Unsolved No More: Axiom's AI Breakthrough in Mathematics

A Breakthrough in Mathematical AI

Five years ago, mathematicians Dawei Chen and Quentin Gendron grappled with a challenging area of algebraic geometry linked to differentials, crucial elements in calculus that measure distances on curved surfaces. During their explorations, they stumbled upon a perplexing formula from number theory, one that hindered the completion of their theorem. Unable to resolve it, they ultimately published their findings as a conjecture.

The Encounter That Changed Everything

After many fruitless hours prompting ChatGPT for a solution, Chen attended a math conference where he serendipitously reconnected with Ken Ono, a renowned mathematician. Having recently transitioned to Axiom, an AI startup founded by his mentee, Axiom proved to be a game-changer. The following morning, Ono presented Chen with a proof courtesy of their AI, AxiomProver, connecting the dots with unprecedented clarity.

“Everything fell into place naturally after that,” Chen remarked about his collaboration with Axiom, which has now released their findings on arXiv.

Axiom's Proving Power

AxiomProver's success doesn't stop with Chen's conjecture. The AI has made significant strides, solving questions that perplexed human experts across various mathematical domains for years. Though still not tackling the most famous unsolved problems, such developments highlight the steady advancement of AI's reasoning capabilities.

Real-World Applications of Axiom's Technology

Beyond the labyrinth of advanced mathematics, Axiom's techniques have potential applications in cybersecurity. By employing AI to ensure software reliability, we could see significant improvements in robustness against cyber threats. Axiom's CEO, Carina Hong, emphasizes the value of mathematical research as a vital laboratory for broader technological advancements.

“Math is really the great test ground and sandbox for reality,” recalls Hong. “There are a plethora of high-value use cases waiting to be discovered.”

The AI Landscape: A New Paradigm

Axiom's innovative approach combines large language models with its specially designed AxiomProver, which is uniquely adept at navigating and solving mathematical challenges. This stands in stark contrast to existing systems like Google's AlphaProof, which demonstrated promise but fell short in this realm.

Ono remarked on the transition: “The AI-generated proof for the Chen-Gendron conjecture signifies a new paradigm for proving theorems.”

Novel Approaches to Problem Solving

What sets Axiom apart is its ability to utilize a unique mathematical language called Lean to verify proofs, enabling the AI to innovate rather than merely replicate existing solutions. One of the new proofs derived by AxiomProver is a standout example of how AI can solve problems from inception to conclusion independently, displaying a level of mathematical creativity previously thought to be the exclusive domain of human intellect.

Community Perspectives: Mathematicians Weigh In

Experts in the field, such as Scott Kominers from Harvard Business School, express awe at these breakthroughs. Kominers states, “AxiomProver solved a problem fully automated and verified, showcasing an elegance and beauty in the math it produced.”

The new generation of proofs includes a probabilistic model concerning “dead ends” in number theory, and another drawing from historical techniques used to address Fermat's Last Theorem, further emphasizing Axiom's expanding capabilities.

Looking to the Future: AI and Mathematics

Ono's vision extends beyond assisting mathematicians; he aspires to unlock the mysteries behind breakthrough moments in mathematical discovery. “I'm eager to know if we can predict those 'aha' moments,” he says, suggesting a deeper understanding of the mechanics behind innovation.

In reflecting on AI's role in the realm of mathematics, Chen remains optimistic. “Just as calculators did not eliminate multiplication tables, I believe AI will serve as a crucial intelligent tool, or perhaps a partner, opening extensive horizons for mathematical research.”

---

This article reflects insights from the article by Will Knight published in the AI Lab newsletter. Revisit prior editions and insights here.