From Euclid's Elements to the methodology of mathematics. Two ways of viewing mathematical theory

Piotr Błaszczyk

Abstract


We present two sets of lessons on the history of mathematics designed for prospective teachers: (1) Euclid's Theory of Area, and (2) Euclid's Theory of Similar Figures. They aim to encourage students to think of mathematics by way of analysis of historical texts. Their historical content includes Euclid's Elements, Books I, II, and VI. The mathematical meaning of the discussed propositions is simple enough that we can focus on specific methodological questions, such as (a) what makes a set of propositions a theory, (b) what are the specific objectives of the discussed theories, (c) what are their common features.
In spite of many years' experience in teaching Euclid's geometry combined with methodological investigations, we cannot offer any empirical findings on how these lectures have affected the students' views on what a mathematical theory is. Therefore, we can only speculate on the hypothetical impact of these lectures on students.

Keywords


Elements, theory, theorem, triangulation, similar figures, Pythagorean theorem

References


Baldwin, J.: 2018, Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert, Philosophia Mathematica 26, Issue 3, October 2018, 346–374.

Barwise, J.: 1999, Handbook of Mathematical Logic, North Holland, Amsterdam.

Błaszczyk, P., Mrówka, K.: 2013, Euclid, Elements, Books V–VI. Translation and commentary, Copernicus Center Press, Kraków, [In Polish].

Fitzpatrick, R.: 2007, Euclid’s Elements of Geometry. Edited and provided with modern English translation, by Richard Fitzpatrick, http://farside.ph.utexas.edu/.

Hilbert, D.: 1922, Neubegründung der Mathematik. Erste Mitteillung, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 1, 157–177.

Hilbert, D.: 1970, Foundation of Geometry, translated by L. Unger. Open Court, La Salle, Illinois.

Maor, E.: 2007, The Pythagorean Theorem: A 4,000-Year History, Princeton University Press, Princeton, New Jersey.

O’Leary, M.: 2010, Revolutions of Geometry, Willey, New Jersey.

Polya, G.: 1957, How to Solve It, Princeton University Press, Princeton.


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