2000 character limit reached
The paradoxes and the infinite dazzled ancient mathematics and continue to do so today (2401.02806v1)
Published 5 Jan 2024 in math.HO and cs.LO
Abstract: This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were handled arithmetically and the continuous magnitude quantities which were handled geometrically. We look at how approximations and mixed numbers (whole numbers with fractions) helped develop the arithmetization of geometry and the development of mathematical analysis and real numbers.
- George Berkeley. The Analyst: or A Discourse Addressed to an Infidel Mathematician. First printed in 1734. In A. A. Luce and T. E. Jessop, editors, The Works of George Berkeley Bishop of Cloyne, volume 4, pages 53–102. Nelson, London, 1951.
- Heath. The 13 Books of Euclid’s Elements. Dover, 1956.
- David Hilbert. The Foundations of Geometry. The Open Court Publishing Co, 1902.
- A Primer of Mathematical Analysis and the Foundations of Computation. College publications, ISBN 978-1-84890-443-9, October 2023. 434 pages.
- W. R. Knorr. The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. Reidel, Dordrecht and Boston and London, 1975.
- Roshdi Rashed. The development of Arabic mathematics: between arithmetic and algebra. Boston Studies in the Philosophy and History of Science (BSPS, volume 156). 1994.
- Roshdi Rashed. Entre arithmétique et algèbre: Recherches sur l’histoire des mathématiques arabes. Ouvrage publié avec le concours de l’Unesco. Société d’édition ”Les Belles Lettres”. Paris, 1984.