# The Art of Math

*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.

- Eric Temple Bell
- Jacques Hadamard
- 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

Thus

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?

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- Animated Logical Graphs • 50 | Inquiry Into Inquiry

There’s a nice interplay between geometric and logical dualities in C.S. Peirce’s graphical systems of logic, rooted in his discovery of the

amphecksand and flowering in his logical graphs for propositional and predicate calculus. These bear the dual interpretations he dubbedentitativeandexistentialgraphs.Here’s a Table of Boolean Functions on Two Variables, using an extension of his graphs from trees to cacti, illustrating the duality so far as it affects propositional calculus.

Actually mathematics history is fascinating. I was interested in Mathematics from the Bell trilogy that I won from a math competition. It turns out most of his history is a bit flawed. Then I read Stillwell’s “Mathematics and Its History” and Dunham’s “The Calculus Gallery” and they were both marvelous. Another great source is not exactly history but close “Prime Numbers and the Riemann Hypothesis” by Mazur and Stein.

All of these give context to Lipton’s et.al. critique of formal verification of programs – Mathematics is indeed a social phenomenon with all of the warts that come with it such as the exclusion of women until recently. (As an aside, I think that some of Lipton, et al are less convincing with the rise of better AI – but we will see since people want explanatory AI).

Dear Dick & Ken,

Another way of looking at Peirce duality is given by the following Table, which shows how logical graphs denote boolean functions under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the boolean functions denoted under the entitative interpretation and Column 3 shows the boolean functions denoted under the existential interpretation.

Dear Dick,

What tragically might be the history of Mathematics if Ramanujan instead of finding Hardy in England, he might have to deal with someone like Edison? Or how different will be the history of Mathematics if the plagiarism to Perelman would never have happened? What would be the end of our journey if we could finish to check and review the whole proof of the Riemann Hypothesis:

https://doi.org/10.5281/zenodo.3648511

or the whole proof of P versus NP:

https://doi.org/10.5281/zenodo.3355776

However, thanks for all your help.

Best,

Frank

What tragically might be the history of Mathematics if Ramanujan instead of finding Hardy in England, he might have to deal with someone like Edison? Or how different will be the history of Mathematics if the plagiarism to Perelman would never have happened? What would be the end of our journey if we could finish to check and review the whole proof of the Riemann Hypothesis:

https://doi.org/10.5281/zenodo.3648511

or the whole proof of P versus NP:

https://doi.org/10.5281/zenodo.3355776

However, thanks for all your help.

A few clean drawings on a tabula rasa …A logical concept represented by a boolean variable has its

extension, the cases it covers in a designated universe of discourse, and itscomprehension(orintension), the properties it implies in a designated hierarchy of predicates. The formulas and graphs tabulated above are well-adapted to articulate the syntactic and intensional aspects of propositional logic. But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic. Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts. All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.A more graphic picture of Peirce duality is given by the next Table, which shows how logical graphs map to venn diagrams under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

https://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/

Oops, wrong link. Here is the Table …

Logical Graphs • Entitative and Existential Venn Diagrams

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist. To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes. What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements. In the jargon of transformation groups these classes are called

orbits. We find the sixteen rows partition into seven orbits, as shown below.Venn Diagrams and Logical Graphs on Two Variables • Orbit Order

In the last of our six ways of looking at the Peirce duality between entitative and existential interpretations, we consider the previous Table of Logical Graphs and Venn Diagrams sorted in Orbit Order.