University Seminar- Computability, Complexity, and Algebraic Structure-The Area of a Rectangle: Euclid, Eudoxus, and the Earliest Construction of the Real Numbers

Time: Wednesday, December 17, 9:00–10:00 am

Place: Rome Hall, Room 352

Speaker:  Keshav Srinivasan, GWU

Title: The Area of a Rectangle: Euclid, Eudoxus, and the Earliest Construction of the Real Numbers

Abstract:  In calculus and analysis classes, it is often stated that Riemann integral and Lebesgue measure are how an area is defined.  This is only partially true, however; they presuppose the fact that the area of a rectangle is length times width.  This fact is often taken for granted, but the proof is considerably more complicated than it appears.  We will discuss the Eudoxian theory of proportions, an ancient Greek definition of geometrical ratios that gave rise two millennia later to the Dedekind Cut construction of the real number system.  We will go over Euclid's proof of the area of a rectangle using the Eudoxian theory of proportions.  We will make use of logician Dana Scott's rewriting of Euclid's arguments into the modern language of mathematics.  In the final part of the talk, we will discuss neologicist attempts to use the Eudoxian theory of proportions to revitalize Gottlob Frege's foundational work on defining the real numbers.