Math contains a lot of stuff

AI is getting good at math lately. Maybe we we will soon see AI that is much better than humans on all math tasks ("Math ASI"). What could this mean?

There are at least three maths:

• Math A (Ancestral) is when you study natural mathematical objects. To be considered natural, there has to be motivation from the real world, other math, or intuition. A big part of this is proving theorems.
• Math B (Benchmarky) is when you resolve conjectures from Math A. This is often used for benchmarking academics, and nowadays AIs.
• Math C (Computational) is when you prove things in a formal system such as ZFC or Lean.

Math B is the set intersection of maths A and C.

Math A is weird and mysterious, what exactly counts as natural? On the other hand, math C is perfectly verifiable non-physical tasks. Maybe the successes of AI in math B are due to it being a subset of math C, and we should soon see math C ASI? But math C contains a lot of stuff!

Demonstration/Hmm: factoring

Consider the question for some integers : is there a factor of below ? It's considered that the only way to answer this in general is:

1. Invent a scalable quantum computer.
2. Run Shor's algorithm on it to find the factors of .
3. Check if the smallest factor is below .

According to experts, humans might be able to execute this soon-ish. And this is a math C-ematical question, so maybe a real math C ASI has to have a quantum computer ¯\_(ツ)_/¯

Demonstration 2: algorithms

Consider the question: is there an algorithm for sorting/matrix multiplication/etc that runs like super fast?

This might appear to not be a mathematical question, but let's examine it:

• "algorithm for sorting/matrix multiplication/etc" - algorithms can be represented as code in some programming language, it is a mathematical statement to say that a program computes some function.
• "runs" - models of runtime cost on given hardware are mathematical objects. Maybe it's hard to get a perfect model because there is a lot going on with some of it secret, but a simplified model should not be too far off.
• "super fast" - models of expected input distribution are mathematical objects. We can operationalize "super" as on average 10% faster than the current best known algorithm.

When humans invent super fast algorithms they rarely prove such theorems, but it seems this wouldn't be very hard if tried, just very annoying and not useful. The theorem statement could also be formalized before actually inventing the algorithm.

So a math C ASI should also be an algorithm optimization ASI.

There is a wrinkle because, though it sounds unlikely, it could be possible to prove the existence of such an algorithm without actually inventing it. We could solve this by admitting problems of the form "give an explicit satisfying ..." to our definition of math.

Demonstration 3: AI development

Consider the question: is there an algorithm that implements a super smart AI?

OK, I don't know how to make that question mathematical. Famously ML is a weird science thing.

But consider the question: is there an algorithm that runs reasonably fast and compresses [giant blob of data from the internet] very well?

Giant blobs are mathematical objects after all. You also need a good definition of lossy compression, and maybe the landscape of time/space/training/inference cost is complex, but still it seems easier to ask the question than invent the answer, which as currently known is LLMs. Plausibly a significant proportion of work that people do and have done in improving LLMs is answering perfectly stateable questions ("can we improve some parameter of this a bit").

So a math C ASI should be kinda good at parts of AI development.

Conclusions

So this looks to be an AI bull case because AI is improving at math and I'm saying being good at math is being good at improving AI.

For an AI bear case, we should focus our cope crystals on ways that the improvements might not generalize to all of math C. I'm hearing the current most impressive results are within a small part of math B (counterexamples something), so there is potential here.