I write about the subtasks that we should solve along the way as we reach superintelligence in mathematics.

Please take note that I am not an expert in mathematics.

Achieve superintelligence in competitive programming

I have detailed this in a separate post.

Compared to mathematics, superintelligence in competitive programming is much easier than mathematics because

  • Problem statements in competitive programming are easy to understand, even for the most difficult problems in the world. However, the most difficult problems in research mathematics require deep mathematical expertise to even understand.
  • It is easy to invent problems in competitive programming, interesting problems can be invented just by making slight modifications to an existing problems. In mathematics, trivial modifications rarely make interesting questions.
  • It is easy to check correctness in competitive programming. You can use an inefficient algorithm to check the output of a proposed efficient algorithm. Checking the correctness of mathematical proofs also requires deep mathematical expertise.
  • Code in competitive programming is expected to run within seconds. The feedback received is fast. Code can be written to sanity check mathematical ideas but most mathematical ideas cannot be checked with code that can be run in a reasonable amount of time.

Demonstrating success in this subtask of achieving superintelligence in competitive programming will show us what a superintelligent community will look like. This will help inform the design and implementation of the superintelligent mathematical community.

Achieve superintelligence in verifiable mathematical tasks

There are some mathematical tasks where the result is verifiable

  • Taking integrals, which can be validated with differentiation
  • Simple counting problems that can be validated with code run for a reasonable amount of time
  • Olympiad geometry, where there exist symbolic systems that check the correctness of proofs

Similar to how we train superintelligence in competitive programming, we can achieve superintelligence in these easily verifiable mathematical tasks.

Perfect retrieval in mathematical knowledge

We expect a superintelligent AI system to be able to recall and build on existing mathematical knowledge. We need to be able to solve this subtask.

Perfect performance on well-known contest problems

Mathematical contest problems are usually inspired by a similar problem, or use a combination of well-known techniques. This subtask requires the AI system to be able to solve all problems that can be solved by reading and understanding the editorial of a similar problem.

There might be contest problems where the necessary mathematical knowledge has not been publish - solving this milestone is not required for the subtask. I am not sure if there exist such problems, maybe some Putnam problems belong to this category.

Reverse engineer mathematical discoveries

Given a substantial mathematical discovery, reverse engineer the mathematical process to achieve the discovery - what was the key idea behind the discovery that was not mentioned in the paper? Even if the ideas are mentioned in the paper, can the AI system infer the key ideas?

The AI system can propose a combination of ways of how the discovery happened and validate with the author. Solving this subtask should so we can elucidate the chain of thought behind all the mathematical discoveries we have in our history.

Recreate the history of mathematics

Given 17th-century mathematics (Newton, Leibniz, Fermat), can you derive 18th-century mathematics (Euler, Lagrange, Laplace)? Can you derive 19th-century mathematics? 20th-century mathematics? What are some of the ideas in mathematics that were most unlikely to be discovered?

If we want to discover new mathematics, we need to solve this subtask for re-discovering old mathematics.

Identify errors in published research

Not all published mathematics research is correct. Can the AI system figure out what was incorrect?

Before AI proposes a solution to an open problem, it needs to be able to check whether its solution is correct. If the AI has been proven to be able to consistently identify mistakes in drafts and in published research, then it is worth it to spend human time to evaluate the solution from the AI.

When we solve this subtask, we have a good validator and we can achieve superintelligence in mathematics.