Two Mathematical Revolutions Separated by More than Time
Benjamin Skuse
Two hundred years to the day since German mathematician Bernhard Riemann was born, it is worth reflecting on how his work revolutionised mathematics permanently, particularly in light of the revolution happening in the field today as a result of the influence of AI.
It can be argued that both revolutions represent a fundamental transition, not just in the problems that mathematicians solve, but in how they think, reason, and define mathematical progress. Yet their differences are informative too, highlighting key questions concerning the future of mathematics.
A Crash Course in Riemann
Born in 1826, Riemann made profound contributions to mathematics in the 39 years he was on this Earth. Much of this was down to how he approached mathematics. Instead of viewing the subject as a huge library of formulas requiring routine algebraic manipulation, he took a step back to look at the abstract geometric properties of mathematical objects to understand (and solve) mathematical problems at a much deeper level. Riemann essentially shifted focus from computation to conceptualisation – a way of conducting mathematics that is as relevant today as it was back then.
His first major contribution to mathematics perfectly encapsulates this approach and was also one of his most profound ideas: Riemann surfaces. In his 1851 doctoral dissertation, Riemann introduced these novel surfaces to provide a geometric foundation for the theory of functions of a complex variable.
In more detail, Riemann found the idea unsettling that if you plot out many such functions (such as \(f(z) = \sqrt{z}\)) on a 2D plane, there are points which hold two or more different answers simultaneously (e.g. \(+2\), \(-2\)). So instead of a single flat plane, he imagined a 3D spiral surface where every point on the surface gives exactly one answer. This meant that every complex function could have its own natural Riemann surface where it becomes well-behaved and single-valued.
Just three years later in 1854, Riemann delivered a lecture that introduced his theory of higher dimensions entitled On the Hypotheses which lie at the Bases of Geometry. Though not universally acknowledged as ground-breaking in his lifetime, this theory freed human thought from the prison of rigid Euclidean geometry and the three-dimensional space we experience.
Introducing the concept of an \(n\)-dimensional manifold – a space that looks flat locally, but can be warped, curved, and higher-dimensional globally – Riemannian geometry, as it became known, would go on to form the basis of Albert Einstein’s general relativity, where gravity is framed as the curvature of \((3+1)\)-dimensional spacetime, and is the foundational basis for the entire field of topology.
Alongside other famous contributions, including the first rigorous formulation of the integral and his work on Fourier series, what many people today know Riemann for is his eponymous hypothesis. This is because the Riemann hypothesis is one of the Millennium Prize Problems, seven fiendishly difficult questions in mathematics that the Clay Institute challenged the mathematical community to solve in the year 2000. As described in far more detail in this earlier post, the Riemann hypothesis comes from an observation by Riemann that the frequency of prime numbers is very closely related to the behaviour of the Riemann Zeta function:
\[ \zeta(s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + … .\]
In the Clay Institute’s lay formulation of the problem, the Riemann hypothesis asserts that all interesting solutions of the equation \(\zeta(s) = 0\) lie on a certain vertical straight line. If true, this would illuminate how the primes are distributed – a mystery that has captivated mathematicians for centuries – while also instantly confirming thousands of advanced results across physics, cryptography, and number theory.
AI Starting to Live Up to Its Promise
Up until very recently, only one of the Millennium Prize Problems had been solved: the Poincaré conjecture by Grigori Perelman (2006 Fields Medal). But on 8 September, OpenAI claimed that it employed 10,000 AI agents to solve the Navier–Stokes existence and smoothness problem. Though not yet acknowledged as solved by the Clay Institute, the achievement represents a watershed moment for AI, and by far the most significant contribution to mathematics by a machine.
Whether the claim proves true, and whether AI can solve the Riemann hypothesis and the rest of the Millennium Prize Problems, remains to be seen. But one thing seems fairly clear: AI is starting to generate a huge methodological and philosophical shift in mathematics, akin to that which Riemann himself caused.
The main difference is that this revolution leaves mathematicians questioning where they fit into the mathematical process. Where Riemann freed mathematicians to think beyond calculation and instead conceptualise mathematics in their minds, AI is freeing mathematicians from laborious literature searches, theorem proving, aspects of conjecture generation ,and even the limitations inherent to the human mind.
This raises a serious concern: Aside from tending to the models and interpreting their outputs, if an AI-driven approach becomes the default, what does the work of a human mathematician look like? And will this change be as dramatic as that which Riemann prompted (in the traditional sense of the word).
The panel session “AI in Mathematical Research” at the 13th Heidelberg Laureate Forum offered a platform for some of the most influential thought leaders in this space to debate this central issue. Jacob Tsimerman (2026 Fields Medal), Peter Scholze (2018 Fields Medal), Michael Harris (Columbia University, USA) and Geordie Williamson (University of Sydney, Australia) were all present to offer their thoughts, and they varied considerably.
Scholze, for instance, played down the significance of OpenAI’s claim to have solved the Navier–Stokes existence and smoothness problem because the solution to this singular problem alone is not why mathematicians want to solve it: “The goal of those problems is really to illuminate the landscape around those problems,” he said. Moreover, Scholze added: “[Navier–Stokes] was a problem where we could see that it might be within reach, which I think is not currently true at all for the other [Millennium Prize] problems.”
For Scholze, the idea that an AI-driven approach will likely become the default is by no means inevitable. He still sees room for the slow, deliberative mathematics that has served humanity well for centuries. “The important thing was always what you, as a person, learn about mathematics,” he argued. “How to think deeply about a problem, how to be stuck, and yet find the right idea.”
Similarly, Michael Harris argued for the purpose of mathematics to remain as a shared human endeavour. “A number of colleagues have told me that PhD students are coming to them and saying, “Is there a future for human mathematics?”, he recalled. “For me, since I see mathematics as an integral part of human culture, I understand this to mean that there’s no future for human culture. I consider such [a position] to be… irresponsible.”
The Immediate and Deep Consequences of AI
Seeing AI’s influence on mathematical practice as being of more immediate and deeper consequence were Williamson and Tsimerman. “What we want as a mathematical community is understanding, but we measure this against unsolved problems, and the problem is that these two measurements are very, very quickly becoming uncorrelated,” argued Williamson. “So I think that we have to diversify enormously what we value as a community.”
A key skill Williamson sees as being of central value to mathematics going forward is communication. This is why he, alongside others, formed Mathematical Discourse, a journal that celebrates and publishes high-quality mathematical talks.
The other panellists also acknowledged how significant human communication will be as a role and skill for human mathematicians in the age of AI. But in addition, Tsimerman argued that human mathematicians can play a critical role in bringing mathematics back from obscurity to a more central place (as it once occupied) in human society. “Assuming the future goes the way that I expect it to … we’re going to get superhuman mathematicians very soon,” he exclaimed. “And I think that many topics that we consider out of reach for pure mathematics are going to suddenly become in reach as we get access … to mathematics moving at a superhuman rate.”
This is why Tsimerman would like to see human mathematicians gearing up for the arrival of their superhuman colleagues “to tackle new areas that we currently consider applied science” by making frameworks and infrastructure that steer AI systems to filter for ideas in these areas, and by translating AI-derived concepts into human-digestible information that can be used for real-world benefit.
So, from one angle, AI is unlikely to have the kind of impact Riemann made in the 19th Century. From another, today’s AI revolution threatens to undo Riemann’s revolution, plunging mathematics into an era in which deep human understanding is no longer the primary goal of the field. And viewed from yet another different perspective, it’s bringing mathematics back to its original position as a central human cultural endeavour, where solutions to tough mathematical problems are applied to help solve the global challenges of our time. Which stance you take is personal. But as the Forum debate showed, each is equally valid.
The post Two Mathematical Revolutions Separated by More than Time originally appeared on the HLFF SciLogs blog.