A third-year mathematics undergraduate here, already scared to tears… 🥲🥲🥲🥲🥲

I spent three hours working through the first two problems. They both feel very natural and interesting.

I partly agree with the view that AI’s mathematical ability lies within the convex hull of human techniques—at least for now. That may not remain true.

For general-purpose models produced only through LLM training algorithms, I think their reasoning stays within that convex hull. Such models tend to draw on large amounts of historical material to summarize mathematical techniques, then find solutions through feedback and error correction over a broad but finite set of trial-and-error paths. There is no innovation at the methodological level, which I see as a limitation of LLMs.

But if we continue developing training algorithms specifically for mathematical models, these could make mathematical thinking more efficient: fewer tokens for more meaningful results. Such algorithms would emphasize logic, construction, completeness, and innovation, with chains of thought and token-output mechanisms better suited to mathematical research. A useful example is the logic of Lean 4 formal verification—an organizational structure like Lean 4. Under scaling laws, AI should then generalize to mathematical intelligence that humans have not reached.

I do not know whether OpenAI and Anthropic have internally trained models specifically for mathematics, or whether that specialization goes far enough.

Some broader thoughts on this era, the future of mathematics, and my choices as a mathematics undergraduate

  1. In the near future, the abilities we expect of mathematicians will change, and the distribution of interests within mathematics will have to adjust to the new rules. Greater emphasis may fall on asking good questions, using AI efficiently, and access to computing power.
  2. We have all seen how vigorously code harnesses are developing. After the code harness comes the math harness. Mathematics also involves deductive reasoning under fixed rules, so it needs a good harness to improve models’ abilities.
  3. Mathematics as a field will expand over the next few years. Mathematicians will not lose their jobs in the short term; we may even need more people who combine mathematics, AI, and engineering. Similarly, coding agents have increased demand for senior engineers in the IT industry in the short term.
  4. I will not change fields because of this, nor do I think I have wasted three years. I choose to keep studying undergraduate mathematics well, and I am optimistic about infrastructure work at the intersection of AI and mathematics. Even if I really am displaced, studying mathematics is a wonderful interest in itself!

I very much hope funding for mathematics will increase. Mathematics is like a lottery: there is a small chance of a result that multiplies human productivity several times over. More funding can compound those small probabilities into a larger one. After sufficient internal experimentation, I hope OpenAI and Anthropic will release useful tools and interfaces for thousands of mathematicians. Together, mathematicians could produce an extraordinary abundance of results in a short time, ushering in exponential growth in mathematics!

As for the claim that most results have no concrete meaning for human life or production, I would look at it probabilistically. Suppose we spend $100,000 on electricity and obtain 1,000 results. If even one leads to something like:

  1. Fourier analysis in communications, sound, images, and signal processing;
  2. complex analysis in electrical engineering, fluid mechanics, and control;
  3. information theory in digital communications, compression, and the internet;

or operations research, game theory, statistics… then the return to humanity cannot be measured in money.

And might the world’s ultimate secrets be hidden in mathematics? What if our generation discovers the ultimate secrets of the universe?

How do we determine whether a theory really benefits human life and production?

We do not have enough people to verify every conclusion and think through whether each one is meaningful.

But I believe this is what the second stage of the AI × Math industry will address, perhaps after 2035. We will need to keep designing AI that helps us put mathematical results into practice. That would close the loop for the field and make mathematics a perpetual engine for humanity!

The second stage, however, needs the first to produce a large volume of mathematical results before people have an incentive to work on what comes next.

Stay optimistic, everyone. This is an extraordinarily exciting era!