The Art of Math | Gödel's Lost Letter and P=NP

Art, history, and controversy

Jemma Lorenat is an assistant professor at Pitzer College in Los Angeles. She teaches and does research on the history of mathematics.

Today I thought we’d look at some of her work.

History of math is one topic that we have focused on many times before. More on that in a moment.

Her Art

But before we do that I wish to present Lorenat’s art work. My late father, Jack Lipton, was an artist and so perhaps I have a genetic interest in art. Lorenat is an artist besides being a mathematician. You can see some of her drawings of famous mathematicians here. Her elegant style I find appealing. See if you like it as much as I do. My dad taught me:

A clean drawing is more difficult to execute than a busy one. It is hard to hide flaws when your art is clean.

Her drawings are clean indeed.

Here are three of Lorenat’s drawings of the following three famous mathematicians in some order: Which is which? Prizes will not be given to those with correct answers.

1. Eric Temple Bell
2. Jacques Hadamard
3. Henri Lebesgue

Her Research

Lorenat’s research is on the history of mathematics. My first choice is to create math, but I am intrigued by the history of who did what, when, and why. We must understand history—at least in broad strokes—if we are to continue to make progress. History helps us understand how progress was made and how it was not. History teaches us much about our field, about mathematics.

Several online sources show the field’s breadth and scope. Among issues and topics, we note:

• Who is a result named for?
• Who gets the credit for a result?
• What is the strongest result known?
• Is this result correct?

It is fun to see the process in action. One example I have been involved in for a long time is the study of vector addition systems and reachability problems. There continues to be exciting news, for instance, a paper last year showing that a central reachability problem is vastly harder than had been conjectured. I will discuss, however, an issue from two centuries ago that Lorenat has illuminated.

The Duality Controversy

Lorenat has a talk on the geometric theory of duality. It was a prime example of a controversy in the discover of a basic math idea. Here duality means: Given a statement from projective geometry we can flip points and lines and still leave its correctness invariant. This is the duality:

In complexity theory we have our own duality. Instead of flipping points and lines we can exchange boolean operations

and also exchange

Lorenat’s talk highlights a controversy between: Joseph Diaz Gergonne, Jean-Victor Poncelet, and Julius Plucker. Her work is here.

A plagiarism charge in 1827 sparked a public controversy centered between Jean-Victor Poncelet (1788-1867) and Joseph-Diez Gergonne (1771-1859) over the origin and applications of the principle of duality in geometry. Over the next three years and through the pages of various journals, monographs, letters, reviews, reports, and footnotes, vitriol between the antagonists increased as their potential publicity grew. While the historical literature offers valuable resources toward understanding the development, content, and applications of geometric duality, the hostile nature of the exchange seems to have deterred an in-depth textual study of the explicitly polemical writings. We argue that the necessary collective endeavor of beginning and ending this controversy constitutes a case study in the circulation of geometry. In particular, we consider how the duality controversy functioned as a medium of communicating new fundamental principles to a wider audience of practitioners.

A further comment is here:

Of this feud, Pierre Samuel has quipped that since both men were in the French army and Poncelet was a general while Gergonne a mere captain, Poncelet’s view prevailed, at least among their French contemporaries.

Open Problems

Did you see which drawing was which?