The epic of Galois

Hi Everyone,

Thank you for forwarding along the request for artists I posted last week! It did manage to surface some great contributors who can help.

I've sprinkled several vague mentions of "a project about Galois" here and there in past posts this year, but it might be worth laying out the full plan here. It's grown to be the largest project this channel has worked on, and you are, in effect, the producers.

How it started

It all started over a year ago when I was tentatively planning to do a video on a lesser-known proof of the unsolvability of the quintic by Vladimir Arnold in 1963. The traditional way to teach this topic is to use a piece of math known as Galois theory, which requires a hefty bit of background material to build up to it (most notably group theory and abstract linear algebra). The charm of Arnold's proof is that it is, just barely, doable in a single lecture, with no heavy machinery required.

A funny thing happened, though, where one of the entries to SoME1 covered the same proof I had in mind. My own contest scooped me! Thereafter it was still kicking around in my mind, as I thought about ways to tweak Arnold's approach or ways I might motivate the key ideas more clearly.

In the tentative plans I had, I was going to briefly mention the history of the problem. After several renaissance Italians had found a general method for solving cubic equations, then shortly after another one for solving quartic (degree 4) equations, centuries passed with no one managing to find a general method for solving quintics. "Solving", here, is being used in a very restricted sense to mean writing an explicit root of a polynomial using only the operations of addition, subtraction, multiplication, and nth roots.

The young Norwegian mathematician Niels Abel was the first to provide a complete proof that this is actually impossible for polynomial equations with degree 5 or more. A few years later, the even younger French mathematician Évariste Galois built the beginnings of a more powerful and general framework to answer a more refined version of this question, which is to be able to look at a specific degree 5 equation, like x^5 -2 = 0, or x^5 - x - 1 = 0, and determine whether its roots can be written down using radicals (the first one clearly can, the second one, as it happens, cannot).

Then, about a century and a half later, Vladimir Arnold comes in and offers a completely different perspective, one using a topological approach, which generalizes Abel's result in a different way. It shows that even if you allow for a wide swath of alternate functions to be used in an attempted-quintic-formula, such as exponential functions, trig functions, logarithms, etc., such a formula is still impossible. Moreover, Arnold's approach is much easier to explain to an audience with a high school level background in math.

But here's the thing, nobody actually cares about this question. 

Solving polynomial equations is important, yes, but writing those solutions using radicals is, I think it's fair to say, almost completely useless. For engineering purposes, numerical methods like Newton's method are much more convenient (and fast!). And for theoretical purposes, it's very unclear what describing a root of a polynomial using nested radicals ever buys you that studying the polynomial expression directly would not. Many of Abel's and Galois's contemporaries viewed the question as a stale problem, and I would agree. 

The reason mathematicians these days care so much about it has less to do with its result, and more to do with its place in history. To study it, Galois invented a new kind of math, most notably one of the first well-formed articulations of group theory. It was rough around the edges, and incomplete in many ways, but it was one of the first footsteps taken in a dramatic paradigm shift through the 19th century which saw abstract algebra come to dominate more classical "solve for x" style algebra.

So with all that in mind, there's something funny about doing a whole video about the Arnold approach, when its charm is that it avoids the use of Galois theory, if the whole reason we care is that this is the moment in history that motivated the invention of Galois theory (and with it, group theory)!

On top of that, Galois himself is the center of one of the most famous stories in the history of math. The headline fact is that he died in a duel at the age of 20, but the more I read into everything about his life leading up to this point, the more completely enraptured I was. Almost every mathematician knows about the duel, but much less well-known is that before the duel he had been imprisoned for political dissidence, that he'd been expelled from university, that he was rejected (twice) from the École Polytechnique, along with dozens of others moments of drama and intrigue in his tragically short life.

How it's going

All this considered, I started mapping out plans for a project on the life and math of Galois at the beginning of this year. What began as a 30-second aside in a video about Arnold's proof has since grown to something more like a 2-hour documentary.

The plan is to break it up into chapters that oscillate between the life story, and the mathematical story. The life story parts involve a very new style for the channel, characterized by artwork that I hope gives the feeling of wandering through a well-illustrated picturebook covering the tale. We even did a two-day photo shoot with volunteer actors and a professional photographer friend to get reference shots for the artwork. Kurt has been leading the efforts on that front, and more recently I've added a few contract artists to help parallelize the efforts given the monumental scope which has emerged.

Researching this history was also a new, and shockingly time-consuming, process on my end. Many biographies play fast and loose with the facts, sometimes just fabricating them with no actual reference. Conflicting facts, unsubstantiated anecdotes, and exaggerations abound. With ordinary math videos, it's easy enough to know if what I'm saying is correct (just prove it!), but history is quite different.

I'm equally excited about the mathematical chapters. When I read the work of Lagrange from the 1770s which motivated the work of Abel and Galois, it was the first time I felt like I really understood why group theory exists. That is, what was the sequence of questions asked that led to people wanting to systematize the study of symmetry into a new kind of mathematical object, as opposed to just regularly leveraging symmetry as a useful problem-solving tool? Lagrange asks a highly accessible question, which any high schooler can appreciate, which motivates the need for group theory much better than vague statements about "studying symmetry" ever can.

Thanks for the patience!

Every month when there is no new video, I feel a pang of guilt. It will still be a while before this project is fully done, but I believe we should (hopefully) have at least the first chapter out to you by the end of this month, with decent progress into chapter 2 happening in parallel.

This project absolutely could not happen without this Patreon support, and I hope by the end we can produce something you'll be proud to have played a critical role in creating.