omnicognate
2 is an interesting property and I agree that it's not possible for LLMs to possess it because it's a property of natural language itself, not the model. Not possessing it doesn't mean that LLMs can't be used to write software (obviously), but it does mean that using them isn't like using a compiler, which does have property 2. This is usually brought up in response to people claiming using LLMs is just another step in the process of increasing abstraction that led from punch cards to assembler to low level languages to high level languages*. That point stands: LLM use is qualitatively different to those steps because LLMs aren't "deterministic" in the sense of property 2.
There must be a better word for this, but I can't think of one right now.
* Even Linus Torvalds used that cockamamie framing just now: https://lore.kernel.org/lkml/CAHk-=whTs09_ozN+e09GU68N1wPP56...
(I assume you mean explain itself to humans here.)
I keep seeing people saying this as fact. It might, but it might not, and there are various reasons one might be skeptical that it will, using current techniques. Not least among these is that explaining something well to a human requires a degree of empathy and sophisticated theory-of-mind that LLMs have not yet exhibited yet*, and that is an area where humans are uniquely advantaged by being humans.
Someone will almost certainly reply by just stating as fact that they will achieve this goal, or pointing out that past predictions that LLMs wouldn't be able to do particular things have failed, and that's all very well. I certainly wouldn't be rash enough to claim it's impossible, but there are still models of all this available that are consistent with all the successes so far but don't predict "LLMs can do anything and will do everything". We'll find out in due course.
(I should clarify that the benchmark here is good human explanations. Of course many humans are capable of producing "explanations" worse than Markov chain text, let alone LLM output.)
* My idle speculation is that a lack of decent theory-of-mind is why LLMs still suck at using the "it's not X it's Y" construct. I doubt they actually use it any more frequently than humans when writing on the same topics, but they use it wrong and it's jarring. This particular rhetorical flourish works when there's some reason to think that X is an idea that might actually be present or forming in the mind of someone reading the passage (or in some other mind that they in turn might be imagining, eg. a fictional character), and that explicitly calling it out and negating it will have a particular dramatic effect. LLMs probably use it with about the expected frequency, but they use it for that reason (frequency) and then come up with something to put in the X and Y slots. The result often jars because the X it uses isn't something pertinent to the reader's thought process and/or the Y isn't a convincing, persuasive or relevant counterpoint to it. I suspect the result is that they stick out and seem more frequent than they are.
If I were a mathematician working on the Riemann Hypothesis, I'd probably be rather pissed off right now but continuing my work, i.e. the same as I am as a software developer.
That's why I'm not an artist.
In hindsight, along with illustrating the potential of computers as a bicycle for the mind it also gave the first indication that if such proofs were to become very easy to produce they could also be used as a bulldozer to bury the mind.
Grocery store owner A's revenue last year was €500,000. Its profit was €0. It does not have an IT department because it's a corner shop that employs 2 people and has one computer.
Grocery store owner B's revenue last year was €50,000,000,000. Its profit was €0. It has an IT department because it's a huge fucking corporation with a thousand massive stores, multiple office blocks, 150,000 employees and all of the networking and IT infrastructure an operation of that size needs. It's considering opening a bank.
The number that tells you the difference between the scales of these two companies is the revenue, not the profit.
I need to stop wasting my time on the pit of ignorance this site has become.
> Companies try to cheap out all the time, in incredibly stupid ways.
Yeah, and sometimes that results in someone turning their freezers into ovens. I don't think we disagree on this bit.
The post I replied to (currently the top post on the article) is trying to paint a picture of some harried manager of a struggling grocery store having to fire up networking tools, which is ridiculous. A company doing tens of billions in revenue employs or contracts some of the "laptop class" (to use that post's terminology) to do that, regardless of what their profit margin is. They don't get a "shift-work labor class" employee to take a break from stacking shelves and check the firewall config.
This is a very weird thread.
Intermarché - €44.5bn revenue (2024)
Cooperative U - €28.35bn revenue (2025)
These aren't struggling grocery stores. They are massive businesses collectively running several thousand stores and employing hundreds of thousands of people. They have IT departments.
They are proving useful for the parts of those things that have that verification, but in neither case is that the whole of the field. In maths, for example, we have seen LLMs find answers to some interesting problems/conjectures, but haven't yet seen them pose any similarly interesting problems/conjectures themselves because the criteria for a "good conjecture" are unclear whereas the criteria for a resolution of an existing conjecture are usually supplied by the conjecture itself.
The criteria for statement C (which OpenAI targeted) only require a C-infinity smooth force, whereas the paper considers constructions of the same kind as the OpenAI one but with a real analytic force. All real analytic functions are C-infinity smooth, but not vice versa [2]. Wikipedia says the Navier-Stokes problem for real analytic forces is still unsolved, so the question of whether a similar approach can be used there is presumably of some interest, but it doesn't appear to affect whether the Prize criteria have been satisfied and the paper doesn't explicitly make any claim that it does.
Re the problem still showing as "active": The Clay Institute don't consider problems solved until the solution has been fully digested by the mathematical community and is well established. In the case of the Poincare Conjecture that wasn't until 2010, 7 years after Perelman put his papers on the arXiv. Nobody is expecting them to mark Navier-Stokes solved any time soon.
Your comment was the first I've heard of that paper, BTW, so thanks for that. However, that's another indication that this probably doesn't bear on the Millennium Prize criteria. The lead author, Peter Constantin, is a leading expert on the problem and the Clay Institute's page on it has a video of a lecture by him. If he had come out and said OpenAI hadn't solved the problem as stated I'm sure there would have been a lot of noise about it. The paper is dated 17th of September so presumably there has been plenty of time.
[1] https://www.claymath.org/wp-content/uploads/2022/06/navierst...
[2] https://en.wikipedia.org/wiki/Non-analytic_smooth_function
Not entirely sure what you mean by this, but every interpretation I can think of is AFAIK (I am not a mathematician) indeed inconceivable by Gödel's incompleteness theorem. As the article says the only thing it's mathematically possible for us to get out of formal verification is a proof of consistency relative to (i.e. assuming the soundness of) some other system. As far as formal verification of "math itself" is concerned it's turtles all the way down.
Edit: Oh, do you mean that we could come to trust formal verification as much as we trust (well established) maths generally? That's a different speculation entirely and not an objective or well specicified one. There are plenty of people on this site who already trust a Lean proof more than a traditional one, without understanding either.
The ruling on this is here and is perfectly readable: https://coa1.azcourts.gov/Portals/1/OpinionFiles/Div1/2026/S.... The reporting of it has been abysmal.
Also note the obvious (but usually glossed over in news articles) fact that vacating the sentence doesn't mean he doesn't get a sentence. It means he gets remanded for resentencing (by a different judge, I imagine, but IANAL). Edit: Reading about resentencing processes it sounds like it could go back to the same judge, actually.
I'm not. To the extent my taxes are used for maths research (which is minute as a proportion of them) I want them to be used for the development of human mathematical understanding, culture and education, not trawling through a mountain of slop mechanically generated by a Silicon Valley startup that's about to IPO for trillions in the hope there might be some nuggets of insight hiding in there. If the latter is an activity with some value the startup in question can pay for it.
This morning I consiged my old underwear to the trash heap of history and strode forward into a new age of fresh boxer shorts. I'm off to write a blog about how to query databases in the age of my new pants.
Cool exoskeletons, though.
Edit: If you'd like a better medicine based one, look to radiology, where AI is an omnipresent tool but claims that radiologists are no longer needed, based on an ignorant view that a radiologist's job is "classify images according to what diseases they indicate" have only contributed to a crippling worldwide shortage of radiologists.