Grant Sanderson on AI and the Future of Mathematics

AI Progress in Mathematics as a Global Proxy

AI is advancing faster in mathematics than in almost any other field, making the domain a concrete preview of how AI progress will manifest across other sectors of the economy. This acceleration is driven by two primary factors: verifiability (the ability to objectively prove a result is correct) and grindability (the ability to run thousands of parallel, deterministic rollouts in a containerized environment to solve credit assignment problems).

While AI has reached a level where it can perform exceptionally well on International Math Olympiad (IMO) problems—specifically in geometry—it still faces a "spiky" frontier. For example, it struggles more with combinatorics, which often requires a more playful, puzzle-like creativity. This suggests that AI capability does not advance as a uniform wave but as a series of jagged breakthroughs in specific domains.

Theorem Proving vs. Conceptual "Mountain Building"

There is a fundamental distinction between solving a known problem (theorem proving) and creating the conceptual frameworks that make those problems solvable (mountain building).

The Three Paths to Solving Hard Problems

Grant Sanderson identifies three potential ways an AI might solve a major challenge like the Riemann Hypothesis:

1. The Lightning Bolt: Connecting two existing, deep domains of expertise (e.g., analytic number theory and random matrix theory). This is the most likely path for LLMs given their superhuman breadth of knowledge.
2. Mountain Building: Creating an entirely new theory or a new way of thinking that crystallizes a subject. This requires a level of intelligence that would likely permeate all white-collar work if achieved.
3. Raw Hustle: A brute-force, thousand-page proof that is logically correct but lacks conceptual elegance and is difficult for humans to digest.

The Galois Example and the Verification Loop

Conceptual breakthroughs often have incredibly long verification loops. Using Évariste Galois and the birth of group theory as an example, Sanderson notes that the utility of Galois's insights was not immediately apparent. It took nearly a century for the mathematical community to recognize the value of group theory, eventually leading to breakthroughs in physics (such as the prediction of quarks). This highlights a critical limitation of current RLVR (Reinforcement Learning from Verifiable Reward) environments: they reward immediate correctness, not the long-term productivity of a new conceptual framework.

The Future Role of the Mathematician

As AI automates the "theorem economy"—the process of proving theorems once a definition has been established—the human role in mathematics is shifting.

From Prover to Curator

Sanderson suggests that mathematicians may evolve into roles similar to art museum curators. While an AI can generate a proof or even explain it clearly, humans are still needed to navigate the nearly infinite space of ideas and decide which ones are worth pursuing. This curation is a social phenomenon based on trust and shared human interest.

The Stability of Education

Teaching and mentoring are viewed as some of the most stable post-AGI roles because they are deeply relational. A great educator does not just provide an explanation (which an AI can do) but "jujitsus" a student's specific misconceptions, reframing their mental models in real-time—a level of theory of mind that current AI lacks.

Technical Constraints and the "Theory of Mind" Gap

Autoregression and Unpredictability

The autoregressive nature of LLMs (predicting the next token) can be a hindrance to high-level creativity. True insight often requires a deliberate, unpredictable move that contradicts the most likely next token. While parallelizing agents with different biases or "refreshing" their context can mitigate this, the inherent nature of token prediction remains a constraint.

The Writing Problem

AI struggles with high-quality writing not because it cannot explain concepts, but because it lacks a sophisticated theory of mind. Effective writing requires the author to constantly project the reader's mental state—predicting what they know and how they will react to a specific phrasing. Sanderson compares this to the "Botox effect," where people who cannot physically mimic facial expressions struggle to read emotions in others; similarly, AI cannot "mimic" the human cognitive experience, making deep empathy in writing difficult.

Practical Takeaways for Students

For those pursuing mathematics or coding in the age of AI, Sanderson offers several pieces of advice:

• Focus on Value Creation: Understand where the funding comes from and what specific value you add to the ecosystem (e.g., brand value, teaching, or basic science).
• Leverage Human Curation: Use LLMs to prune branches of knowledge, but rely on human-authored artifacts (books, lectures, papers) to provide the initial motivation and conceptual organization.
• Embrace the Relational: Lean into the coaching and mentoring aspects of the field, as these are the least likely to be automated.