What enabled the production of mathematical knowledge in complex analysis?

José Gerardo Piña-Aguirre; Rosa María Farfán Márquez

doi:10.29333/iejme/12996

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José Gerardo Piña-Aguirre ¹ ^* , Rosa María Farfán Márquez ¹

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¹ Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav-IPN), México City, MEXICO^* Corresponding Author

Abstract

With the objective of identifying intrinsic forms of mathematical production in complex analysis (CA), this study presents an analysis of the mathematical activity of five original works that contributed to the development of Cauchy’s integral theorem. The analysis of the mathematical activity was carried out through the identification of the types of expressions used and the way they were used by the historical subjects when communicating their results, to subsequently identify transversal elements of knowledge production. The analysis was refined by the notion of confrontation, which depicts the development of mathematical knowledge through the idea of building knowledge against previous knowledge. As a result of the study we established epistemological hypothesis, which are conceived as conjectures that reveal ways in which mathematical knowledge was generated in CA.

Keywords

mathematics education
epistemological analysis
complex analysis
qualitative documentary research
epistemological hypothesis
confrontation

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 18, Issue 2, May 2023, Article No: em0734

https://doi.org/10.29333/iejme/12996

Publication date: 01 Apr 2023

Online publication date: 25 Feb 2023

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References

Bachelard, G. (1934). La formation de l’esprit scientifique: Contribution a une psychanalyse de la connaissance objective [The formation of the scientific mind: Contribution to a psychoanalysis of objective knowledge]. Librarie Philosophique J. Vrin.
Bak, J., & Popvassilev, S. (2017). The evolution of Cauchy’s closed curve theorem and Newman’s simple proof. The American Mathematical Monthly, 124(3), 217-231. https://doi.org/10.4169/amer.math.monthly.124.3.217
Bottazzini, U., & Gray, J. (2013). Hidden harmony–Geometric fantasies. The rise of complex function theory. Springer. https://doi.org/10.1007/978-1-4614-5725-1
Bottazzini, U., & Tazzioli, R. (1995). Naturphilosophie and its role in Riemann’s mathematics. Revue d’Histoire des Mathématiques [Journal of Mathematics History], 1(1), 3-38.
Bråting, K., & Pejlare, J. (2015). On the relations between historical epistemology and students’ conceptual developments in mathematics. Educational Studies in Mathematics, 89(2), 251-265. https://doi.org/10.1007/s10649-015-9600-8
Briot, C., & Bouquet, J. (1875). Théorie des fonctions elliptiques. Parigi.
Brousseau, B. (2006). Theory of didactical situations in mathematics: Didactique des mathématiques [Mathematics didactics], 1970-1990. Kluwer.
Cantoral, R., & Farfán, R. (2004). La sensibilité à la contradiction: Logarithmes de nombres négatifs et origine de la variable complexe [Sensitivity to contradiction: Logarithms of negative numbers and origin of the complex variable]. Recherches en Didactique des Mathématiques [Research in Didactics of Mathematics], 24(2-3), 137-168.
Cantoral, R., Farfán, R., Hitt, F., & Rigo, M. (1987). Epistemología del concepto de función logarítmica [Epistemology of the concept of logarithmic function]. Matemática Educativa, Revista del Programa Nacional de Formación y Actualización de Profesores de Matemáticas [Educational Mathematics, Magazine of the National Program for the Training and Updating of Mathematics Teachers], 1(1), 11-17.
Cauchy, A. (1814). Mémoire sur les intégrales définies [Memory on definite integrals]. Gallica-Math: Œuvres Completès.
Cauchy, A. (1825). Mémoire sur les intégrales définies, prises entre des limites imaginaires [Dissertation on definite integrals, taken between imaginary limits]. Gallica-Math: Œuvres Completès.
Cauchy, A. (1849). Mémoire sur les quantités géométriques [Memory on geometric quantities]. Gallica-Math: Œuvres Completès.
Cauchy, A. (1853a). Mémoire sur les fonctions des quantités géométriques [Memory on the functions of geometric quantities]. Gallica-Math: Œuvres Completès.
Cauchy, A. (1853b). Mémoire sur les fonctions continues des quantités algébriques ou géométriques [Dissertation on continuous functions of algebraic or geometric quantities]. Gallica-Math: Œuvres Completès.
Cid, E. (2015). Obstáculos epistemológicos en la enseñanza de los números negativos [Epistemological obstacles in teaching negative numbers] [Doctoral thesis, Universidad de Zaragoza].
D’Azevedo-Breda, A., & Dos Santos, J. (2021). Learning complex functions with GeoGebra. Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales [Journal of the Colombian Academy of Exact, Physical and Natural Sciences], 45(177), 1262-1276. https://doi.org/10.18257/raccefyn.1504
De Toffoli, S., & Giardino, V. (2015). An inquiry into the practice of proving in low-dimensional topology. In G. Lolli, M. Panza, & G. Venturi (Eds.), From logic to practice Italian studies in the philosophy of mathematics (pp. 315-336). Springer. https://doi.org/10.1007/978-3-319-10434-8_15
Dittman, M., Soto-Johnson, H., Dickinson, S., & Harr, T. (2016). Game building with complex-valued functions. PRIMUS, 27(8-9), 869-879. https://doi.org/10.1080/10511970.2016.1234527
Farfán, R. (2012). Socioepistemología y ciencia. El caso del estado estacionario y su matematización [Socioepistemology and science. The steady state case and its mathematization]. Gedisa.
Garcia, S., & Ross, W. (2017). Approaching Cauchy’s theorem. PRIMUS, 27(8-9), 758-765. https://doi.org/10.1080/10511970.2016.1234525
Gascón, J. (1993). Desarrollo del conocimiento matemático y análisis didáctico: Del patrón análisis-síntesis a la génesis del lenguaje algebraico [Development of mathematical knowledge and didactic analysis: From the analysis-synthesis pattern to the genesis of algebraic language]. Recherches en Didactique des Mathématiques [Research in Didactics of Mathematics], 13(3), 295-332.
Gascón, J. (2014). Los modelos epistemológicos de referencia como instrumentos de emancipación de la didáctica y la historia de las matemáticas [The epistemological models of reference as instruments of emancipation of didactics and the history of mathematics]. Educación Matemática [Mathematics Education], 25, 99-123.
Goursat, E. (1884). Démonstration du théoréme de Cauchy: Extrait d’une lettre adressée à M. Hermite [Proof of Cauchy’s theorem: Extract from a letter addressed to M. Hermite]. Acta Mathematica [Journal of Mathematics], 4, 197-200. https://doi.org/10.1007/BF02418419
Goursat, E. (1900). Sur la definition generale des fonctions analytiques, d’apres Cauchy [On the general definition of analytic functions, according to Cauchy]. Transactions of the American Mathematical Society, 1(1), 14-16. https://doi.org/10.1090/S0002-9947-1900-1500519-7
Grattan-Guinness, I. (2004). History or heritage? An important distinction in mathematics and for mathematics education. The American Mathematical Monthly, 111(1), 1–12. https://doi.org/10.1080/00029890.2004.11920041
Gray, J. (2000). Goursat, Pringsheim, Walsh, and the Cauchy integral theorem. The Mathematical Intelligencer, 22(4), 60-66. https://doi.org/10.1007/BF03026773
Hanke, E. (2020). Intuitive mathematical discourse about the complex path integral. In T. Hausberger, & M. Bosch (Eds.), Proceedings of the 3^rd Conference of the International Network for Didactic Research in University Mathematics (pp. 102-111). University of Bizerte and INDRUM.
Martínez, F. (2017). Caracterización del pensamiento matemático de alumnos y alumnas de ingeniería, relativo al origen de la variable compleja. El caso de logaritmos de números negativos [Characterization of the mathematical thought of engineering students, relative to the origin of the complex variable. The case of logarithms of negative numbers] [Master’s thesis, Cinvestav].
Moore, E. (1900). A simple proof of the fundamental Cauchy-Goursat theorem. Transactions of the American Mathematical Society, 1(4), 499-506. https://doi.org/10.1090/S0002-9947-1900-1500551-3
Nahin, P. (1998). An imaginary tale: The story of -1 . Princeton University Press.
Needham, T. (1997). Visual complex analyisis. Oxford University Press.
Neuenschwander, E. (1981). Studies in the history of complex function theory II: Interactions among the french school, Riemann and Weierstrass. Bulletin of the American Mathematical Society, 5(2), 87-105.https://doi.org/10.1090/S0273-0979-1981-14923-5
Oehrtman, M., Soto-Johnson, H., & Hancock, B. (2019). Experts’ construction of mathematical meaning for derivatives and integrals of complex-valued functions. International Journal of Research in Undergraduate Mathematics Education, 5(3), 394-423. https://doi.org/10.1007/s40753-019-00092-7
Papadopoulos, A. (2017). Cauchy and Puiseux: Two precursors of Riemann. In L. Ji, A. Papadopoulos, & S. Yamada (Eds.), From Riemann to differential geometry and relativity (pp. 209-232). Springer. https://doi.org/10.1007/978-3-319-60039-0_7
Piña-Aguirre, J. G., & Farfán, R. (2022). Sobre los procesos de demostración y el contexto de producción de la memoria sur les intégrales définies de Augustin-Louis Cauchy [On the processes of demonstration and the context of production of the memory on definite integrals by Augustin-Louis Cauchy]. Revista de Historia Da Educação Matemática [Magazine of History of Mathematics Education], 8, 1-22.
Ponce, J. (2019). The use of phase portraits to visualize and investigate isolated singular points of complex functions. International Journal of Mathematical Education in Science and Technology, 50(7), 999-1010. https://doi.org/10.1080/0020739X.2019.1656829
Puiseux, V. (1850). Recherches sur les fonctions algébriques [Research on algebraic functions]. Gallica-Math: Œuvres Completès.
Puiseux, V. (1851). Nouvelles recherches sur les fonctions algébriques [New research on algebraic functions]. Gallica-Math: Œuvres Completès.
Remmert, R. (1991). Theory of complex functions. Springer. https://doi.org/10.1007/978-1-4612-0939-3
Riemann, B. (1851). Grundlagen für eine allgemeine theorie des functionen einer veränderlichen complexen grösse [Foundations for a general theory of the function of a variable complex quantity] [Doctoral thesis, Göttingen].
Smithies, F. (1997). Cauchy and the creation of complex function theory. Cambridge University Press. https://doi.org/10.1017/CBO9780511551697
Soto, H., & Oehrtman, M. (2022). Undergraduates’ exploration of contour integration: What is accumulated? The Journal of Mathematical Behavior, 66, 1-19. https://doi.org/10.1016/j.jmathb.2022.100963
Soto-Johnson, H. (2014). Visualizing the arithmetic of complex numbers. International Journal for Technology in Mathematics Education, 21(3), 103-114.
Soto-Johnson, H., & Hancock, B. (2019). Research to practice: Developing the amplitwist concept. PRIMUS, 29(5), 421-440. https://doi.org/10.1080/10511970.2018.1477889
Soto-Johnson, H., & Troup, J. (2014). Reasoning on the complex plane via inscriptions and gesture. The Journal of Mathematical Behavior, 36, 109-125. https://doi.org/10.1016/j.jmathb.2014.09.004
Soto-Johnson, H., Hancock, B., & Oehrtman, M. (2016). The interplay between mathematicians’ conceptual and ideational mathematics about continuity of complex-valued functions. International Journal of Research in Undergraduate Mathematics Education, 2(3), 362-389. https://doi.org/10.1007/s40753-016-0035-0
Troup, J., Soto-Johnson, H., Karakok, G., & Diaz, R. (2017). Developing students’ geometric reasoning about the derivative of complex valued functions. Digital Experiences in Mathematics Education, 3(3), 173-205. https://doi.org/10.1007/s40751-017-0032-1
Weil, A. (1979). Riemann, Betti and the birth of topology. Archive for History of exact Sciences, 20, 91-96. https://doi.org/10.1007/BF00327626

How to cite this article

APA

Piña-Aguirre, J. G., & Farfán Márquez, R. M. (2023). What enabled the production of mathematical knowledge in complex analysis?. International Electronic Journal of Mathematics Education, 18(2), em0734. https://doi.org/10.29333/iejme/12996

Vancouver

Piña-Aguirre JG, Farfán Márquez RM. What enabled the production of mathematical knowledge in complex analysis?. INT ELECT J MATH ED. 2023;18(2):em0734. https://doi.org/10.29333/iejme/12996

AMA

Piña-Aguirre JG, Farfán Márquez RM. What enabled the production of mathematical knowledge in complex analysis?. INT ELECT J MATH ED. 2023;18(2), em0734. https://doi.org/10.29333/iejme/12996

Chicago

Piña-Aguirre, José Gerardo, and Rosa María Farfán Márquez. "What enabled the production of mathematical knowledge in complex analysis?". International Electronic Journal of Mathematics Education 2023 18 no. 2 (2023): em0734. https://doi.org/10.29333/iejme/12996

Harvard

Piña-Aguirre, J. G., and Farfán Márquez, R. M. (2023). What enabled the production of mathematical knowledge in complex analysis?. International Electronic Journal of Mathematics Education, 18(2), em0734. https://doi.org/10.29333/iejme/12996

MLA

Piña-Aguirre, José Gerardo et al. "What enabled the production of mathematical knowledge in complex analysis?". International Electronic Journal of Mathematics Education, vol. 18, no. 2, 2023, em0734. https://doi.org/10.29333/iejme/12996