The equations for describing how fluids flow were first derived around 200 years ago. Since then, understanding the Navier-Stokes equations – named after French mathematician Claude-Louis Navier and English mathematician George Gabriel Stokes – has been a long-running quest for mathematicians. Last month, American corporation OpenAI stunned the mathematics world by announcing that its AI agents had answered a longstanding question regarding these equations, namely: do they sometimes result in infinite solutions? The answer, according to these agents, is: yes. Whilst mathematicians had always suspected that such solutions were possible, nobody had never able to demonstrate this – until now.
The way this discovery has come about is likely to have a dramatic effect on the mathematics world. OpenAI began working on the problem after discovering that a pair of mathematicians had made progress on it. A few days later, their agents had solved it. This is not an isolated case either: over the past few months, AI has made breakthrough after breakthrough in mathematics. It is not clear at this stage how many other longstanding mathematical problems will fall to AI, but it is likely to be many. Not all mathematicians are excited about this though. Some argue that AI does not really understand mathematics in the way humans do. Others contend that AI tends to produce convoluted, hard-to-follow arguments which give little insight into the underlying problem.
Where does this leave the world of mathematics? These are unprecedented times, although it should be noted that mathematics has gone through upheaval before. One such upheaval occurred when French mathematician and philosopher René Descartes made a link between geometry and algebra in the 17th century. Mathematicians of the time worried that this discovery would turn them into drones who simply manipulated symbols and came up with no new insights. But 300 years later most would argue that making the link between algebra and geometry was a huge step forward. This suggests that future generations might say the same about AI. I doubt many present-day mathematicians will be convinced by this argument though.
On the face of it, the reticence of mathematicians to embrace AI-generated proofs of longstanding mathematical problems seems quite strange. You’d think they’d be glad that these problems were finally being solved! In any other scientific field that would indeed be the case. I expect most physicists would be quite happy if AI solved a longstanding mystery about the nature of the universe. And if AI discovered a cure for cancer then I’m sure all medical professionals would be overjoyed. So why are mathematicians apparently so averse to AI solving problems in their field? I think this reticence to embrace AI-generated solutions reveals something interesting about the nature of both mathematics and mathematicians.
In his 1940 essay A Mathematician’s Apology, the English mathematician G. H. Hardy defended the pursuit of mathematics for its own sake. One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics is that which has no practical applications. He saw real mathematics as essentially a creative activity, rather than an explanatory or expository one. Hardy’s essay came in for a lot of criticism from his fellow mathematicians, many of whom pointed to applications of mathematics to justify their existence. But I think Hardy’s essay represents an honest assessment of what actually drives mathematicians to do what they do.
For mathematicians, the value of mathematics comes not from the results they derive but from the process by which they derive them. The obvious difference between mathematics and creative pursuits like painting is that whereas a painter’s output can be appreciated by the layperson, fully appreciating a mathematician’s output requires detailed technical knowledge that most laypeople do not possess. In fact the mathematics world has become so fractured now that it is probably fair to say that most new mathematics is not understood by most mathematicians. The process of doing mathematics is better compared to that of creating some obscure avant-garde jazz that only a handful of people in the world are able to appreciate. In which case you rather have to wonder what the point is.
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