesgue, Integrirovanie i otyskanije primitivnych funkzij (Leçons sur l’intégration et la recherché des fonctions primitives), Moscow–Leningrad, 1934.

C. Méray, Nouveau précis d’analyse infinitésimale, Par Charles Méray. Publication, F. Savy, XXIII, Paris, 1872, p. 310.

H. Putnam, What is mathematical truth?, Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Math. 2 (1975), no. 4, 529—533, MR 479920.

B. Riemann, Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. t. XIII, 1868, 87–132.

Ph. L. Seidel, Note über eine Eigenschaft der Reihen, welche discontinuirliche Functionen darstellen, Abhandl. Der Math. Phys. Klasse der Kgl. Bayerschen Akademie der Wissenschaften V, München, 1847, 381–394.

G. Sinkevich, Concepts of a Numbers of C. Méray, E. Heine, G. Cantor, R. Dedekind and K. Weierstrass, Technical Transactions, Kraków, 2014, p. 211– 223.

G. Sinkevich, Heinrich Eduard Heine. Teoria funkzij (Heinrich Eduard Heine. Function theory), Matematicheskoye modelirovanie, chislennyie metody i komplexy programm (Mathematical simulation, calculus of approximations and program system), St.-Petersburg, 18(2012), p. 6-46.

G. Sinkevich, K istorii „epsilonistyki” (To the history of epsilonics), Mathematics in Higher Education (Matematika v vyshem obrasovanii), No. 10 (2012), p. 149—166.

G. Sinkevich, Rasvitie poniatija nepreryvnosty u Ch. Méray (The development of notion of continuity in Ch. Méray), Trudy X Miejzdunarodnych Kolmogorovskich chtenij (Proceeding of X Kolmogorov’s reading. Jaroslavl, 2012, p. 180–185.

G. Sinkevich, Uliss Dini I poniatie nepreryvnosti (Uliss Dini and the notion of continuity), Istoria nauki i techniki (The history of science and technics), Moscow, 10(2012), p. 3–11.

G. G. Stokes, On the Critical values of the sums of Periodic Series, Transactions of the Cambridge Philosophical Society, Vol. VIII, p. 533-583.

O. Stolz, Bolzano’s Bedeutung in der Geschichte der Infinitesimalrechnung (German), Math. Ann. 18, no. 2 (1881), 255-–279, MR 1510103, JFM 13.0203.01.

O. Stolz, Vorlesungen über allgemeine Arithmetik: Nach den neueren Ansichten, Bd. I. Leipzig, 1885, p. 156—157.

J. Wallis, The arithmetic of Infinitesimals (1656), translated by J. A. Stedall, USA, Springer, 2004.

K. Weierstrass, Ausgewählte Kapitel aus der Funktionenlehre, Vorlesung gehalten in Berlin 1886 mit der Akademischen Antrittsrede, Berlin 1857 und drei weiteren Originalarbeiten von K. Weierstrass aus den Jahren 1870 bis 1880/86. Teubner- Archiv für mathematic. Band 9, Reprint 1989.

A. Yushkevich, Istoria matematiki (The History of Mathematics), Moscow: Nauka, Vol. 3 (1972).

A. Yushkevich, L. Carnot i konkurs Berlinskoj akademii nauk 1786 na temu o matematicheskoj teorii beskonechnogo (L.Carnot and the competition of Berlin academy of Sciences 1786 on the mathematical theory of infinite), Istorikomatematicheskie issledovania (Historical-mathematical researches), XVIII (1973), p. 132—156.

A. Yushkevich, Chrestomatija po istorii matematiki. Matematicheskij analis (Reading book on the history of mathematics. Analysis), Moscow: Prosveschenije, 1977.

A. Yushkevich, Razvitije ponjatija predela do K. Weierstrassa (The development of the notion of limit till K. Weierstrass),Istoriko matematicheskie issledovania (Historical-mathematical researches), Moscow: Nauka, XXX (1986), p. 1—81.


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