# Geometry as made rigorous by Euclid and Descartes

A general seminar talk, October 31, 2013.

- Abstract
For Immanuel Kant (born 1724), the discovery of mathematical proof by Thales of Miletus (born around 624 BCE) is a revolution in human thought. Modern textbooks of analytic geometry often seem to represent a return to prerevolutionary times. The counterrevolution is attributed to René Descartes (born 1596). But Descartes understands ancient Greek geometry and adds to it. He makes algebra rigorous by interpreting its operations geometrically.

The definition of the real numbers by Richard Dedekind (born 1831) makes a rigorous converse possible. David Hilbert (born 1862) spells it out: geometry can be interpreted in the ordered field of real numbers, and even in certain countable ordered fields.

In modern textbooks, these ideas are often entangled, making the notion of proof practically meaningless. I propose to disentangle the ideas by means of Book I of Euclid's

*Elements*and Descartes's*Geometry.*- Introductory notes
- Discussed before the slides, 8 pp., size A5 paper, dated October 31, 2013
- Slides used in talk
- ii + 23 + 3 pp., size A6 paper, landscape orientation, dated October 31, 2013
- Original slides
- Too many for talk: ii + 42 + 3 pp., size A6 paper, landscape orientation, dated October 31, 2013
- Related report
Analytic geometry

- Related article
Model-theory of vector-spaces over unspecified fields

(2009; more on the technicalities of interpretations, building on Descartes's interpretation of a field)