Generative AI has changed mathematics forever. Where to from here?

Less than a week ago, the artificial intelligence firm OpenAI unveiled ten groundbreaking advances in mathematics and computer science, all accomplished using their unlaunched model Astra. These discoveries span various mathematical domains, such as geometry, cryptography, and coding theory. OpenAI also released a statement addressing the responsibilities towards the mathematical community, acknowledging concerns related to attribution, accuracy, and the evolving nature of mathematical discovery.

The rise of generative AI systems, like OpenAI's Astra, has sparked a wave of mathematical breakthroughs. The company's announcement came with a statement on "responsibility to the mathematical community," highlighting the growing recognition that AI's technical advancements are just one aspect of a larger narrative about the future of mathematics itself.

AI is reshaping the mathematical landscape in profound ways. University departments are confronting the ethical implications of these tools in research and teaching. Peer-reviewed journals face a surge of AI-generated papers, forcing editorial boards to grapple with assessment and disclosure policies. Funding agencies are also formulating guidelines for acceptable use of AI in mathematical applications and evaluation.

As the impact of AI on mathematics becomes more apparent, the philosophical questions surrounding the nature of mathematical discovery are becoming increasingly prominent. If a large language model (LLM) can create a proof, construct an example, or formulate a new question, what does this say about the value of human mathematical creativity? Is mathematics primarily about generating new theorems, or is it about cultivating understanding? When a machine contributes to a discovery, who should be credited?

The mathematical community is divided on the answers to these questions. Some mathematicians express genuine concern about the future of the discipline, even suggesting a "profound spiritual crisis" has been triggered by the rapid advancement of AI-assisted mathematics. Others have attempted to establish principles for integrating AI into mathematical research in a responsible manner.

The Leiden Declaration, signed by thousands of mathematicians worldwide, advocates for AI to augment rather than replace human creativity in mathematics, emphasizing transparency, accountability, and proper attribution.

During the International Congress of Mathematicians, the world's most prestigious mathematical gathering, Terence Tao, a Fields medallist, referred to the "age of AI." He urged mathematicians to contemplate how AI systems could be incorporated into research to strengthen the discipline's culture and values.

Two contrasting approaches to AI in mathematics have emerged. My colleague Saul Freedman refused to use AI in his recent collaboration due to ethical concerns about the technology's environmental and social impacts. Consequently, the work proceeded without the use of AI tools. In contrast, another colleague, Aluna Rizzoli, successfully utilized an OpenAI model and a supercomputing cluster to discover an object his team had spent years searching for.

The computation took only 43 hours, a fraction of the time it would have taken his collaborators and him to develop the method independently.

While these collaborations resulted in significant mathematical advancements, they exemplify the contrasting perspectives on the role of generative AI in the research process. One approach advocates for AI exclusion due to ethical concerns, while the other embraces AI as an indispensable research partner, maintaining the ethical standards of mathematical research.

Neither approach is inherently incompatible with producing excellent mathematics. Instead, mathematicians are still grappling with the best way to integrate AI into their work while adhering to the core values of collaboration, transparency, and intellectual integrity that define their discipline.