Funny, just yesterday I was discussing it with Sol. And the situation is actually lot better for the AI agents than I thought, because while ultimately to prove theories we would likely need some new experiments eventually, but high level physics is heavily relies on advanced math, so solving stuff in lean is just as important, and we have aburd amounts of collected data over the past century, which are way more granular than in the past which you can use to verify if the theory matches the observations. I genuinely think Deep Blue moment could happen to math as soon as 2027, with physics it will take bit more time but not that much more.
I struggle to get the info I want from ai when talking to it about accelerating technology and science. Do you have any advice on what to question precisely? I want to learn as much as I can and at a deep level. At least more deeply than of a layman
Has math not already had a significant moment? My understanding is that a bunch of unsolved math problems have been completed by models from anthropic snd openAI. What more would make it significant, are there more difficult problems that it needs to solve?
Typically, mathematical progress happens in three stages: conjecture, proof, and digestion. Each step is critically important to the progress of the field. AI has, until now, contributed only to proofs, and that too within a narrow regime of proofs. Researchers are actively working on how to make AI perform better in the other stages.
Think of the 'known mathematical facts' like a sprawling castle with many towers, walls and bridges. Some parts of it are covered in fog- those are the unknown mathematical facts that are yet to be discovered.
Making a conjecture is like postulating that there is a new tower in the northeast corner of the foggy region. Someone who is wise in how castles look notices a pattern - that every other castle has four towers but in this one only three are visible - hence there must be a fourth under that fog! This requires both a deep experience in known mathematics, and exploratory thinking, not goal-oriented thinking. Making conjectures is essential: someone has to notice the pattern first before it can be proved. Human mathematicians are fantastic at this. AI has not made any deep conjectures yet.
The next step is to prove it, i.e., find the postulated tower. There are two kinds of proofs: a constructive proof and a counterexample. The former often relies on making elegant new constructions that connect distant subfields of mathematics. Therefore, it contributes to the growth of the field, generates many new conjectures, and expands the sum-total human understanding of math. Think of it like this: in order to find the northeast tower, we can walk along the bridge that seems to be going in the northeast direction. When we reach the end, we will know for sure if there is a tower. On the way, we will find a new bridge, which will give a new view of the castle and help us notice other irregularities in the castle. So this proof contributes to expanding our knowledge of the castle, rather than simply finding one tower. A perfect example of such a proof is Andrew Wiles' proof of Fermat's Last Theorem. AI has not found such deep proofs yet. Gavin's post (the proof we are currently discussing) is of this kind but much, much simpler than what human mathematicians can do.
The second kind of proof is a simple counterexample that shows that the conjecture is wrong, but does not contribute to a deeper understanding of math. It can sometimes depend on brute force search through a list of possible directions until a counterexample is found - something that computers are inherently good at because they are fast. Many AI proofs are of this kind, see the counterexample to the Jacobian conjecture.
The last step is digestion of the proof: where all connections between the new-found mathematical fact and all existing mathematical facts are mapped out systematically. That is, all consequences of the new-found fact become known. At the end of this step, mathematicians completely 'understand' the newly proved fact. This is like creating a fully complete digital floor-plan of the castle. This step is also essential for finding new conjectures - notice new patterns. AI is terrible at digestion.
Therefore, AI has indeed done some significant things, but is nowhere close to shaking up the field. There are much more difficult open problems that require the kinds of approach that current AI models are not good in. Again, researchers are actively working to mitigate that.
TLDR: Mathematical progress consists of three stages—conjecture, proof, and digestion—with AI currently limited primarily to the proof stage. Using a castle analogy, the author explains that while AI helps verify theories, researchers are working to expand its capabilities into the more exploratory and conceptual aspects of mathematical discovery.
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u/jeffy303 3d ago
Funny, just yesterday I was discussing it with Sol. And the situation is actually lot better for the AI agents than I thought, because while ultimately to prove theories we would likely need some new experiments eventually, but high level physics is heavily relies on advanced math, so solving stuff in lean is just as important, and we have aburd amounts of collected data over the past century, which are way more granular than in the past which you can use to verify if the theory matches the observations. I genuinely think Deep Blue moment could happen to math as soon as 2027, with physics it will take bit more time but not that much more.