The Sense of Proportion in Euclid
A general seminar talk at Gebze Institute of Technology, Friday, May 8, 2015, at 3:00 p.m. I did not use slides, but wrote on the whiteboards. Here are my notes:
A proportion is an identification of ratios. In the Elements, Euclid (c. 300 b.c.e.) gives two definitions of a proportion: a clear definition for arbitrary magnitudes, and an unclear definition for numbers. A positive real number, as defined by Richard Dedekind (1831–1916), can be understood as a ratio of magnitudes in Euclid's sense. However, unlike Euclid, Dedekind establishes the existence of all of the so-called real numbers: this has been overlooked, at least by some of Dedekind's contemporaries. It has also been thought that Euclid's ratios of numbers are just fractions in the modern sense; but this makes Euclid wrong in ways that he is not likely to be wrong. Euclid is more careful than we often are today with the foundations of number theory. He proves rigorously that in an ordered ring whose positive elements are well-ordered, multiplication is commutative. Seeing this can be helped by treating the reading of Euclid as an instance of doing history: history in the sense worked out by the philosopher and historian R. G. Collingwood (1889–1943) in several of his books.