| When a task or problem feels overwhelming, the best thing you can do is to try breaking it up into smaller pieces. Suddenly things don't feel so insurmountable. The same is true in mathematics. The greatest mathematical mysteries — questions that have stood open for centuries and shaped mathematicians' understanding of a field — are too hard to tackle at once. Mathematicians need to approach them in incremental steps, using each step as a new foothold as they climb toward their ultimate goal.
So instead of trying to prove some massive, consequential statement, mathematicians often start by studying that statement's implications. If the statement is true, they ask, then what else must be true? Each of those consequences might be less sweeping, less radical than the original statement — but sometimes, one of them is an easier problem to solve. It can act as a sort of foundation. Proving that it's true won't immediately prove the truth of the original statement. But disproving it will: If a consequence of the original statement is false, then the original statement must be false too.
Mathematicians build towers of implications in this way. In such a tower, the grandest statements are at the top, and their consequences lie below. Proving the topmost statement would prove all the ones beneath it, and finding a counterexample to the consequence at the bottom would demolish the whole edifice.
And even though a proof of a statement lower in the tower can't directly prove statements above it, such a proof can still be seismic. It gives mathematicians more confidence in the structure's integrity. And they can try using the techniques they developed for the proof to tackle the tower's loftier statements.
This approach to math is similar to how any of us might break a problem into many smaller pieces — but in this case, the pieces have a distinctive hierarchy. That hierarchy can guide mathematicians for decades.
What's New and Noteworthy
I've thought about math towers a lot this year, after I got to write about one of the biggest math breakthroughs of 2025. The Kakeya conjecture asks how to rotate a needle in the air so that it points in every direction but sweeps through as little space as possible. It sounds like a brainteaser from a magazine for nerdy kids, but it's one of pure math's biggest open problems.
Or it was, until Joshua Zahl and Hong Wang showed the minimum space this motion could take up, proving the Kakeya conjecture in three dimensions. The result ended a century-old story, but it also marked the beginning of an even bigger one. The conjecture is the foundation of a true math tower. Three other important conjectures live above it. They all have to do with the study of waves — functions that oscillate up and down — and their connection to the Kakeya conjecture has been a surprising and fruitful one.
At the top of many towers lies one of the biggest problem in mathematics: the Riemann hypothesis, a conjecture about a special function called the Riemann zeta function, which determines how the primes are distributed along the number line. Mathematicians aren't close to solving it. But if it's true, it will automatically imply hosts of other conjectures. Intriguingly, physicists have managed to build an even higher floor above it, showing that the existence of a kind of quantum system would imply the Riemann hypothesis. In 2017, researchers narrowed the search for this strange beast.
Meanwhile, the mathematician Hugh Woodin dares to dream of a single, giant tower of truths that can describe the mathematical universe. Since the early days of set theory in the 1870s, mathematicians have been contending with a hodgepodge of exotic infinities of different sizes. To make sense of them, and to prove more conjectures of interest, set theorists have added assumptions, or axioms, to "ZFC," the nine core axioms that underlie modern mathematics. Each of these new axioms defines a different infinity. Set theorists then try to show that these additional infinities are still consistent with ZFC.
Woodin and other mathematicians believe that all these infinities exist in a nice, orderly hierarchy: If they were to prove that one infinity is consistent with ZFC, it would imply the consistency of all the infinities below it. But earlier this year, some of Woodin's former disciples went rogue, unearthing evidence that the mathematical world is not so orderly. They found strange infinities that seem to be consistent with ZFC (though that has yet to be proved) — but these new infinities exist outside the tower. If they can prove their claim, they'll have shaken up the whole mathematical landscape. | 
