Terence Tao on AI and the Depletion of Fruitful Mathematical Problems

The Crisis of Non-Renewable Mathematical Problems

AI is currently mining the collection of fruitful open mathematical problems in a non-renewable fashion. While the total number of possible mathematical questions is infinite, the subset of problems that are actually "worth" solving—those that reveal deep insights, uncover new connections, or drive the field forward—is finite and scarce.

Terence Tao likens this to a region suffering from a critical shortage of drinking water while surrounded by a massive ocean: one can generate infinite trivial problems (e.g., calculating the $10^{10^{10}}$th digit of pi), but the problems that provide genuine mathematical progress are rare and are now being consumed by automated solution-extraction tools.

Flattening the Difficulty Landscape

Every technological advance in mathematics reduces the difficulty of solving problems, which typically expands the horizon of what is reachable. However, the current AI era is characterized by a "flattening" of the difficulty landscape without the creation of clear new frontiers.

The Loss of Mathematical Geometry

In mathematics, the "difficulty landscape"—the understanding of which questions are easy, which require effort, and which are currently impossible—acts as a compass for researchers. When AI tools solve problems indiscriminately, they flatten this landscape, making it impossible for humans to discern the geometry of a field and identify where the most promising new questions lie.

The Absence of Defined Boundaries

Unlike previous tools, there are no clear boundaries separating "AI-feasible" problems from "AI-hard" problems. This is exacerbated by AI companies refusing to disclose negative results or the specific processes used to reach solutions, leaving researchers in the dark about the limits of the technology.

The "Scorched-Earth" Risk to Open Science

The ability of AI to rapidly solve problems creates a systemic risk to the culture of open science and the development of new mathematicians.

The Scooping Effect

Because AI can be deployed at a scale far beyond individual human capacity, the mere rumor of a human researcher working on a promising problem can trigger a massive AI-powered effort to solve it first. This creates a "scooping" incentive that may drive mathematicians to become secretive about their research directions, reversing centuries of tradition in open scientific collaboration.

Impact on Early-Career Researchers

This environment is particularly damaging to PhD students and early-career researchers. The speed of AI-generated results contributes to "publication inflation," leaving young mathematicians anxious about the value of their work and the level of expertise required to secure tenure in a world where raw solutions are cheap.

Redefining Mathematical Value: Results vs. Insights

Tao argues that the mathematical community must shift its value system from the act of solving a problem to the extraction of insight from the solution process.

The Problem with Raw Solutions

A raw solution—especially one generated by an AI that may be an uninterpretable "pile of Lean" code—does not necessarily advance the field. If a solution is found but the underlying mechanism remains a mystery, the "entrance" to new mathematics has been closed without the interior being explored.

Shifting the Goalposts

As the cost of obtaining answers drops toward zero, mathematical value must move upward:

From solving a problem $\rightarrow$ Understanding a structure
From proving one theorem $\rightarrow$ Explaining why a class of phenomena occurs
From competing over an entrance $\rightarrow$ Exploring the world behind it

Synthesis of Community Perspectives

Discussion among the mathematical and technical community highlights several counterpoints and extensions to Tao's thesis:

"If a problem is 'solved' (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession." — @dvt

"Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play." — @olalonde

Key Points of Contention

The Chess Analogy: Some argue that mathematics is following the path of chess, where engines initially threatened the game but eventually became tools that elevated human play to a higher level of understanding.
The Resource Argument: Some disagree that problems are non-renewable, suggesting that AI can be used to pose new, fundamental questions, thereby "renewing" the supply of open problems.
The Institutional Response: Suggestions have been made for institutions (like the Clay Mathematics Institute) to refuse recognition for formalizations that lack human-readable, insightful proofs to incentivize AI labs to produce usable knowledge rather than just "glory" results.