Mathematics and AI

[Ves]

I am a math PhD student. Because of this, I thought I'd talk about the recent achievements of AI from the perspective of someone within the mathematics community. I was originally going to type some kind of essay, but I don't think I can do better than the open letter to the Mathathon signed by a large number of mathematicians:
https://docs.google.com/document/d/1IL0b2oG2KvvSnxn_DuXsNxuHaCefDJ9hNY1k9BP3kKQ/view?tab=t.0

There are also good talks given by Terence Tao on this subject. Here is one at the ICM:

And here is a talk intended for a more general audience:

The basic idea, which applies to many fields, such as math and software engineering, is that the real bottleneck is human understanding.

For math, that is essentially the entire point of the field. If you have a machine that can generate a proof of some result by cobbling together existing knowledge, especially if it's done in an inelegant way that is difficult for humans to understand, then human understanding is not advanced very much, and no new mathematics was developed. That will only happen if someone reads the proof, takes time to extract the methods that were used to solve it, communicates them to others, and passes along the knowledge to "the next generation."

For software engineering, the bottleneck is not in writing code (once you're sufficiently experienced with this), it's in understanding the problem you're trying to write code to solve and understanding the current state of your codebase so you know what changes to make and how to design your software on a high level. Even if you use AI to write code for you, you have to read the code the AI has generated and understand what it does, which takes time.

Now, this is assuming we actually care about human welfare, that we want to have the humans understand what the AIs are doing, and that we want to have humans "directing things". It seems like many of the people at top AI companies assume that humans are not going to be in charge of the future in the next decade, but they're all in a race to the bottom and so they can't help but destroy everything (from their POV). I know people at some software companies in the bay area where everyone is just using AI to write thousands of lines of code each day, and nobody can peer-review the code because nobody knows what's happening anymore.

I also have concerns for the funding of math. If funding primarily comes from people with money who only care about math for new applications that can make them more money, and if AI can give them results with none of the human understanding (since they don't care about that), the field as a whole could dry up if there are no resources provided for it. One practical way this would be bad (if we care about humans knowing what is going on) is if we lose a huge number of people that understand math well, we lose a huge number of people that can understand what the AIs are doing when they are doing math. Personally, though, it would also be very sad to see the field die.

I could say more, but maybe I'll just leave it at it seems the field of mathematics is going to have to undergo a huge paradigm shift, but nobody exactly knows how.

[Ves]

Update: It seems the open letter partially worked!

https://www.businessinsider.com/openai-caltech-ai-math-hackathon-backlash-anthropic-2026-9

[Xonas Pitfall]

On 11/09/2026 at 0:07 AM, Ves said:

I also have concerns for the funding of math. If funding primarily comes from people with money who only care about math for new applications that can make them more money, and if AI can give them results with none of the human understanding (since they don't care about that), the field as a whole could dry up if there are no resources provided for it. One practical way this would be bad (if we care about humans knowing what is going on) is if we lose a huge number of people that understand math well, we lose a huge number of people that can understand what the AIs are doing when they are doing math. Personally, though, it would also be very sad to see the field die.

Super interesting! Although I doubt the companies would actually want people who understand math to disappear. Most likely, they’ll just end up working more on correcting, evaluating, and guiding AI. I have a feeling that’s where a lot of jobs will be heading in some form.

For example, with coding/software development, systems design and having good "taste" in UI/UX/design seem more important than ever as AI gets better at actually writing the code. I’d imagine something similar could happen with math.

But I know very little about this subject, so it’s really interesting to hear about. Thanks for sharing!

I’d love to hear more of your thoughts in future threads if you feel like sharing more 

[Leo Gura]

On 9/10/2026 at 3:07 PM, Ves said:

nobody knows what's happening anymore.

Tagline of the 21st century

[Leo Gura]

A few serious points:

1) AI can be used to find interesting correlations between all the math humans know so far. There are probably interesting low-hanging fruit to pick just from connecting unconnected dots in existing knowledge. But at some point this easy fruit will run out.

2) Some kinds of math problems can be solved with a brute force approach. Not nearly all, but some. So AI could play a role there, similar to how it did with the protein folding problem with AlphaFold.

3) AI companies are definitely betting that humans won't even need to understand things anymore in the future because AI will magically handle everything. However, my position is that this is a sci-fi fantasy, not how it will really work. There is no good evidence yet that AI can replace human understanding. If it can, that is the paradigm shift, but it may never come.

Edited yesterday at 01:34 AM by Leo Gura

[JTL]

Have you seen this? @Leo Gura I just saw someone mentioning it on reddit, but I found a article aswell so I think it is legit. This is insane.

https://futurism.com/artificial-intelligence/drama-openai-supposed-mathematical-breakthrough

https://www.reddit.com/r/singularity/comments/1wafkt4/breakthrough_for_navierstokes_from_tristan/

It seems openai reached out to him and offered him that he could be lead author on a rewrite of OpenAI’s Navier-Stokes proof.

 

Edited 22 hours ago by JTL

[Butters]

9 minutes ago, JTL said:

Have you seen this? @Leo Gura I just saw someone mentioning it on reddit, but I found a article aswell so I think it is legit. This is insane.

https://futurism.com/artificial-intelligence/drama-openai-supposed-mathematical-breakthrough

https://www.reddit.com/r/singularity/comments/1wafkt4/breakthrough_for_navierstokes_from_tristan/

It seems openai reached out to him and offered him that he could be lead author on a rewrite of OpenAI’s Navier-Stokes proof.

 

You shouldn't take those things seriously. OpenAI is a commercial operation run by a scam artist, not some sort of scientific institution lol.

[Uddi]

True intelligence requires awareness and AI doesn't have that so it cannot be trusted for critical systems without human supervision and intervention. I am so tired of this AGI hype.

[Ves]

On 9/12/2026 at 8:14 PM, Leo Gura said:

1) AI can be used to find interesting correlations between all the math humans know so far. There are probably interesting low-hanging fruit to pick just from connecting unconnected dots in existing knowledge. But at some point this easy fruit will run out.

Yes. There is so much published mathematics that each person only knows a tiny fraction of what exists, even in their own field. I mean, I study algebraic geometry, which was revolutionized by Grothendieck in the 60s. He published a ton of papers laying out the new foundations of the field ( EGA, FGA and SGA). I haven't read almost any content from any of these, only textbooks which are ultimately derived from these papers, and I only know some of what's contained in the textbooks. My advisor certainly does not know most of what is contained in Grothendieck's papers. And this is only the "foundations" of the field, let alone the thousands of problems that have been solved in AG.

So, it absolutely seems like AI will be used to sort of fill in the "convex hull" generated by our current knowledge, and this is mostly what will happen first (by people unwilling to spend $10 million+ on compute to solve a hard problem like OpenAI did for Navier-Stokes). The question is, who is going to read the massive influx of papers that are being created and submitted to the arXiv? People are considering splitting the arXiv into two different sites to handle this.

 

On 9/12/2026 at 8:14 PM, Leo Gura said:

3) AI companies are definitely betting that humans won't even need to understand things anymore in the future because AI will magically handle everything. However, my position is that this is a sci-fi fantasy, not how it will really work. There is no good evidence yet that AI can replace human understanding. If it can, that is the paradigm shift, but it may never come.

Yeah, this is one of the big incoming societal issues I think. There's likely going to be a huge division between people who actually know things and can read, interpret, and understand what an AI is doing to help guide it (if they use it for whatever reason) vs. people that sort of just use it to do what they can't and remain ignorant. Also in this second camp are probably those that are too easily fooled into believing what it's saying is wise/a prudent response because of some general pro-AI biases, or because the output is different than what they would have thought of, etc.

My girlfriend is a software developer, and she already sees this. There are people not on her team that use AI heavily, but also clearly don't understand the goal they're trying to accomplish or the details of their situation as they use it. It's very frustrating to try to work with them. Recently there was a series of meetings, it's somewhat of a long story, but in the last one these people were still confused about very basic details of the project (and it is not at all a complicated project, it's just deciding the best way to allocate some group IPs). At the conclusion, they asked for information about the project details so that they could send it to Claude to make a short summary. She was like "...I can just summarize what we are doing for you...".

A more relatable (but less specific) example is Suno, the AI music generator. Adam Neely has a great video on it. I just think about how many people are going to *avoid* learning an instrument or any music theory because they have access to an AI music generator that can generate whatever they want. It's unfortunate. Of course, there is also the huge crisis in education...

[Ves]

21 hours ago, JTL said:

Have you seen this? @Leo Gura I just saw someone mentioning it on reddit, but I found a article aswell so I think it is legit. This is insane.

https://futurism.com/artificial-intelligence/drama-openai-supposed-mathematical-breakthrough

https://www.reddit.com/r/singularity/comments/1wafkt4/breakthrough_for_navierstokes_from_tristan/

It seems openai reached out to him and offered him that he could be lead author on a rewrite of OpenAI’s Navier-Stokes proof.

 

I think this isn't a very clear-cut situation with the information that is publicly available, it seems mostly like a breakdown of communication to me. You can read Tristan's statement here, and OpenAI's response here.

In my opinion, there was some skullduggery by OpenAI in that they heard a rumor that these two collaborating mathematicians (one of which works at Anthropic) had made a breakthrough in this Euler problem, and then they decided to drop a ridiculous amount of money on Navier-Stokes as a result. It seems that the resolution of Navier-Stokes use different that the method that Tristan and Levent used to construct finite time blowups of Euler equations (something about "forced" vs. "unforced"; I don't know much about the field of PDEs in general), but I sort of wonder if OpenAI just got lucky that their model took a different route than Tristan and Levent.

Tristan sort of jumped to conclusions that OpenAI had accessed his conversations with Codex, for reasons that seem reasonable in his account of what happened. There was also some pettiness with OpenAI not wanting Levent to be an author on the paper because he worked at Anthropic. I have a pretty low opinion of Sebastien and his colleagues after seeing some of them mock a heartfelt response to Levent on X, but whatever.

The effect of this drama that I see is that mathematicians are concerned that OpenAI is training models off of their conversations (it is, unless you "opt out", who knows if that actually does anything), which would result in their arguments being plagiarized if AI regurgitates them in future solutions. This is a general issue with AI in math: it doesn't know who was responsible for the data it's trained on, so it can't cite any sources properly.