What we lose in the pursuit of insight

When I decided to study math in earnest, I imagined myself at the origin of a vast landscape, sloping upwards away from me in all directions and full of interesting features and formations. Every early course I took felt like ascent up one of its slopes, with the benefit of a knowledgeable tour guide to show me the most navigable path and point out peculiarities along the way.

In Linear Algebra, systems of equations led to matrices, vector spaces and linear transformations, and ultimately eigenvalues and eigenvectors. In Real Analysis, sequences led to limits and continuity, differentiation and integration, and ultimately the ideas of completeness and compactness. The end of each course always felt like cresting a summit, only to realize I was at the foothills of an even grander and more treacherous mountain. I’d then prepare for the next ascent, better at climbing mountains than when I started.

I knew the Millennium Prize problems as iconic peaks in this landscape which older, braver climbers had dedicated their lives towards conquering. Occasionally, I would hear whispers about an active summit attempt. If I was lucky, I’d catch a glimpse of one of these peaks myself after finishing an undergraduate-level course.

Terence Tao’s A Severe Misalignment of AI in Mathematics criticizes the efforts by frontier labs to solve difficult problems because they make no attempt to glean comprehensible ideas from their formalizations and connect these ideas to literature. Consequently, they cannot use these insights to formulate new questions, which may be even more important than the solutions themselves. In our analogy, they are airlifted to the summits of our tallest known peaks without finding navigable paths that human mathematicians can follow, and lacking the perception to identify even greater peaks further in the distance. Rebuttals argue that producing a verified proof of a difficult problem can only be helpful, as it does not preclude mathematicians from interpreting the solution and integrating the ideas into mathematical canon. And if frontier models have only just become able to solve our most difficult math problems, it is only a matter of time before they will be able to extrapolate new conjectures as a human mathematician would.

These arguments tacitly acknowledge the pursuit of insight as the core objective of mathematics. I argue this is an incomplete representation of why students choose to study the subject. Some students simply love mountaineering.

I am now five years removed from the active study of mathematics, working in an entirely unrelated field. Despite this, reading about OpenAI’s progress on Navier-Stokes felt incredibly bittersweet. It is what I imagine a 20th century amateur explorer might have felt if they learned Mt. Everest had been conquered for the very first time via helicopter. Yes, the achievement does not preclude human climbers from continuing to attempt the summit, even benefitting from information about summit conditions. Yes, that humanity was able to reach the summit at all is undeniably an epic accomplishment. However, the method employed feels partially irreverent to the history of human struggle against the mountain.

Mathematics is as much an art as a science, with notions of beauty and structure not unlike painting or filmmaking. And like other creative arts, medium and methodology are often as important as the final result itself. Furthermore, the struggle of a difficult problem is not just an obstacle to insight. It is meaningful in its own right, as a way of feeling connected with all those that struggled with it previously. Mathematicians follow the paths of predecessors, benefit from the trail markers left behind, and occasionally leave footholds of their own. In doing so, they participate, however modestly, in humanity’s millennia-old effort to find order in the universe.

We are entering a new paradigm of mathematical discovery in which not every problem will be solved by careful human ascent. Researchers will grow accustomed to being dropped off at a summit and charting a course back down to our existing body of knowledge. The resulting advances in our sciences will be tremendous. However, this progress may not come for free. When I imagine my teenage self at the origin of that vast landscape, I wonder whether I would still feel excited to climb.