Did AI Just “Solve” Math? (Let’s Take a Closer Look) | AI Reality Check
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Did AI just solve a significant math problem?
Last week, OpenAI published a press release titled, An OpenAI Model Has Disproved a Central Conjecture in Discrete Geometry. They were talking specifically about the planar unit distance problem, which was first posed by Paul Erdos in 1946.
What did OpenAI achieve with their recent announcement?
Now this is actually a pretty simple problem to state. It basically says, What is the maximum number of pairs of points in a set of endpoints in a flat plane that can be exactly one unit of distance apart? Now, back in the 1940s, Erdos proposed an answer to this question.
Is the result of AI's math breakthrough truly important?
He couldn't prove it, but he thought he knew what the answer was.
Are LLMs now smarter than human mathematicians?
Last week, OpenAI essentially announced that they had used an LLM to prove that Erdos' proposed answer was, in fact, incorrect. The OpenAI press release was accompanied by a video that featured dramatic music and a group of researchers writing earnestly on a comically small blackboard as they explained why this was a big deal. Here, let's play a clip of that video.
This is the first mathematical breakthrough due to an AI. It's been described as the most well-known problem in combinatorial geometry. So for a whole subfield of mathematics, it's like maybe the best known problem there is.
The mainstream press soon picked up on this story with enthusiasm. Here's the new scientist headline. Mathematicians stunned by AI's biggest breakthrough in mathematics. People on X predictably went even more wild.
Will AI conquer all equally hard mathematical challenges?
Peter Diamandis tweeted the following. An open AI model just proved an 80-year-old math conjecture from Paul Erdos, one of the most prolific mathematicians in history. We're going to solve everything. All right, so what's actually going on here? Did AI just reach genius level? Has math as a discipline just been automated?
What is the future of mathematics in the age of AI?
As a theoretical computer scientist myself who has published a lot of applied mathematics research in my days and someone who proudly boasts an Erdos number of three, which you can look up if you don't know what that means, I am, for obvious reasons, particularly interested in these questions. Well, it's Thursday, which means it's time for an AI Reality Check episode of this show, which is the perfect opportunity to seek some answers. So that's exactly what we're going to do. As always, I'm Cal Newport, and this is Deep Questions, the show for people seeking depth in a distracted world.
All right, so we need to start by getting more specific about what exactly OpenAI actually did, and then we can get into the implications of what that means for the rest of us.
How does this AI breakthrough affect mathematicians?
All right, so we're looking at this unit distance, planar unit distance conjecture. Erdos was convinced that he had identified the answer to the question.
What are the implications of AI on mathematical reasoning?
I don't want to get too mathy here, but just to say it quickly, Erdos thought that if you were placing endpoints into the plane, the maximum number of points that you could get to be a unit distance apart would be upper bounded by n raised to the power of 1 plus some constant c divided by the double log of n. Now, as you're probably noticing as you listen to me, that second term in the sum is going to tend towards 0 as n increases, asymptotically speaking. So this result, the answer... asymptotically is going to approach plain linear as the point set increases. That's a really elegant answer. Erdos was convinced that was right. A lot of other mathematicians just assumed that that was right because Erdos is usually right.
And so people tried for a long time to prove that was indeed the fundamental limit. Now, what OpenAI did was they released a paper that said, no, that's wrong. We actually have a counterexample. We have a way of placing points that That is going to have more points. We're going to feature more points at unit distance than that limit, even as n increases. I believe the actual bound is something like n plus 1 plus some small fixed constant epsilon that stays fixed as you increase n as opposed to approaching 0. So they had a counterexample. They didn't say... Here's the right answer. Here is what the limit is. Here's the best. Here's the most possible point. So they didn't replace Erdos' conjecture with a better conjecture.
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Chapters
8 chapters
1
Did AI just solve a significant math problem?
0:00–0:17
2
What did OpenAI achieve with their recent announcement?
0:17–0:34
3
Is the result of AI's math breakthrough truly important?
0:34–0:37
4
Are LLMs now smarter than human mathematicians?
0:37–1:34
5
Will AI conquer all equally hard mathematical challenges?
1:34–1:58
6
What is the future of mathematics in the age of AI?
1:58–2:48
7
How does this AI breakthrough affect mathematicians?
2:48–2:56
8
What are the implications of AI on mathematical reasoning?
2:56–31:41
Speakers
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