The Larson Research Center

Peter Woit’s current entry is entitled King of Infinite Space, and it’s about H.S.M. Coxeter and the new biography about him, by Siobhan Roberts. I’m definitely going to read Roberts’ book. I hope she gives the correct pronunciation of his name in it. If not, can someone here give me a clue as to how to pronounce his name that doesn’t sound obscene?

I’d also like to observe that, unlike Peter, Danny, and others who have commented on Peter’s post, I had no interest in these things as a child. My interests were in sports (marbles, tetherball, kickball, softball, basketball, football) and airplanes. I loved aviation, and daydreamed about flying in middle school, when these guys were thinking of polytopes!

I vaguely remember being taught something about sets, I think, but my seventh grade algebra teacher told me that I probably ought not to consider a career in mathematics. I sort of had a mild version of the “crying jags” over the meaning of the “two roots” of the number one, and the necessity for the imaginary number it conjured up, only my problem wasn’t trying to imagine negative cabbages, but just trying to understand what the statement of the problem meant. I couldn’t understand how to relate the concepts of “roots,” the radical, and imaginary numbers, and do it all within the confines of a 7th grade class period.

Now I realize how little foundation for understanding numbers, magnitudes, and the operations that relate them, I actually had. So, while those really smart guys were having fun with all this, we dummies were so lost and perplexed that we grew to hate the stuff.

Now, with a more mature perspective, I want to understand what I didn’t understand as a child, but it’s tough going. The experience of learning that Coxeter’s career was sort of a counter-revolutionary one, viz-a-viz the new math Bourbaki, is kind of like having to have someone explain a joke to you. It helps you understand what the joke was about, but it doesn’t make you laugh, and you still feel left out in a sense.

I’m really a late comer. I didn’t even know about Coxeter, or the Bourbaki, until Peter posted his experiences with the man’s ideas. Now, thanks to the Internet, I can see that he is credited with saving the subject of geometry from being completely abandoned by educators, who were convinced that geometry was nothing but an impediment to learning algebra. The Bourbaki were a group of French mathematicians whose ralling cry was Jean Dieudonne’s rant “Down with Euclid, death to the triangles!” (the French!!!). As Roberts explains:

In the 1930s, a secret society of crème de la crème French mathematicians sat in a Parisian café and devised an outlandish prank: They invented a mathematician by the name of Nicolas Bourbaki and decided to write under this nom de plume, publishing an encyclopedic treatise that aimed to restructure the discipline of mathematics axiom by axiom. Mathematical folklore has it that, for a time, the international mathematical community was duped; papers appeared by Nicolas Bourbaki, yet no one knew this man who never turned up at conferences as promised.

Writing in the Boston Globe, Roberts gives us the motivation for her book:

Crying `Death to Triangles!’ a generation of mathematicians tried to eliminate geometry in favor of algebra. Were it not for Donald Coxeter, they might have succeeded…Through his lifelong work as geometry’s apostle, Coxeter, who died in 2003 at 96…, became known by his followers around the world as “the man who saved geometry” in a mathematical era characterized by all things algebraic, abstract, and austere.

I guess I’ll have to read the book to learn the details of this “New Math” melodrama, but Peter gives us a flavor of what it entailed from his personal experience:

One theme of [Roberts’] book is to set Coxeter, as an exemplar of the intuitive, visual and geometric part of mathematics, up against Bourbaki, exemplifying the formal, abstract and algebraic. Bourbaki is blamed for the New Math, and I certainly remember being subjected by the French school system in the late sixties to an experimental math curriculum devoted to things like set theory and injective and surjective mappings. On the other hand, I also remember a couple years later in the U.S. having to sit through a year-long course devoted to extraordinarily boring facts about triangles, giving me a definite sympathy for the Bourbaki rallying cry of “A bas Euclide! Mort aux triangles!”. To this day, both of these seem to me like thoroughly worthless things to be teaching young students.

Here’s where it starts to get interesting, because, as we see over and over again, the real behind-the-scenes struggle is between the discrete and the continuous. Peter makes a comment that reveals this, when he writes:

Actually Bourbaki and Coxeter ended up having a lot in common. They both pretty much ignored modern differential geometry, that part of mathematics that has turned out to be the fundamental underpinning of modern particle physics and general relativity. Coxeter’s most important work probably was the notion of a Coxeter group, which turns out to be a crucial algebraic construction, and ended up being a main topic in some of the later Bourbaki textbooks. A Coxeter group is a certain kind of group generated by reflections, and Weyl groups are important examples. Coxeter first defined and studied them back in the 1930s, part of which he spent in Princeton. Weyl was there at the same time giving lectures on Lie groups, and used Coxeter’s work in his analysis of root systems and Weyl groups.

Coxeter groups and associated Coxeter graphs pop up unexpectedly in all sorts of mathematical problems, and Roberts quotes many mathematicians (including Ravi Vakil, Michael Atiyah and Edward Witten) on the topic of their significance.

Peter’s complaint that “They both pretty much ignored modern differential geometry, that part of mathematics that has turned out to be the fundamental underpinning of modern particle physics and general relativity,” is a complaint that the mathematics of discrete principles of symmetry, such as Coxeter’s groups, seem to have no relevance to physics. As he explained earlier in his comment:

Coxeter’s main interest was in “classical” geometry, the geometry of figures in two and three dimensional space and he wrote a very popular and influential college-level textbook on the subject, Introduction to Geometry. Much of this subject can be thought of as group theory, thinking of these figures in terms of their discrete symmetry groups. This subject has always kind of left me cold, perhaps mainly because these groups play little role in the kind of physics I’ve been interested in, where what is important are continuous Lie groups, both finite and infinite-dimensional, not the kind of 0-dimensional discrete groups that Coxeter mostly investigated.

Of course, zero-dimensional, vectorial, magnitudes, which are really 1, 2 and 3 “dimensional,” scalar, magnitudes, is what we are all about here at the LRC.  The idea of abstract algebra, based on group theory, is all the rage in physics today, but I think the difference between the n-dimensional, continuous, Lie groups and the zero-dimensional, discrete, Lie groups that Coxexter championed, may have more to do with today’s fundamental crises in theoretical physics than people realize.

The continuous Lie groups have to do with rotations, and the continuum of smooth manifolds, with the associated connections, covers, fiber bundles, etc, which wouldn’t be possible without imaginary numbers, and the complex and hypercomplex numbers that they make possible, even though these concepts have their equivalents in the three-dimensional Clifford algebra, which treats rotations in a more efficient manner than can be done with complex and hypercomplex numbers.

However, the key aspect to understand in all this is the role of rotation in the concept of higher dimensions.  Is rotation, and all the baggage it brings with it, really necessary to define magnitudes of higher dimensions, or is there another way, a way that might lead to resolving the continuum - discrete dilemma that is perplexing modern theoretical physics?

I’ll start discussing this next.