EL 118 Final Project: The Tragic History of Evariste Galois

A study of the fabled life of a brilliant mathematician using discrete models of mathematical logic

Dylan Cashman

DISCLAIMER: The author of this work did not intend for this to be a factual biography. It is in no way to be used as factual evidence in the study of Galois' life. It also is not a formal treatise, in any way, of modern mathematics. It is simply an exploratory project.The author of this work did not intend for this to be a factual biography. It is in no way to be used as factual evidence in the study of Galois' life. It also is not a formal treatise, in any way, of modern mathematics. It is simply an exploratory project.

I wrote the following explanation for the theory of this project. It refers repeatedly to Eastgate System's program Storyspace, a spatial hypertext medium.

When I first began work on this project, I knew that I wanted to combine two subjects that I feel people often overlook the connections between. I feel like writing, and storytelling, often follows some algorithmic structure in its flow and narrative. If we construct our stories, or arguments, around logical progressions and syllogisms, then what happens if we construct our narratives on other mathematical structures? This was the basis for the idea of using Storyspace to demonstrate the geometry of logic. When discussing it with peers in George Landow's class, someone suggested that the narrative be based on mathematics, to motivate the form. I knew I had to write about Evariste Galois – the conflicts he felt between his passion for mathematics and the severity of his public and political life perfectly paralleled the confusion created by this project.

I began with the idea of having a biography of his life laid out in discrete (meaning singular points) Cartesian coordinates. I figured that each horizontal line would be a different representation of his life, written in a different style. The further negative the narrative goes, the more abstract the storytelling would be, and if we move in the positive direction, we go in the opposite of abstract, which I defined as literal. I quickly found problems in executing this proposal. There was a lot of space for my own judgment to confound the work. I had to define for myself what abstract meant. I originally was motivated by different fictional styles; I had thought that I would use William Faulkner as my example for abstract thinking and writing. I found out quickly that writing like Faulkner is a lot more difficult than it looked. Instead, I found my motivation for abstract versus literal in two places.

The first inspiration was in the essay The Storyteller by Walter Benjamin. I encountered it in a class in Modernism, and I felt like it defined polar styles of writing, or communicating, very explicitly. In the essay, Benjamin suggests that the most pure form of communication, face-to-face storytelling, is being replaced by the introspection of novels and the ever-increasing flow of information from the media. He sets up a dichotomy between communicating facts for the sake of knowing facts, and communicating ideas for the sake of sharing experience. His judgment is that our information-driven world is killing our ability to communicate experience, and since reading his essay I have agreed with him. I decided that this was more easily defined, so I considered literal to be information-based, and abstract to be more experience-based.

I wanted to subtly inject as much mathematical influence as possible. Galois was the founder of modern abstract algebra, so I decided it would be appropriate to write with a more and more mathematical tone as the entries became more abstract. In mathematics, abstraction is usually defined as using more and more general terms. Abstract Algebra is designed around generalizing properties between numbers, or more generally, elements of a set. We may think that algebra means adding and subtracting, but Galois suggested that adding and subtracting numbers is just one form of algebra, and that there are many other forms of algebras (like matrices and matrix operations).

There are a few other mathematical notions at play that probably don't seem apparent at first. When we study functions, especially solutions to differential equations, we are basically finding a path that will get us from point a to point b, whether it is a path for a robot to walk on or the simulated path of blood in a heart-modeling program. Often, even if we execute all the algorithms correctly, our solution will fail as time goes on. This brings up the mathematical notion of stability. The idea is that the further away from our correct solution we get, the more unstable the values we obtain are. In this manner, the further away from the original narrative we get, i.e. on the extreme top or bottom, the less stable the narrative becomes. This is why the extremes basically fail at describing the life of Galois.

One of my original plans did make it through to the completed project; that is, the geometric functions within the storyspace spatial web. I had originally planned to explore what a narrative would sound like if it followed a non-linear progression, and it was this that originally motivated the coordinate system of documents. It is interesting to experience the narrative going through a circular motion; it first delves into the abstract representation of the information, then comes again to normal representation, and then heads backwards into the literal, until we arrive at where we started.

Some of the math influences I had not planned for at all, but found that the project was shaping itself in their directions. I noticed that the extreme abstract view and the extreme literal view were more in common with each other than any other views. I understood this to be a comment on infinities. In certain schools of mathematics, the numbers positive infinity and negative infinity are actually the same point. Somehow, the project seemed to represent this phenomenon – even though they are farther apart, the two extremes are similar in their deviation from the origin. I also found it interesting that my attempts to use mathematical analysis and literary analysis overlapped, even though they are considered two completely different modes of thinking. In mathematical analysis, we take the very few things that we know to be true, and we speculate and explore possibilities. I had not seen the commonality with other forms of analysis before this project.

Hopefully, the work will make some more sense. However, it is exploratory in nature – and I feel that, just as this will not prove any earth-shattering claims, it is a somewhat unusual combination of disciplines. I hope that it might demonstrate that academics, and even the narratives that we remember life by, are often more related than is readily apparent.