| When most people think of math, they think of carrying out calculations or solving equations. But professional mathematicians devote most of their time to a completely different practice: coming up with proofs. A proof is a sequence of logical steps that build on each other, culminating in the declaration that some statement is true. (Sometimes mathematicians take this to the meta level, proving statements about what's provable, as I discussed in my last newsletter.) Research mathematics is rarely a blind exploration. Mathematicians typically know what they want to prove. They have also built a large and varied tool kit to help them prove it. But sifting through those tools and getting them to work together in the right ways can be enormously challenging. When I interview a mathematician, it can take an hour just to get a 10,000-foot view of how their proof works. Sometimes we'll get to a particular step — one seemingly tiny mathematical link in the proof's long chain of arguments — and I'll learn that it took an entire year and 100 pages to prove it. I first learned about proofs when I took a geometry class in 10th grade. We would be given an arrangement of intersecting lines, for example, and asked to prove that two angles were the same. To craft the proof, we'd have to write each logical sub-statement alongside its justification in a two-column format. It felt very different from what I had thought math to be. But at the end, the conclusion hit harder, its truth made stark by the documentation of the journey. My high school proofs always took a straight and obvious path to their destination. Each statement got closer to what I wanted to prove, until I was able to write and justify the assigned prompt. Research mathematicians have to get far more creative. Their proofs hew to many different styles and methods, depending on the mathematician's taste and the requirements of their goal. Proofs are ultimately more about logic and intuition than about calculation. A mathematician has a chosen destination and a set of tools for getting there. What that path ultimately looks like can range from perfectly linear to downright loopy. But many follow one (or more) of a few time-tested approaches. What's New and Noteworthy As the mathematician Fan Chung recounted to Quanta in a recent Q&A, the key to any big proof is often figuring out how to frame the question in the right way. For instance, mathematicians might not know how to prove a statement directly. But if they can show it's equivalent to another statement, they can focus on proving that instead, which might be an easier task. Regardless, a proof comes down to showing that a series of smaller sub-statements are true. One straightforward way to do this is to divide the proof into cases: If you want to prove a statement about all whole numbers, you might prove it for all even numbers first, then all odd numbers. Each case might require a different approach. Similarly, mathematicians sometimes imagine the various kinds of counterexamples that pose a threat to a given conjecture. Then they rule them out one by one, until they can definitively claim the conjecture is true. This is how two mathematicians proved the 3D Kakeya conjecture earlier this year, a "once in a century" result in geometry. The 3D Kakeya conjecture says that if you point a needle in every direction in three-dimensional space, it must carve out a particular volume (for a certain definition of "volume"). To prove the conjecture, the mathematicians Hong Wang and Joshua Zahl figured out what the possible counterexamples would have to look like. In 2022, they showed that one category of counterexample couldn't exist. This year, they ruled out the others. This meant they'd solved the problem: There couldn't be counterexamples to the Kakeya conjecture, so it had to be true. Another common method that mathematicians learn early in their career is called a proof by contradiction. These proofs work by assuming the opposite of what you want to prove, then showing that the assumption leads to a contradiction. Take, for example, a groundbreaking 2022 proof that black holes are mathematically stable. Mathematicians started by assuming that a solution to the equations that describe a black hole will not be stable. They then showed that, no matter how long it takes for the solution to break down, it's possible to extend the solution beyond that time. This contradicts the initial assumption — that the solution is not stable — meaning that the opposite has to be true. Another proof technique that students often learn early on in their math careers is the proof by induction. It's a fun one, whose beauty Erica Klarreich demonstrates in her detailed walkthrough of two famous mathematical proofs. With induction, she explains, a particular proof of a statement about all whole numbers requires just two steps. First show that the statement is true for the number 1. Then show that if the statement is true for any given number, it must also be true for the next number, ad infinitum. It turns "the intuition behind the domino effect into a powerful mathematical principle," Klarreich wrote. And then there are existence proofs. Lots of mathematical questions boil down to whether an object with specific features exists. Are there graphs (collections of vertices and edges) that have relatively few edges but are still highly connected? Are there functions that are continuous everywhere but differentiable nowhere? To prove a statement, mathematicians can construct the object in question mathematically by writing down the right equations — or, in the case of one intrepid trio this summer, physically. Or they can prove that it must exist with a probability greater than zero, even if they can't provide a concrete example. For instance, the mathematician Boaz Klartag recently proved the existence of more efficient sphere packings in high dimensions. He devised a random process to do so. But he didn't actually have to carry out the process he'd prescribed — he just had to show that it had a nonzero probability of resulting in an optimal packing. These methods aren't the only arrows in a mathematician's quiver, but they're among the most well worn. Modern math proofs tend to be enormously complex and lengthy. The several-hundred-page tomes that fill the scientific preprint site arxiv.org every day invariably rely on a rich mixture of techniques to get the job done. But when you break down any one of them, it often comes down to some creative combination of these logical games. | 
