Ratio Then and Now
An invited talk at Model Theory and Mathematical Logic: Conference in honor of Chris Laskowski’s 60th birthday, University of Maryland, College Park, June 21–3, 2019
Title: Ratio Then and Now.
“Heraclitus holds that the findings of sense-experience are untrustworthy, and he sets up reason [logos, ratio] as the criterion” (Sextus Empiricus)
“It is necessary to know that war is common and right is strife [eris] and all things happen by strife and necessity” (Heraclitus, according to Origen)
1. Strife has arisen between the historian of mathematics and the mathematician who thinks about the past. One must be both, to understand Euclid’s obscure definition of proportion of numbers. Proportion is sameness of ratio. When this occurs between two pairs of numbers, something should be the same about each pair. In Book VII of the Elements, this can only mean that the Euclidean Algorithm has the same steps when applied to either pair of numbers. From this, despite modern suggestions to the contrary, Euclid has rigorous proofs, not only of what we call Euclid’s Lemma, but also of the commutativity of multiplication.
2. Apollonius of Perga gives three ways to characterize a conic section: (i) an equation, involving a latus rectum, that we can express in Cartesian form; (ii) the proportion whereby the square on the ordinate varies as the abscissa or product of abscissas; (iii) an equation of a triangle with a parallelogram or trapezoid. The latter equation holds in an affine plane. With the advent of Cartesian methods in 1637, the equation seems to have been forgotten, because it is not readily translated into the lengths (symbolized by single minuscule letters) that Descartes has taught us to work with. With the affine equation, Apollonius can give a proof-without-words of what today we consider a coordinate change, performed with more or less laborious computations.
3. By interpreting the field where algebra is done in the plane where geometry is done, Descartes does inspire new results. An example still builds on work of an ancient mathematician, Pappus of Alexandria. The model companion of the theory of Pappian affine spaces of unspecified dimension, considered as sets of points with ternary relation of collinearity and quaternary relation of parallelism, is the theory of Pappian affine planes over algebraically closed fields.
- size A5, 12 point type