The Mathematics of Uncertainty

Terence Tao, widely regarded as one of the greatest living mathematicians, recently discussed his experience with LLMs in a revealing interview. The analogy he offers frames the relationship between human mathematicians and AI beautifully, and it has profound implications for how we think about mathematical reasoning itself.

When a human faces a mathematical problem, Tao suggests, they climb a wall of uncertain height. They choose a path, test holds, sometimes fall, and the process takes time. The uncertainty is inherent. You don't know if the wall is climbable until you reach the top or run out of options.

LLMs work differently. They don't climb. They jump. They see an entire field of possible approaches and make probabilistic leaps toward what training data suggests might work. Sometimes these leaps land precisely where intended. Sometimes they miss spectacularly.

What LLMs Actually Do in Mathematics

Tao's mathematical community has been experimenting with LLMs extensively. The results are instructive, and they don't match the hype cycles.

For straightforward problems with established solution patterns, LLMs perform reasonably well. Ask an LLM to solve an integral from a calculus textbook, and it will often succeed. These problems have clear methods, well-documented approaches, and numerous training examples.

But mathematics at the research frontier is different. These problems lack known solutions. The approach isn't documented in the mathematical literature. The wall's height is genuinely uncertain, and sometimes the wall doesn't exist at all.

LLMs struggle here. They can propose approaches based on superficial pattern matching, but they can't engage in the deep, multi-step reasoning that research mathematics demands. They jump confidently toward solutions that don't exist.

The Collaboration Question

One of Tao's most interesting observations concerns collaboration. Mathematics is intensely collaborative. Proofs are refined through peer review. Ideas develop in conversation. The community advances together.

LLMs, despite being called assistants, don't particularly excel at this collaborative dimension. They can answer questions and suggest approaches, but they can't maintain the sustained, nuanced dialogue that productive mathematical collaboration requires.

Chatbots respond to prompts. They don't propose counterarguments unprompted. They don't notice when a colleague's approach has implications for their own work. They don't follow research programs over months and years.

This matters because it suggests LLMs may remain tools for individual mathematicians rather than collaborators in the human sense, regardless of how sophisticated they become.

What Changes and What Stays the Same

Tao's perspective cuts through both AI hype and AI skepticism. He acknowledges that LLMs have genuinely changed some aspects of mathematical practice. Finding specific formulas, checking routine calculations, exploring known territory, these tasks have become faster.

But the fundamental nature of mathematical discovery remains intact. The hard part isn't computation. It's knowing what to compute, which conjectures are worth pursuing, which approaches might connect distant fields.

LLMs will help mathematicians move faster across well-mapped terrain. The uncharted territory still requires human judgment, creativity, and the willingness to climb walls without knowing if they lead anywhere.

The Future of Mathematical Tools

Looking forward, Tao sees a role for math-specific tools rather than general-purpose LLMs. Proof assistants like Lean have already changed how some mathematicians work. They formalize mathematical reasoning in ways that allow computers to verify correctness.

LLMs trained specifically on mathematical content might eventually offer more reliable assistance than general chatbots probed with mathematical prompts. The key is specialization. A model trained on mathematical literature, proof databases, and formal methods can develop domain-appropriate intuitions.

For now, Tao's advice to mathematicians using LLMs is pragmatic: use them for what they're good at, don't expect them to be collaborators, and always verify their outputs. The walls of mathematics remain to be climbed, and while jumping looks faster, it doesn't guarantee reaching the top.

Implications Beyond Mathematics

This jumping versus climbing framework extends beyond mathematics. Any domain where expertise means knowing which approaches might work, where reasoning chains are long and uncertain, and where collaboration shapes outcomes, faces similar questions.

Software architecture. Scientific research. Legal analysis. Strategic planning. In all these fields, LLMs can make impressive leaps but lack the climbing skills humans develop through sustained practice.

The question isn't whether AI will replace mathematicians. It's whether mathematicians, and other experts, can develop workflows that leverage AI's jumping ability for appropriate tasks while preserving the climbing skills that remain uniquely human.

Frequently Asked Questions

Does Terence Tao use LLMs in his own mathematical work?

Tao has experimented with LLMs and discussed their capabilities publicly. He finds them useful for certain routine tasks but notes that they don't yet contribute meaningfully to the creative, collaborative aspects of research mathematics.

What is the difference between jumping and climbing in Tao's analogy?

Climbing refers to the human approach, gradual, methodical, uncertain, where you discover the path as you go. Jumping represents LLMs making probabilistic leaps toward potential solutions based on pattern matching without truly understanding the terrain.

Will LLMs ever become true mathematical collaborators?

Current LLMs lack the sustained reasoning, counterargument development, and long-term research memory that mathematical collaboration requires. Whether future models will overcome these limitations remains an open question, both technically and philosophically.