turkmath.org

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 book