Artificial Intelligence Transforms Mathematics, Raising Questions About Goals and Values

The field of mathematics is undergoing significant changes as artificial intelligence (AI) increasingly tackles complex problems once thought beyond its reach. This shift has mathematicians reevaluating their role in the discipline and considering how it will evolve alongside rapidly advancing technology.

At the recent International Congress of Mathematicians in Philadelphia, 2006 Fields Medalist Terence Tao reflected on this pivotal moment for mathematics during a talk titled ‘Mathematics in the Age of AI.’ While many conversations focus on the capabilities of future AI systems, Tao argued that mathematicians need to ask more fundamental questions about their goals and values.

Tao assumed that soon, AI will be able to perform a reasonable fraction of mathematical tasks successfully. He then posed a crucial question: If this is the case, what does it mean for our understanding of successful mathematics? In other words, why do we do mathematics in the first place?

Mathematicians have long focused on problem-solving as a core aspect of their work. However, with AI now impacting this area significantly, Tao chose to focus on this facet during his talk. He identified several stages involved in mathematical problem-solving: providing and verifying proofs, evaluating and explaining these proofs, and making them accessible to the broader community.

AI has made significant strides in generating candidate proofs or crucial ideas, as well as rigorously checking very long or intricate proofs. However, Tao emphasized that AI’s limitations become apparent when it comes to human evaluation and explanation of these proofs. He noted that even large models can produce 100,000-line proofs that are difficult for humans to understand.

The problem lies in the fact that if no one understands a proof, its likelihood of being accepted by the community becomes slim. A successful result is not just about generating a correct answer but also about making it valuable and useful to others. AI might be good at finding results, but it often fails to provide context that makes people care.

Tao argued that without human evaluation and explanation, a proof cannot reach its full potential – namely, becoming canonicalized or widely accepted as part of the mathematical canon. This process involves not just generating correct answers but also providing context, describing how one arrived at those results, and making them accessible to others.

The value of mathematics is often measured by its applications in other fields such as engineering and physics. However, these applications are typically built on top of canonicalized knowledge – the kind found in textbooks rather than recent research papers. Tao noted that AI itself relies heavily on this existing body of work for its success.

So where do we go from here? Tao encourages mathematicians to engage in discussions about their goals and values regarding problem-solving, education, publication, and community culture. He challenges them to take the initiative in integrating AI into mathematics rather than passively reacting to it.

The integration of AI will require mathematicians to set clear rules on what is acceptable or not, rather than letting external actors define these standards for us. This involves considering how we can use AI tools effectively while maintaining our values and goals as a community.