The (R) evolution of the limit in differential and integral calculus
DOI:
https://doi.org/10.56162/transdigital403Keywords:
mathematical analysis, historical development of limits, mathematics education, history of calculus, pedagogy of limitsAbstract
Throughout the 18th and 19th centuries, some people considered that the fundamental changes produced in mathematical and geometric analysis were disguised by expressions such as limit or function, by mathematicians. Therefore, the concept of limit comes from the ancient reasoning of operational equality as allowing from the definition of a variable a certain scope of mathematical maneuvers that affect an unknown process but always leading from n to d using a metaphorical expression that contains a gnoseological background. In our opinion, this way of proceeding conceals a terminological ambiguity that leads to a misleading interpretation. On the one hand, you have an endless procession, but on the other, you always try to get close to a number on the limit. It is seen that there seems to be a resistance with the notion of infinitesimal, especially from a gnoseological perspective and with implications in teaching.
References
Burscheid, H., & Struve, H. (2002). Die Integralrechnung von Leibniz–eine Rekonstruktion. Studia Leibnitiana, 34(2), 127-160.
Carmine, B. (2022). Cauchy an Innovative Mathematician: the Fundamentals of Infinitesimal Calculus, the Theorem of Finite Increments and the Functions of an “Imaginary” Variable. International Journal of Advanced Engineering and Management Research, 7(2), 187–195. https://doi.org/10.51505/ijaemr.2022.7215
Cauchy, A. L. (1821). Cours d’analyse de l’Ecole royale polytechnique; par m. Augustin-Louis Cauchy ... 1.re partie. Analyse algébrique. de l’Imprimerie royale.
Choudary, A. D. R., & Niculescu, C. P. (2014). Differential Calculus on R. Real Analysis on Intervals, 215–280.
Coltuc, D. (1995). Curve generation on discrete lattices. Vision Geometry IV, 2573, 137–147.
De Fabia Abreu, L. A., De Almeida, A. M. F. B., Ferreira, M. L., De Oliveira, C. A. A., & Schubring, G. (2020). The history of mathematics in the texbooks of Cajori, Eves, Boyer, and Struik. Revista Brasileira de Historia da Ciencia, 13(2). https://doi.org/10.53727/rbhc.v13i2.39
De Faria Rezende, B. L. (2023). Teoremas limite para variáveis aleatórias de Bernoulli dependentes [Tesis de doctorado, Universidad de São Paulo]. https://doi.org/10.11606/T.104.2023.TDE-24082023-084945
Duchesneau, F. (1989). Leibniz and the Philosophical Analysis of Science. Studies in Logic and the Foundations of Mathematics, 126, 609–624.
Earl, R. (2023). All change… the calculus of Fermat, Newton, and Leibniz. En R. Earl (Ed.), Mathematical Analysis: A Very Short Introduction (pp. 17–38). Oxford University Press.
Edwards, H. M. (2007). Euler´s definition of the derivative. Bulletin of the American Mathematical Society, 44(4), 575–580.
Erlichson, H. (1999). Johann Bernoulli’s brachistochrone solution using Fermat’s principle of least time. European Journal of Physics, 20(5), 299. https://doi.org/10.1088/0143-0807/20/5/301
Euler, L. (1755). Institutiones Calculi differentialis. Petropolis. https://archive.org/details/bub_gb_sYE_AAAAcAAJ/page/n13/mode/2up
Grabinsky, G. (2007). Euler, El Prestidigitador de las Series. Miscelánea Matemática (45), 55-66.
Kutateladze, S. S. (2007). Excursus into the history of calculus. arXiv. https://arxiv.org/abs/math/0701068v1
Nauenberg, M. (2011). Barrow and Leibniz on the fundamental theorem of the calculus. arXiv. http://arxiv.org/abs/1111.6145
Ng, N.-F., & Yeung, S.-K. (2020). Limit of Weierstrass Measure on Stable Curves. arXiv. http://arxiv.org/abs/2012.09642
Parker, A. E. (2013). Who Solved the Bernoulli Differential Equation and How Did They Do It? The College Mathematics Journal, 44(2), 89–97.
Penagos Vega, C. (2013). Aportes realizados por Leibniz a la consolidación del Cálculo Diferencial. Universidad Pedagógica Nacional.
Pérez-Fernández, F. J., & Aizpuru, A. (1999). El Cours d’Analyse de Cauchy. SUMA, 30, 5–25. https://revistasuma.fespm.es/sites/revistasuma.fespm.es/IMG/pdf/30/005-026.pdf
Perkins, D. (2012). Calculus and Its Origins. Mathematical Association of America, Inc.
Pradhan, J. B. (2020). Artefatos culturais como uma metáfora para comunicação de ideias matemáticas. Revemop, 2, e202015. https://doi.org/10.33532/revemop.e202015a
Sezgin Memnun, D., Ayd?n, B., Özbilen, Ö., & Erdo?an, G. (2017). The abstraction process of limit knowledge. Kuram ve Uygulamada Egitim Bilimleri, 17(2), 345–371. https://doi.org/10.12738/estp.2017.2.0404
Vargas Elizondo, C., (2008). Euler: su contexto matemático, filosófico y científico. Ingeniería. Revista de la Universidad de Costa Rica, 18(1-2), 111-119.
Wang, N. L., Agarwal, P., & Kanemitsu, S. (2020). Limiting values and functional and difference equations. Mathematics, 8(3), 407. https://doi.org/10.3390/math8030407
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