A New Kind of Mathematical Partner
For decades, computers have been essential tools in mathematics, but primarily for brute-force calculations and data processing. The new generation of AI, particularly large language models (LLMs), is different. These systems are not just executing instructions;
they are generating novel ideas and identifying patterns that humans might miss. This shift is transforming AI into an assistant that can handle literature reviews, verify proofs, and even act as a sounding board for new theories. Researchers are finding that AI can accelerate discoveries, accomplishing in days what might have previously taken months.
Cracking Decades-Old Problems
A prime example of this new partnership is the work done by Google's DeepMind. Its AI system, FunSearch, made a significant discovery related to the 'cap set problem,' a notoriously difficult puzzle in combinatorics that has stumped experts for decades. The problem involves finding the largest group of points in a multi-dimensional grid where no three points form a straight line. FunSearch, which pairs a creative LLM with an automated evaluator to weed out incorrect ideas, discovered a new construction for large cap sets, finding a solution previously unknown to mathematicians. This marked one of the first times an LLM was used to find a verifiable new piece of knowledge for a famous scientific problem.
Untangling Knots and Finding Connections
It's not just about solving puzzles. AI is also revealing deep and unexpected connections within pure mathematics. In 2021, DeepMind's AI was used to explore the complex fields of knot theory and representation theory. In knot theory, which has applications in understanding everything from DNA to fluid dynamics, mathematicians study invariants—properties that remain the same even when a knot is twisted or deformed. By analyzing millions of knots, the AI was able to detect a surprising new relationship between different types of invariants. This guided mathematicians toward a completely new theorem, establishing a bridge between two areas of knot theory that were long suspected to be related but never formally linked.
A Tool, Not a Replacement
Despite these breakthroughs, the consensus is that AI will not make human mathematicians obsolete. Instead, it is becoming a powerful collaborator. The most successful applications have involved a synergy between machine and human. The AI excels at searching through massive datasets and finding subtle patterns, but human mathematicians are still needed to provide the initial direction, interpret the results, and formulate rigorous proofs. In many cases, the AI generates thousands of potential patterns, and it's the mathematician's intuition that guides the process and identifies which leads are worth pursuing. Models can still make errors or 'hallucinate' incorrect arguments, making expert oversight crucial.
