AI Advances in Mathematics: Proofs, Conjectures, Navier‑Stokes

Yang-Hui, a mathematical physicist and fellow at the London Institute for Mathematical Sciences, has been deeply involved since 2017 in exploring how AI can assist in solving and generating new problems in mathematics. He highlights the rapid advancements in AI-guided mathematics, noting that the field is evolving so quickly that he needs to check its progress weekly.

AI's Growing Capabilities in Mathematics

Yang-Hui categorizes mathematical research into three approaches:

• Top-down mathematics: Guided by intuition and experience to identify problems or potential solutions.
• Bottom-up mathematics: Building mathematics from fundamental principles, akin to constructing a language sentence by sentence.
• Meta-mathematics: Examining the nature of mathematics itself, particularly how AI can solve mathematical problems, sometimes through what is colloquially termed "proving by vibe."

The field of AI-guided mathematics has seen significant growth, with independent AI companies like Axiom AI, Google DeepMind, and Harmonic developing specialized language models to solve and reason through mathematical problems.

Breakthroughs and Solved Problems

• International Math Olympiad: As of summer last year, DeepMind's AlphaGo, AlphaGo 2, and AlphaProof could solve problems at a gold medal level, which are extremely challenging high school-level problems.
• Research-level problems:
  • Frontier Math: A project involving OpenAI, which Yang-Hui was part of, aimed to benchmark AI against 700 research-level problems. Approximately 20% of these have been solved.
  • Erdős problems: Harmonic's system has solved about 6% of these research-level problems proposed by the legendary mathematician Paul Erdős, which have intrigued mathematicians for about a century. These solutions were achieved purely unguided by humans.
  • Standing Conjectures: Axiom AI recently solved four long-standing conjectures in mathematical literature, which were open problems proposed by authors and previously understood only by experts in those specific fields.

The Millennium Prize Problems

At the turn of the century, the mathematics community identified seven crucial open problems. One was solved within a decade, leaving six. These problems span all areas of mathematics and theoretical physics. DeepMind is reportedly working on the Navier-Stokes problem, one of these Millennium Prize Problems.

Yang-Hui initially predicted that DeepMind would crack Navier-Stokes within a couple of years. However, in a surprising turn of events, it was Anthropic and OpenAI that announced a solution to this problem within six months of his prediction. This period has been described as the most dramatic year in the history of mathematics since Euclid, with major problems being resolved at an unprecedented rate.

Rumors suggest that two more Millennium Prize Problems have been resolved. OpenAI also announced the resolution of 100 more open problems across different branches of mathematics. An emergency committee of Fields Medalists has been formed to announce these solutions in batches, allowing for human and expert evaluation.

The Role of AI in Mathematical Research

While AI is making incredible strides, Yang-Hui emphasizes that we are not yet at a stage where one can simply ask an AI to "generate paper" or "generate proof." However, the integration of math verifiers could significantly increase reliability.

Yang-Hui shared a personal experience where GPT suggested a correct, unknown technique for a technical problem in finite group theory and algebraic geometry that he and his collaborators were working on. This technique was verified manually and will be acknowledged in their published work. Such instances are becoming more common in research papers.

Trusting AI Outputs

For most professional mathematicians, the primary concern is the correctness and reliability of AI-generated output, rather than the internal workings of the AI. Yang-Hui advises caution, operating under the hypothesis not to trust anything an AI generates without verification.

Two reliable uses of AI in mathematics include:

1. Citation verification: Asking for citations or links to papers to confirm their existence and relevance.
2. Code generation: Requesting AI to write code (e.g., in Python, Mathematica, C++) to test mathematical ideas. This code can then be run externally to verify results.

The Importance of Mathlib

A significant development is the coupling of AI systems like DeepMind's AlphaProof and Harmonic's Aristotle with Mathlib. Mathlib represents "formalized mathematics," a bottom-up approach where every step of every theorem and lemma is typed into a computer language called Lean. If the code compiles, the proof is considered correct.

The game-changer in recent months has been the ability to auto-translate human mathematical language into Mathlib's computer code. This allows AI-generated proofs, which might otherwise be prone to hallucination, to be rigorously checked for logical reasoning. If the translated code compiles, the reasoning is deemed correct, forming the basis of this new frontier in mathematics.

Conjecture Formulation

While much attention has been on proof formalization and verification, Yang-Hui's initial work in 2017 focused on using AI to formulate conjectures by analyzing mathematical data. This "top-down" approach, akin to experimental scientists forming hypotheses, is crucial for identifying new problems.

AI is now capable of suggesting methods of attack and formulating numerous conjectures. The challenge lies in discerning "good" problems from "bad" ones, as unlike proof verification, evaluating the interest or significance of a conjecture is more complex. However, Yang-Hui believes this capability will soon be refined.

The Human Element: Understanding and Insight

Despite AI's ability to provide verified proofs, Yang-Hui stresses that understanding remains the single most important aspect of mathematical research. A correct proof, even if verified by a system like Lean, does not automatically equate to an advanced piece of mathematics or usefulness to the broader mathematical community.

The formation of expert committees to evaluate AI-generated solutions highlights the need for human digestion of these results to gain new insights. Without this, the purpose of doing mathematics itself could be questioned. The current situation is likened to an alien species providing true or false statements, where the ability to verify truth values through formalization is amazing, but the human task is to derive understanding from these truths.