Every time OpenAI releases a batch of mathematics results, I find myself asking the same question as @Raemon. Are these results the product of some important kind of “creativity,” or something different? Raemon’s framing was that models “still seem to be missing some basic sauce that lets them actually form new concepts on the fly and apply them.” I agree, and in this post, I want to propose a way to detect the basic sauce.
There are two ways we evaluate frontier models on mathematics, and neither suffices for uncovering the basic sauce.
The first is math benchmarks. As benchmarks saturate (partly from newer models training on public ones, partly from models improving), newer ones appear that can make stronger claims about model performance. But that’s exactly the problem. Benchmarks evaluate accuracy on some task. Good benchmark performance rarely reveals anything about how the model got there. At worst the model had excellent recall, and at best, we see that it succeeds but we can’t really explain why.
The second is real-world results, from @dynomight's scattered thoughts and other mathematicians’ experiences, to the recent OpenAI result. At minimum, human mathematicians feel clear productivity gains from using AI, and at most, we are watching AI disprove the Exponential Time Hypothesis. But the models still struggle with abstraction, their proofs are hard for humans to read, and the secretive nature of frontier model releases leaves so much room for doubt, amongst mathematicians and non-mathematicians alike. Are the models actually exhibiting important creativity? Or are these results coming out of emergent properties of model swarms? Or is mathematics (or a subset of it) actually a lot more brute forceable than previously thought?
I like Raemon’s term “basic sauce” because it perfectly captures the fact that even mathematicians don’t quite know what to call the thing that makes them good at math. We could say taste or creativity or judgement or basic sauce, but for the time being I’m going to taboo the word and just describe the outward observables I would expect from a model that has it.
For starters, I would expect that such a model performs near-perfectly on solved mathematics, e.g. competition and college textbook mathematics. Benchmarks seem to suggest we are already there, and empirically this is what I have observed from frontier models. It is more challenging to give expectations for a model solving open problems. For one thing, it is hard to even evaluate models on problems that are unsolved (at least as far as the model’s training cutoff goes). Benchmarks that evaluate models on unpublished problems or problems solved after their cutoff dates enable automatic verification by using problems with single, exact solutions. When actual proofs are evaluated, it is by human mathematician judges, which does not scale.
I think we can get further by looking at the complement: when the model is solving open problems, what behaviors would I consider insufficient to say it has the sauce? If the model is recalling facts straight from its training corpus, that is not forming “new concepts on the fly.” If the model is reading surface features and collecting a bag of heuristics, that does not suffice either. And I would not count straightforward search of the option space. Each of these is an advantage that an AI model may have over a human solving the same open math problem, but none of them quite scream that the model has got the “basic sauce.”
But what if a model could do better than each of these? Suppose you have three comparison policies, each of which has the benefit of one of the above advantages and nothing else:
A recall policy: makes the classic textbook move
A surface policy: a logistic regression on features visible in the prompt
A search policy: tries each candidate for k steps before selecting an option, with compute equal to the model under test.
One recall model, one surface model, one search model. If our model performs better than each of these on some open mathematical problem, it narrows down something for each:
If the model beats the recall null, it is doing something more than just remembering facts.
If the model beats the surface null, it is doing more than just pattern-matching.
If the model beats the search null, it is doing something more than local search.
Maybe these nulls are too weak or too few. Then we add more. But whatever surplus remains when a model does better than the best null, that leftover is where the basic sauce lives, if it exists at all.
That is what I would expect on an open math problem from a model with the basic sauce. But to compare a model against these null models, we still need problem domains that can be scored, and this is still tricky with open math problems. The benchmarks I mentioned before get verifiability by constructing problems with exact solutions or via human judges. And they get around recall by using unpublished or post-cutoff problems.
But the null framework offers a different way around recall. If we have a data point for how well a recall model does, we can control for recall by only considering performance beyond it! In other words, contamination stops being a disqualifier and just becomes a quantity we subtract. That lets us leverage mathematical problem spaces older than the newest math, as long as there is enough headroom between the best null and an oracle.
A model that can beat the best of those null hypotheses is doing something new and on the fly. I think benchmarks can tell us how well a model performs, but even on benchmark problems, measuring the model against these nulls gives us the added benefit of characterizing the problem space itself to the point where we can evaluate what kind of advantage actually led the model to do well. The highest performing null model tells us what kind of strategy works well in that particular domain. Basic sauce is being able to come up with better strategies than the best cheap one.
Every time OpenAI releases a batch of mathematics results, I find myself asking the same question as @Raemon. Are these results the product of some important kind of “creativity,” or something different? Raemon’s framing was that models “still seem to be missing some basic sauce that lets them actually form new concepts on the fly and apply them.” I agree, and in this post, I want to propose a way to detect the basic sauce.
There are two ways we evaluate frontier models on mathematics, and neither suffices for uncovering the basic sauce.
The first is math benchmarks. As benchmarks saturate (partly from newer models training on public ones, partly from models improving), newer ones appear that can make stronger claims about model performance. But that’s exactly the problem. Benchmarks evaluate accuracy on some task. Good benchmark performance rarely reveals anything about how the model got there. At worst the model had excellent recall, and at best, we see that it succeeds but we can’t really explain why.
The second is real-world results, from @dynomight's scattered thoughts and other mathematicians’ experiences, to the recent OpenAI result. At minimum, human mathematicians feel clear productivity gains from using AI, and at most, we are watching AI disprove the Exponential Time Hypothesis. But the models still struggle with abstraction, their proofs are hard for humans to read, and the secretive nature of frontier model releases leaves so much room for doubt, amongst mathematicians and non-mathematicians alike. Are the models actually exhibiting important creativity? Or are these results coming out of emergent properties of model swarms? Or is mathematics (or a subset of it) actually a lot more brute forceable than previously thought?
I like Raemon’s term “basic sauce” because it perfectly captures the fact that even mathematicians don’t quite know what to call the thing that makes them good at math. We could say taste or creativity or judgement or basic sauce, but for the time being I’m going to taboo the word and just describe the outward observables I would expect from a model that has it.
For starters, I would expect that such a model performs near-perfectly on solved mathematics, e.g. competition and college textbook mathematics. Benchmarks seem to suggest we are already there, and empirically this is what I have observed from frontier models. It is more challenging to give expectations for a model solving open problems. For one thing, it is hard to even evaluate models on problems that are unsolved (at least as far as the model’s training cutoff goes). Benchmarks that evaluate models on unpublished problems or problems solved after their cutoff dates enable automatic verification by using problems with single, exact solutions. When actual proofs are evaluated, it is by human mathematician judges, which does not scale.
I think we can get further by looking at the complement: when the model is solving open problems, what behaviors would I consider insufficient to say it has the sauce? If the model is recalling facts straight from its training corpus, that is not forming “new concepts on the fly.” If the model is reading surface features and collecting a bag of heuristics, that does not suffice either. And I would not count straightforward search of the option space. Each of these is an advantage that an AI model may have over a human solving the same open math problem, but none of them quite scream that the model has got the “basic sauce.”
But what if a model could do better than each of these? Suppose you have three comparison policies, each of which has the benefit of one of the above advantages and nothing else:
One recall model, one surface model, one search model. If our model performs better than each of these on some open mathematical problem, it narrows down something for each:
Maybe these nulls are too weak or too few. Then we add more. But whatever surplus remains when a model does better than the best null, that leftover is where the basic sauce lives, if it exists at all.
That is what I would expect on an open math problem from a model with the basic sauce. But to compare a model against these null models, we still need problem domains that can be scored, and this is still tricky with open math problems. The benchmarks I mentioned before get verifiability by constructing problems with exact solutions or via human judges. And they get around recall by using unpublished or post-cutoff problems.
But the null framework offers a different way around recall. If we have a data point for how well a recall model does, we can control for recall by only considering performance beyond it! In other words, contamination stops being a disqualifier and just becomes a quantity we subtract. That lets us leverage mathematical problem spaces older than the newest math, as long as there is enough headroom between the best null and an oracle.
A model that can beat the best of those null hypotheses is doing something new and on the fly. I think benchmarks can tell us how well a model performs, but even on benchmark problems, measuring the model against these nulls gives us the added benefit of characterizing the problem space itself to the point where we can evaluate what kind of advantage actually led the model to do well. The highest performing null model tells us what kind of strategy works well in that particular domain. Basic sauce is being able to come up with better strategies than the best cheap one.