# Descartes as Model Theorist

This was a talk in the Istanbul Model Theory Seminar, May 16, 2013. Revised in April, 2016, my notes for the talk are now 35 pages (size A5, 12-point type, dated April 19):

Some of the ideas were continued in Spaces and Fields,

April 2016.

For the record, the original notes are at the bottom of this page. Meanwhile, the new introduction says in part:

The present document is an exposition of three related topics:

- The interpretation of a field in a Euclidean plane, which allowed Descartes to create what we call analytic geometry. Descartes did not supply the details, but we can do it by Pappus’s Hexagon Theorem, as well as other means.
- The interpretation of the scalar field in a vector space (of dimension at least 2) over the field by means of par- allelism. This leads to the model-theoretic results of the paper “Model-theory of vector-spaces over an unspecified field.”
- The interpretability, by existential formulas, of fields with several derivations in certain Lie rings equipped with a group endomorphism. This leads, through the paper “Fields with several commuting derivations,” to the existence of model-complete theories of Lie rings with group endomorphism.
I alluded to the last item at [§1, p. 424, of the vector-spaces paper]; but the present document is so far the most thorough account.

The following abstract is reproduced in the notes, as is a transcript of the talk itself.

## Abstract

In his *Geometry* of 1637, Rene Descartes gave a geometric
justification of algebraic manipulations of symbols. He did this by
interpreting a field in a vector-space with a notion of parallelism.
At least this is how we might describe it today. I alluded to this in
the abstract for my February 28 seminar, but did not actually talk
about it. Now I want to talk about it.

By fixing a unit, Descartes defines the product of two line segments as another segment. He relies on a theory of proportion for this. Presumably this is the theory developed in Book V of Euclid's Elements—the theory that inspired Dedekind's definition of real numbers as “cuts” of rational numbers.

But this theory has an “Archimedean” assumption: for any two given segments, some multiple of the smaller exceeds the larger.

In fact this assumption is not needed, as Hilbert observed in
*Foundations of Geometry.* Hilbert uses instead Pappus's
Theorem. This work may be known now as “interpreting a field in a
projective plane”.

I tracked down Pappus's original argument (from the 4th century), and yesterday I wrote an account of it on Wikipedia.

As for model theory, another result that comes out of these considerations is that there are model-complete theories of Lie-rings equipped with an endomorphism of the abelian-group-structure.

## Notes

The following documents been superseded by the version given above:

16 pp., A5 paper (I said something about all sections of the notes, mainly in the order 2, 3, 1):

In fact the foregoing notes have been edited and corrected from the following version, posted the day after the seminar (also 16 pp., A5 paper):