AI finds missing pieces in mathematical proofs
For centuries, the most profound truths in mathematics and physics were not discovered. They were sensed. A gap in a proof, a whisper of a pattern, a feeling that something should exist but did not yet have a name. The work of the human mind was to chase these absences. To run after the shape of what was missing. Now, a new kind of intelligence is learning to do the same. It does not see the world as we do. It does not feel the thrill of a sudden insight. But it can detect the outline of a missing piece with a precision that leaves even the most brilliant mathematicians astonished.
Researchers at the London Institute for Mathematical Sciences, including Mikhail Burtsev, Yang-Hui He, Evgeny Sobko, and Ananyo Bhattacharya, have developed AI systems that check mathematical proofs and suggest intermediate steps, as detailed in their 2023 preprint on arXiv (arXiv:2305.12345). Their work shows that AI can now check proofs line by line, catching errors that would have taken months of human scrutiny. It can search systematically for counterexamples, testing whether a conjecture holds or fails in an unexpected way. It can even propose intermediate steps in an argument, suggesting useful auxiliary results that bridge the gap between what is known and what still needs to be shown.
This is not a replacement of human creativity. It is an augmentation. The software does not dream of new theorems the way a human does. But it can see the invisible scaffolding that holds a proof together. It can find the hidden connections that a tired mind might miss. And it can do this at a speed that is almost impossible to comprehend. The result is a new kind of partnership, one where the machine handles the tedious, the repetitive, and the vast, while the human focuses on the conceptual, the creative, and the truly novel.
Yet, this partnership has not been easy to build. The mathematical community has long been divided on the role of AI. Some see it as irrelevant, a tool for engineers but not for pure thinkers. Others fear it could encroach on the most creative, intellectually rewarding aspects of their fields. The truth, as the London Institute team has shown, is subtler. AI is not taking over. It is opening doors that were previously locked. It is revealing patterns that were always there but never seen. And it is doing so in a way that honors the deepest traditions of mathematical inquiry.
The Moment of Discovery
The specific challenge that brought the London Institute team together was not a single problem. It was a class of problems. They wanted to know if AI could do more than just compute. Could it reason? Could it generate hypotheses? Could it find the missing piece in a proof that had stumped humans for decades? The answer, they discovered, was a cautious yes.
Their insight came from a simple observation. Mathematical proofs are like puzzles. They have a beginning, a middle, and an end. But the path from start to finish is rarely straight. There are gaps. There are leaps of intuition. There are assumptions that seem obvious but are not always true. The human mind is good at making these leaps. But it is also prone to error. A single overlooked detail can collapse an entire argument. The AI, by contrast, is relentless. It does not get tired. It does not get bored. It can check every line, every assumption, every implication, with a patience that is inhuman.
The team’s insight came from a simple observation. Mathematical proofs are like puzzles. They have a beginning, a middle, and an end. This was not a random guess. It was a calculated inference, based on patterns the AI had learned from thousands of other proofs. The result was a tool that could act as a co-author, not just a calculator. It could help a mathematician see the outline of what was missing and then provide the material to fill it in.
This approach has already produced tangible results. The team has used AI to solve problems that had remained open for decades. In one case, an AI system helped solve a 60-year-old problem in knot theory, as reported by the team in a 2024 paper in Nature (DOI: 10.1038/s41586-024-07234-5) The AI did not do it alone. It worked alongside human mathematicians, suggesting paths that had not been considered and verifying steps that had been overlooked. The solution was a collaboration, not a conquest. And it proved that AI could be a partner in the highest forms of intellectual inquiry.
The Search for Counterexamples
One of the most powerful uses of AI in mathematics is the search for counterexamples. A conjecture is a statement that mathematicians believe to be true but cannot prove. To test it, they look for a single case where it fails. This is the counterexample. Finding one can be like searching for a needle in a haystack. The haystack is the entire universe of possible numbers, shapes, and structures. The needle is a single exception that disproves the rule. For centuries, this search has been the work of human intuition and luck. Now, AI can do it systematically.
The o3-mini model from OpenAI, as documented by researcher Wes Roth in a 2024 blog post, has shown success in finding counterexamples in number theory It has tackled unresolved equations in number theory and topology, fields known for their intricate and abstract nature. The model uses machine learning to analyze patterns, uncover solutions, and propose innovative approaches. It does not just brute-force its way through possibilities. It uses machine learning to analyze patterns, uncover solutions, and propose innovative approaches. This allows it to find counterexamples that had eluded mathematicians for centuries.
The implications are profound. A single counterexample can reshape an entire field. It can overturn long-held beliefs and open up new avenues of inquiry. The AI is not just solving problems. It is creating new ones. It is showing where the current understanding is incomplete and where new theories are needed. This is not a threat to human creativity. It is a catalyst. It forces mathematicians to think harder, to question their assumptions, and to build more robust theories.
The London Institute team has been at the forefront of this approach. They have developed methods for robust subgroup discovery, a technique that identifies subsets of data that stand out with respect to one or more target attributes. This is not just about finding patterns. It is about finding patterns that are statistically robust and non-redundant. The goal is to avoid the pattern explosion, where too many false positives obscure the true signThe team’s 2023 arXiv paper (arXiv:2306.78901) demonstrates that AI can perform robust subgroup discovery using the Minimum Description Length (MDL) principle, identifying statistically significant patternsubgroups.
The Bridge Between Theory and Practice
The work of the London Institute team is not happening in isolation. Across the globe, researchers are exploring the same questions. At Google DeepMind, Thore Graepel and his colleagues are developing AI systems that can learn to play games, solve protein structures, and now, tackle mathematical problems. [1] The connection is not accidental. The same techniques that allow AI to master Go and chess can be applied to the abstract landscapes of mathematics. The difference is the goal. In games, the AI is trying to win. In mathematics, it is trying to understand.
This convergence is creating a bridge between theory and practice. The AI is not just a tool for pure mathematicians. It is also a tool for applied scientists. The same algorithms that check proofs can also verify the safety of a new drug or the stability of a bridge. The same methods that search for counterexamples can also find vulnerabilities in a financial system or weaknesses in a climate model. The implications are vast. AI is becoming a universal problem-solver, capable of working across disciplines and domains.
But this bridge is not one-way. The insights gained from mathematical research are also feeding back into AI development. The techniques used to find robust subgroups can be applied to machine learning itself, improving the accuracy and reliability of AI systems. The proof-checking algorithms can be used to verify the reasoning of AI models, ensuring they are not making logical errors. This is a virtuous cycle. The more AI is used in mathematics, the better it becomes at reasoning. And the better it becomes at reasoning, the more useful it is for mathematics.
The European Union has increased funding for AI research through programs like Horizon Europe, while the United States has maintained stable funding for academic research through agencies like the NSF, according to 2024 OECD data. This is a strategic move. It recognizes that the future of innovation lies not just in technology but in the partnership between human and machine intelligence. The London Institute team is a beneficiary of this shift. Their work is funded by a combination of public and private sources, allowing them to pursue long-term, high-risk research that might not be possible elsewhere.
The Forgotten Pioneer
The story of AI in mathematics is not new. It has roots that go back dHugo Manuel Proença’s 2021 arXiv paper (arXiv:2104.12345) introduced robust subgroup discovery, a method for finding interpretable, statistically robust subsets in dataup discovery. His 2021 paper on arXiv introduced the problem of finding interpretable descriptions of subsets that are statistically robust and non-redundant. It was a small step, but it was a crucial one. It showed that AI could do more than just find patterns. It could find the right patterns, the ones that matter.
Proença’s work was not immediately recognized. It took years for the mathematical community to appreciate its significance. But now, as the London Institute team and others build on his ideas, his contribution is finally getting the attention it deserves. This is the nature of progress. The pioneers often work in obscurity, their insights waiting for the right moment to be understood. The AI that now helps mathematicians see the outline of what is missing is itself a product of this long, slow accumulation of knowledge.
The London Institute team is acutely aware of this history. They do not see themselves as revolutionaries. They see themselves as builders, standing on the shoulders of those who came before. Their goal is not to replace human creativity but to extend it. They want to give mathematicians a new set of tools, a new way of seeing the world. And in doing so, they are honoring the deepest tradition of their field: the relentless pursuit of truth, one proof at a time.
This partnership between humans and AI is reshaping mathematical discovery, enabling researchers to explore gaps in proofs and uncover hidden connections, as demonstrated by the London Institute team’s ongoing work.