Gorgon Zee
Legend
Starting a new thread to give a little clarity on this news item. There has been a lot of heated discussion about this and especially its impact on mathematicians. I am a trained mathematician by trade who now works to use AI effectively and ethically within a not-for-profit in a US hospital system, so I had some interest!
What did the AI actually do?
Quoting science magazine: "Nearly 200 years ago, Claude-Louis Navier and George Gabriel Stokes sought to describe the chaotic motion of fluids mathematically, writing down equations that are still routinely used by everyone from aircraft designers to climatologists. But the formulas they developed weren’t guaranteed to describe all possible conditions. Mathematicians have spent decades trying to either prove the equations can do that or, alternatively, that there are situations in which the equations break down and give a nonsensical result."
What OpenAI did is to find an example of a situation where the equations do indeed break down and gave a nonsensical result.
How impressive is that?
Well, the equations are complex and people have been trying to do it for a long time, so it's not easy to do. But finding an example where things don't work is one of the easier mathematical tasks. Because the equations are complex, people rely on intuition to try and find an example, guided by their expertise. But another way of doing it is simply to try a lot of ideas and see if any work. The OpenAI approach was to use an enormous amount of computing power to find the example. It was guided first by experts who decided that this was the problem that they should work on, and then by a trial experiment for simpler cases, before finally setting 10,000 agents to try stuff out.
Honestly, I'm not that impressed. I expect computers to be better than people at trying out a huge range of possibilities and evaluating them. Finding a specific case is really what they should be best at. If you asked me to disprove the statement "All published 18th century English poems have the word 'the' in them" without a computer, I'd have the dickens of a time trying to work out which poets are most likely not to use that word, searching their works, etc. It would take me forever. But a computer does that fast because it is good at searching and testing rapidly.
This is not the first time this has happened
In 1976 a computer was used to solve the four color problem (Four color theorem - Wikipedia). It produced a proof that detailed 1,936 configurations, tested each of them and showed the theory held for them. This was a really big deal at the time -- many mathematicians refused to beleive the proof until each of the configurations was checked personally by a human mathematician.
The 4 color problem is a much simpler one than the Navier-Stokes problem. But the basic idea is the same -- creating examples and checking to see if they fit the theory. In the 1976 paper, the authors were able to list all possible examples, and so the computer proved the theory held. In the 2026 paper, they could not list all possible examples, and so had to set agents to search the enormous space for reasonable examples, eventually finding one that invalidated the theory.
Are mathematicians out of a job?
Yeah, like there are a ton of great jobs available anyway! But seriously, both this and some previous AI successes argue that when it comes to generating, combining and modifying examples, and then evaluating them at scale, AI is a potential rival for human mathematicians. I'm not convinced that it can come up with truly unique solutions as it is at heart a mechanism for repackaging known information, but let's be really clear here -- an awful lot of science is repackaging known information into new and useful forms, so this is still a big deal.
If AI can do so well at mathematics, which is really tricky, can't it do even better in simpler areas?
LLMs don't process concepts directly, like people do (caveat: I am not an expert in the area of human cognition). They process concepts only through their expression in a syntax. So LLMs work best for subjects where concepts are routinely expressed in a formal, structured language. Yup, mathematics -- and also software coding, for which LLMs are extremely effective. It is actually because mathematics is so "tricky" for us that it is easy for LLMs.
If a mathematician (AI or human) says something in math-speak, there is an excellent chance it is provably right or wrong. Or if not, there are known ways to translate that statement into others that are provably correct translations. That isn't remotely true in many other fields.
Closing thoughts
GenAI is a phenomenal tool for combining information in ways that are well-structured, fit logically and aimed at a given goal. Some believe that as they become better at this, we will see results that are indistinguishable from top-tier human creativity. Others feel their inability to directly handle concepts and reliance on language is a hard limit on their ability to be truly creative. But for a large class of useful situations, second-class creativity by combining known information in helpful ways is valuable, and likely to remain so.
Aside: Did the OpenAI system use other scientist's prompts and so steal their ideas?
I don't know and frankly, do not really care. I feel pretty confident it could have found the example eventually so whether or not it "cheated" is not important from a viewpoint of understanding AI capability. We'll see more examples of this sort in mathematics, bioscience and pharmaceutical research and other areas where we have formal languages and a desire to find point examples.
What did the AI actually do?
Quoting science magazine: "Nearly 200 years ago, Claude-Louis Navier and George Gabriel Stokes sought to describe the chaotic motion of fluids mathematically, writing down equations that are still routinely used by everyone from aircraft designers to climatologists. But the formulas they developed weren’t guaranteed to describe all possible conditions. Mathematicians have spent decades trying to either prove the equations can do that or, alternatively, that there are situations in which the equations break down and give a nonsensical result."
What OpenAI did is to find an example of a situation where the equations do indeed break down and gave a nonsensical result.
How impressive is that?
Well, the equations are complex and people have been trying to do it for a long time, so it's not easy to do. But finding an example where things don't work is one of the easier mathematical tasks. Because the equations are complex, people rely on intuition to try and find an example, guided by their expertise. But another way of doing it is simply to try a lot of ideas and see if any work. The OpenAI approach was to use an enormous amount of computing power to find the example. It was guided first by experts who decided that this was the problem that they should work on, and then by a trial experiment for simpler cases, before finally setting 10,000 agents to try stuff out.
Honestly, I'm not that impressed. I expect computers to be better than people at trying out a huge range of possibilities and evaluating them. Finding a specific case is really what they should be best at. If you asked me to disprove the statement "All published 18th century English poems have the word 'the' in them" without a computer, I'd have the dickens of a time trying to work out which poets are most likely not to use that word, searching their works, etc. It would take me forever. But a computer does that fast because it is good at searching and testing rapidly.
This is not the first time this has happened
In 1976 a computer was used to solve the four color problem (Four color theorem - Wikipedia). It produced a proof that detailed 1,936 configurations, tested each of them and showed the theory held for them. This was a really big deal at the time -- many mathematicians refused to beleive the proof until each of the configurations was checked personally by a human mathematician.
The 4 color problem is a much simpler one than the Navier-Stokes problem. But the basic idea is the same -- creating examples and checking to see if they fit the theory. In the 1976 paper, the authors were able to list all possible examples, and so the computer proved the theory held. In the 2026 paper, they could not list all possible examples, and so had to set agents to search the enormous space for reasonable examples, eventually finding one that invalidated the theory.
Are mathematicians out of a job?
Yeah, like there are a ton of great jobs available anyway! But seriously, both this and some previous AI successes argue that when it comes to generating, combining and modifying examples, and then evaluating them at scale, AI is a potential rival for human mathematicians. I'm not convinced that it can come up with truly unique solutions as it is at heart a mechanism for repackaging known information, but let's be really clear here -- an awful lot of science is repackaging known information into new and useful forms, so this is still a big deal.
If AI can do so well at mathematics, which is really tricky, can't it do even better in simpler areas?
LLMs don't process concepts directly, like people do (caveat: I am not an expert in the area of human cognition). They process concepts only through their expression in a syntax. So LLMs work best for subjects where concepts are routinely expressed in a formal, structured language. Yup, mathematics -- and also software coding, for which LLMs are extremely effective. It is actually because mathematics is so "tricky" for us that it is easy for LLMs.
If a mathematician (AI or human) says something in math-speak, there is an excellent chance it is provably right or wrong. Or if not, there are known ways to translate that statement into others that are provably correct translations. That isn't remotely true in many other fields.
Closing thoughts
GenAI is a phenomenal tool for combining information in ways that are well-structured, fit logically and aimed at a given goal. Some believe that as they become better at this, we will see results that are indistinguishable from top-tier human creativity. Others feel their inability to directly handle concepts and reliance on language is a hard limit on their ability to be truly creative. But for a large class of useful situations, second-class creativity by combining known information in helpful ways is valuable, and likely to remain so.
Aside: Did the OpenAI system use other scientist's prompts and so steal their ideas?
I don't know and frankly, do not really care. I feel pretty confident it could have found the example eventually so whether or not it "cheated" is not important from a viewpoint of understanding AI capability. We'll see more examples of this sort in mathematics, bioscience and pharmaceutical research and other areas where we have formal languages and a desire to find point examples.
