# Euclid Seminar

Seminars on Euclid’s *Elements* in a world literally made of triangles. Euclid’s text is the wellspring of both geometry and logic and we are interested in both.

Why read Euclid?

**For a geometer** Euclid is the historical root of their subject, and from Hilbert and Riemann to Grothendieck and Deligne many important modern geometers have drawn inspiration from a close analysis of the classic work.

**For a logician** the *Elements* is perhaps the most influential example of a coherent body of mathematical proof, and it continues to be interesting from the point of view of modern logic (see *geometric logic* and their classifying topoi, as in the work of Avigad-Dean-Mumma cited below).

**For a learning theorist** the process by which our perceptual experience of space has been distilled into the axioms and deductions of the *Elements* is a key example of the capacity of learning systems to arrive at abstract mathematics as a result of a general quest to “fit the true distribution of experience”.

From Hilbert’s “Foundations of geometry”.

Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry. The choice of the axioms and the investigation of their relations to one another is a problem which, since the time of Euclid, has been discussed in numerous excellent memoirs to be found in the mathematical literature. This problem is tantamount to the logical analysis of our intuition of space.

Here is an extract from R. Hartshorne’s “Geometry: Euclid and Beyond”.

It is for this logical structure, perhaps even more than for its mathematical content, that Euclid’s

Elementsis famous. Theaxiomatic methodof sequential logical deduction, starting from a small number of initial definitions and assumptions, has become the basic structure of all subsequent mathematics. Euclid’sElementsis the first great example of this method.

From J. Avigad, E. Dean, J. Mumma “A formal system for Euclid’s elements”

On the surface, it might seem that there is a straightforward cognitive explanation as to why some of Euclid’s diagrammatic inferences are basic to geometric practice, namely, that these inferences rely on spatial properties that are “hardwired” into our basic perceptual faculties. In other words, thanks to evolution, we have very good faculties for picking out edges and surfaces in our environment and inferring spatial relationships; and these are the kinds of abilities that are needed to support diagrammatic inference.

O’Keefe and Nadel in “The hippocampus as a cognitive map” on Kant:

His resolution of these problems, reached in the Dissertation of 1770 (cf. Handyside 1928) and the Critique of pure reason (1787), was that space was indeed absolute, but that it was not a property of the physical world. Rather, it was an innate organizing principle of the mind, by which the sensations derived from the physical world were constructed into a conscious manifold.

Space was a way of perceiving, not a thing to be perceived.

## Resources

- Bryne’s Euclid (link).

## Difficulties

- See p.32 of Langlands.