Dydak’s axiomatization of Euclidean geometry

Jerzy Dydak sent me an interesting paper a couple of months ago, which you can find on his website here.   Entitled “A Topological Approach to the Axiomatization of Geometry”, it proposes a new way of building Euclidean plane geometry from he ground up.

Readers will know that although Euclid was the first to propose an axiomatic foundation for geometry, his axiom system is not precise enough by modern standards.  In the 19th century Hilbert gave a complete system of axioms for geometry, and other mathematicians followed in his steps.  In 1932 G.D.Birkhoff published a paper called A system of axioms for plane geometry based on scale and protractor.    This appeared in the Annals of Mathematics!  Birkhoff’s innovation was to assume the real numbers as given: his axioms stated that certain geometric quantities could be “measured” by real numbers.   This was of course quite different from Euclid’s approach, where the “theory of proportion”, equivalent to what we would call today the theory of the real numbers, was developed as part of geometry (Elements, book 5).   Birkhoff’s approach has been followed by many later writers of textbooks (such as my own Elementary Geometry for example).

Dydak wants to get back to the Euclidean order of business where the real numbers are developed concurrently with the foundations of geometry.  The fundamental undefined notion in his theory is the ternary relation of betweenness (as in “C is between A and B”).   This allows one to define line segments, rays, and so on.  What would conventionally be called “the completeness of the real numbers” is expressed in terms of connectedness and compactness properties of line segments (themselves defined in terms of betweenness, of course); and the “arithmetic” of real numbers appears as the result of adjoining and subdividing line segments, very much in the Euclidean manner.

It would be interesting to try teaching geometry this way.  You spend a lot of time thinking about one-dimensional geometry in this development; but, as a bonus, you understand the foundations of the real number system in an intuitively appealing way.  Dydak also suggests that the concept of lines as defined by an abstract “betweenness” might help make models of other geometries, such as the Klein model or Poincare model of hyperbolic geometry, less mysterious: no more “we are going to call this a ‘line’ even though it doesn’t look like one.”

 

Contract signed for “Winding Around”

So I signed the contract last week for “Winding Around”, my book based on the course I taught in the MASS geometry/topology track last year.  It will appear in the American Mathematical Society’s Student Mathematical Library series, and the manuscript is due to be delivered to them on April 1st – I leave it to you whether or not you think this is an auspicious day!   The book centers around the notion of “winding number” (hence “Winding Around”) and uses that as a peg on which to hang a variety of topics in geometry, topology and analysis — finishing up, in the final chapter, with the Bott periodicity theorem considered as one possible high-dimensional generalization of the winding number notion.

The intended audience is an undergraduate one (there was skepticism from some of the AMS readers about this, but I told them the MASS students made it through okay!) and the tone is, I hope, entertaining and discursive.  As I say in the introduction, “Winding around is a description of the book’s methodology as well as of its subject-matter.”

 

Metric approach to limit operators VI

I thought I’d finished this sequence of posts with number five, but then I spent a little time talking with Jerry Kaminker and Rufus Willett and I think that I understood two things: first, how to formulate the limit point construction more cleanly and, second, the “symmetry breaking” role of the ultrafilters which is not clear in what I had written so far.  Read on. Continue reading

Metric approach to limit operators V

In the previous post I sketched out the condensation of singularities argument which finishes the proof under the assumption that the underlying metric space \(X\) is a group.  In this case all limit operators act on the same Hilbert space, namely \(\ell^2(X)\), and the weak compactness of the set of all limit operators plays a critical role.

In the more general situation described by Spakula and Willett, each limit operator (say at a boundary point \(\omega\)) acts on its own Hilbert space \(\ell^2(X(\omega))\).   In order to bring this situation under sufficient control to continue to make the weak compactness argument, we are going to need some kind of bundle theory. Continue reading