Cardano founder Charles Hoskinson says the rapid progress of artificial intelligence in advanced mathematics has gone far beyond what he once expected. 

This is particularly noticeable when it comes to AI systems producing and formally verifying complex mathematical proofs.

In a recent YouTube broadcast, Hoskinson discussed claims surrounding an AI-generated approach to the Navier-Stokes Millennium Prize problem, one of mathematics’ most famous unsolved challenges.

Hoskinson said the apparent mathematical capabilities demonstrated by modern AI systems were extraordinary.

"We never anticipated the extent to which AI would come in," Hoskinson said.

He explained that his earlier expectation was that formal mathematical systems would primarily allow larger groups of mathematicians to collaborate more effectively.

"The idea of the AI itself would fully write the proof, it was pretty far out," he said. "LLMs really surprised us."

"And it’s pretty remarkable to see how far they’ve gotten and what they’ve been able to achieve and do."

Navier-Stokes controversy

Hoskinson has long been interested in formal mathematics. He noted that he previously founded a center for formal mathematics at Carnegie Mellon University. 

He admitted that the emergence of LLMs capable of potentially generating the proofs themselves was not something he had fully anticipated.

"It’s pretty remarkable to see how far these things have gotten," he said.

Hoskinson’s comments came as he discussed claims about an AI-generated solution related to the Navier-Stokes equations.

The Navier-Stokes existence and smoothness problem is one of the Clay Mathematics Institute’s Millennium Prize Problems. It concerns whether sufficiently well-behaved solutions always exist for the equations describing three-dimensional fluid motion.

Hoskinson stressed the importance of the problem for both mathematics and physics.

"For OpenAI to claim that they have solved this, this is something that would fundamentally change the mathematics paradigm," he said.

He called solving Navier-Stokes "a big deal in mathematics" and noted its relevance to fluid dynamics, aerospace engineering, mechanical engineering and physics.

Hoskinson compared the situation to one mathematician finding another researcher’s notes. 

"For an academic, if you’re an entrepreneur, know that your ideas, if you share them in AI with these frontier models in the cloud, they’re not your ideas anymore," he warned.

Hoskinson used the controversy to promote the importance of private AI environments, arguing that researchers should be able to use powerful models without exposing confidential research logs or intellectual property to centralized providers.

Despite questioning the provenance of the mathematical work, Hoskinson said the AI system’s work was itself striking.

"What this does mean is that a model is sufficiently advanced that it knows how to steal a smart person’s work, take that work, improve it, iterate it, and formalize it to the extent that it actually solves a hard problem," he said.

"That is very humanlike behavior," he noted. 

"Everything is stolen. Nothing is truly original. And that’s the beauty of it. You can take what came before and you can make it better," the Cardano founder added. 

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