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An OpenAI model has disproved a central conjecture in discrete geometry
641–650 of 1001 posts
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#642Earlier quoted context omitted.
Personally I don't find this to be true anymore! It's not always great and does still will often tend towards unneeded complexity (especially if not pushed a bit), but I often find GPT 5.5 writing code I would have written myself. This was very much not true with earlier models (who make something that worked, but I'd always have to rewrite to make it "good code").
Personally I found 5.5 a massive step back from 5.4. Both of them still use way too many fallbacks and unnecessary checks, especially if you're having it output php. It's fine if you're just one person and checking everything and able to catch and correct. But it's really bad when you have a team all using it, not checking the output and trusting it's output leading to spaghetti code. Technically works, but very mess…
Yeah, it has a tendency to default to "smallest local hack that will work" and code as defensively as possible.
BUT I have had great success using AGENTS.md and becoming better at prompting to get it to not be like this.
Basic approach in AGENTS.md: don't code defensively, yada yada, we have a validation layer at X, no need to check for anything behind that layer. Works well.
An approach I've found helpful when prompting: What would be the best architecture for this change? If you say "do X" it'll tend to just do the hackiest, shortest path thing. If you say, "what's the best way to do X?" it will think more holistically.
That said, who knows, maybe when it's PHP it just really wants to hack ;-)
(Also, yes, you still need to review the code -- it will still do stupid things, so you can't just be pure hands off w/o ending up with quality degredations. The same is true of humans too though in my experience...)
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#643As someone starting grad school for pure mathematics, this has me both excited and nervous, but mainly the latter...
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#644Earlier quoted context omitted.
Today I generated the equivalent of two LOTR books just to fix three missing rows in my SQL models (and open a PR), so +1
or put differently, you melted x cubic meters of polar ice
Based on some napkin math, that would be about ~100 watt hours of electricity on an H100 cluster, or, roughly the same amount of energy needed to boil a kettle for a cup of tea.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#645Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…
> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.
Always, always always, the problem with research and development is leadership, not insufficient supportive technology. It is a political problem, there is absolutely, positively no shortage of technologies to support research. Your optimism is totally misplaced. The NSF funding cuts have negatively impacted math more than AI has benefitted it. And guess who supports the administration that cut NSF funding? The people who ousted the PhDs from OpenAI.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#646I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)
PhDs used to mean publishing a novel mathematical result: when has that changed?
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#647The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…
What you describe here has always been true in all sciences, but also in medicine. But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers. So the crossdomain pollination that used to exist in scientists is not only not encouraged. It's also actively punished by society.
> But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers.
Can you explain more what you're referring to, because this has not been my experience at all. Heck, when I went to college, cross disciplinary majors were all the rage.
I think the thing that is just factually difficult is to actually become skilled in multiple different domains, precisely because the level of study/practice/rehearsal to become proficient in any individual domain keeps going up.
A long time ago you could be a Renaissance man by essentially dabbling in different fields. But today, as this article points out, you need extremely deep expertise in any one area just to understand the status quo - this proof required extremely deep expertise in two separate areas that mathematicians were surprised to be related at all.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#648Earlier quoted context omitted.
Not necessarily. Obviously playing Kasparov on the board requires more planning ability than managing a McDonald's but look at where chess bots are now. There's much more to being human than our "cognitive abilities"
Conjecture: the first AI to successfully manage a McDonald’s will be a Gemini.
They no longer have to limit themselves to forking software but can do a global Google Burgers in a single prompt. It will no doubt be a huge success before shut down.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#649Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…
I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a…
I want to push back against the notion that the math already exists in the weights, both in the practical and the philosophical sense. The LLM had to do an enormous amount of computation to find the counterexample. We know it wasn't looking up the answer from its internal representation, because the conjecture was unproven. The proof came into being when the model output it, and if they'd run it for less time or asked it something else then the conjecture would still be unsolved.
I'm also afraid of a world where AI completely replaces human mathematicians, but if we remain collaborators, then that's a world I can still feel excited about.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#650This HN thread depressed me. I’m still thinking about why. Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers…
I think part of it is that one side throws rocks and so it never even matters was is in the article. It becomes a battle if the article is good or the article is shit. Yes, I'm tired too. I want you have real discussions about these things. But the problem is everyone believes their reality is real and anyone's reality that disagrees is fake. It just escalates. I take long breaks from HN because I realize I just come…
IMO it's because people have fragile egos / worldviews they don't want shaken. Pretty much any opinion I type on this website gets instantly downvoted, unless it reaffirms the popular / mainstream narrative, because I happen to see the world differently than most.
This website is quite awful, and I also don't know why I spend any time on it. It's definitely not a website intended for meaningful discourse. It's a website where you can reaffirm whatever opinion is already established, and if your opinion is at all controversial or even just out of the box, you'll be punished for it.