A teenager's rejected equations became symmetry math that helped predict particles
His academy called the work no good, and even his supporters needed decades to make sense of it.
Dwarkesh Podcast · Grant Sanderson (@3blue1brown) – AI disproved a famous math conjecture. Now what?He was a teenager in prison, writing mathematics that the experts had already rejected.
Galois, a young mathematician obsessed with equations, had submitted his papers and been turned down. What he was writing was not very coherent. It was not a complete proof, and it did not clearly explain what his theory actually was. The verdict from the academy, in effect, was no good.
But he kept writing. In prison he produced a sweeping argument about how mathematics changes, from thinking only about numbers to fluency with algebraic expressions. His instinct was that there was another layer of abstraction to reach, where instead of studying formulas themselves, mathematicians would study the symmetries that underlie those formulas.
That instinct became group theory, the mathematics of symmetry. It later proved useful in physics and in cryptography, among other fields. Judging whether it was a productive idea involved what amounted to a hundred year wait.
To see why, go back to a problem every schoolchild meets. The quadratic formula solves equations with x squared. That formula was worked out long ago, and later Italian mathematicians, competing through intellectual challenges rather than real duels, secretly found formulas for equations with x cubed and then x to the fourth power. The formula for degree four is enormously complicated, and is usually broken into steps rather than written out in full.
For hundreds of years, no one answered the next question: was there a formula for degree five, the quintic? Mathematician Lagrange had the first useful instinct, that solving equations was related to symmetry, noticing for example that if you add four variables, shuffling their order changes nothing, while with more complicated expressions some shuffles change the value and some do not. That pattern mattered because such expressions had an unexpected relationship to reducing a harder equation to a simpler one, so symmetry looked like a possible path to a formula.
He never solved it. Abel, a young and precocious Norwegian mathematician, later proved that no such general formula using roots is possible. He died at 26 of tuberculosis.
Galois pushed the ideas of Lagrange and Abel toward abstraction. Because Abel had already proved impossibility, it is hard to say exactly what problem Galois solved. He did not even exhibit one specific quintic that could be proved unsolvable.
Then he died after a duel. The story that he wrote everything the night before is mostly myth, because he had tried five times to get published. He asked his brother and a friend to pass his papers along to leading mathematicians of the day, including Gauss. Even then, the notes did not catch on.
About 20 years passed before Liouville, another mathematician, saw something in them and tried to clean them up. It took about 20 more years before Jordan put together something like the modern treatment of group theory attributed to Galois.
The payoff came still later. In the twentieth century, Gell-Mann, a physicist thinking about what matter is built from, used reasoning about how certain groups break down to anticipate quarks, the particles that make up protons and neutrons. Predicting particles from pure questions about symmetry was one of group theory's finest late applications, arriving more than a century after a rejected teenager sensed that something was there.
Questions people ask
What is group theory, in plain terms?
It is the mathematics of symmetry, a way to study all the shuffles or transformations that leave some object essentially unchanged. For equations, it tracks how swapping the unknown solutions preserves certain relationships. Physicists later used the same machinery to classify symmetries of particles and crystals.
What does it mean that the quintic is unsolvable?
It does not mean fifth-degree equations have no answers. They do, and particular ones are easy, like x to the fifth power minus two. It means there is no single formula, using only ordinary arithmetic and roots, that gives the solutions for every quintic, the way the quadratic formula does for degree two.
Why did it take so long for Galois to be recognized?
His manuscripts were incomplete and hard to read, and the central concept was unfamiliar. Joseph Liouville needed about two decades to see value and publish an edited version, and Camille Jordan needed about two more decades to present the ideas in modern form. Those later uses, from cryptography to particle physics, lay far in the future and could not have been checked at the time.