A little over a week ago, artificial intelligence company OpenAI announced ten advances in mathematics and computer science made with their as-yet unreleased model Astra. The discoveries cover a wide range of mathematical fields, including geometry, cryptography and coding theory.
Author
- Melissa Lee
Senior Lecturer, School of Mathematics, Monash University
These ten are just the latest in a rapidly growing list of mathematical breakthroughs made by generative artificial intelligence (AI) systems.
OpenAI accompanied the announcement with a statement on "responsibility to the mathematical community". Here, the company acknowledges concerns about attribution, accountability for the correctness of AI-assisted results, and the changing nature of mathematical discovery.
The statement reflects a growing understanding that the remarkable technical advances due to AI are only part of a bigger story about what mathematics means, and what its future will look like.
AI is raising big questions
Large language models (LLMs) such as ChatGPT and Claude are creating disruptions in all parts of the mathematical community.
University departments are grappling with the ethical implications of these tools in their research and teaching. Editorial boards of many peer-reviewed journals are being inundated with AI-written papers and must decide how to assess and disclose the use of generative AI. The arXiv preprint repository has seen a sharp increase in mathematical submissions in recent months.
Funding agencies are starting to develop policies governing acceptable use of AI in applications and assessment.
Broader philosophical questions about how and why mathematics is done in the first place are hard to avoid.
If an LLM can discover a proof, construct an example, or formulate a new question, is there anything uniquely valuable about human mathematical creativity? Is mathematics mostly about producing new theorems, or is it about developing understanding? When a machine contributes to a discovery, who should get the credit?
Growing concern - but little consensus
Among working mathematicians, there is little consensus about the answers. Some have genuine anxiety about the future of the discipline. One mathematician wrote that the advance of AI-assisted mathematics had triggered a "profound spiritual crisis" .
Others have tried to articulate principles for integrating AI into research in a responsible way. The recent Leiden Declaration , signed by thousands of mathematicians from around the world, argues AI should augment rather than replace human mathematical creativity. At the same time, it emphasises transparency, accountability, and proper attribution.
Two weeks ago at the International Congress of Mathematicians, the largest and most prestigious meeting in the field, Terence Tao spoke about " the age of AI ". The famous Fields medallist urged mathematicians to think ahead. In his view, the question is how AI systems might be incorporated into research to strengthen the culture and values of mathematics.
Two attitudes to AI
Two collaborations from my own research over the past few weeks show just how varied these attitudes have become.
My colleague Saul Freedman and I announced an answer to the "semiregularity problem" about highly symmetric networks. This had been the subject of dozens of peer-reviewed papers over several decades.
From the outset, Saul made it clear he did not want AI used in any way during our collaboration due to his concerns about the environmental and social impacts of the technology. Therefore, the work proceeded entirely without the use of these tools.
After we announced our result, two colleagues told me they had independently tried to use LLMs to solve the same problem without success.
AI-powered discovery - with human connection
In contrast, a couple of weeks ago another colleague, Aluna Rizzoli, emailed to tell me he had found an object that collaborators and I had spent over two years searching for. We had used considerable computing resources, but Aluna had used an OpenAI model and a supercomputing cluster. His computation took a mere 43 hours.
The method ChatGPT had generated, working within the intriguingly named " Monster group ", relied on a much more intricate version of an existing algorithm. It would have taken my collaborators and me months to develop this generalisation.
Perhaps the most striking aspect is what happened next. Rather than "scooping" our ongoing work, Aluna invited us to write a joint paper based on the discovery, which appeared on arXiv last week.
Working out where AI fits
Each of these collaborations produced important mathematical advances, yet they embody starkly contrasting attitudes to generative AI.
One deliberately excluded the technology on ethical grounds, while the other embraced it as an indispensable research partner, without abandoning the etiquette of mathematical research. Neither approach appears inherently incompatible with doing excellent mathematics.
That is perhaps the defining feature of the current moment. Rather than converging on a single view, mathematicians are still working out where AI fits within the research process.
The conversation is no longer about whether LLMs are capable of contributing to mathematics research. Instead, we are talking about how their use can be reconciled with the values of collaboration, transparency, and intellectual integrity that form the foundation of our discipline.
Melissa Lee receives funding from the Australian Research Council. The collaborations described in this article were conducted as part of the programme 'Algebraic groups, geometry, invariants and related topics', funded by Isaac Newton Institute for Mathematical Sciences at the University of Cambridge under the EPSRC grant EP/Z000580/1.
