[fred123123]

The reason i am asking is about research mathematics. In this application what is really important is continual intuition and theory building, this takes time and trial and error. However this is very fruitful in solving problem. Currently the LLM can do mathematics that is somehow already in their weight very well. See the recent things. But i suppose true research would require continual learning that goes beyond the context window. Thats why i am wondering if this solvable. My intuition is that it is easily solvable, hence we could make a true mathematical continuously building theory to prove things.

[HarHarVeryFunny]

For that sort of application then maybe a chunked vector store memory would work if 1M context is too small to hold the entire history? In Terrance Tao's ChatGPT discussion it seemed that the LLM was making good use of the in-context information.

I think for true, especially open-ended, research you really also need things like curiosity, but perhaps a lot can still be done with the same sort of prompting that got the Jacobian result - just tell the model to keep on going, investigate anything that seems interesting/unexplained, etc?

[fred123123]

Yes maybe you could scheme it, like think of 10 things you could try, spawn subagents, try them, perhaps one would succseed and yes that think that worked could be placed in a memory system. But there is also value in things that dont work ect. So true continual learning would be really good. I am currently reasearching in mathematics and some things it cant get, i really have to think of them myself. But other things where it is easier to "build" from already existing knowledge works really well. I kinda wanne know how this will continue. It can do things that are "in distribution", but can it build its own new distribution and continue building. Mabye ill try this using a normal RAG system. I would concider it a succsess if it create a new "theory" like 100 pages genuinely new stuff, and then applies it to a problem. But my intuition is this will not work lol

[HarHarVeryFunny]

Yes, that would be the dream - new scientific/mathematical discovery - but I share your intuition that we are not there yet!

I don't know what it would take to have an AI that might make a brand new discovery/theory like this... part of it would be the mechanics of continual learning, curiosity, etc, but maybe also part education/mentorship?. How/when does a human mathematician, tackling a problem and getting nowhere (or maybe via some other motivation), end up inventing a whole new framework/paradigm to make progress ?!

I think it would be very surprising to see today's AI (LLMs) do this, because there is just too much missing. LLMs were designed only to be next token predictors - i.e. copying/automation machines. There are trained to predict, hence copy, what they have seen before, not to innovate (other than by combining what they have already seen). Humans may learn and process language in a way similar to an LLM, based on prediction, and our brains are certainly evolved for prediction, but there is a lot more to us than that. At least half of our cortex is dedicated to feedback and learning - what to do when our predictions are wrong. LLMs don't have this. When an LLM prediction "fails" it hallucinates. When a human prediction fails other mechanisms kick in (curiosity, boredom, frustration), some purely cognitive, some emotional, and we potentially explore, innovate and overcome.

Presumably if you try the RAG (vector store) approach, then one necessary component would be for the system to continually review what it has already learnt and consolidate/expand that by looking for generalizations, exceptions, contradictions, implications, etc. In fact maybe this would be the core of the system, with subagents assigned to each of these (and more) tasks. You would just "seed" the memory with the task you wanted it to work on, some hints on direction, difficulties, etc, then let it get to work. This would be rather like a GOFAI "blackboard" system where a bunch of loosely collaborating agents make progress on a problem by using a shared blackboard to get new work and post results (which then become inputs to other agents).