After Math

On September 8th, 2026, OpenAI declared it had crafted an AI-generated resolution to the Navier–Stokes existence and smoothness problem, an esteemed Millennium Prize Problem. This statement ignited discourse surrounding credit distribution and the collaborative roles of humans and machines in achieving the feat. Furthermore, it heightened existing parallels with previous AI triumphs in fields often considered bastions of human intellect.

Tristan Buckmaster, a mathematician involved in the Navier–Stokes debate, remarked: "This is a Deep Blue–Kasparov moment." Such existential inquiries for mathematics then follow, like ripples in a pond. If AI can now deliver answers to the most profound mathematical questions, does the field risk becoming "solved," as some have prophesied for chess and Go?

The response of "keep playing" holds merit. As philosopher C. Thi Nguyen contends, the essence of playing a game extends beyond victory; it encompasses the journey. This sentiment extends to intellectual pursuits like chess and Go. For mathematicians, playing remains vital; it is not merely about winning but about the process, the very essence of mathematical inquiry.

However, a stronger course lies in rejecting the notion that mathematics is a game, which could necessitate retreat. The question then shifts: what emerges after AI addresses mathematics' most challenging questions? Moreover, what fuels our pursuit of mathematics itself? The claim that AI "solved" mathematics rests on two assumptions, both tempting yet misguided: AI truly solved a mathematical problem.

The first assumption is flawed because merely answering a problem, even with formal validation, falls short of genuine resolution. A proof that human mathematicians can comprehend and build upon is crucial. If AI were to furnish such a proof, the story would not conclude, for the second assumption—mathematics is solely about problem-solving—is also erroneous.

Mathematicians strive to forge new concepts, theories, connections, unify disparate domains, inspire future generations, and produce work cherished for its beauty and profundity. Therefore, we must reject the narrative of AI overpowering humans in mathematics and delve into the true nature of mathematics and our aspirations for it.

OpenAI provided an answer to the Navier–Stokes query: yes, singularity does exist. In The Hitchhiker's Guide to the Galaxy, Deep Thought provided an answer to life, the universe, and everything: 42. Neither response fully satisfies. OpenAI offered more than a simple "yes." It delivered two proofs, one in Lean formalization and a manuscript purported to contain the corresponding informal proof.

Yet, this still leaves us unsatisfied. The issue lies in the nature of proof itself. Two proofs exist: a logical one and an intelligible one. While a Lean formalization adheres to the logical notion of proof, meeting verification criteria through mechanical means, it may not satisfy the intelligible notion of proof—understanding the mathematical argument's essence.

The intelligible proof aims to grasp the proposition's truth, connect with existing knowledge, and pave the way for further advancements. OpenAI's result may not have achieved this intelligible proof. Genuine proofs embody both logical and intelligible qualities. Historically, they have intertwined, as mathematicians could not construct intricate logical proofs without first grasping the underlying ideas.

However, with AI, these notions can diverge. Formal proofs can exist independently of intelligible proof, posing a dual challenge. An intelligible mathematical argument may convey profound insights yet fail to validate the result's truth, potentially becoming a "roadblock rather than an inspiration." Hales's Flyspeck project exemplifies this dilemma, striving to provide rigorous proofs for complex theorems.