The path to mathematical superintelligence | Tudor Achim

**Elise Hu** (0:04)
You're listening to TED Talks Daily, where we bring you new ideas and conversations to spark your curiosity every day. I'm your host, Elise Hu. As AI becomes more intertwined in every aspect of our lives, humanity and science are facing a huge new truth problem. The tools we're increasingly relying on to accelerate discoveries such as generative AI and large language models.
Well, they have a habit of completely making things up.

**Tudor Achim** (0:31)
We simply don't have the human bandwidth to review all these proofs. Are we resigned to drown in a sea of unverified claims where we can't really tell truth from fiction?

**Elise Hu** (0:40)
That was computer scientist Tudor Achim, and yes, he's talking about AI hallucinations. For most applications, these far-fetched inventions might be inconvenient or confusing, but for science, they could be catastrophic. In this talk, he shares why he thinks a 400-year-old idea holds the fix. The German mathematician Leibniz dream of a logical framework where errors are simply impossible. He makes the case that grounding AI in this formal mathematical verification could transform it from unreliable chatbots into rigorous partners for scientific discovery. The talk is coming up right after a short break.
And now, our TED Talk of the day.

**Tudor Achim** (1:29)
Let's take a look at this clay tablet. It might not look like much, but it's actually some of the oldest mathematics we have. It's a 4,000 year old message in a bottle from ancient Babylon, a precursor to the quadratic equation. And for four millennia, people have been doing math basically the same way.
Someone will have a brilliant idea, they'll write it down, and their peers will discuss and check it.
It's a process built on creativity, communication, and most importantly, trust between people. And what might seem like a humble or simple process is anything but. It's not just been successful, it's been, as the physicist Eugene Wigner famously put it, unreasonably effective. Wigner was pondering and trying to unravel a deep mystery. Why should the abstract, creative, and often bizarre ideas that spring from a mathematician's imagination so often be the perfect language with which we understand the universe? Why should the strange laws of non-Euclidean geometry, which were originally conceived of as a thought experiment in the 19th century, turn out to be the exact mathematics that Einstein needed for general relativity? Why should the esoteric math of group theory, which was originally designed to study the abstract nature of symmetry, be fundamental to understanding everything from particle physics to the patterns in crystals? Well, there's no logical reason it has to be this way. This strange connection between pure mathematical thought and the real world has actually been the invisible engine driving human progress.
Every piece of technology that defines our lives was ignited with a mathematical spark. If you take the device in your phone, its brain is based on the quantum mechanics of semiconductors, and that's a theory built on linear algebra and complex numbers. The wireless signals that get data to it, they're just a concrete manifestation of Maxwell's equations. And finally, the security that protects your data online is based on number theory, which for a long time was truly considered the most pure and least applicable possible branch of mathematics. And now, it safeguards trillions of dollars in the global economy. And now we come to AI.
Modern AI is not just built with math, it's forged from it. A neural network is just a monumental structure of applied mathematics, and when AIs learn, they're using the tools of calculus to navigate vast landscapes of possibilities with billions of dimensions. So, AI is, in its soul, a mathematical idea that's given life through computation.
So we agree that math is the foundation that modern civilization is based on.
But that foundation is starting to show some signs of strain. The very process of human-led discovery that's gotten us to this point is nearing a breaking point buckling under the weight of its own success. And now AI, which is one of mathematics' greatest creations, is accelerating us towards that breaking point faster than the world is ready for. So let's just look at some evidence.
Consider the Poincaré conjecture. This is a legendary problem. It's a fundamental question about the nature of three-dimensional shapes originally posed in 1904 And for nearly a century, it stood as an unconquered Everest of mathematics. Until in 2002, a Russian mathematician working in isolation named Grigori Perelman posted a series of three short cryptic papers online. He didn't bother submitting them to a journal. He just put them on the internet and walked away. His fellow mathematicians had to stop what they were doing and try to decipher it. And several teams working independently of the best apologists in the world took the next four years to try to unpack the arguments, fill in the logical gaps, and eventually at the end, after they really reviewed it, declare that yes, he did it. He proved the Poincaré conjecture. But that's interesting because what it took one person to write a proof and a global multi-year intellectual mobilization to check it. And that's in the best case, when the proof is correct. Consider Andrew Wiles' proof of Fermat's Last Theorem. With the electrifying announcement in 1993 in Cambridge, the world celebrated. But during the peer-review process, deep in it, a single thread was found out of place in that magnificent tapestry of a proof, and when we started to pull on it, the proof started to unravel. And this wasn't a small mistake. Andrew Wiles and his collaborator, Richard Taylor, took two years of heroic secret effort to try to fix it. And that effort included some insights that Andrew Wiles said were among the most important in his life. And that's before we throw AI into the mix.