The paradoxes and the infinite dazzled ancient mathematics and continue to do so today
Fairouz Kamareddine, Jonathan Seldin
TL;DR
The paper surveys how ancient Greek mathematics confronted infinity by maintaining two numerical cultures: discrete numbers and continuous magnitudes, and traces the path toward real numbers and analysis. It traces the progression from natural numbers to $Z$ and $Q$, and onward to the real numbers $R$, through constructions, approximations, and geometric reasoning that underpin the arithmetisation of geometry. Key themes include anthyphairesis, the method of exhaustion, and the debate over infinitesimals, culminating in the formalization of the real-number system via completeness. The historical perspective clarifies the foundations of analysis and computability, revealing how ancient methods underlie modern mathematical rigor.
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.