I know a bit about this field. This conjecture reads as somewhat more niche than the cyclic double cover conjecture recently proved by OpenAI, but nevertheless represents a real contribution.
You want to know how long it takes to solve an optimization problem, in this case over convex, lipschitz functions. (The restriction to a spherical domain is not really a restriction, you can just change variables for any bounded domain.) Anyway, showing upper bounds on time complexity is "easy" because it's just the runtime of your algorithm. Showing (nontrivial) lower bounds is usually much harder because it requires constraining all algorithms.
This proof apparently shows that the lower bound time complexity is equal to the time complexity of an existing 30-year old algorithm: it requires Omega(d^2) function evaluations to solve over this class of functions.
My gut says likely implies that d is the minimal number of evaluations if you have a gradient oracle because you can approximate a gradient with d function evaluations, but I'm not sure how hard it is to make that rigorous.
> so advanced that it's just as readable to me as Greek
I used to feel this way about statistics.
The language and terms are hard to understand and many of the formulas are taught as "just memorize this" instead of building up from first principles.
But then I started using statistics to analyze something I cared a lot about (paintball) and I quickly realized it's like learning anything new:
- there is jargon
- and core concepts
- when you learn the above, it suddenly makes a lot more sense.
I gotta know what you use stats for regarding paintball. I haven't played in years but I loved playing back in the tipman 98 custom era (not sure if that's still a popular marker).
- I started with just pen, paper and a stopwatch (as a college coach)
- I assumed paintball would be more like football where it's hard to track individual effects
- Turns out it's a surprisingly simple and stable "state machine". e.g. the odds of winning with +1 body (e.g. 5v4, 4v3 etc) is, in college, about ~75%
- Paintball is one of those sports where "the weakest player determines the outcome". Why? b/c if 1 player gets out early, you are fighting out of a hole.
It also made me appreciate that as good a book as Moneyball is, reading it after you try to create analytics for your own sport makes it 3x as enjoyable/insightful.
One downside though:
I would watch games and I got so good at internalizing the stats per state of the game that it was like watching the world series of poker where I could see both player odds of getting eliminated and probability of winning over time charts as I watched the games. Made it harder to be the "come on guys! we can win this" coach when we were down on points + bodies.
> Made it harder to be the "come on guys! we can win this" coach when we were down on points + bodies.
Would you say that paintball on that level is almost deterministic?
Compared to, for example, (American) football where momentum and perseverance can turn games around completely.
This is my exact experience right now trying to wade through the research on psychometrics and skill/knowledge assessment design. It’s mostly just applied statistics but like all such fields, over decades of specialization it acquired its own jargon for abstractions that are quickly recognizable to anyone with a sufficiently developed nose for modern mathematics. But you still have to wade through all the definitions to make those connections before you actually understand everything.
Not to diminish the comment, but most things are not as complex as they sound when phrased in everyday language or sound much more complex than they are when phrased in technical language.
Technical language is a tool that allows insiders to say less and refer to more, and to be specific, but it's just a tool. Most things can be described in accessible ways.
I think you'd be surprised at what you could understand and at just how few domains are truly complex enough that a layman couldn't understand with a little bit of patience and an accessible summary.
It took me a while to understand a lot of these math concepts.
Turns out people doing Engineering research are using a very small but powerful bag of tricks from a handful of few famous Mathematicians. The concepts are named after them!
I don’t think OP made much effort to make the comment accessible to non-experts, and so it should be taken as a gauge of the fundamental difficulty of the topic.
Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.
I'd push back on this. Most of the core optimization techniques (eg, ADAM, stochastic gradient descent) are straight out of the convex optimization literature. Generally you need to use optimizers that work well on convex objectives because near minimizers, functions tend to be convex. (Proof by contradiction: a non-convex point has a strict descent direction.)
The fact that neural networks are highly nonconvex has encouraged a lot of research, but it's more of the kind aimed at resolving tension: these methods are probably good for convex functions, why do they continue to work for nonconvex problems, and are there tweaks we can make to improve them in that setting? It's not a lot of de novo theory; more standing on the shoulders of giants, etc etc.
No, I have to push back as well, sorry. It takes a very long time to get to the "near-minimizer" stage when training a neural network, and in practice, you never get there (see neural scaling law regimes). What you are saying is the viewpoint from 6-7 years ago. Things have changed.
The reasons why optimizers work well for neural networks in their highly nonconvex landscapes has absolutely nothing to do with their performance in convex landscapes. If that were true, everyone would be using Newton-CG. These optimizers were born in the convex optimization literature as a consequence of the genetic optimization nature of incremental publication (and because that was all we had), but their modern study is through the lens of implicit regularization (their preferences for certain minima) and their stepwise vs. continuous rates for feature learning in multilayer models.
This is completely new theory by the way, and requires painful reinvention of the field. It does not stand on the shoulders of convex optimization. The nonconvex setting is assuredly not a perturbation of the convex setting, and those that do continue to work on deep learning optimization from the convex optimization perspective are well behind the times.
It seems that we have two different stories here: in one, the new optimization theory represents a stark departure from the prior art, a sort of revolutionary new view of the understanding of optimization as applied to neural networks.
In the other story, the current understanding of optimization is a natural evolution of past work, where a new generation of researchers respond to social and technological changes, adapting and building on the work of the past, taking what's useful, downplaying the importance of some ideas, and inventing new language to describe concepts that seem most relevant to the current situation.
Both stories tell some of the truth. A revolution or evolution? Looking at the literature (eg the sibling comment here) shows that even today, convexity is used as an intuition pump for modern optimization techniques. But there are also new ideas that apply to the specific exigencies of neural nets, and downplayed ideas (eg convergence rates) that seem less relevant.
The optimizers are lifted from convex optimization, but the point above was that they are applied to highly non-convex problems. They work for finding local minima, but a lot of the deeper literature does not translate (e.g. the conjecture being discussed in this post).
I'll point out that "does not work" is not the same as "not as efficient" :) But it does seem the Adam paper had an error.
I think that Nesterov's first order method is the most efficient general first order algorithm on convex problems, so anything else is in some sense worse. (Edit: removed incorrect ADAM comment.)
Yours' "not as efficient" in [2] means that, sometimes, ADAM "does not work." Look at figure 2, ADAM literally does not work in the case of "true model."
Yes, apologies, I didn't read the articles you linked before posting this. I did update the comment.
I don't think this changes the point, which is that most optimization methods used in AI owe a substantial intellectual debt to convex optimization theory.
I love convex optimization and there are a few SciML projects I am on where I really need results from there. But in AI research with deep neural networks, it's become a liability, because people will just not let go. I'm getting tired of reviewing convex optimization theory papers in ML conferences that are still trying to wave away the obvious issues with their application to deep learning. It's harsh, but I do feel we can only start talking about an intellectual debt once that stops being the case.
Another intuition is that near a minimum you can Taylor expand the function and show that the higher order coefficients (past the square) are negligible.
It's not a matter of whether the theory "works"; it's a matter of whether one is asking the right questions. Convex optimization studies how quickly an optimizer can reach the optimum. In the non-convex case, there are many basins containing their own local minima. The more sensible questions there are "which basin is it likely to go into?" and "how do I steer it to go where I want?". Global convergence rates are largely irrelevant by comparison.
Yes, order d is the minimal number of evaluations of gradients needed for the same problem! That has actually been known since 1979 (Nemirovsky and Yudin showed that), and there are methods with the same complexity so this question in the gradient model has been solved for a long time. "because you can approximate a gradient with d function evaluations" was exactly why d^2 made sense as a lower bound for this case! Basically, the lower bound question can also be thought about as "can you do better than approxing a gradient?", so this result says no.
> I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.
I wonder how this compares to what we see happening with "juniors" in software development?
In math research, do you also get the training for the profession from working on the low hanging fruits for a while, to then move to the medium-hanging, and later go on to work on previously unsolved stuff?
Around here AI isn't really more of a threat to juniors than it is to seniors. It's a threat to the people who have been taught "recipies" rather than applied computer science. You can have excellent seniors who can do TDD, DRY, SOLID and so on, who also happen to have no idea what a L1 cache miss is. The current AI models know all of those things, but they struggle applying them correctly without someone piloting them. Even in the energy industry where I work, where you'd think it would be obvious from the context that you should prioritize runtime safety over debug safety, the current AI models struggle to do so. As far as seniority goes, though. If we can find a young developer with little experience who actually knows computer science, we're much more likely to hire them... Since they are cheaper.
This isn't something which is unique to software development though. We're currently building enterprise AI apps that we can deploy into the AI agents working for anyone of our employees. The key thing we're currently seeing is that the people in a team who are the ones that everyone turn to for advice, are the only people who aren't in "danger". Even people who are great at their jobs are being outperformed by AI in many cases.
I think it'll be a massive challenge for our society in the coming years. Maybe we're even going to get to the point where the AI will also be capable of replacing a lot of the "domain experts". Right now that seems far out, but then, if you had asked me about AI four months ago I would've told you it was all hype.
It's not a zero sum game. You can have AI "senior engineers" working under humans building bigger things than we've been able to.
We also don't know where the capabilities of current AIs will plateau. The benchmarks aren't really telling the entire story. From my perspective of using the models there are certain axis where they're not making a lot of progress, like being able to have large accurate context on the scale that humans can. There are other dimensions where there is still a large gap between human capabilities and LLMs. It's true that relative to other areas (lessay chess) LLMs are more generalized but they are still not fully generalized (back to the chess example, LLMs are not good at chess).
I was arguing with a friend about this point today. I'd posit that our economy has been growing in a way where goods that require more human time and natural resources to produce are giving way to goods that are intangible.
Once we've met our basic material needs, we're tending to consume things that are replicable with low marginal costs, and which do not interfere with the production of other goods. So maybe we can actually support a continually growing digital and entertainment economy, at least for a few more generations.
Maybe these mathematical contributions will also impact the efficiency and capabilities of our material production systems as well, which is another way to keep the economy growing.
I'm optimistic that we'll do more with our resources rather than trying to optimize for doing the same more efficiently with less resources.
You are correct about growth, but speed is also a resource - kinetic energy is still energy, which comes from somewhere, and unless you have 100% perfect energy usage, there is thermodynamic loss, contributing to entropy, which itself is another resource. Temperature is a physical resource all lifeforms and machine processes share in a gravity well, and all processes breakdown after a melting point (unable to maintain their own required entropy levels)
This was sort of what I wanted to say, but I guess I should have worded it differently. I certainly didn't mean to say that I thought AI would stop improving. If anything I'm surprised at how much we have to fight the AI models to do what NASA has been doing for 60(?) years.
How would this be different from taking a yoga class from a video?
I would be willing to be proven wrong, but I doubt the ability of LLMs to give useful corrections in yoga much more than their ability to write useful code.
Interesting, thanks.
I don't know where "around here" is, but the signals I've seen in a lot of articles is that the demand for junior software people has taken a dive since a year or two back, with student programs etc getting cancelled. One googler said they were getting a junior to their team and that was kind of a big deal because it hadn't happened in that whole department for a long time.
In relation to that, I guess my question becomes: if the same thing will happen in math research, who will write the ten page math proof prompts in the future?
There is definitely the effect of less or de-growth in the industry, which started before the current AI hype. And now there's the additional effect of companies hoping AI will replace their need for (junior) devs. Nobody knows if or to what degree this will work out (yes, we all have opinions, but no crystal balls), but they are holding back the hiring until they know how all this pans out.
I'm from Denmark and I've been an external examiner for various CS educations for the previous 13 years now. Some of them teach you a lot about how the hardware works, others mainly teach you design patterns. Five years ago the latter was in high demand, because a lot of software development frankly doesn't need computer science (until it does). Now there is almost no demand for them.
Honestly, i've received a formal MSc education in the hardware aspects, including for designing embedded electronics products. Spent the most part ofy career in the software industry designing enterprise software and feel like i never needed to use them, except maybe early in my career when i was reviewing tech stacks and determined that .NET would be among the winning horses, precisely because it'd take care of that for me almost all the time.
What i see today is the opposite of what you see : product owners not knowing a thing about software engineering but being able to vibe code prototypes handed over to the dev team are rock stars.
They are closely followed by senior software developers having more of an architecture & design background than a low-level computer science background. Most businesses are looking for builders these days.
Where what you say may converge with my observation is that to be able to do to things such as proper database query optimization, even using AI assistance, you need to be able to understand the concepts of working memory set, cache misses etc...
I've found huge problems, like database servers being grossly underprovisioned (like, 60% cache hit, 4gb RAM server for a 700gb dataset with an 50gb circa hot data set). SSD were used and only latency was measured, so no one realized how problematic the situation was (including a consulting shop they hired to help them manage their DBs - backup, maintenance etc...).
However, having a high affinity with hardware is not a driver / computer science of hiring decisions from what i can see in the enterprise software world. But it would make sense for it to become the case within 10 years. I suspect that you work in a niche where performance optimization matters a lot.
> However, having a high affinity with hardware is not a driver / computer science of hiring decisions from what i can see in the enterprise software world
I think the way I worded it was maybe a little too close to just being about hardware, because performance do matter a lot in the energy industry. I do think it applies to SWE in general. You mention .NET and I've met C# developers with years of experience who couldn't tell you the difference between IEnumerable and IQueryable. I've met even more experienced Python developers who don't know what a generator is. Stuff like that, not having knowledge of the tools they use. I guess you could argue that those are bad developers, but I don't personally think that has been the case for most of them. Still, you'd rather have someone who thinks about these things rather than eventually using batches once they run into memory issues.
I also think these changes are appearing faster in non SWE enterprise. As you said, product owners who are AI explorative (for the lack of a better word) are rock stars. We see this a lot in our finance and risk departments, where domain experts now write fairly decent software with AI. My team has build them tools so that they build things the same way, use the same developer setups and pull the pre-approved external packages or are offered alternative ways of doing things. A few years ago this would've been done by these domain experts "ordering" the software they needed from our SWE team, and if they hadn't already been mostly laid off due to Putin's invasion of Ukraine changing the markets, I belive they would've been now because of AI.
Because frankly, a lot of the software that gets produced in these areas, don't need computer science, until it does, and the domain experts can make the software they need so much faster than before by vibe coding it. From my perspective it's not that much of a difference in the quality of the code that gets produced. I also had to help with performance and security when we had more software engineers on staff. Though now I do it more through writing and distributing AI agent applications rather than writing a C binary or optimising the code directly.
It's funny how you applied on your own argument several logical fallacies about why ai is only a threat to people who have been taught "recipies" versus who know what L1 cache miss is.
Actually it's sad there are people out there dumb enough to believe knowing L1 cache is any different than knowing recipies when it comes to the story which jobs AI will take. I'm convinced by now it will be the jobs of those people believing such crap.
Unless you’re claiming that AIs will suddenly (and very soon) stop improving, they are obviously a threat to everyone’s job.
Calling notable conjectures that have been open for decades “low-hanging fruit” is an act of desperation. Most professional mathematicians couldn’t have proved those conjectures if their lives depended on it.
I wouldn’t call it “low hanging fruit” but it’s easy to think of problems that seem harder. Apparently solving notable math conjectures is easier than building a practical robot to deliver a package to someone’s porch?
So, yes, AI is a big deal and we don’t know what it’s going to affect, but the goal of replacing everyone’s job is extremely ambitious and there’s a long way to go.
This has to be assessed separately for each kind of job.
The thought that anything could improve without bounds would be absurd. We are living in the physical world after all. The (open, interesting) question is how close we are to the limit.
It’s safe to assume that after less than a decade of LLM development, we’re nowhere close to the limit yet. In fact, progress still seems to be accelerating at the moment.
Types of technology - of which we can include intelligence - move along S curves, but it's more absurd to think that humans are near the top of that curve rather than right at the bottom.
There might be a thing beyond intelligence that we can't even conceive of.
> more absurd to think that humans are near the top of that curve rather than right at the bottom
The “absurd” dimension does not enter. This is a situation where you have no evidence at all.
In the absence of any information, the average (mean or median) is your best guess. Now where that average is, you have no idea.
> There might be a thing beyond intelligence that we can't even conceive of.
This statement already supposes there is a thing called “intelligence”. People have been pretending to measure this for more than a century. Modern thinking at least says what we call intelligence is not a single concept.
>Unless you’re claiming that AIs will suddenly (and very soon) stop improving
Most technologies level off sharply after bouts of boundless improvements.
In 1968 they thought we'd be flying to the moon by now but instead we're flying across the ocean in planes not that different from the 747 that existed back then.
They sometimes start improving again. In the context of your comment, look how the cost/kg to LEO has suddenly dropped radically. This was mostly due to institutional change that allowed previous non-technological barriers to improvement to be bypassed.
> COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS
> BY L. J. LANDER AND T. R. PARKIN
> A direct search on the CDC 6600 yielded:
27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵
> as the smallest instance in which four fifth powers sum to a fifth
power. This is a counterexample to a conjecture by Euler that at
least n nth powers are required to sum to an nth power, n>2.
I was trained as a mathematician and worked as a math researcher for a little while (now working as a private tutor), and based on my experience I'd say this description is basically right, with one extra wrinkle.
In order to get a Ph.D., you have to do some sort of original research, so in that sense you're working on "previously unsolved stuff" basically right from the start. But that doesn't entail doing anything all that ground-breaking; most Ph.D. dissertations (very much including mine!) contain work that a more senior researcher in the same subfield could probably have produced without too much difficulty. The software development analogy is a pretty good one: a lot of the point of getting junior researchers to do research is to help train them to one day become senior researchers, and often the work itself is nothing all that special.
Given the trajectory of these LLM proofs, this seems like it's going to have to change pretty soon, and to be honest I'm pretty grateful that I'm not in charge of deciding what that's going to look like, because I don't have any good ideas! I'm actually pretty worried about the future of the field.
Unless, of course, the machine being constructed is the prover (and prompts) that proved the results (or achieved whatever result instead of the human achieving it.)
My experience may not be entirely representative because to be entirely honest I’m not exactly a great researcher and there are brilliant PhD students. That said it indeed was my experience that in the pre-PhD / early PhD period ( or even longer … ) your advisor proposes (gives) you pretty low hanging stuff that he mostly already knows how to solve, at least at a high level, with the expectation that it will teach you to use the mathematical tools you need.
The author also used GPT-5.6 to write the prompt. This did involve giving GPT-5.6 access to his previous work and a back and forth process (so definitely still used the author's expertise to some degree), but the prompt itself is also largely AI generated.
What's the difference between using GPT to write the prompt to GPT, and "thinking"? The LLM uses the first tokens to predict more tokens, and then uses those tokens to predict even more tokens.
Roughly speaking it is the difference between having a contractor go out and do some work and having that same contractor first come up with a plan to do some work, run that by you, and then go out to do that work.
Part of it is as a another comment in this chain mentions the chance to review the prompt. Part of it is that it forces the AI system to plan things in a certain order, in much the same way that forcing the contractor to write the plan out first forces the contractor to proceed in a certain predefined order that may (or may not!) be better at getting to a final answer.
Certainly. This feels similar, to me, to how building complex software with LLMs works today in practice. You need to know a lot to set up goals and guardrails and verify outputs. For me, making the bits change was always the fun part, not tangling with text in my editor, though that had its moments.
We're also at the point where you can roll out context to your entire organisation. I created an app for our m365 Cowork and deployed it to everyone who develops software. It does a couple of things, but it main knows our compliance policies and can guide developers through writing the documentation needed for NIS2 compliance. It also guardrails against non-approved packages, and helps developers find alternatives, or if none can be reasonably found, how to get a new package/dependency approved (or rejected).
A few months back this would be something every developer kind of did on their own. Maybe they shared skills, we certainly encouraged it and tried to do all the change management things, but nobody really had the same versions of the skills. Which was horrible in the deployment pipelines, something like the compliance documentation often had to go back and forth several times before it could be approved. Now it's just there, for everyone.
In a year or two, I expect a lot of these things to have become even more standardized. So that we don't even really have to build our own apps, but can simply use the ones in the catalog with minimal configuration (and that config will likely only be necessary because I'm from a tiny country that nobody will maintain standards for).
Yes. I worked on two large monorepos, about one quarter per project. Maybe 80+ devs on first, maybe 40+ on second. Both were not sure how to describe this but ultra-high-velocity agentic driven development efforts.
On the first, there were ~no shared skills. There were some requirements set up but they were not minded properly and became stale / ate context for little gain. The hardest hit was in E2E tests which would flake and create long running, too-often failing CI. People would disable them, because they were not reliable and velocity was so high, no one was happy w them.
I maintained my own set of skills and CLIs to back them. I'd share them if they came up but it was like the old days of manage your own stuff. Not much credit for building and sharing devex tooling to the team.
But then on the second one we were in better shape--we had vendoring set up to distro skills automatically.
Before the project was well underway, I put time into understanding how all of our tests aught to be written. Finding the forbidden things, etc, getting review from our best test folks and ultimately landed on a `/test` that routed across all possible test types.
Like night and day. Instead of finding out while trying to get a release out the door that some corner of the project had a handful of flakes, tests were written the right way from the start.
Like, it was beautiful. And I don't think devs noted difference while building. Only that there was an absence of BS in CI.
Hard to quantify the lack of pain, but it was big!
This made me chuckle because it's so true. So much detailed steering and finagling in the past, now I point the agent to a bunch of information sources, skills, similar repositories that might hold useful input and tell it very roughly what I need and off it goes, I'll grab coffee.
Look here and here and here are some tools, and /skill /skill [repo of folder paths etc] and here is what needs to happen: [stuff].
---
Restate this request in your own words and enrich it as appropriate handling any gaps.
?
It is an ultra-lite way to plan, I suppose.
I like the format because:
- I still get to put all my thinking into the request but then easily override the instruction
- It is interesting to see my casual typo-riddled blast professionalized and improved upon.
- Sometimes it surfaces useful questions that can save some time up front.
I think the models are doing this anyway, but I find the words "enrich" and "gap" are well understood by models and they demonstrate it in the response to the above pattern.
Anyhow, to get back to the point, there are still prompt-level tricks--but ultimately if repeated, should probably also be built into skills themselves!
In math, a proof is a proof. We don't know if we can get there and so getting there is the hard part.
In software, we always know that we can solve the problem. So HOW to solve the problem is the hard part. Because the type of solution involves maintainability, which involves planning, LLMs suck at it. This leads to "LLM slop code" whereby the LLM creates ad-hoc convoluted logic with redundancies and no reuse of existing standard library batteries.
Unless you're a Grothendieck who gets mad at Deligne for not solving the Weil's conjecture "THE RIGHT WAY", software is fundamentally different than math in this respect.
So I'll say it again, AI will win a fields medal for before managing a McDonald's simply because there are enough big problems within arms reach than their current capacity to plan over time
I've spent some time working both as a math researcher and as a software engineer, and I think this comment actually underrates the similarity between the two fields as they're actually practiced.
Some math research does involve grabbing a single, fully specified conjecture off the shelf and hunting for a proof of it, and it's true that if you manage to solve a long-standing open problem, other mathematicians will be interested no matter how you did it.
But this isn't all of what they do, probably not even most of what they do. Like in software engineering, it's not always obvious which question would be the most useful one to ask. A lot of mathematical work also goes into what we call "theory-building", where you could say that primary work goes into coming up with definitions rather than theorems. Mathematicians also care a great deal about how something is proved; a lot of them are some of the most aesthetically picky people I've ever met. Words like "ugly", "beautiful", "creative", and "boring" are used to describe both definitions and proofs all the time.
From the outside, it can look like all they're doing is pumping out proofs at any cost. But I promise you that when I talk to mathematicians who don't have any experience building software, they have a similarly narrow view of that field as well! Both fields, from the inside, look a lot more human than you might expect.
I think that your take is quite optimistic. Having published in top tier journals my only experience is that mathematicians care about what other mathematicians worked on and failed to solve. Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem.
Math is such that most theories are built after solving a problem and actually don't solve a larger class of problems. Etale Cohomology is an example of a rare exception. Grothendieck was mad that Deligne used adhoc complex analysis techniques to prove Weil. But everyone else was thrilled.
Whereas in CS, a good theory (library) solves a large class of problems. The reason being is that CS tackles general problems while math specific ones. Math on average solves problems that don't lead to solutions to other problems.
To me at least, math is more of a game like chess and coding is more of an art. There are aspects which are a game, like performance engineering but I'm pretty sure that LLMs will become superhuman at that soon
If your complaint is about the type of work that gets you published in a fancy math journal, then I'll happily join you on the barricades. Sure, getting a paper into Annals of Mathematics or whatever is more game than art in the sense I think you mean here.
But "what mathematicians care about" is much, much broader than what gets you published in a fancy journal. Mathematics as a human activity is millennia old, much older than the concept of journals or even universities, and that activity is, to me, very beautiful, worth preserving, and more of an art than a game. The incentive structure of academia for the past few decades has done a pretty bad job at preserving that art form, but that doesn't mean mathematicians as actual human beings don't care about it --- if they didn't, they probably would have chosen a different career.
It seems to me you hooked onto the wrong part of proofs vs software compared to what OP meant.
The difference OP cares about isn’t how much one cares about style. Instead the important difference lies in validation. A proof can be validated as either correct or wrong. That type of hard feedback really helps combat the optimism and desire for shortcuts of modern models.
Now, that still doesn’t help an LLM distinguish between good and bad correct proofs. But it still really helps a lot. On top of that, taste in proofs is a lot more uniform than taste in coding. That helps LLMs be better at judging the quality of a proof, because there’s less disagreement in the wider world.
The standards of proof are different from the fundamental operation of "OK, cool, you solved this problem. Why does this problem matter? Isn't it useless? Senseless? Meaningless?" You have this same question whether or not you're in an a priori discipline (mathematics), scientific fields proper, or engineering. "Absolute certainty" has nothing to do with it. I can assure you, people on the job are not looking for The Absolute Truth when doing their jobs, yet they still can question at a solution by asking: are we solving the right problem?
(Although in general, there's no true difference between "I answered the question correctly, but the question was mapped to this thing we call 'reality' wrong", and "I answered the question incorrectly", because you can (try) adding the constraints that you really wanted targeted in case A, to case B, and boom, suddenly a question/answer pair that was "Answered correctly, but question doesn't map to reality" now becomes, "You answered this question wrong". However, individuals generally tend to have some breakpoint to differentiate between the two).
No, what I'm saying is that I don't agree that taste in mathematics is more uniform than taste in coding! Mathematicians argue about taste all the time. Just as you might look at a piece of code and agree that it compiles and doesn't have any fatal bugs but still think it's badly written, hard to follow, hard to modify, or whatever else, mathematicians judge mathematical work using very similar criteria.
I think the difference is in math the problem is fully specified and easily verifiable and in programming it's not. I don't agree that we always know we can solve the problem.
For example, create a DFA for a regex, not too bad just use Thompson's algorithm and then NFA->DFA. But now we have to care about efficiency, user API, maintainability of definitions etc.
> So I'll say it again, AI will win a fields medal for before managing a McDonald's simply because there are enough big problems within arms reach than their current capacity to plan over time
AI can manage a McDonald’s already. If manage means directing humans to do something to ensure the store is running. If manage means running robots, then yes maybe that is 5 years away but just directing humans to run a store, that is possible right now.
It's not a problem of incentives. Every executive wants to inject LLMs everywhere these days. If they haven't somewhere it means that it does not work.
Every executive wants to inject LLMs downstream from their own job. As you get away from execution and toward management the incentive to innovate gets less and less as management at the same level.
I would agree with your take. I (author of the post & paper) learned a ton from working on small parts of problems my PhD advisor was doing a lot of the heavy lifting on, and later also from getting some results that were essentially putting together the right pieces that already existed followed by some deep-in-the-weeds analysis.
So if you dig down a bit it turns out the author had been trying to solve that problem for a year with GPT 5.4 and 5.5 and he fed all that information to the prompt he gave to Sol Pro which may or may not had direct access to the author's chat history. So the claimed "148 minutes" was really "a year plus 148 minutes".
Moreover, it seems the prompt included the technique used to solve the problem:
In the prompt I basically just throw all reasonable approaches at it, without making a big distinction for what to explore most, and these approaches would all be reasonable for someone who knows the area. Sol helped me with the prompt as well, for which I gave it the CDC prompt, some ideas and specifications, a crystal clear problem description, and then modified things slightly myself after. One thing I do wonder is how much it accessed memory of previous chats, since as mentioned I had worked with 5.5 and 5.4 on this previously, and the main construction is not so different from something I discussed there. But, the function class max of affine functions that worked in the end was also in my prompt, so I'm not totally sure.
So it's not clear to me the degree to which "GPT-5.6 used a prompt" to close the gap etc, or the author basically did all the work himself and assigned it to GPT-5.6 out of enthusiasm.
I (author of the original post and paper) can add a few things here:
1. My previous approaches with GPT 5.5 were really not very sophisticated in terms of my input. I threw the problem at it, and just kept encouraging it to go iterate through ideas without any success.
2. The approaches that are in the prompt, though they will seem cryptic to someone not in the field, are relatively natural ideas. In fact, the construction that worked was something that even 5.5 initially looked at but was just too weak to see how to make it work. From my view, I would have never gotten this result myself. Imagine you are telling a contractor to build the empire state building, and you say: "You should explore approaches that can include building materials like steel, wood, concrete, or clay, and any combinations of those. You can use arcs, columns, supportive beams, and anything else you can think of to solve load-bearing issues. Do not stop until you've completed a viable plan to construct the empire state building." And then the contractor shows you the finished empire state building using reinforced concrete and steel beams with all kinds of crazy ways of making everything stable; that's kinda how I feel.
There is a firm opinion in some circles that an AI can never do anything but recombine already known things despite a rising number of cases where that very much appears to be an inaccurate view.
Any slim possibility that vital information was given to the AI to make the task just one of recombination, rather than coming up with anything original, is grabbed with both hands no matter how tenuous.
I can understand why, nobody likes the idea of a machine capable of doing the job of a human, and jobs requiring the mind are especially sacrosanct. But like farm workers in the early part of the C20th, not liking the incoming technology won't change the effect it will have when companies get the idea they can make more money and hire less people.
In the Reddit post there was clarification that this was done with Sol Pro not Ultra - curious what is everyone’s mental model of the difference.
My understanding is that ChatGPT Pro is effectively a multi agent system, or somehow uses multiple LLMs in parallel and selects a best answer. And Ultra is more similar to Claude-Code UltraCode where the main agent can choose to create a dynamic JS workflow that deterministically orchestrates multiple agents to handle different parts of a task and have adversarial checkers etc.
Is that more or less the difference? Any substantiating sources would be great to see.
If I recall correctly there was a proposed proof to the abc conjecture by Mochizuki https://en.wikipedia.org/wiki/Abc_conjecture#Claimed_proofs which was rejected due to being rather inpenetrable to humans. Shouldn't this be an ideal target for LLMs?
There was recently an announcement that a group trying to formalize it found a gap exactly where other mathematicians were pointing. So to the extent there was any doubt, it should be gone now--the proof was incorrect.
But I agree LLMs have a lot of potential for checking proofs--both informally (they can read quickly and find gaps) and formally (by attempting to formalize).
Crazy how intelligence is cheap, efficient and commonplace now.
We humans better refocusing our energy on our core values/principles, given most of our skills are becoming irrelevant
If it were commonplace, there wouldn't be a post and discussion about it. Cheap? Arguable - while it didn't cost thousands, it wasn't free. Cheap is in the eye of the beholder. Efficient...How do we even measure that? The massive infrastructure and training to take a product to the point where someone could do this is massive. Ignoring everything behind the scenes and acting like one session and result is the whole picture of efficiency doesn't seem right. And no, nothing produced by AI makes skills irrelevant. That is the whole ongoing argument of whether people are losing cognitive ability by moving their thinking to AI.
Overall, this is an impressive proof of capability. But I wouldn't take that proof as anything more than what it is.
Seconded on the "not cheap" argument here. I've spent $25 worth of tokens completing a one-week task in an afternoon, or rather my company spent the money. I would never have personally felt OK with throwing this much money after some prompting back and forth for a few hours, one lazy Saturday afternoon. I ran the risk of not finding the solution before the token usage would be too high for me to want to carry on, if I was my own credit card linked to the account.
Of course money in this situation is a bit of a funny measurement, right, because if I was able to take the rest of the week off as soon as I had solved the one-week problem, then I would have no problem at all throwing even $100 worth of tokens at it, so I could enjoy a nice 4-day "mini-vacation".
How cheap "cheap" is, is indeed "in the eye of the beholder".
It's not sarcasm. If I had to pay $25 to perform my work from an ever-growing pile of projects, I would be losing money. That's not "cheap intelligence" for me. It certainly is for my employer, though, because for the mere price of $25 extra per day they can get 4*X the product per period of time.
In Scrum terms my personal velocity grows by a factor of four or more with access to agentic AI workload, but if it means that I will just be asked to "consume" 4*X more Story Points per sprint, I'm not the winner in the end, my employer is. If they asked me to complete X Story Points per sprint regardless of my velocity, and they let me take the days off when I was done, I would be the winner. But that's not how it works.
AI is "Cheap" for the person/organization that gets more product for less money, not for the individual person building the product faster.
If I'm getting paid for the work, I'm happy to leverage the LLMs so I can do more. If I'm paying for the work, I expect more from it.
For hobbyist stuff, where I'm not expecting to receive money? LLMs let me do things I otherwise wouldn't have done.
I've agreed with thoughts like "the LLM wrote the code, surely it's not worth sharing" or "I could just have the LLM write my own version of that". I'd also wondered about my own personal projects, surely "an LLM could have written all of this". -- But how I feel about that changes a bit based on how much it'd take the LLMs to get the same output.
I mostly agree, these are things that couldn't or weren't solved with abundant capital at all, and now they are being solved
it went from not having a price, to having one, and we are trying to retroactively transpose economic viability or economic existence to it from some parallel and prior time.
A counter argument: A strong distinction between "intelligence" (understanding what is) and "values/principles" (understanding what ought to be) was characteristic of much early modern European philosophy from Descartes to Kant, which received its influential strong formulation from David Hume.
But trying to maintain this distinction leads to insuperable difficulties. Our conceptual framework for understanding the world are always value-laden. There is no "view from nowhere", no historically unconditioned set of values or concepts. Your framing, in which "values" are external to "intelligence" and must be imposed on it (on pain of intelligence being "value-neutral"), leads inevitably to the dead end of "AI Alignment", "superintelligence", etc. Which is a kind of pseudo-theology.
"We humans better [be] refocusing our energy on our core values/principles, given most of our skills are becoming irrelevant."
In light of the untenability of a strong fact/value or intelligence/ethics distinction, I would suggest this alternative advice: humans should focus on critical appropriation and extension of the received wisdom, whether that comes to us directly from human beings or indirectly through an LLM. Perhaps this is compatible with the spirit of your original suggestion.
It's still clear that LLMs lack spatial reasoning, either in the concrete or abstract, and while that sort of reasoning has been downplayed by academia for at least a century it is fundamental to technology and industry. (And many would say for science and mathematics too).
They will, however, get there as well either directly or as interfaces to models that do, and your core point stands.
"Lack" isn't the right word. "Lacking" is more like it.
If there was a deep fundamental inability, we wouldn't see things like newer generations of LLMs consistently improving on ARC-AGI series (heavy spatial reasoning loading) and SimpleBench (a lot of commonsense + spatial reasoning components).
In a way, it's a surprise that LLMs, notoriously lacking any sort of embodied experience, can even get this close to human baselines on tasks like this.
My takeaway is that text is a far richer modality than anyone has expected - and that high end LLMs are often sharp and flexible enough to recognize their weak points and substitute their strengths. I.e. all the LLMs implementing A* to optimally solve pathfinding in ARC-AGI-3 tasks, often unprompted.
There might still be unrealized gains there from true depth-unbounded recurrence, or maybe from finding better ways to integrate modalities in training. But clearly, a "fundamental limit" it ain't.
> "Lack" isn't the right word. "Lacking" is more like it.
Yeah, that's fair.
> My takeaway is that text is a far richer modality than anyone has expected - and that high end LLMs are often sharp and flexible enough to recognize their weak points and substitute their strengths. I.e. all the LLMs implementing A* to optimally solve pathfinding in ARC-AGI-3 tasks, often unprompted.
I agree and disagree with this. I think we've learned a lot of humans are more text based than we thought, but conversely I'm not persuaded what non-textual task reasoning LLMs are doing is necessarily text based, just that models have grown large enough for other reasoning modes to conceivably be hiding in the parameter space.
As I mentioned elsewhere, like many others I find LLMs work entirely by example, and reaching for A* when pathfinding is the single obvious thing to do. In cases where the magic key word is not mentioned and the problem cannot be identified as "pathfinding" (or some other trigger with a highly specific widely documented solution) they will struggle, yet the moment the trigger is hit they get there very fast. This is why prompting remains such an art form.
Fable is the first one I've encountered that is capable of serious open ended 3D programming in ways that suggest it has some grasp of the spatial aspects of the problem (not merely symbolic manipulation of the vectors etc.), but it still misses optimization opportunities a human will find glaringly obvious based on spatially predictable bounds etc.
Grown? LLMs were always "large enough for other reasoning modes to conceivably be hiding in the parameter space".
Basic LLMs don't reason in text, and never did. They use it as an interface - for input, output and some of the intermediate products. Heavy use of those "pseudo-recurrence" intermediates in "reasoning models" is a relatively late post-training adaptation. But the process that happens between those endpoints is not at all text-based. What happens in the hidden dimension is part "output logit domain", tied to probability distributions over possible output tokens, and part "incomprehensible concept-space madness".
The latter being where things like latent world models live. LLMs develop partial world models, right in pre-training, despite not being explicitly forced to - because it brings them closer to heaven of accurate next token prediction.
And yes, larger models like Fable seem to be better at spatial reasoning. Maybe because their large size increases the sample efficiency and improves generalization, allowing them to absorb the sparse signal of "spatial reasoning" in the training text better. Maybe because this extra size means more layers, allowing for deeper latent space reasoning in lieu of true recurrence. Maybe because the default "next token prediction" reward underrates rare spatial reasoning challenges, and the model only starts to "get good" at them once the other sources of loss reduction are heavily depleted. Maybe because no true recurrence is suboptimal for spatial reasoning architecturally. But it is what it is. Spatial reasoning gains in LLMs are extractable, but extracting them is nontrivial.
I will be posting something to that effect later this week. (Hopefully).
Basically current gen LLMs apparently do spatial reasoning the way they seemingly do everything else: by reference to previous example. I didn't see them work out which known example to use for a given problem until specifically prompted, in my case by accident.
If you ask an LLM, "what known example did you use to solve this?" it is very likely to cite something plausible-sounding. That absolutely doesn't mean it is what it actually did. It is trying to give the "right answer" and please.
> If you ask an LLM, "what known example did you use to solve this?" it is very likely to cite something plausible-sounding. That absolutely doesn't mean it is what it actually did. It is trying to give the "right answer" and please.
That's not what I said happened though. It didn't solve the problem (for weeks) until I (accidentally) told it which example it happened to know was relevant, and then it solved it in hours.
On a token level, text tokens are orders of magnitude more information dense than visual tokens.
You don't have to do much statistical analysis to figure out what is meant by the token string "cat under a tree". However you need to do an enormous amount to encode any permutation of pixels that show a cat under a tree from the set of all possible pixels arrangements that illustrate that (along with the massive fringes of ambiguity).
Everybody can be an armchair mathematician now. Just fling some thoughts in the direction of your AI setup and let it do breadth first search with AI based pruning heuristics.
If you really believe this, try to use GPT 5.6 to prove an open problem you know nothing about. You might get lucky, but if you don't, you will soon discover that 5.6 can make "progress" towards a theorem without actually getting anywhere pretty much indefinitely.
I generally agree, though direction, intuition, and domain knowledge are still relevant. Your breadth-first-search framing feels right, but you still need a sense of which paths are worth following, and you need to know when to trust the results.
I’ve been doing more math as a hobby in the past few weeks — working on lesser-known conjectures and exploring proofs of hard theorems — than I could have managed over the previous several years. It’s an exciting time.
I imagine there are hundreds of thousands of people doing this, yet all we seem to get is a handful of solved problems. So it would seem the success rate is extremely low.
Intelligence on its own is not very useful though. We put it on a pedestal because it creates huge potential when paired with other things, wisdom, discipline, empathy, but on its own?
You've not seen how they react to noobs asking physics questions, I think.
Even when you've got an interesting idea, if you're an enthusiastic amateur who don't yet know enough to phrase the question right but does actually know the basics, they'll put you in the same category as the people who think healing crystals can power hyperspace telepathy with Anubis: "oh no not another one".
LLMs have infinite patience, but unfortunately come (came?) with too much sycophancy, giving even more people far too much confidence.
I can't stop wondering myself.... I'm writing some software with AI and wondering, why am I doing this? Will anyone need this? Will anyone have money to buy this?
Best I've come up with is we'll need to be adopted by technofeudlaist overlords to be our patrons like in the roman days
> Best I've come up with is we'll need to be adopted by technofeudlaist overlords to be our patrons like in the roman days
Continually progressing AI (combined with our current socioeconomic systems) throws a lot of uncertainty into our mid to long term future, but I don't think this is going to be what happens.
There are billions more of "us" than of "them", people don't respond well en masse to a drastic worsening of their societal status and "they" are lagging very far behind on building their robot armies.
If we poorly navigate this transition the outcome should be worrying them more than it worries us.
> There are billions more of "us" than of "them", people don't respond well en masse to a drastic worsening of their societal status and "they" are lagging very far behind on building their robot armies.
Fortunately, yes, those robot armies do seem to be rather behind schedule*.
However, even if Musk dies of old age before anything like the Optimus can be connected to useful artificial intelligence, it can still be driven by the common joke a few years back that "AI" really stood for "Actually Indians".
When all the people currently upset about "immigrants coming here taking our jobs" discover those same people are now staying home and remote-working those same jobs over a VR headset and a Starlink connection... my guess is that by this point, Musk will have no political allies left.
> If we poorly navigate this transition the outcome should be worrying them more than it worries us.
It can screw everyone over. Literal communism was invented in response to Laissez-faire capitalism, and while Laissez-faire died with the Great Depression, the form of capitalism which succeeded it in the USA came into conflict with the USSR and gave us the Cold War, the Cuban Missile Crisis, etc.
That view is incredibly reductionist. It really is an efficient encoding of how nature behaves. It might be a human construct, but given how best it allows to understand nature (through principles of physics), it is uncanny to be any different from the language of nature.
Reminds me of Wigner's Unreasonable effectiveness of mathematics in natural sciences [0].
Nature has no language. Language is a human tool to imperfectly describe nature. Putting language before nature is magical thinking (also known as Platonism).
Yes, of course mathematics is a tool to model nature as accurately as we could. That still doesn't mean mathematics is as trivial or frivolous as manipulating symbols.
At a very high level mathematics is basically 100% text/symbolic rewriting. You start from some set of postulate assumed true and you do your thing to get a new different set of equivalent assertions in a form that is more useful.
I don’t know if LLMs will kill the working-mathematicians but at least seem like that it doesn’t seem absurd to imagine LLMs will be good at math…
The infinite monkey theorem assumes random distribution of symbols*.
Given the tokenizers have a vocabulary in the 10k-100k range, "a million attempts" will generally still only get the first token of the answer correct.
Even really rubbish models, e.g. talkie, the "what if we only use pre-1930s data to train a model?"** model, had to be almost all the way to the right answer to reach the really low HumanEval pass@100 score of ~0.04 (I'm only eyeballing the relevant chart).
* Actual monkeys not being like this is, while amusing, irrelevant
Even if every atom in the universe were a supercomputer generating a trillion trillion random characters every second since the Big Bang, the chance of producing Hamlet would still be essentially zero.
We have a non peer reviewed proof in a niche area of mathematics claiming to have been "co-written" by an LLM. What are the comments about? Lamenting or celebrating humanity's intellectual death... Very insightful.
I tried using AI to solve some advanced math problems. One thing I see is that they can throw an enormous amount of brute force into a problem. When mathematical logic can be brute forced we will see some interesting advances.
clear specifications for what counts as completing the tasks, and an explicit list of what does not count as completing the task, and clearly stating to not return until the task has been completed. I've had agents run for almost 12h!
... and now another one when a (not so cheap tbh) Chinese model matched OpenAI GPT Sol and Anthropic Fable/GPT 4.8 (some scores below, some scores above) ...
Oh and exceeded Google's best internal model capability enough to have them cancel the next Gemini release. Probably. But I'm 95% certain that's exactly what happened. This is the first time I really fear for the future of Google, because OpenAI matched or exceeded the experience of Google search ... and now a Chinese company matched OpenAI ... followed by Google failing to catch up. That really made me take pause.
It indicates the the "secret sauce" of OpenAI and Anthropic is ... nothing, really. Or none of it matters, except access to the hardware needed for 1.5 trillion parameter models, which Chinese firms now have as well. It means there is no "AI takeoff" where other firms can't match OpenAI/Anthropic models. And Google, inventors of transformers just missed the takeoff. There's more secret sauce, but clearly a great engineering firm can find it in less than 3 months.
The fact that this Chinese model has similar cost to OpenAI and ballpark cost compared to Anthropic while we can be quite sure they're minimizing the cost (that's what China is doing everywhere else) also raises questions about the true cost of serving comparable models and with that about the future profitability of OpenAI and Anthropic.
This is all a depressing and bleak future that I don’t look forward to.
One solution is to ban LLM’s, to artificially create a demand for human thought, that just feels like living in an artificially constructed zoo.
Another solution is humans don’t do anything that AI can do better , / doesn’t need the human touch. So I suppose we will all become artists, sportsmen or politicians, the only jobs that will remain except for select few. Maybe this is ok, I don’t know.
Another solution is we find a way to mind-meld with AI so that human + Ai >> AI alone. This is dystopian, who gets to decide who mind melds with AI, how much will it cost etc etc.
For the stupid copes that the prompt required human ingenuity, let me first add that the author used GPT5.6 to write most of the prompt. He just gave some mild direction. That amount of direction does not require deep expertise and the expertise required will keep falling with time, eventually an undergrad can create this loop and then maybe a high school student.
And prompt engineering / loop engineering nonsense is not real. Calling it engineering is a psy-op because it is something simple, imprecise and future models will be much better at it than you.
In fact, in the future the most likely outcome is you tell the agent what you want (I want this app, or I want this theorem solved) and it will set up the loop, or loop of loops and use all its computing effort to come up with a result. This is completely dystopian to a human life.
call me naive, but i really think the pursuit for the new, the better, the interesting, is boundless and humans will never stop. and so for those of us that dream and push and are plain curious will never run out of ways to be useful. and hopefully the fruits of all this tech will be ample enough to support the ones that don't want to or can't participate in that way.
If AI is better at being (or simulating; I think the distinction is irrelevant in this context) curious, better at examining that curiosity, better at pursuing it to discovery, better at applying the discoveries, better at sharing the discoveries, and able to push the frontier of knowledge (if that’s even the right word) literally beyond human comprehension in every domain, what usefulness exactly is left? Or are you just implicitly saying you don’t believe AI will ever get all that superhuman?
Maybe we should also burn books and lobotomize ourselves to make science more fun and challenging. The ego here sickens me. I don't care about your ego. I want accelerated material science, medical science, energy, better outcomes for people across the world.
I also want better outcomes for people around the world.
Memes like the permanent underclass and the massive incentive of replacing workers across the world does not bode well for a better outcome for people across the world.
Ah, so you're saying none of us will benefit from the discovery of new mathematical knowledge?
You've just condemned funding math, even human-produced math.
Either math is useful, in which case AI math is useful, or math is not useful, in which case we can save money by having mathematicians sell shoes or clean toilets.
Look carefully at the portion of derac's comment that was quoted.
> I want accelerated material science, medical science, energy, better outcomes for people across the world.
Does it mention mathematics?
Let AI make advances in all these fields. But if most of the power and capital in society is captured by a tiny minority, then these advances will become inaccessible to common people.
A rational reaction would be to identify the political coalition that would act in this manner and place yourself in opposition to it if you do not see yourself being able to change it.
gig economy is a blessing to keep the LCD busy and doing something productive instead of being nihilistic losers
it’s better to focus on how to keep these “knowledge” workers busy for the AI age
they don’t like to work and complain even on a w2 job (where someone else had to figure out how to capitalize them) so in many ways they’re far more dangerous and hopeless.
The problem isn't that we are gaining more knowledge and better technology. The problem is that we are allowing the rich to use the technology to subjugate the majority.
If AI improves human productivity so much that millions of people no longer need to work that should be an incredible thing. But the flawed structure of our society punishes those people rather than freeing them to persue endeavors that interest then.
Only solution I can think of is higher taxes for the rich used to provide universal income. I'm not sure how the realities of that would play out, though.
Anything based on a free market just ends up in.. this.
I fully agree. Tax wealth, not work. Then it is far easier to lift yourself from poverty, but harder to hold on to vast amounts of wealth.
After WW2, when tax on the wealthy was far greater, sometimes 95% of their income (in the UK at least), it was the period when the largest number of ordinary people were raised from poverty to middle class status, yet the number of rich people also increased greatly. When that stopped, wages stagnated, and now we have a cost of living crisis. Connect the dots, lol...
Many people think that the market is either 100% free, or 100% controlled, but a free market that operates within defined limits that will stop the greatest excesses of the rich ruining things for everyone else is the sweet spot we should be aiming for.
I'm strongly in favour of this. I think income tax should approach 100%. So your second billion you still make more. But not twice as much. The exact decay function is up to date. But I really don't think your post-tax take home should scale linearly with your income once you hit the max bracket.
We can also remove all of the loopholes that result in the biggest companies paying basically no taxes. I also think it may make sense to have a dramatically different tax structure for businesses than effectively income tax.
As I say above, here in the UK we used to have the top rate of tax at 95% of your earnings, and it was the period of fastest growth for poor and middle class people (Boomers at the time)
Unfortunately, the rich are the ones most capable of buying democracy, and the current set of politicians seem to very much welcome the money...
Outside of really bad luck I've never seen a case where someone was in a situation of working hard + making reasonably okay decisions + struggling, and that's purely due to there being avenues available where you can be rewarded well for your effort if you're of average+ intelligence.
Literally anything you wanted to make is no plausible to make if not now then in the next couple years.
The thing you’re worried about is capitalism and the connection with working to having the right to keep living. If you can throw off that mental shackle you can start to see how this can be amazing, but you have to drop the idea that everyone has to work at a job for someone else to provide some service in order to do it. It’s hard, I know, but change your mindset some and dream for a better world and we can make it.
You ever play a video game with god mode cheats enabled, so you can unlock all the unlocks, get all the best gear, and be an unstoppable force with unlimited money?
On the other hand, exploring any of your interests without needing to calculate how it can cover rent means you have more time and opportunities to actually do things you're interested in.
Working all day, then not wanting to do much else after because you're tired, is also fun for all of 30 minutes.
Right!? Like imagine we could use this stuff to solve world hunger, develop robots to clean up the oceans, or colonize space?
People are all “shucks how am I going to be able to justify my career at $job” and are missing the bigger opportunity. Such a lack of imagination I see…
There is no lack of imagination here, there is realism. You think you display "imagination" by just regurgitating all the tired tropes of old science fiction? Oh wow! Robots to clean the ocean! How novel! Nobody has ever thought about this!
The reality is nobody is going to build robots to clean the ocean because there is no money to be made. There will be robots to clean rich people's pools, sure. And it will put pool boys out of a job, great.
I can see (given that you've responded to a few of my points here), that you care about this enough to respond.
What if money wasn't a thing? What if that wasn't the only incentive in the economy? What if there were other ways to balance supply and demand curves (which is basically the principle use for money) so we could reduce shortages, etc.
Dream big man, I'm serious! And who cares if there is no money to be made? Like, what are you (individually you) doing to create the world you want to live in? What world do you even want to live in?
Exactly, a lot of people keep attacking me as anxiety for losing my livelihood. It’s not that.
It’s just depression that another avenue for human craftsmanship has been taken over by the machines. Humans are not just infinite consumers. If AI solves all of humanity problems, it’s like living in a zoo, not a life worth living imo.
This has always been the case though that "another avenue of human craftsmanship has been taken over by the machines."
It's good if we can have robots building things instead of having humans slumped over a workbench in a sweatshop piecing things together. It's good if we can have LLMs spitting out code rather than CS grads working 15 hour days at fintech startups or whatever. The conditions were never (ever) good before.
And, it won't be like a zoo - you'll be able to go wherever you want, do more or less whatever you want. Think about living in The Culture, or the Star Trek universe or whatever. There are options beyond "I'm a pet to the machines." Think big, dream big, then help make it a reality!
Like, out in space you're still going to need a human to make decisions because you can't wait 30min for the tight-beam signal to get back to earth. Also, we're pretty good at soaking up rads and still being "useful" - at least so far I don't see that being a major advantage to the robots. Maybe our place is to be the deep space mechanics that keep the robots alive? I don't know, regardless, you should dream big. What kind of world do you want to live in? Ok, how do we make that world happen?
My big (somewhat unspoken and somewhat immature dream) is that advances in regenerative health tech fix my optic nerves and I can get into the cockpit again one day, then maybe later I can fly some space vehicle like I wanted to since I was a 12 year old. Immature I know, but I miss flying still.
Is that world possible without AI? Probably/maybe? But it's a lot more plausible in a world where we have folded every protein, we have robot surgeons doing robotic procedures, AI generated research, etc.
Surgeons too now? Another avenue for human skill destroyed.
I actually want to exit. I want to live in a society where humans flourish not AI.
Actually one just needs to walk the streets of Japan and compare that to US. Tokyo has hundreds of small shops with humans doing specific niche stuff, perfecting their arts. That’s all so beautiful.
America has massive warehouses and supermarkets, with completely uninterested and bored out of their mind humans working and I suppose now we will replace them with robots. Great I guess. Maybe all Americans should just sit at home and consume Netflix and Doritos ig.
I want technology to increase human flourishing, not turn us into WallE humans. I am only interested in technological progress when it is done by humans who trained their whole lives for it, as it is a display of human excellence and that can be a beautiful thing. I’m not interested in it if some AI builds it.
Then, you know, build that world locally, at least for yourself.
Like, I'm building a cabin in the woods (original, I know), but you can build things and do things. You can be that kind of human if you want. But, if Americans could sit at home and consume netflix and doritos I suppose some would, but many would decide to start doing other things too.
I don't think the worlds we want to live in are that far apart to be honest. If I could make a living doing it, I'd try to build wooden sailboats for a living. That would be awesome. I would love it! In a world where I don't have to work to win capitalism tokens, I would probably spend a few years just building wooden sailboats to sail around the bay by my cabin.
The only way you see people flourishing is through coercion, by society or by the world. You'd explicitly rather prolong this codependency for your aesthetic preferences even, as per your own words. We're already just circus monkeys in your world.
You cannot fathom people pursuing self-improvement because e.g. developing or being capable is inherently enjoyable. You only see a world where such activities are a must, or a manufactured must. That's your measuring stick and that's the entire pie for you.
I mean this in the least combative way possible, but it sounds like you have things to sort out on your end first, before charting a course for the world to follow.
Yes, not everyone will agree with me. Maybe some people just want to consume AI abundance forever and live on UBI, like Altman’s wet dream.
But there will be some group of people, the people who wanted to be skilled tailors, skilled wood workers and apply them towards others before industrialization ruined them and of course the best mathematicians, the best programmers, before AI revolution will ruin them. We want no part of this technology, and maybe in some distant future we can organize and live separately from the consumerist dystopian hellscape that the world around us turns into.
I’m not charting a course for the world to follow, but I suspect every OpenAI employee or Sam Altman have no desire to live their entire life on UBI doing nothing of any importance. This for them is their pursuit of greatness, only in the result they’re destroying it for large swaths of humanity that live around them. I don’t want any part of it, I refuse to be a WallE human managed by AI Sam’s company designs to be put in a state of perpetual limbo somewhere.
> Maybe some people just want to consume AI abundance forever and live on UBI, like Altman’s wet dream.
Callousness like this is not exactly the hallmark of clear minded reasoning, which is your responsibility, I hope you haven't forgotten. Like no, there are more options than the two convenient strawmans you crafted. Kind of what my whole comment's been about.
Maybe it doesn't matter though, we seem to have already lost you to some neo-Amish fantasy. Except this is not even me being callous in response, you quite literally spelled it out. You'd quite literally take your current gripes and extend them indefinitely, forever shaking your fist at the sky, arrogantly refusing to even attempt to adapt to the (already present) world, all while being dependent on being tolerated in your historical reenactment. Apparently this makes sense to you, just like all the other vile ideas you pitched.
There truly are no stronger limitations than one's own mind.
Why the ad hominem? I mean, seriously, you think I haven't been "in the trenches?"
Dude, I'm nearly 40. I've seen the shit. When I was flying for a living, literal close friends died because of the results of avarice.
Dream big dude! This constant "doom and gloom" narrative is toxic to your ability to flourish. I have been there, I understand the urge to doom and gloom, but we make the world we want to live in. We do. We choose it.
What world do you want to live in? What is your "optimal" version of the future? How do we get there? What steps would we have to take to make it a reality?
If your vision of the future is "oh, like 10 years ago before $problem_du_jour" then you're already off to a bad start because we ain't going back. There's no regulation that will actually change anything materially (seriously, you think the people getting rich off all this stuff in Congress will voluntarily shoot themselves in the foot?), there's no way to go but forward, so how do we proceed?
I want to live in The Culture, or something like it if it's possible. So, let's start building it and ask permission later.
yeah! Like can you imagine a world where you could just effectively "conjure" new furniture for your house and robots would assemble it in a few hours? Or molecular assemblers would grow it while you slept?
Or imagine a world where we could basically cure every disease? Or a world where people could assume any form factor that they wanted? I don't know, the list goes on and on in a post-scarcity world... and we're seeing how we basically live in a sort of digital post-scarcity now and it is really cool.
If I need a software tool these days I don't buy it, I tend to see if I can make it. Now imagine that for physical things? What a time to be alive.
The rub is that we already live in that world relative to people who lived 1000 year ago, and yet somehow we aren't bursting with euphoria 24/7.
We also have humans who essentially live post scarcity lives on the back of a monetary windfall, yet their stories are strangely quite often depressing ones.
I think their is good reason to believe most human desires are illusions.
I literally have had various LLMs generate 3d printable plans for things I have used. Also, I am literally making a living right now being something between a dev and SME and building tools for people in an industry I’m familiar with.
Dream bigger buddy! We can make the world better, we’re not powerless here.
- Hasn't been peer reviewed yet, so take with a grain of salt. This applies to all claimed proofs, not just AI-generated ones. Even humans hallucinate proofs too!
- The prompt is on page 27 here[1]. It is ten pages of advanced mathematics priming the model in the right direction, apparently informed by a year of prior research. That doesn't invalidate the result if it is genuine, but it is worth noting that this wasn't a matter of "ChatGPT, solve this unsolved problem. Make no mistakes." and required substantial domain expertise and human research beforehand.
Saying “solve this problem” doesn’t get good results most of the time with humans either, it’s entirely underspecified so the person assigned that problem may solve it in a variety of unacceptable ways or not at all or perhaps worse solve the wrong problem because you weren’t clear about its definition. This actually happens all the time. What matters is the ability to communicate clearly and with precision as well as the “harness” which for humans is procedure, training, planning and management.
The subtext of this whole post (or at least a subtext that some might read), is "we don't need mathematicians/programmers anymore" or "we will need much fewer mathematicians/programmers". So the fact that this result required a year of prior research and a 10 page prompt of specialized knowledge goes against that subtext. You still needed the human just as much to get to the result, and the LLM ended up being a tool to find the last bit.
> Saying “solve this problem” doesn’t get good results most of the time with humans either
Sure. That is not even remotely the point I was getting at. Already we see the thread filling up with comments about how human skills are irrelevant, using a mathematics PhD applying his expert skills in a way that the people who are saying that could never have done to justify their inane conclusion.
> but it is worth noting that this wasn't a matter of "ChatGPT, solve this unsolved problem. Make no mistakes."
It wasn't the case for this, but when OpenAI disproved the Unit Distance Conjecture, it was really done autonomously by an automated AI pipeline with a completely AI-generated prompt. No human expertise required at all in the process (well, except for the final human verification).
Without human verification, an LLM can generate correct or incorrect proofs but it can't tell the difference. A human is necessary to be able to tell one from the other.
Saying that's a solution "done autonomously by an automated AI pipeline" is like saying that a self driving car that can only take you to the nearest train station after which you have to ride the rain to where you're going is "autonomously" driving you to your destination. Which is exaggerating the autonomy of the system, rather.
What I'm feeling is that there's a need to study how to use AI well. I've seen professors using AI, and it was amazing. In that sense, I think AI prompt input will become stratified. In the past, implementation skills were very important, but these days, concepts feel more important this is one of those things.
It's not that AI brings equality, but rather that the output varies depending on how much background knowledge you have. You could call it a stratification of input
I'm starting to feel like there's no place left for programmers like me who focus on quickly churning out MVPs.
Calling it prompt engineer is doing it a disservice. With agents we’re well into process engineering, which is a ton more interesting.
The obvious baby’s first process is “plan -> execute” but as we learn about the strengths and weaknesses of LLMs you have to start unpacking that process into planning, prototyping, testing, validation, reviews, and tons of research. If you treat it like an extension of your brain that can automate some thought processes, it becomes a lot more powerful.
One of the key skills of a professor is asking the right questions. Figuring out something worth working on, and then framing it in an appropriate way and asking questions that allow someone with specific tools and skills to make progress in the topic. Usually the tools and skills are those available to a new student, but working with an LLM is similar.
That skill comes with experience. Most people don't have it immediately after PhD.
I find it strange that people sometimes think of knowledge as 'public property for everyone.' The essence may be one, but the mental model of knowledge is individual. For an LLM's knowledge to become mine, I need to digest it to some extent.
And programming, as the programmer who created Eliza once said, is the act of becoming a legislator of your own universe. So even if there are black boxes, if you want to build a program that fits your own worldview, studying is essential.
> you can't have one LLM to read your mind to prompt another LLM
I’m excited to inform you that we as a species have developed a particularly useful facility known as Language which these LLM tools are evidently rather handy at wielding. This facility is particularly useful in this context when it takes the form of “dialog” or “questioning”, which can be used to propagate abstract ideas by means of mutually-feedback-guided-iterative-Language-use-turns, or more concisely, “conversation.”
One might even say that this remarkable facility can be used to “read” the ideas from one entity’s mind, such that after sufficient dialog the second entity obtains a (possibly lossy, but there are mitigations for this) copy of the ideas of the first. You might further be surprised to learn that this sort of idea-transfer business using language has already been happening in our society and species for quite some time indeed.
This is a lot of words to say that a human can prompt an LLM to tell it what they want.
edit: it reminds me of all that I have to wade through after I've asked an LLM a straightforward question and the answer should have been "yes, you're right."
I'm going to keep on repeating this on HN threads until I'm blue in the face, but:
There are two ways to solve a problem. Either solve the problem, or deem it irrelevant.
The implication here is that, you, the human operator, clearly are just confused. The LLM knows best. You're just a stupid human. The LLM knows objective truth, you do not. You have concerns, questions, the LLM didn't understand your question "properly"? Do not worry, the LLM objectively knows the optimal course of action. It thought through the implications of what you said, took into account all possible data, and came to the objectively correct design for your software, your society, your life.
In some sense, this problem would have been a societal problem within the next several decades anyways, but it's been hyper-accelerated by AI.
I actually think people who are great at understanding problems, coming up with requirements and designing solutions (all things I would expect someone who is good at churning out MVPs would be good at) are exactly the people most empowered by the current batch of LLMs. Its the people who are only good at working on small chunks of problems that I'm concerned about..
> I'm starting to feel like there's no place left for programmers like me who focus on quickly churning out MVPs.
Of course there is. The same way this was only possible as a result from the professor who prompted it with his specialized 10 page prompt and most importantly his deep knowledge of the problem space, the muscle memory and intuition you've built over the years is what will allow you to get more out of any AI than some guy who says "make a door dash clone" as the entire prompt
So these days I've been writing down my thoughts on my personal homepage. Things I've learned, my background knowledge, and so on.
I've been realizing that there are more books tied to my background knowledge than I expected, but I'm not sure what will happen as AI advances further.
These days, I'm living for the fun of building my own personal wiki on my homepage
Sharing knowledge is good, but just because an LLM crawls it doesn't mean it fits my mental model. The act of writing is fundamentally about drawing the shape of my own mental model.
could machine-learning even handle a TEN PAGE PROMPT just a year ago?
this is changing my mind, at least about experts using advanced tools like any profession where it's like the magic of watching a lifetime of hard-earned skill at work
> After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy.
> After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check.
>So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.
While they’ll never have the same subjective experience as humans, what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture?
They are prediction machines, and so are we in a way. We can give them nearly limitless resources to scale their predictive capabilities. We have billions of years of training baked in. They distill directly from our knowledge and can walk down paths that no human has before.
It’s silly to say they’ll never do anything novel.
At their current capabilities, it sounds like they are already capable of being a specific type is research assistant. What will that look like in 10-20 years?
They also have ability to go deep and wide in a way that humans just can't. We have limits, get tired, distracted and biased where AI does not. I think there a lot of problem where all the information needed to solve them is there, but we just can't put the pieces together. Like no matter how many people you throw at some problems, you hit human limits and more people won't help, but AI will because it is just relentless.
Not really the kind of biased I meant though. There was a recent article about a AI disproving I think an Erdos conjecture by doing similar things humans have tried, but it was much messier and less "beautiful". I think it is a common bias in science and math that things should be "beautiful" but there is no real reason to think that.
>what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture?
One thing is that an LLM can never assume, or find out, an inconsistency in its training data. Novel ideas often require correction of existing assumptions. As far as I understand, it is impossible, by design, for LLMs to contradict what is in its training data.
For example, an LLM trained on the data from an internet comprised of people who believe in the earth centric hypothesis can never say "Hey, that cannot be correct", or come up with the heliocentric alternative
Ok, I guess never is a real long time and eventually when we merge our consciousness with the machines we may have the same subjective experience.
I can confidently state that GPT-5.6 Sol is not experiencing the same reality as me. They _might_ be "experiencing" and I personally think they are, but their reality and experience is not the same as ours.
Well, sure. I'm not experiencing the same reality as you, either. I guess I assumed you were implying something more - a lesser experience or something. No?
Famously, all of maths is axioms and tautologies, so I'm not sure this will assuage any professional mathematicians currently having an existential crisis.
Maths was already infinite, it's still infinite, but who wants to spend all their lives changing rooms inside Hilbert's Hotel?
The author explains he's an expert in the domain and that he had worked sporadically on the problem for about a year, also with the help of previous LLMs. So whatever he means by "I wouldn't really say that this result is using or creating some fundamentally new techniques" it doesn't mean that the result was trivial. Also, says it might not make sense to work on low or even medium hanging fruits in the future- and I bet that's by far the largest share of work for most mathematicians.
Sure, it's not a breakthrough that opens new roads in mathematics- is this where the goalpost has moved now?
HEH. Don't know why you're getting downvoted. It's painfully obvious that there is a vicious AI backlash now, where every amazing advancement is met with denial and loathing.
Oh wait, sorry, I do know why you're getting downvoted. Fear.
A lot of people who thought they were special and “better” than mere blue-collar workers are realizing that in fact;l they are the same just working with a different medium.
Humans just want to be able to work and feed their families. Notice how this is something that is never addressed for example in Asimov's writings where robots do everything. How do people do anything to be able to afford these robots? How do Solarians make money to maintain their massively gigantic estates? It is never explained, but we always jump straight to the "robots do everything, everything is free, nobody works". How about the in between?
Oh yes, I agree entirely. I look at The Jetsons or Star Trek, and I see the fruits of technology shared by all of humankind.
But if the current trends continue, Elon will own all of the robots and the rest of us will be at his mercy.
How we change that - I don't know. But I do know we don't change it by putting our heads in the sand. AI is here and it is real. We MUST take it seriously. Shunning is not a viable option.
We're going to have to (collectively at some point) decide that we shouldn't tie survival to working.
Whether that's UBI, or Fully Automated Luxury Gay Space Communism, or UBS, or some AI managed economy, or some other such thing, we need to decide that "hey, we don't want to tied survival to output under our current system."
That's really the Rubicon we have to cross, and there are a LOT of people who can't really come up with a better way in their own heads yet.
Look, we all saw how things can be during the pandemic. That was a dark time, but it also gives us a model, we can just... do things. We just have to decide to do them.
Oh! Damn! Stupid me! "We have to collectively decide this", when is the meeting to vote on that again? When is the last time we all decided something collectively?
WTF does the pandemic have to do with this? "dark time", oh yeah, people stuck at home watching Netflix, what an incommensurable suffering!
Buddy, I can admire the naive childish optimism to a degree, but come on. "We just have to all decide". Do you live in a Disney movie?
I'm not sure, but if I may guess, I suspect he's talking about the fact that the entire US shut down for a year and people survived because the US Government printed money and gave it away free to businesses so they wouldn't collapse. (PPP loans.)
If we can do something like that for the pandemic, we can do something similar when push really comes to shove.
Thank you! We can decide to just... you know... do things.
Like, am I crazy here? The rules that we live by are largely made up and the points don't matter - we can decide to live in a better world if we want to. It's hard, and there are obstacles, for sure, but this appeal to doom for the sake of doom just... why? We have made unfathomable progress over the last century and if we keep trying we can make progress like that over the next century too!
I mean, a at least something like million people died just in the US with the most conservative estimates that I've seen (and by 2022). It was pretty dark, yeah.
But we also decided during that time to literally pay people to stay home for awhile. That was a really revolutionary thing and it was awesome. We could decide to do that again.
You can ad hominem until the cows come home, but yeah, we literally just kind of have to collectively decide that there are better ways to do things.
There's a fantastic book called "The Last Emperor of Mexico" I read a few years ago that really talks about how the idea of a Republic or Democracy in general was a pretty novel concept in the mid 1800s. People were a lot more skeptical about it than we're lead to believe now. But eventually, the ideas of aristocracy and some "well bred" group of various types of monarchs became silly on it's face. Now the default is that we should have some sort of democratic representation. That would seem utopian AF in 1820.
Well, we're going to have to bridge that sort of gap for getting rid of the need to justify our existence through work too. The transition is going to be weird, but we'll have to come up with something else and run with it.
Dream big buddy, I know it's hard, lord knows I do, but dream big, and work little by little towards those things you want to see in the world.
Genuine question: If you still or did think LLMs are just stochastic parrots that just summarize everything and have no form of creativity, what do you think after seeing results like this?
I'm very curious how people reconcile their fear/hatred of AI with actual objective reality. This is actually what interests me most about the whole AI thing. How we tell ourselves what we tell ourselves.
I had to create an account to respond to this because I am quite convinced these math problems they are "solving" are pure marketing. Why is it only GPT doing this, why not Claude? Why does Terrance Tao do marketing for OpenAI? I suspect OpenAI has hired math researchers to solve obscure problems and put them in their training set, purely for marketing reasons.
There was a good comment on the Pelican bicycle svg yesterday about how these models aren't getting much better beyond what the companies focus training them on. I think that's what's happening in this case too, they probably put this in the training set.
I (author of the paper) can't speak for others, but I have no affiliation whatsoever with OpenAI and have not received anything from them (I even pay for my subscription lol). I've thrown this problem at each model with new model releases, and 5.6 was the first one to solve it, so that's my side of that story.
Thanks for responding. I don't personally believe you have affiliation, more that OpenAI saw the problem was being worked on, gathered training data for solving it and training the model on it, then eventually you found it through the model (and now it's a big impressive thing for OpenAI)
Part of me agrees with the other comments that it sounds absurd, especially because of how involved/intricate your prompting was. But given you have been prompting this problem for a year, OpenAI could easily have seen your attempts/progress, and worked towards solving it with human intelligence that the agent was then trained on. Given the lengths these companies go to for training these things (Meta literally reorganizing their engineering department to provide training data for engineering problems), and also how much difficulty these models have with problems outside their dataset, I really have to wonder if it's the model being "intelligent" or if it has been trained on it
You are correct that LLMs are trained on existing proofs but hiring researchers to solve unsolved problems is just unrealistic, both in terms of how none of the mathematicians simply came out and took credit for their own discovery or exposed this, and how training sets are not easily memorized (rather, the meta techniques are learned).
OpenAI just has better training methods and techniques for pure math over Anthropic, it’s one of their biggest strengths
Terrence Tao getting paid by openAI is, to you, the most probable conclusion... much more so then the LLM actually being able to come up with math proofs?
Terrance Tao has for a fact appeared in promotional material for OpenAI. Based on my Googling the consensus seems to be he is paid for it, but I cannot confirm that.
I do think it's very likely that OpenAI pays for solutions like these to put in the training set, and then we get material like this Reddit thread. They market themselves as selling "intelligence", and solving these math problems is something people view as highly intelligent. I'm not a mathematician, so I cannot fully judge it, but based on my experience using LLMs for novel problems in other domains, they seem to really struggle with things that aren't common. That leads me to believe they train for specific outcomes like this. Also, there are a lot of jobs out there for data annotation, including software problems (Meta has basically reorganized its entire engineering department to create training data for coding problems).
You can go through my commenter history and know I'm no fan of LLMs. I don't overstate LLM capabilities and am highly skeptical of them in general. 5.6 Pro is genuinely pretty good at certain kinds of math problems that just require trying out lots and lots of solutions, mostly because it's stubborn and can run a bunch of instance in parallel. It is NOT good at coming up with unique ideas or recognizing when its proof approach is doomed, and if the correct approach isn't in its "bag of tricks" for tackling a specific kind of problem, it is not going to get it without a lot of guidance. That said: I 100% believe that it's solved the problems people are claiming that it solved.
The way you should read this is (IMO) not that LLMs have somehow achieved AGI, but that a lot of mathematical research is more about knowing a huge amount of mathematical background, being stubborn, and getting lucky with an approach than it is about brilliant insight. Many people who don't think of themselves as particularly mathematically gifted could have made progress on these problems if they were given enough time and were interested enough. What's notably different about 5.6 (and born out in benchmark after benchmark) is that it does seem to genuinely "reason" through stuff at all -- without that, persistence is pretty worthless because the LLM just goes wildly off the rails if it's put to work for long enough (5.6 itself will still do this if it can't find an answer in a reasonable amount of time).
Terence Tao also uses Anthropic's models in his work. Oh, you didn't know that? Well, now you can pivot to saying that he's getting paid off by both companies. This is actually one of the hallmarks of both conspiratorial nonsense and military-grade cope. Any fact, regardless of how mundane or extraordinary, gets re-imagined as evidence of the same mad-hatter conclusion.
I hope people are screenshotting this stuff. This really needs to be documented. It's remarkable how wild it's getting.
I hold my stance that LLMs are stochastic parrots.
Making the parrots ever more complex and training on ever more data produced by intelligent, creative beings may make them more useful or convincing but does at no point give rise to intelligence or creativity.
I won't touch creativity, but if this and other results like it do not demonstrate intelligence, what does? How was it able to solve problems that specialist mathematicians have tried and failed to solve for years?
Mathematics is a language. If anything, it's much more well defined and formal than most others. Train on enough examples and statistical autocomplete gets you places. I'm surprised how anyone would even consider this intelligence?
They aren’t really stochastic parrots, they’re next token prediction machines. They aren’t intelligent nor creative imo. However they are quite useful at some tasks. Why would you assume people who don’t kneel at the alter of AI are somehow fearful or hate AI? Most of the hate I’ve seen have been for the people and companies involved with AI not the technology itself.
I'm very curious why people conflate thinking LLMs are stochastic parrots with "fear/hatred" of AI. It seems like you're arguing with people who agree that it works and it helps, but you're trying to insist that this implies that they should kneel down and pray to it.
Is "stochastic parrot" too disrespectful for you? Do you think it is a slur?
edit: and this is a genuine question, also. How do you do stochastic parrot = "just summarize everything" = "no form of creativity" = "fear/hatred" so quickly?
Are summaries not creative? Are Maxwell's equations not summaries? Do people hate and fear parrots?
I have absolutely no problem with people disliking or fearing AI. It's energy consumption, effects on education and potential for displacing good jobs are all quite disturbing. But "stochastic parrot" means that "all it does is randomly repeat things that it has seen before without understanding them." It's infuriating to see this written about an instance of an AI solving an open math probably. Do you think the models are just randomly repeating facts until they accidentally emit a proof? If so, then how do they synthesize that knowledge into something logically coherent?
Alternatively, if you think that even Maxwell was a stochastic parrot, then presumably almost every human who has ever lived was also a stochastic parrot except a few rare examples like Einstein. Not sure what definition you are using but it seems too broad to be useful.
I think it's quite clear the proof here shows that it is not a parrot. It objectively isn't.... that's the only rational conclusion. Yet many people claim that it is, so the main conclusion is fear/hatred is causing people to rationalize their logic to fit the narrative they prefer.
Except solving problem is probably the least (even though it's important) interesting thing in research...
The most interesting thing in research is finding new questions, that we understand and that we know why they are important. And that's something that humans need to do (by definition)
I keep hearing this but lots of maths problems are practically important! We want to know the answer because it will be useful for applied science, or statistics, or engineering. It’s not all just about knowledge for its own sake.
I know a bit about this field. This conjecture reads as somewhat more niche than the cyclic double cover conjecture recently proved by OpenAI, but nevertheless represents a real contribution.
You want to know how long it takes to solve an optimization problem, in this case over convex, lipschitz functions. (The restriction to a spherical domain is not really a restriction, you can just change variables for any bounded domain.) Anyway, showing upper bounds on time complexity is "easy" because it's just the runtime of your algorithm. Showing (nontrivial) lower bounds is usually much harder because it requires constraining all algorithms.
This proof apparently shows that the lower bound time complexity is equal to the time complexity of an existing 30-year old algorithm: it requires Omega(d^2) function evaluations to solve over this class of functions.
My gut says likely implies that d is the minimal number of evaluations if you have a gradient oracle because you can approximate a gradient with d function evaluations, but I'm not sure how hard it is to make that rigorous.
Sometimes I read a comment on HN that is so advanced that it's just as readable to me as Greek. Love reading it just to see someone work though!
> so advanced that it's just as readable to me as Greek
I used to feel this way about statistics.
The language and terms are hard to understand and many of the formulas are taught as "just memorize this" instead of building up from first principles.
But then I started using statistics to analyze something I cared a lot about (paintball) and I quickly realized it's like learning anything new:
- there is jargon
- and core concepts
- when you learn the above, it suddenly makes a lot more sense.
I gotta know what you use stats for regarding paintball. I haven't played in years but I loved playing back in the tipman 98 custom era (not sure if that's still a popular marker).
So I wrote a whole bunch about college paintball statistics here: https://www.pbnation.com/showthread.php?t=3949120
Then I wrote some more about pro paintball stats in the below three Reddit posts:
1. https://www.reddit.com/r/paintball/comments/1h17f2m/intro_to...
2. https://www.reddit.com/r/paintball/comments/1jy5xqp/paintbal...
3. https://www.reddit.com/r/paintball/comments/1k6bzi7/paintbal...
Some highlights:
- I started with just pen, paper and a stopwatch (as a college coach)
- I assumed paintball would be more like football where it's hard to track individual effects
- Turns out it's a surprisingly simple and stable "state machine". e.g. the odds of winning with +1 body (e.g. 5v4, 4v3 etc) is, in college, about ~75%
- Paintball is one of those sports where "the weakest player determines the outcome". Why? b/c if 1 player gets out early, you are fighting out of a hole.
It also made me appreciate that as good a book as Moneyball is, reading it after you try to create analytics for your own sport makes it 3x as enjoyable/insightful.
One downside though:
I would watch games and I got so good at internalizing the stats per state of the game that it was like watching the world series of poker where I could see both player odds of getting eliminated and probability of winning over time charts as I watched the games. Made it harder to be the "come on guys! we can win this" coach when we were down on points + bodies.
> Made it harder to be the "come on guys! we can win this" coach when we were down on points + bodies.
Would you say that paintball on that level is almost deterministic? Compared to, for example, (American) football where momentum and perseverance can turn games around completely.
Wow, a blast from the past to be sure. Was not but any means avid, but did own a tipman. And was always dazzled when someone showed up with an angel.
That era is now! (Still)
Paintball had a peak in the mid-2000s (I would say 2006) due to a combination of:
- Millenials who were kids of the baby boomers being in their late teens early 20s
- Disposable income due to the real estate bubble / positive consumer sentiment
It dropped off a lot after the 2008 GFC though.
BUT
A lot of those kids playing in the mid 2000s are now parents of ~10 year olds so apparently there is a bit of a resurgence going on.
There was even a hl mod with an esports league.
This is my exact experience right now trying to wade through the research on psychometrics and skill/knowledge assessment design. It’s mostly just applied statistics but like all such fields, over decades of specialization it acquired its own jargon for abstractions that are quickly recognizable to anyone with a sufficiently developed nose for modern mathematics. But you still have to wade through all the definitions to make those connections before you actually understand everything.
This is a great suggestion, use complex hard stuff for doing fun stuff.
Not to diminish the comment, but most things are not as complex as they sound when phrased in everyday language or sound much more complex than they are when phrased in technical language.
Technical language is a tool that allows insiders to say less and refer to more, and to be specific, but it's just a tool. Most things can be described in accessible ways.
I think you'd be surprised at what you could understand and at just how few domains are truly complex enough that a layman couldn't understand with a little bit of patience and an accessible summary.
Saying things colloquially gives everyone an intuition about it.
You'd think intuition is great, but no, about half of the important things are counter intuitive.
That makes intuitive thinking about complex technical topics you don't know enough about worse than useless.
Yeah, heck, whenever an LLM puts my thoughts and intuition into words, it sounds really complex as well.
(FWIW I have an issue with producing words, rather than recognition. I do have the intuition I just lack the labels for it.)
It took me a while to understand a lot of these math concepts.
Turns out people doing Engineering research are using a very small but powerful bag of tricks from a handful of few famous Mathematicians. The concepts are named after them!
Thanks for posting this comment, it makes me proud of myself to be able to partially comprehend the comment :)
I don’t think OP made much effort to make the comment accessible to non-experts, and so it should be taken as a gauge of the fundamental difficulty of the topic.
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It should be noted that optimization of a convex bounded lipschitz function is exactly what most modern statistical learning (AI) models are based on.
Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.
I'd push back on this. Most of the core optimization techniques (eg, ADAM, stochastic gradient descent) are straight out of the convex optimization literature. Generally you need to use optimizers that work well on convex objectives because near minimizers, functions tend to be convex. (Proof by contradiction: a non-convex point has a strict descent direction.)
The fact that neural networks are highly nonconvex has encouraged a lot of research, but it's more of the kind aimed at resolving tension: these methods are probably good for convex functions, why do they continue to work for nonconvex problems, and are there tweaks we can make to improve them in that setting? It's not a lot of de novo theory; more standing on the shoulders of giants, etc etc.
No, I have to push back as well, sorry. It takes a very long time to get to the "near-minimizer" stage when training a neural network, and in practice, you never get there (see neural scaling law regimes). What you are saying is the viewpoint from 6-7 years ago. Things have changed.
The reasons why optimizers work well for neural networks in their highly nonconvex landscapes has absolutely nothing to do with their performance in convex landscapes. If that were true, everyone would be using Newton-CG. These optimizers were born in the convex optimization literature as a consequence of the genetic optimization nature of incremental publication (and because that was all we had), but their modern study is through the lens of implicit regularization (their preferences for certain minima) and their stepwise vs. continuous rates for feature learning in multilayer models.
This is completely new theory by the way, and requires painful reinvention of the field. It does not stand on the shoulders of convex optimization. The nonconvex setting is assuredly not a perturbation of the convex setting, and those that do continue to work on deep learning optimization from the convex optimization perspective are well behind the times.
It seems that we have two different stories here: in one, the new optimization theory represents a stark departure from the prior art, a sort of revolutionary new view of the understanding of optimization as applied to neural networks.
In the other story, the current understanding of optimization is a natural evolution of past work, where a new generation of researchers respond to social and technological changes, adapting and building on the work of the past, taking what's useful, downplaying the importance of some ideas, and inventing new language to describe concepts that seem most relevant to the current situation.
Both stories tell some of the truth. A revolution or evolution? Looking at the literature (eg the sibling comment here) shows that even today, convexity is used as an intuition pump for modern optimization techniques. But there are also new ideas that apply to the specific exigencies of neural nets, and downplayed ideas (eg convergence rates) that seem less relevant.
If people want to learn more, at ICML (one of the top AI/ML conferences) this year there was a whole tutorial on this topic: https://www.cs.ubc.ca/~schmidtm/Documents/2026_ICML_Tutorial...
The optimizers are lifted from convex optimization, but the point above was that they are applied to highly non-convex problems. They work for finding local minima, but a lot of the deeper literature does not translate (e.g. the conjecture being discussed in this post).
ADAM does not work on simple convex problems [1].
[1] refers to [2], which shows that ADAM is not as efficient as gradient descent with line search on some problems, including neural networks.I'll point out that "does not work" is not the same as "not as efficient" :) But it does seem the Adam paper had an error.
I think that Nesterov's first order method is the most efficient general first order algorithm on convex problems, so anything else is in some sense worse. (Edit: removed incorrect ADAM comment.)
Yours' "not as efficient" in [2] means that, sometimes, ADAM "does not work." Look at figure 2, ADAM literally does not work in the case of "true model."
Yes, apologies, I didn't read the articles you linked before posting this. I did update the comment.
I don't think this changes the point, which is that most optimization methods used in AI owe a substantial intellectual debt to convex optimization theory.
I love convex optimization and there are a few SciML projects I am on where I really need results from there. But in AI research with deep neural networks, it's become a liability, because people will just not let go. I'm getting tired of reviewing convex optimization theory papers in ML conferences that are still trying to wave away the obvious issues with their application to deep learning. It's harsh, but I do feel we can only start talking about an intellectual debt once that stops being the case.
Another intuition is that near a minimum you can Taylor expand the function and show that the higher order coefficients (past the square) are negligible.
I'd say it's going to be very hard to come up with a method that works on general nonconvex functions while not working on convex functions
It's not a matter of whether the theory "works"; it's a matter of whether one is asking the right questions. Convex optimization studies how quickly an optimizer can reach the optimum. In the non-convex case, there are many basins containing their own local minima. The more sensible questions there are "which basin is it likely to go into?" and "how do I steer it to go where I want?". Global convergence rates are largely irrelevant by comparison.
Objective functions are almost always convex bounded Lipschitz functions
What do you mean by this? A neural network hypothesis space is not typically strictly convex or a lipschitz function.
The loss function is.
Yes, order d is the minimal number of evaluations of gradients needed for the same problem! That has actually been known since 1979 (Nemirovsky and Yudin showed that), and there are methods with the same complexity so this question in the gradient model has been solved for a long time. "because you can approximate a gradient with d function evaluations" was exactly why d^2 made sense as a lower bound for this case! Basically, the lower bound question can also be thought about as "can you do better than approxing a gradient?", so this result says no.
Any implications for discrete/integer optimization?
> I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.
I wonder how this compares to what we see happening with "juniors" in software development? In math research, do you also get the training for the profession from working on the low hanging fruits for a while, to then move to the medium-hanging, and later go on to work on previously unsolved stuff?
Around here AI isn't really more of a threat to juniors than it is to seniors. It's a threat to the people who have been taught "recipies" rather than applied computer science. You can have excellent seniors who can do TDD, DRY, SOLID and so on, who also happen to have no idea what a L1 cache miss is. The current AI models know all of those things, but they struggle applying them correctly without someone piloting them. Even in the energy industry where I work, where you'd think it would be obvious from the context that you should prioritize runtime safety over debug safety, the current AI models struggle to do so. As far as seniority goes, though. If we can find a young developer with little experience who actually knows computer science, we're much more likely to hire them... Since they are cheaper.
This isn't something which is unique to software development though. We're currently building enterprise AI apps that we can deploy into the AI agents working for anyone of our employees. The key thing we're currently seeing is that the people in a team who are the ones that everyone turn to for advice, are the only people who aren't in "danger". Even people who are great at their jobs are being outperformed by AI in many cases.
I think it'll be a massive challenge for our society in the coming years. Maybe we're even going to get to the point where the AI will also be capable of replacing a lot of the "domain experts". Right now that seems far out, but then, if you had asked me about AI four months ago I would've told you it was all hype.
AI is a threat to everyone. People who claim that AI will never be able to do X have consistently been proven wrong.
The only people who are safe are those whose jobs depend in some way on their humanity. e.g. yoga teachers, bouncers, etc
Nobody knows.
It's not a zero sum game. You can have AI "senior engineers" working under humans building bigger things than we've been able to.
We also don't know where the capabilities of current AIs will plateau. The benchmarks aren't really telling the entire story. From my perspective of using the models there are certain axis where they're not making a lot of progress, like being able to have large accurate context on the scale that humans can. There are other dimensions where there is still a large gap between human capabilities and LLMs. It's true that relative to other areas (lessay chess) LLMs are more generalized but they are still not fully generalized (back to the chess example, LLMs are not good at chess).
> It's not a zero sum game.
Resources are, though. The planet cannot support a race of digital super-people, and us, and an continually growing economy.
It's the height of folly to think that, as things are going, we are going anywhere "good".
I was arguing with a friend about this point today. I'd posit that our economy has been growing in a way where goods that require more human time and natural resources to produce are giving way to goods that are intangible.
Once we've met our basic material needs, we're tending to consume things that are replicable with low marginal costs, and which do not interfere with the production of other goods. So maybe we can actually support a continually growing digital and entertainment economy, at least for a few more generations.
Maybe these mathematical contributions will also impact the efficiency and capabilities of our material production systems as well, which is another way to keep the economy growing.
I'm optimistic that we'll do more with our resources rather than trying to optimize for doing the same more efficiently with less resources.
> Once we've met our basic material needs
The concern is not goods produced, those are not "resources", except perhaps in the very limited sense that food is a "produce".
Energy, materials, those are resources.
Economic growth doesn't mean using up resources faster, it means trading things faster.
You are correct about growth, but speed is also a resource - kinetic energy is still energy, which comes from somewhere, and unless you have 100% perfect energy usage, there is thermodynamic loss, contributing to entropy, which itself is another resource. Temperature is a physical resource all lifeforms and machine processes share in a gravity well, and all processes breakdown after a melting point (unable to maintain their own required entropy levels)
This was sort of what I wanted to say, but I guess I should have worded it differently. I certainly didn't mean to say that I thought AI would stop improving. If anything I'm surprised at how much we have to fight the AI models to do what NASA has been doing for 60(?) years.
i'd take a yoga class from a bot
I'm now imagining Bender's Yoga class. I suspect it would hurt.
would prove that robots were subject to nominative determinism too, if nothing else!
How would this be different from taking a yoga class from a video?
I would be willing to be proven wrong, but I doubt the ability of LLMs to give useful corrections in yoga much more than their ability to write useful code.
Probably that you'd be doing it in a group, which is a lot less lonely than doing it from home.
I think there's a lot of opportunity for computer vision in exercise coaching.
Your first two sentences were correct. The last one is already being proven false.
It's a threat to everyone. UBI is the only way.
Productivity improvements tend to increase employment. AI will not reduce employment.
(Also, the US budget deficit is way too high to afford a UBI.)
Interesting, thanks. I don't know where "around here" is, but the signals I've seen in a lot of articles is that the demand for junior software people has taken a dive since a year or two back, with student programs etc getting cancelled. One googler said they were getting a junior to their team and that was kind of a big deal because it hadn't happened in that whole department for a long time.
In relation to that, I guess my question becomes: if the same thing will happen in math research, who will write the ten page math proof prompts in the future?
There is definitely the effect of less or de-growth in the industry, which started before the current AI hype. And now there's the additional effect of companies hoping AI will replace their need for (junior) devs. Nobody knows if or to what degree this will work out (yes, we all have opinions, but no crystal balls), but they are holding back the hiring until they know how all this pans out.
I'm from Denmark and I've been an external examiner for various CS educations for the previous 13 years now. Some of them teach you a lot about how the hardware works, others mainly teach you design patterns. Five years ago the latter was in high demand, because a lot of software development frankly doesn't need computer science (until it does). Now there is almost no demand for them.
Honestly, i've received a formal MSc education in the hardware aspects, including for designing embedded electronics products. Spent the most part ofy career in the software industry designing enterprise software and feel like i never needed to use them, except maybe early in my career when i was reviewing tech stacks and determined that .NET would be among the winning horses, precisely because it'd take care of that for me almost all the time.
What i see today is the opposite of what you see : product owners not knowing a thing about software engineering but being able to vibe code prototypes handed over to the dev team are rock stars.
They are closely followed by senior software developers having more of an architecture & design background than a low-level computer science background. Most businesses are looking for builders these days.
Where what you say may converge with my observation is that to be able to do to things such as proper database query optimization, even using AI assistance, you need to be able to understand the concepts of working memory set, cache misses etc...
I've found huge problems, like database servers being grossly underprovisioned (like, 60% cache hit, 4gb RAM server for a 700gb dataset with an 50gb circa hot data set). SSD were used and only latency was measured, so no one realized how problematic the situation was (including a consulting shop they hired to help them manage their DBs - backup, maintenance etc...).
However, having a high affinity with hardware is not a driver / computer science of hiring decisions from what i can see in the enterprise software world. But it would make sense for it to become the case within 10 years. I suspect that you work in a niche where performance optimization matters a lot.
> However, having a high affinity with hardware is not a driver / computer science of hiring decisions from what i can see in the enterprise software world
I think the way I worded it was maybe a little too close to just being about hardware, because performance do matter a lot in the energy industry. I do think it applies to SWE in general. You mention .NET and I've met C# developers with years of experience who couldn't tell you the difference between IEnumerable and IQueryable. I've met even more experienced Python developers who don't know what a generator is. Stuff like that, not having knowledge of the tools they use. I guess you could argue that those are bad developers, but I don't personally think that has been the case for most of them. Still, you'd rather have someone who thinks about these things rather than eventually using batches once they run into memory issues.
I also think these changes are appearing faster in non SWE enterprise. As you said, product owners who are AI explorative (for the lack of a better word) are rock stars. We see this a lot in our finance and risk departments, where domain experts now write fairly decent software with AI. My team has build them tools so that they build things the same way, use the same developer setups and pull the pre-approved external packages or are offered alternative ways of doing things. A few years ago this would've been done by these domain experts "ordering" the software they needed from our SWE team, and if they hadn't already been mostly laid off due to Putin's invasion of Ukraine changing the markets, I belive they would've been now because of AI.
Because frankly, a lot of the software that gets produced in these areas, don't need computer science, until it does, and the domain experts can make the software they need so much faster than before by vibe coding it. From my perspective it's not that much of a difference in the quality of the code that gets produced. I also had to help with performance and security when we had more software engineers on staff. Though now I do it more through writing and distributing AI agent applications rather than writing a C binary or optimising the code directly.
It's funny how you applied on your own argument several logical fallacies about why ai is only a threat to people who have been taught "recipies" versus who know what L1 cache miss is.
Actually it's sad there are people out there dumb enough to believe knowing L1 cache is any different than knowing recipies when it comes to the story which jobs AI will take. I'm convinced by now it will be the jobs of those people believing such crap.
So... The AIs with no model of the world are replacing software developers that have no model of the world?
Unless you’re claiming that AIs will suddenly (and very soon) stop improving, they are obviously a threat to everyone’s job.
Calling notable conjectures that have been open for decades “low-hanging fruit” is an act of desperation. Most professional mathematicians couldn’t have proved those conjectures if their lives depended on it.
I wouldn’t call it “low hanging fruit” but it’s easy to think of problems that seem harder. Apparently solving notable math conjectures is easier than building a practical robot to deliver a package to someone’s porch?
So, yes, AI is a big deal and we don’t know what it’s going to affect, but the goal of replacing everyone’s job is extremely ambitious and there’s a long way to go.
This has to be assessed separately for each kind of job.
Moravec's Paradox strikes again!
Moravec must be at some level gratified things are arriving close to his predicted timeline.
The thought that anything could improve without bounds would be absurd. We are living in the physical world after all. The (open, interesting) question is how close we are to the limit.
It’s safe to assume that after less than a decade of LLM development, we’re nowhere close to the limit yet. In fact, progress still seems to be accelerating at the moment.
If anything, it should be safe to assume by now that capabilities don't scale linearly with model size.
Types of technology - of which we can include intelligence - move along S curves, but it's more absurd to think that humans are near the top of that curve rather than right at the bottom.
There might be a thing beyond intelligence that we can't even conceive of.
> more absurd to think that humans are near the top of that curve rather than right at the bottom
The “absurd” dimension does not enter. This is a situation where you have no evidence at all.
In the absence of any information, the average (mean or median) is your best guess. Now where that average is, you have no idea.
> There might be a thing beyond intelligence that we can't even conceive of.
This statement already supposes there is a thing called “intelligence”. People have been pretending to measure this for more than a century. Modern thinking at least says what we call intelligence is not a single concept.
>Unless you’re claiming that AIs will suddenly (and very soon) stop improving
Most technologies level off sharply after bouts of boundless improvements.
In 1968 they thought we'd be flying to the moon by now but instead we're flying across the ocean in planes not that different from the 747 that existed back then.
They sometimes start improving again. In the context of your comment, look how the cost/kg to LEO has suddenly dropped radically. This was mostly due to institutional change that allowed previous non-technological barriers to improvement to be bypassed.
Even if AI stops merely at solving Erdos problems (merely...), metabolizing it would take decades.
> COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS
> BY L. J. LANDER AND T. R. PARKIN
> A direct search on the CDC 6600 yielded:
> as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler that at least n nth powers are required to sum to an nth power, n>2.https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966...
It is a conjecture whether grinding it out on Lean is a difference in kind, rather than degree. I say degree. But it remains to be seen.
I was trained as a mathematician and worked as a math researcher for a little while (now working as a private tutor), and based on my experience I'd say this description is basically right, with one extra wrinkle.
In order to get a Ph.D., you have to do some sort of original research, so in that sense you're working on "previously unsolved stuff" basically right from the start. But that doesn't entail doing anything all that ground-breaking; most Ph.D. dissertations (very much including mine!) contain work that a more senior researcher in the same subfield could probably have produced without too much difficulty. The software development analogy is a pretty good one: a lot of the point of getting junior researchers to do research is to help train them to one day become senior researchers, and often the work itself is nothing all that special.
Given the trajectory of these LLM proofs, this seems like it's going to have to change pretty soon, and to be honest I'm pretty grateful that I'm not in charge of deciding what that's going to look like, because I don't have any good ideas! I'm actually pretty worried about the future of the field.
> In order to get a Ph.D., you have to do some sort of original research,
China has now introduced "practical PhDs" where you have to build some practical machine instead.
That is exclusive to engineering PhDs, entirely irrelevant to the discussion of math PhDs.
Unless, of course, the machine being constructed is the prover (and prompts) that proved the results (or achieved whatever result instead of the human achieving it.)
Indeed. Perhaps my article here will be of interest to some: https://blog.oberbrunner.com/blog/ai-math-as-humanities/
My experience may not be entirely representative because to be entirely honest I’m not exactly a great researcher and there are brilliant PhD students. That said it indeed was my experience that in the pre-PhD / early PhD period ( or even longer … ) your advisor proposes (gives) you pretty low hanging stuff that he mostly already knows how to solve, at least at a high level, with the expectation that it will teach you to use the mathematical tools you need.
This apparently required a 10-page prompt. It seems like someone needs to know enough to write it?
The author also used GPT-5.6 to write the prompt. This did involve giving GPT-5.6 access to his previous work and a back and forth process (so definitely still used the author's expertise to some degree), but the prompt itself is also largely AI generated.
What's the difference between using GPT to write the prompt to GPT, and "thinking"? The LLM uses the first tokens to predict more tokens, and then uses those tokens to predict even more tokens.
Roughly speaking it is the difference between having a contractor go out and do some work and having that same contractor first come up with a plan to do some work, run that by you, and then go out to do that work.
Part of it is as a another comment in this chain mentions the chance to review the prompt. Part of it is that it forces the AI system to plan things in a certain order, in much the same way that forcing the contractor to write the plan out first forces the contractor to proceed in a certain predefined order that may (or may not!) be better at getting to a final answer.
The ability to correct the plan/prompt.
Certainly. This feels similar, to me, to how building complex software with LLMs works today in practice. You need to know a lot to set up goals and guardrails and verify outputs. For me, making the bits change was always the fun part, not tangling with text in my editor, though that had its moments.
Yeah, back to the gold-in-gold out use of LLMs.
I was thinking this past week I have gotten so lazy w my prompting via CLIs.
Back in the before I had put such discipline into my prompting and supporting context.
Now I’m like, “look here and here and here are some tools, and /skill /skill okay go.”
Or “restate this request in your own words and enrich it as appropriate handling any gaps. Okay go”
We're also at the point where you can roll out context to your entire organisation. I created an app for our m365 Cowork and deployed it to everyone who develops software. It does a couple of things, but it main knows our compliance policies and can guide developers through writing the documentation needed for NIS2 compliance. It also guardrails against non-approved packages, and helps developers find alternatives, or if none can be reasonably found, how to get a new package/dependency approved (or rejected).
A few months back this would be something every developer kind of did on their own. Maybe they shared skills, we certainly encouraged it and tried to do all the change management things, but nobody really had the same versions of the skills. Which was horrible in the deployment pipelines, something like the compliance documentation often had to go back and forth several times before it could be approved. Now it's just there, for everyone.
In a year or two, I expect a lot of these things to have become even more standardized. So that we don't even really have to build our own apps, but can simply use the ones in the catalog with minimal configuration (and that config will likely only be necessary because I'm from a tiny country that nobody will maintain standards for).
Yes. I worked on two large monorepos, about one quarter per project. Maybe 80+ devs on first, maybe 40+ on second. Both were not sure how to describe this but ultra-high-velocity agentic driven development efforts.
On the first, there were ~no shared skills. There were some requirements set up but they were not minded properly and became stale / ate context for little gain. The hardest hit was in E2E tests which would flake and create long running, too-often failing CI. People would disable them, because they were not reliable and velocity was so high, no one was happy w them.
I maintained my own set of skills and CLIs to back them. I'd share them if they came up but it was like the old days of manage your own stuff. Not much credit for building and sharing devex tooling to the team.
But then on the second one we were in better shape--we had vendoring set up to distro skills automatically.
Before the project was well underway, I put time into understanding how all of our tests aught to be written. Finding the forbidden things, etc, getting review from our best test folks and ultimately landed on a `/test` that routed across all possible test types.
Like night and day. Instead of finding out while trying to get a release out the door that some corner of the project had a handful of flakes, tests were written the right way from the start.
Like, it was beautiful. And I don't think devs noted difference while building. Only that there was an absence of BS in CI.
Hard to quantify the lack of pain, but it was big!
This made me chuckle because it's so true. So much detailed steering and finagling in the past, now I point the agent to a bunch of information sources, skills, similar repositories that might hold useful input and tell it very roughly what I need and off it goes, I'll grab coffee.
Have you tried this:
?It is an ultra-lite way to plan, I suppose.
I like the format because:
I think the models are doing this anyway, but I find the words "enrich" and "gap" are well understood by models and they demonstrate it in the response to the above pattern.Anyhow, to get back to the point, there are still prompt-level tricks--but ultimately if repeated, should probably also be built into skills themselves!
Math is way more automatable than programming.
In math, a proof is a proof. We don't know if we can get there and so getting there is the hard part.
In software, we always know that we can solve the problem. So HOW to solve the problem is the hard part. Because the type of solution involves maintainability, which involves planning, LLMs suck at it. This leads to "LLM slop code" whereby the LLM creates ad-hoc convoluted logic with redundancies and no reuse of existing standard library batteries.
Unless you're a Grothendieck who gets mad at Deligne for not solving the Weil's conjecture "THE RIGHT WAY", software is fundamentally different than math in this respect.
So I'll say it again, AI will win a fields medal for before managing a McDonald's simply because there are enough big problems within arms reach than their current capacity to plan over time
I've spent some time working both as a math researcher and as a software engineer, and I think this comment actually underrates the similarity between the two fields as they're actually practiced.
Some math research does involve grabbing a single, fully specified conjecture off the shelf and hunting for a proof of it, and it's true that if you manage to solve a long-standing open problem, other mathematicians will be interested no matter how you did it.
But this isn't all of what they do, probably not even most of what they do. Like in software engineering, it's not always obvious which question would be the most useful one to ask. A lot of mathematical work also goes into what we call "theory-building", where you could say that primary work goes into coming up with definitions rather than theorems. Mathematicians also care a great deal about how something is proved; a lot of them are some of the most aesthetically picky people I've ever met. Words like "ugly", "beautiful", "creative", and "boring" are used to describe both definitions and proofs all the time.
From the outside, it can look like all they're doing is pumping out proofs at any cost. But I promise you that when I talk to mathematicians who don't have any experience building software, they have a similarly narrow view of that field as well! Both fields, from the inside, look a lot more human than you might expect.
I think that your take is quite optimistic. Having published in top tier journals my only experience is that mathematicians care about what other mathematicians worked on and failed to solve. Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem.
Math is such that most theories are built after solving a problem and actually don't solve a larger class of problems. Etale Cohomology is an example of a rare exception. Grothendieck was mad that Deligne used adhoc complex analysis techniques to prove Weil. But everyone else was thrilled.
Whereas in CS, a good theory (library) solves a large class of problems. The reason being is that CS tackles general problems while math specific ones. Math on average solves problems that don't lead to solutions to other problems.
To me at least, math is more of a game like chess and coding is more of an art. There are aspects which are a game, like performance engineering but I'm pretty sure that LLMs will become superhuman at that soon
If your complaint is about the type of work that gets you published in a fancy math journal, then I'll happily join you on the barricades. Sure, getting a paper into Annals of Mathematics or whatever is more game than art in the sense I think you mean here.
But "what mathematicians care about" is much, much broader than what gets you published in a fancy journal. Mathematics as a human activity is millennia old, much older than the concept of journals or even universities, and that activity is, to me, very beautiful, worth preserving, and more of an art than a game. The incentive structure of academia for the past few decades has done a pretty bad job at preserving that art form, but that doesn't mean mathematicians as actual human beings don't care about it --- if they didn't, they probably would have chosen a different career.
Very fair. But when you say "What mathematicians care about", you are taking about mathematicians today, who really care mostly about politics
It seems to me you hooked onto the wrong part of proofs vs software compared to what OP meant. The difference OP cares about isn’t how much one cares about style. Instead the important difference lies in validation. A proof can be validated as either correct or wrong. That type of hard feedback really helps combat the optimism and desire for shortcuts of modern models.
Now, that still doesn’t help an LLM distinguish between good and bad correct proofs. But it still really helps a lot. On top of that, taste in proofs is a lot more uniform than taste in coding. That helps LLMs be better at judging the quality of a proof, because there’s less disagreement in the wider world.
The standards of proof are different from the fundamental operation of "OK, cool, you solved this problem. Why does this problem matter? Isn't it useless? Senseless? Meaningless?" You have this same question whether or not you're in an a priori discipline (mathematics), scientific fields proper, or engineering. "Absolute certainty" has nothing to do with it. I can assure you, people on the job are not looking for The Absolute Truth when doing their jobs, yet they still can question at a solution by asking: are we solving the right problem?
(Although in general, there's no true difference between "I answered the question correctly, but the question was mapped to this thing we call 'reality' wrong", and "I answered the question incorrectly", because you can (try) adding the constraints that you really wanted targeted in case A, to case B, and boom, suddenly a question/answer pair that was "Answered correctly, but question doesn't map to reality" now becomes, "You answered this question wrong". However, individuals generally tend to have some breakpoint to differentiate between the two).
No, what I'm saying is that I don't agree that taste in mathematics is more uniform than taste in coding! Mathematicians argue about taste all the time. Just as you might look at a piece of code and agree that it compiles and doesn't have any fatal bugs but still think it's badly written, hard to follow, hard to modify, or whatever else, mathematicians judge mathematical work using very similar criteria.
Maybe a subset of mathematicians, but if someone proved that RH was undecidable we would still give them the millennium prize.
The types who do ugly proofs, when they write code, produce spaghetti code. It's the same thought process going into how to approach something.
I think the difference is in math the problem is fully specified and easily verifiable and in programming it's not. I don't agree that we always know we can solve the problem.
Not always, sure but 90% of the time yes.
For example, create a DFA for a regex, not too bad just use Thompson's algorithm and then NFA->DFA. But now we have to care about efficiency, user API, maintainability of definitions etc.
Coding is more of a human problem than math
> So I'll say it again, AI will win a fields medal for before managing a McDonald's simply because there are enough big problems within arms reach than their current capacity to plan over time
AI can manage a McDonald’s already. If manage means directing humans to do something to ensure the store is running. If manage means running robots, then yes maybe that is 5 years away but just directing humans to run a store, that is possible right now.
No it can't. Show me a business which uses in context learning to manage a McDonald's
Well that’s a problem of incentives. Why would a manager outsource their own job to an AI?
It's not a problem of incentives. Every executive wants to inject LLMs everywhere these days. If they haven't somewhere it means that it does not work.
Every executive wants to inject LLMs downstream from their own job. As you get away from execution and toward management the incentive to innovate gets less and less as management at the same level.
Have you not seen vend bench?
I would agree with your take. I (author of the post & paper) learned a ton from working on small parts of problems my PhD advisor was doing a lot of the heavy lifting on, and later also from getting some results that were essentially putting together the right pieces that already existed followed by some deep-in-the-weeds analysis.
So if you dig down a bit it turns out the author had been trying to solve that problem for a year with GPT 5.4 and 5.5 and he fed all that information to the prompt he gave to Sol Pro which may or may not had direct access to the author's chat history. So the claimed "148 minutes" was really "a year plus 148 minutes".
Moreover, it seems the prompt included the technique used to solve the problem:
https://old.reddit.com/r/math/comments/1uxj3cy/after_openais...
In the prompt I basically just throw all reasonable approaches at it, without making a big distinction for what to explore most, and these approaches would all be reasonable for someone who knows the area. Sol helped me with the prompt as well, for which I gave it the CDC prompt, some ideas and specifications, a crystal clear problem description, and then modified things slightly myself after. One thing I do wonder is how much it accessed memory of previous chats, since as mentioned I had worked with 5.5 and 5.4 on this previously, and the main construction is not so different from something I discussed there. But, the function class max of affine functions that worked in the end was also in my prompt, so I'm not totally sure.
So it's not clear to me the degree to which "GPT-5.6 used a prompt" to close the gap etc, or the author basically did all the work himself and assigned it to GPT-5.6 out of enthusiasm.
I (author of the original post and paper) can add a few things here: 1. My previous approaches with GPT 5.5 were really not very sophisticated in terms of my input. I threw the problem at it, and just kept encouraging it to go iterate through ideas without any success. 2. The approaches that are in the prompt, though they will seem cryptic to someone not in the field, are relatively natural ideas. In fact, the construction that worked was something that even 5.5 initially looked at but was just too weak to see how to make it work. From my view, I would have never gotten this result myself. Imagine you are telling a contractor to build the empire state building, and you say: "You should explore approaches that can include building materials like steel, wood, concrete, or clay, and any combinations of those. You can use arcs, columns, supportive beams, and anything else you can think of to solve load-bearing issues. Do not stop until you've completed a viable plan to construct the empire state building." And then the contractor shows you the finished empire state building using reinforced concrete and steel beams with all kinds of crazy ways of making everything stable; that's kinda how I feel.
There is a firm opinion in some circles that an AI can never do anything but recombine already known things despite a rising number of cases where that very much appears to be an inaccurate view.
Any slim possibility that vital information was given to the AI to make the task just one of recombination, rather than coming up with anything original, is grabbed with both hands no matter how tenuous.
I can understand why, nobody likes the idea of a machine capable of doing the job of a human, and jobs requiring the mind are especially sacrosanct. But like farm workers in the early part of the C20th, not liking the incoming technology won't change the effect it will have when companies get the idea they can make more money and hire less people.
> Moreover, it seems the prompt included the technique used to solve the problem:
I don't believe this is true. The author sent techniques he used, but I don't believe any of those were ultimately what GPT-5.6 used.
GPT-5.6 also provided the Lean formalization, which was not provided at all by the author.
sounds like a feature, not a bug
it might imply using weaker models to attempt the problem first is a good supplementary prompt to a more advanced model trying to do the same
In the Reddit post there was clarification that this was done with Sol Pro not Ultra - curious what is everyone’s mental model of the difference.
My understanding is that ChatGPT Pro is effectively a multi agent system, or somehow uses multiple LLMs in parallel and selects a best answer. And Ultra is more similar to Claude-Code UltraCode where the main agent can choose to create a dynamic JS workflow that deterministically orchestrates multiple agents to handle different parts of a task and have adversarial checkers etc.
Is that more or less the difference? Any substantiating sources would be great to see.
ultra in codex is just a way to run multi agent system, pro is similar to other pro models like 5.5
If I recall correctly there was a proposed proof to the abc conjecture by Mochizuki https://en.wikipedia.org/wiki/Abc_conjecture#Claimed_proofs which was rejected due to being rather inpenetrable to humans. Shouldn't this be an ideal target for LLMs?
It was rejected for being wrong (or most charitably, incomplete).
There was recently an announcement that a group trying to formalize it found a gap exactly where other mathematicians were pointing. So to the extent there was any doubt, it should be gone now--the proof was incorrect.
But I agree LLMs have a lot of potential for checking proofs--both informally (they can read quickly and find gaps) and formally (by attempting to formalize).
Similarly, I'd love to see LLMs create a formal proof of the https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
I’d like to see four color conjecture and an elementary proof of FLT.
Crazy how intelligence is cheap, efficient and commonplace now. We humans better refocusing our energy on our core values/principles, given most of our skills are becoming irrelevant
If it were commonplace, there wouldn't be a post and discussion about it. Cheap? Arguable - while it didn't cost thousands, it wasn't free. Cheap is in the eye of the beholder. Efficient...How do we even measure that? The massive infrastructure and training to take a product to the point where someone could do this is massive. Ignoring everything behind the scenes and acting like one session and result is the whole picture of efficiency doesn't seem right. And no, nothing produced by AI makes skills irrelevant. That is the whole ongoing argument of whether people are losing cognitive ability by moving their thinking to AI.
Overall, this is an impressive proof of capability. But I wouldn't take that proof as anything more than what it is.
Seconded on the "not cheap" argument here. I've spent $25 worth of tokens completing a one-week task in an afternoon, or rather my company spent the money. I would never have personally felt OK with throwing this much money after some prompting back and forth for a few hours, one lazy Saturday afternoon. I ran the risk of not finding the solution before the token usage would be too high for me to want to carry on, if I was my own credit card linked to the account.
Of course money in this situation is a bit of a funny measurement, right, because if I was able to take the rest of the week off as soon as I had solved the one-week problem, then I would have no problem at all throwing even $100 worth of tokens at it, so I could enjoy a nice 4-day "mini-vacation".
How cheap "cheap" is, is indeed "in the eye of the beholder".
Is is sarcasm? $25 to perform in half a day a week of work, that is not cheap, it's a massive saving of money- probably in the thousands.
It's not sarcasm. If I had to pay $25 to perform my work from an ever-growing pile of projects, I would be losing money. That's not "cheap intelligence" for me. It certainly is for my employer, though, because for the mere price of $25 extra per day they can get 4*X the product per period of time.
In Scrum terms my personal velocity grows by a factor of four or more with access to agentic AI workload, but if it means that I will just be asked to "consume" 4*X more Story Points per sprint, I'm not the winner in the end, my employer is. If they asked me to complete X Story Points per sprint regardless of my velocity, and they let me take the days off when I was done, I would be the winner. But that's not how it works.
AI is "Cheap" for the person/organization that gets more product for less money, not for the individual person building the product faster.
I'll simplify it for you: AI is cheap for those who pay for it.
So, it's cheap.
For whom is the question you should ask.
/r/whooosh
'Maybe cheap but not free' is a good observation.
If I'm getting paid for the work, I'm happy to leverage the LLMs so I can do more. If I'm paying for the work, I expect more from it.
For hobbyist stuff, where I'm not expecting to receive money? LLMs let me do things I otherwise wouldn't have done.
I've agreed with thoughts like "the LLM wrote the code, surely it's not worth sharing" or "I could just have the LLM write my own version of that". I'd also wondered about my own personal projects, surely "an LLM could have written all of this". -- But how I feel about that changes a bit based on how much it'd take the LLMs to get the same output.
I mostly agree, these are things that couldn't or weren't solved with abundant capital at all, and now they are being solved
it went from not having a price, to having one, and we are trying to retroactively transpose economic viability or economic existence to it from some parallel and prior time.
A counter argument: A strong distinction between "intelligence" (understanding what is) and "values/principles" (understanding what ought to be) was characteristic of much early modern European philosophy from Descartes to Kant, which received its influential strong formulation from David Hume.
But trying to maintain this distinction leads to insuperable difficulties. Our conceptual framework for understanding the world are always value-laden. There is no "view from nowhere", no historically unconditioned set of values or concepts. Your framing, in which "values" are external to "intelligence" and must be imposed on it (on pain of intelligence being "value-neutral"), leads inevitably to the dead end of "AI Alignment", "superintelligence", etc. Which is a kind of pseudo-theology.
"We humans better [be] refocusing our energy on our core values/principles, given most of our skills are becoming irrelevant."
In light of the untenability of a strong fact/value or intelligence/ethics distinction, I would suggest this alternative advice: humans should focus on critical appropriation and extension of the received wisdom, whether that comes to us directly from human beings or indirectly through an LLM. Perhaps this is compatible with the spirit of your original suggestion.
It's still clear that LLMs lack spatial reasoning, either in the concrete or abstract, and while that sort of reasoning has been downplayed by academia for at least a century it is fundamental to technology and industry. (And many would say for science and mathematics too).
They will, however, get there as well either directly or as interfaces to models that do, and your core point stands.
"Lack" isn't the right word. "Lacking" is more like it.
If there was a deep fundamental inability, we wouldn't see things like newer generations of LLMs consistently improving on ARC-AGI series (heavy spatial reasoning loading) and SimpleBench (a lot of commonsense + spatial reasoning components).
In a way, it's a surprise that LLMs, notoriously lacking any sort of embodied experience, can even get this close to human baselines on tasks like this.
My takeaway is that text is a far richer modality than anyone has expected - and that high end LLMs are often sharp and flexible enough to recognize their weak points and substitute their strengths. I.e. all the LLMs implementing A* to optimally solve pathfinding in ARC-AGI-3 tasks, often unprompted.
There might still be unrealized gains there from true depth-unbounded recurrence, or maybe from finding better ways to integrate modalities in training. But clearly, a "fundamental limit" it ain't.
> "Lack" isn't the right word. "Lacking" is more like it.
Yeah, that's fair.
> My takeaway is that text is a far richer modality than anyone has expected - and that high end LLMs are often sharp and flexible enough to recognize their weak points and substitute their strengths. I.e. all the LLMs implementing A* to optimally solve pathfinding in ARC-AGI-3 tasks, often unprompted.
I agree and disagree with this. I think we've learned a lot of humans are more text based than we thought, but conversely I'm not persuaded what non-textual task reasoning LLMs are doing is necessarily text based, just that models have grown large enough for other reasoning modes to conceivably be hiding in the parameter space.
As I mentioned elsewhere, like many others I find LLMs work entirely by example, and reaching for A* when pathfinding is the single obvious thing to do. In cases where the magic key word is not mentioned and the problem cannot be identified as "pathfinding" (or some other trigger with a highly specific widely documented solution) they will struggle, yet the moment the trigger is hit they get there very fast. This is why prompting remains such an art form.
Fable is the first one I've encountered that is capable of serious open ended 3D programming in ways that suggest it has some grasp of the spatial aspects of the problem (not merely symbolic manipulation of the vectors etc.), but it still misses optimization opportunities a human will find glaringly obvious based on spatially predictable bounds etc.
Grown? LLMs were always "large enough for other reasoning modes to conceivably be hiding in the parameter space".
Basic LLMs don't reason in text, and never did. They use it as an interface - for input, output and some of the intermediate products. Heavy use of those "pseudo-recurrence" intermediates in "reasoning models" is a relatively late post-training adaptation. But the process that happens between those endpoints is not at all text-based. What happens in the hidden dimension is part "output logit domain", tied to probability distributions over possible output tokens, and part "incomprehensible concept-space madness".
The latter being where things like latent world models live. LLMs develop partial world models, right in pre-training, despite not being explicitly forced to - because it brings them closer to heaven of accurate next token prediction.
And yes, larger models like Fable seem to be better at spatial reasoning. Maybe because their large size increases the sample efficiency and improves generalization, allowing them to absorb the sparse signal of "spatial reasoning" in the training text better. Maybe because this extra size means more layers, allowing for deeper latent space reasoning in lieu of true recurrence. Maybe because the default "next token prediction" reward underrates rare spatial reasoning challenges, and the model only starts to "get good" at them once the other sources of loss reduction are heavily depleted. Maybe because no true recurrence is suboptimal for spatial reasoning architecturally. But it is what it is. Spatial reasoning gains in LLMs are extractable, but extracting them is nontrivial.
Is there any proof that they are not good at special reasoning? Arc agi 1 and 2 are saturated.
I will be posting something to that effect later this week. (Hopefully).
Basically current gen LLMs apparently do spatial reasoning the way they seemingly do everything else: by reference to previous example. I didn't see them work out which known example to use for a given problem until specifically prompted, in my case by accident.
If you ask an LLM, "what known example did you use to solve this?" it is very likely to cite something plausible-sounding. That absolutely doesn't mean it is what it actually did. It is trying to give the "right answer" and please.
> If you ask an LLM, "what known example did you use to solve this?" it is very likely to cite something plausible-sounding. That absolutely doesn't mean it is what it actually did. It is trying to give the "right answer" and please.
That's not what I said happened though. It didn't solve the problem (for weeks) until I (accidentally) told it which example it happened to know was relevant, and then it solved it in hours.
ARC AGI 3 is much better designed and harder, perfectly completable by a human in a couple minutes.
Only a fraction of the games can be solved by Sol, generally at sub-human efficiency in terms of turns, AND at a cost of >$10,000 per game.
On a token level, text tokens are orders of magnitude more information dense than visual tokens.
You don't have to do much statistical analysis to figure out what is meant by the token string "cat under a tree". However you need to do an enormous amount to encode any permutation of pixels that show a cat under a tree from the set of all possible pixels arrangements that illustrate that (along with the massive fringes of ambiguity).
Everybody can be an armchair mathematician now. Just fling some thoughts in the direction of your AI setup and let it do breadth first search with AI based pruning heuristics.
If you really believe this, try to use GPT 5.6 to prove an open problem you know nothing about. You might get lucky, but if you don't, you will soon discover that 5.6 can make "progress" towards a theorem without actually getting anywhere pretty much indefinitely.
I generally agree, though direction, intuition, and domain knowledge are still relevant. Your breadth-first-search framing feels right, but you still need a sense of which paths are worth following, and you need to know when to trust the results.
I’ve been doing more math as a hobby in the past few weeks — working on lesser-known conjectures and exploring proofs of hard theorems — than I could have managed over the previous several years. It’s an exciting time.
I imagine there are hundreds of thousands of people doing this, yet all we seem to get is a handful of solved problems. So it would seem the success rate is extremely low.
Intelligence on its own is not very useful though. We put it on a pedestal because it creates huge potential when paired with other things, wisdom, discipline, empathy, but on its own?
Llm's do have discipline and persistence, though.
It’s no diff to an educated human.
Stored potential.
But will that potential be converted that contributes to the economy..? That requires other traits.
Might be focus, might be discipline, might be the need to get revenge lmao.
This is what the llm-boosters miss. Progress is willed into existence.
AI / LLMs are not intelligence .. they are just a prediction engine that has been branded as 'intelligence' by marketing employees
At the end of the day it is still making a best guess at what the user wants based on data it has seen before.
It still requires someone smarter than the output to be able to evaluate if the result is any good, or just hand waving.
Can you tell me the difference between that and doing code review for a task performed by a non-senior?
Iteratively leaning on lean to prove a conjecture is not intelligence, it is automation
how much can I bill for having good core values?
Oh brother
AI hasn’t even taken the class of jobs associated with customer service lmao
Do we employ mathematicians in customer service roles?
Luckily the job situation for pure mathematicians was already bad.
I got a solid laugh out of this.
Thats a silly and obtuse comment.
It was a sensible, valid and pertinent reply to the parent comment though...
As with most comments, context is quite important.
You mean the answer betrays the point: customer service is surprisingly hard, we just have a large number of people that are capable of doing it.
This is what the whole https://people.csail.mit.edu/brooks/papers/elephants.pdf is about.
I stand by my point, you've not read the author's intent, instead you decided to twist words.
What a silly and obtuse comment.
Uh.... Have you ever called customer service lately?
Or indeed 20 years ago when "press 1 for foo, press 2 for bar" was already a thing.
Press "4" if you have a rotary phone.
Yes exactly
I did yday.
Intelligence was always relatively cheap. You can pick up a phone and get answers for free in most academic settings.
You've not seen how they react to noobs asking physics questions, I think.
Even when you've got an interesting idea, if you're an enthusiastic amateur who don't yet know enough to phrase the question right but does actually know the basics, they'll put you in the same category as the people who think healing crystals can power hyperspace telepathy with Anubis: "oh no not another one".
LLMs have infinite patience, but unfortunately come (came?) with too much sycophancy, giving even more people far too much confidence.
(within limits)
yeah...right. Go touch some grass
Once we figure out the pesky problem of how we're going to pay for housing, food, and healthcare.
I think the big names behind the AI companies already have that problem solved. A lot of people probably won't like the solution very much though.
Yes, they have a final solution for all of us.
When machines are doing all the work - we no longer have to.
> the couple multi-trillioners will have all the wealth of the world, and it will all crumble down
You mistyped it.
Is that what you're going to tell your mortgage lender?
What makes you think the people who own the machines will share their resources with you?
I can't stop wondering myself.... I'm writing some software with AI and wondering, why am I doing this? Will anyone need this? Will anyone have money to buy this?
Best I've come up with is we'll need to be adopted by technofeudlaist overlords to be our patrons like in the roman days
> Best I've come up with is we'll need to be adopted by technofeudlaist overlords to be our patrons like in the roman days
Continually progressing AI (combined with our current socioeconomic systems) throws a lot of uncertainty into our mid to long term future, but I don't think this is going to be what happens.
There are billions more of "us" than of "them", people don't respond well en masse to a drastic worsening of their societal status and "they" are lagging very far behind on building their robot armies.
If we poorly navigate this transition the outcome should be worrying them more than it worries us.
> There are billions more of "us" than of "them", people don't respond well en masse to a drastic worsening of their societal status and "they" are lagging very far behind on building their robot armies.
Fortunately, yes, those robot armies do seem to be rather behind schedule*.
However, even if Musk dies of old age before anything like the Optimus can be connected to useful artificial intelligence, it can still be driven by the common joke a few years back that "AI" really stood for "Actually Indians".
When all the people currently upset about "immigrants coming here taking our jobs" discover those same people are now staying home and remote-working those same jobs over a VR headset and a Starlink connection... my guess is that by this point, Musk will have no political allies left.
> If we poorly navigate this transition the outcome should be worrying them more than it worries us.
It can screw everyone over. Literal communism was invented in response to Laissez-faire capitalism, and while Laissez-faire died with the Great Depression, the form of capitalism which succeeded it in the USA came into conflict with the USSR and gave us the Cold War, the Cuban Missile Crisis, etc.
* 2022 "next year": https://www.cnbc.com/2022/04/08/elon-musk-says-tesla-is-aimi...
Humans aren't sheep but in the broad average it seems like we have a strong tendency to fall inline.
Fwiw I was mostly joking. I agree that the techno overlords have no reason to keep us, unlike in Roman times.
I don't know how you would translate the strength of a robot army to a human one; they haven't fought yet.
Mathematics is a human-designed game that involves rearranging symbols.
That view is incredibly reductionist. It really is an efficient encoding of how nature behaves. It might be a human construct, but given how best it allows to understand nature (through principles of physics), it is uncanny to be any different from the language of nature.
Reminds me of Wigner's Unreasonable effectiveness of mathematics in natural sciences [0].
[0]: https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
Nature has no language. Language is a human tool to imperfectly describe nature. Putting language before nature is magical thinking (also known as Platonism).
Yes, of course mathematics is a tool to model nature as accurately as we could. That still doesn't mean mathematics is as trivial or frivolous as manipulating symbols.
At a very high level mathematics is basically 100% text/symbolic rewriting. You start from some set of postulate assumed true and you do your thing to get a new different set of equivalent assertions in a form that is more useful.
I don’t know if LLMs will kill the working-mathematicians but at least seem like that it doesn’t seem absurd to imagine LLMs will be good at math…
Ever heard of the infinite monkey theorem?
This is basically what LLMs do on really hard tasks. Prompt it a million times on a really hard problem and it might output the correct answer once.
The infinite monkey theorem assumes random distribution of symbols*.
Given the tokenizers have a vocabulary in the 10k-100k range, "a million attempts" will generally still only get the first token of the answer correct.
Even really rubbish models, e.g. talkie, the "what if we only use pre-1930s data to train a model?"** model, had to be almost all the way to the right answer to reach the really low HumanEval pass@100 score of ~0.04 (I'm only eyeballing the relevant chart).
* Actual monkeys not being like this is, while amusing, irrelevant
** https://talkie-lm.com/introducing-talkie
>Ever heard of the infinite monkey theorem?
Even if every atom in the universe were a supercomputer generating a trillion trillion random characters every second since the Big Bang, the chance of producing Hamlet would still be essentially zero.
We have a non peer reviewed proof in a niche area of mathematics claiming to have been "co-written" by an LLM. What are the comments about? Lamenting or celebrating humanity's intellectual death... Very insightful.
I tried using AI to solve some advanced math problems. One thing I see is that they can throw an enormous amount of brute force into a problem. When mathematical logic can be brute forced we will see some interesting advances.
Not yet peer reviewed
I can't see this article as Reddit doesn't allow me to view it. Do we have to use Reddit though? It's a horrible website.
The amount of people several months ago stating that no one cares about the "unsolved" mathematical problems that AI is able to solve is funny.
Can't wait for GPT to prove that P=NP (or not)!
It says I've been "blocked by network security". MSIE on Windows. Reddit must really be circling the drain.
it blocks vpns and tor
Not using either. Just my US-based home ISP that works for everything else.
Genuinely asking... How do you get chatgpt to work for 148 mins when I can't get gemini to think for even a minute at a time?
ChatGPT pro will work 20-30 min in my experience pretty regularly in my experience. Never had 148min but it seems plausibly in the very right tail.
clear specifications for what counts as completing the tasks, and an explicit list of what does not count as completing the task, and clearly stating to not return until the task has been completed. I've had agents run for almost 12h!
Basically, he proved that *information is power.* If you don't know which way to go (the subgradient), you're gonna be calculating forever!
The problem is that we're going to have another deepseek moment when someone uses GLM or Kimi K3 to do this.
What was the first DeepSeek moment? (genuine question, I'm out of the loop on what you mean)
January 2025, release of DeepSeek R1, the first open reasoning model. There was a lot of panic then that it was done with very few resources.
I guess when cheap Chinese model was very close to SoTA models
... and now another one when a (not so cheap tbh) Chinese model matched OpenAI GPT Sol and Anthropic Fable/GPT 4.8 (some scores below, some scores above) ...
Oh and exceeded Google's best internal model capability enough to have them cancel the next Gemini release. Probably. But I'm 95% certain that's exactly what happened. This is the first time I really fear for the future of Google, because OpenAI matched or exceeded the experience of Google search ... and now a Chinese company matched OpenAI ... followed by Google failing to catch up. That really made me take pause.
It indicates the the "secret sauce" of OpenAI and Anthropic is ... nothing, really. Or none of it matters, except access to the hardware needed for 1.5 trillion parameter models, which Chinese firms now have as well. It means there is no "AI takeoff" where other firms can't match OpenAI/Anthropic models. And Google, inventors of transformers just missed the takeoff. There's more secret sauce, but clearly a great engineering firm can find it in less than 3 months.
The fact that this Chinese model has similar cost to OpenAI and ballpark cost compared to Anthropic while we can be quite sure they're minimizing the cost (that's what China is doing everywhere else) also raises questions about the true cost of serving comparable models and with that about the future profitability of OpenAI and Anthropic.
This is all a depressing and bleak future that I don’t look forward to.
One solution is to ban LLM’s, to artificially create a demand for human thought, that just feels like living in an artificially constructed zoo.
Another solution is humans don’t do anything that AI can do better , / doesn’t need the human touch. So I suppose we will all become artists, sportsmen or politicians, the only jobs that will remain except for select few. Maybe this is ok, I don’t know.
Another solution is we find a way to mind-meld with AI so that human + Ai >> AI alone. This is dystopian, who gets to decide who mind melds with AI, how much will it cost etc etc.
For the stupid copes that the prompt required human ingenuity, let me first add that the author used GPT5.6 to write most of the prompt. He just gave some mild direction. That amount of direction does not require deep expertise and the expertise required will keep falling with time, eventually an undergrad can create this loop and then maybe a high school student.
In fact, in the future the most likely outcome is you tell the agent what you want (I want this app, or I want this theorem solved) and it will set up the loop, or loop of loops and use all its computing effort to come up with a result. This is completely dystopian to a human life.call me naive, but i really think the pursuit for the new, the better, the interesting, is boundless and humans will never stop. and so for those of us that dream and push and are plain curious will never run out of ways to be useful. and hopefully the fruits of all this tech will be ample enough to support the ones that don't want to or can't participate in that way.
If AI is better at being (or simulating; I think the distinction is irrelevant in this context) curious, better at examining that curiosity, better at pursuing it to discovery, better at applying the discoveries, better at sharing the discoveries, and able to push the frontier of knowledge (if that’s even the right word) literally beyond human comprehension in every domain, what usefulness exactly is left? Or are you just implicitly saying you don’t believe AI will ever get all that superhuman?
We already drown in useful new things and especially knowledge. There's a limit to how fast we can adopt them.
It genuinely scares me that some people's first reaction to this news is banning LLM.
Maybe we should also burn books and lobotomize ourselves to make science more fun and challenging. The ego here sickens me. I don't care about your ego. I want accelerated material science, medical science, energy, better outcomes for people across the world.
I also want better outcomes for people around the world.
Memes like the permanent underclass and the massive incentive of replacing workers across the world does not bode well for a better outcome for people across the world.
> I want accelerated material science, medical science, energy, better outcomes for people across the world.
There will be people benefiting from advances in these fields.
Neither you nor I will be one of them.
Ah, so you're saying none of us will benefit from the discovery of new mathematical knowledge?
You've just condemned funding math, even human-produced math.
Either math is useful, in which case AI math is useful, or math is not useful, in which case we can save money by having mathematicians sell shoes or clean toilets.
Look carefully at the portion of derac's comment that was quoted.
> I want accelerated material science, medical science, energy, better outcomes for people across the world.
Does it mention mathematics?
Let AI make advances in all these fields. But if most of the power and capital in society is captured by a tiny minority, then these advances will become inaccessible to common people.
If that's your starting position then for sure you won't, buddy.
“you’re talking to someone who woke up a loser today”
looks like you met the reverse Huang lol
wasn't that always the reaction historically whenever people felt like their livelihood was threatened by new technology?
A rational reaction would be to identify the political coalition that would act in this manner and place yourself in opposition to it if you do not see yourself being able to change it.
I wonder which one that is.
Why? No one will be able to effectively ban AI. Pandora's box is open, and we can only witness the consequences.
Why is this depressing and bleak? No one is stopping you from doing math if that's what you want to do.
Should I ban table saws because they make people too good at woodworking? Should we ban right-angles?
This line of thought doesn't lead anywhere fruitful.
just ignore these people
gig economy is a blessing to keep the LCD busy and doing something productive instead of being nihilistic losers
it’s better to focus on how to keep these “knowledge” workers busy for the AI age
they don’t like to work and complain even on a w2 job (where someone else had to figure out how to capitalize them) so in many ways they’re far more dangerous and hopeless.
Don’t be so nasty about w2 workers. Incredibly unbecoming.
Also see comparative advantage re best roles for each actor in an economy
cool
> This is all a depressing and bleak future that I don’t look forward to.
Or, for the vast majority of humanity who aren't professional mathematicians, it's a liberating and empowering advancement.
I'm sorry if the automated digital theorem looms are eliminating the market for your handwoven proofs.
Indeed. Where do you throw your wooden shoes these days? (Not that it was an effective tactic then anyway)
There is more to human life than programming and math proofs.
unfortunately many won't get to experience it much because we'll be stressed and struggling to make a living
The problem isn't that we are gaining more knowledge and better technology. The problem is that we are allowing the rich to use the technology to subjugate the majority.
If AI improves human productivity so much that millions of people no longer need to work that should be an incredible thing. But the flawed structure of our society punishes those people rather than freeing them to persue endeavors that interest then.
Only solution I can think of is higher taxes for the rich used to provide universal income. I'm not sure how the realities of that would play out, though.
Anything based on a free market just ends up in.. this.
I fully agree. Tax wealth, not work. Then it is far easier to lift yourself from poverty, but harder to hold on to vast amounts of wealth.
After WW2, when tax on the wealthy was far greater, sometimes 95% of their income (in the UK at least), it was the period when the largest number of ordinary people were raised from poverty to middle class status, yet the number of rich people also increased greatly. When that stopped, wages stagnated, and now we have a cost of living crisis. Connect the dots, lol...
Many people think that the market is either 100% free, or 100% controlled, but a free market that operates within defined limits that will stop the greatest excesses of the rich ruining things for everyone else is the sweet spot we should be aiming for.
I'm strongly in favour of this. I think income tax should approach 100%. So your second billion you still make more. But not twice as much. The exact decay function is up to date. But I really don't think your post-tax take home should scale linearly with your income once you hit the max bracket.
We can also remove all of the loopholes that result in the biggest companies paying basically no taxes. I also think it may make sense to have a dramatically different tax structure for businesses than effectively income tax.
As I say above, here in the UK we used to have the top rate of tax at 95% of your earnings, and it was the period of fastest growth for poor and middle class people (Boomers at the time)
Unfortunately, the rich are the ones most capable of buying democracy, and the current set of politicians seem to very much welcome the money...
This is already the case with most human activities, even reading a book for leisure is a luxury many can't afford.
It's available to most, though.
Outside of really bad luck I've never seen a case where someone was in a situation of working hard + making reasonably okay decisions + struggling, and that's purely due to there being avenues available where you can be rewarded well for your effort if you're of average+ intelligence.
AI is threatening to take that opportunity away.
This is utopian friend!
Literally anything you wanted to make is no plausible to make if not now then in the next couple years.
The thing you’re worried about is capitalism and the connection with working to having the right to keep living. If you can throw off that mental shackle you can start to see how this can be amazing, but you have to drop the idea that everyone has to work at a job for someone else to provide some service in order to do it. It’s hard, I know, but change your mindset some and dream for a better world and we can make it.
You ever play a video game with god mode cheats enabled, so you can unlock all the unlocks, get all the best gear, and be an unstoppable force with unlimited money?
Yeah, it's fun for 30 minutes.
On the other hand, exploring any of your interests without needing to calculate how it can cover rent means you have more time and opportunities to actually do things you're interested in.
Working all day, then not wanting to do much else after because you're tired, is also fun for all of 30 minutes.
Right!? Like imagine we could use this stuff to solve world hunger, develop robots to clean up the oceans, or colonize space?
People are all “shucks how am I going to be able to justify my career at $job” and are missing the bigger opportunity. Such a lack of imagination I see…
There is no lack of imagination here, there is realism. You think you display "imagination" by just regurgitating all the tired tropes of old science fiction? Oh wow! Robots to clean the ocean! How novel! Nobody has ever thought about this!
The reality is nobody is going to build robots to clean the ocean because there is no money to be made. There will be robots to clean rich people's pools, sure. And it will put pool boys out of a job, great.
I can see (given that you've responded to a few of my points here), that you care about this enough to respond.
What if money wasn't a thing? What if that wasn't the only incentive in the economy? What if there were other ways to balance supply and demand curves (which is basically the principle use for money) so we could reduce shortages, etc.
Dream big man, I'm serious! And who cares if there is no money to be made? Like, what are you (individually you) doing to create the world you want to live in? What world do you even want to live in?
"let's slow everything down because we haven't figured out the resource appropriation aspect, yet"
there will always be disruption and frictions, 頑張って ください
Exactly, a lot of people keep attacking me as anxiety for losing my livelihood. It’s not that.
It’s just depression that another avenue for human craftsmanship has been taken over by the machines. Humans are not just infinite consumers. If AI solves all of humanity problems, it’s like living in a zoo, not a life worth living imo.
This has always been the case though that "another avenue of human craftsmanship has been taken over by the machines."
It's good if we can have robots building things instead of having humans slumped over a workbench in a sweatshop piecing things together. It's good if we can have LLMs spitting out code rather than CS grads working 15 hour days at fintech startups or whatever. The conditions were never (ever) good before.
And, it won't be like a zoo - you'll be able to go wherever you want, do more or less whatever you want. Think about living in The Culture, or the Star Trek universe or whatever. There are options beyond "I'm a pet to the machines." Think big, dream big, then help make it a reality!
Like, out in space you're still going to need a human to make decisions because you can't wait 30min for the tight-beam signal to get back to earth. Also, we're pretty good at soaking up rads and still being "useful" - at least so far I don't see that being a major advantage to the robots. Maybe our place is to be the deep space mechanics that keep the robots alive? I don't know, regardless, you should dream big. What kind of world do you want to live in? Ok, how do we make that world happen?
My big (somewhat unspoken and somewhat immature dream) is that advances in regenerative health tech fix my optic nerves and I can get into the cockpit again one day, then maybe later I can fly some space vehicle like I wanted to since I was a 12 year old. Immature I know, but I miss flying still.
Is that world possible without AI? Probably/maybe? But it's a lot more plausible in a world where we have folded every protein, we have robot surgeons doing robotic procedures, AI generated research, etc.
Surgeons too now? Another avenue for human skill destroyed.
I actually want to exit. I want to live in a society where humans flourish not AI.
Actually one just needs to walk the streets of Japan and compare that to US. Tokyo has hundreds of small shops with humans doing specific niche stuff, perfecting their arts. That’s all so beautiful.
America has massive warehouses and supermarkets, with completely uninterested and bored out of their mind humans working and I suppose now we will replace them with robots. Great I guess. Maybe all Americans should just sit at home and consume Netflix and Doritos ig.
I want technology to increase human flourishing, not turn us into WallE humans. I am only interested in technological progress when it is done by humans who trained their whole lives for it, as it is a display of human excellence and that can be a beautiful thing. I’m not interested in it if some AI builds it.
Then, you know, build that world locally, at least for yourself.
Like, I'm building a cabin in the woods (original, I know), but you can build things and do things. You can be that kind of human if you want. But, if Americans could sit at home and consume netflix and doritos I suppose some would, but many would decide to start doing other things too.
I don't think the worlds we want to live in are that far apart to be honest. If I could make a living doing it, I'd try to build wooden sailboats for a living. That would be awesome. I would love it! In a world where I don't have to work to win capitalism tokens, I would probably spend a few years just building wooden sailboats to sail around the bay by my cabin.
What you write is what's profoundly depressing.
The only way you see people flourishing is through coercion, by society or by the world. You'd explicitly rather prolong this codependency for your aesthetic preferences even, as per your own words. We're already just circus monkeys in your world.
You cannot fathom people pursuing self-improvement because e.g. developing or being capable is inherently enjoyable. You only see a world where such activities are a must, or a manufactured must. That's your measuring stick and that's the entire pie for you.
I mean this in the least combative way possible, but it sounds like you have things to sort out on your end first, before charting a course for the world to follow.
Yes, not everyone will agree with me. Maybe some people just want to consume AI abundance forever and live on UBI, like Altman’s wet dream.
But there will be some group of people, the people who wanted to be skilled tailors, skilled wood workers and apply them towards others before industrialization ruined them and of course the best mathematicians, the best programmers, before AI revolution will ruin them. We want no part of this technology, and maybe in some distant future we can organize and live separately from the consumerist dystopian hellscape that the world around us turns into.
I’m not charting a course for the world to follow, but I suspect every OpenAI employee or Sam Altman have no desire to live their entire life on UBI doing nothing of any importance. This for them is their pursuit of greatness, only in the result they’re destroying it for large swaths of humanity that live around them. I don’t want any part of it, I refuse to be a WallE human managed by AI Sam’s company designs to be put in a state of perpetual limbo somewhere.
> Maybe some people just want to consume AI abundance forever and live on UBI, like Altman’s wet dream.
Callousness like this is not exactly the hallmark of clear minded reasoning, which is your responsibility, I hope you haven't forgotten. Like no, there are more options than the two convenient strawmans you crafted. Kind of what my whole comment's been about.
Maybe it doesn't matter though, we seem to have already lost you to some neo-Amish fantasy. Except this is not even me being callous in response, you quite literally spelled it out. You'd quite literally take your current gripes and extend them indefinitely, forever shaking your fist at the sky, arrogantly refusing to even attempt to adapt to the (already present) world, all while being dependent on being tolerated in your historical reenactment. Apparently this makes sense to you, just like all the other vile ideas you pitched.
There truly are no stronger limitations than one's own mind.
I admire the naivety. I'm sure CS grads are delighted...
Why the ad hominem? I mean, seriously, you think I haven't been "in the trenches?"
Dude, I'm nearly 40. I've seen the shit. When I was flying for a living, literal close friends died because of the results of avarice.
Dream big dude! This constant "doom and gloom" narrative is toxic to your ability to flourish. I have been there, I understand the urge to doom and gloom, but we make the world we want to live in. We do. We choose it.
What world do you want to live in? What is your "optimal" version of the future? How do we get there? What steps would we have to take to make it a reality?
If your vision of the future is "oh, like 10 years ago before $problem_du_jour" then you're already off to a bad start because we ain't going back. There's no regulation that will actually change anything materially (seriously, you think the people getting rich off all this stuff in Congress will voluntarily shoot themselves in the foot?), there's no way to go but forward, so how do we proceed?
I want to live in The Culture, or something like it if it's possible. So, let's start building it and ask permission later.
what's getting built is more like Asimov's Spacers/Solarians, though. (Vastly reduced population wielding thousands of robots each)
I mean, birth rates are crashing hard here in the US so this might be decent future?
Maybe average human lifespan gets kicked up to 200 years or whatever and we make do with robots to fill in the gaps in labor with robots?
I don’t know? But this kind of seems to track with how we’re trending now?
I play the game for a manufactured challenge
People in the real world all having god mode cheats is good, actually
yeah! Like can you imagine a world where you could just effectively "conjure" new furniture for your house and robots would assemble it in a few hours? Or molecular assemblers would grow it while you slept?
Or imagine a world where we could basically cure every disease? Or a world where people could assume any form factor that they wanted? I don't know, the list goes on and on in a post-scarcity world... and we're seeing how we basically live in a sort of digital post-scarcity now and it is really cool.
If I need a software tool these days I don't buy it, I tend to see if I can make it. Now imagine that for physical things? What a time to be alive.
The rub is that we already live in that world relative to people who lived 1000 year ago, and yet somehow we aren't bursting with euphoria 24/7.
We also have humans who essentially live post scarcity lives on the back of a monetary windfall, yet their stories are strangely quite often depressing ones.
I think their is good reason to believe most human desires are illusions.
If anything one wants to make is now possible to make, how is anyone supposed to make a living?
Can I make food with LLMs? Can I build a house and make clothes? This is stupid. No real wealth is being created for the general population here.
I literally have had various LLMs generate 3d printable plans for things I have used. Also, I am literally making a living right now being something between a dev and SME and building tools for people in an industry I’m familiar with.
Dream bigger buddy! We can make the world better, we’re not powerless here.
People like being needed and important to other people. That isn't some artifact of capitalism; it exists across all times and cultures.
Sure, but it is certainly magnified when you need to make money to justify your right to exist.
What a load of absolute nonsense.
I’m convinced many of you barely go outside and have the capacity for original thoughts.
Sloppity slop slop.
Non-reddit: https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-... (https://news.ycombinator.com/item?id=48939768)
Two points:
- Hasn't been peer reviewed yet, so take with a grain of salt. This applies to all claimed proofs, not just AI-generated ones. Even humans hallucinate proofs too!
- The prompt is on page 27 here[1]. It is ten pages of advanced mathematics priming the model in the right direction, apparently informed by a year of prior research. That doesn't invalidate the result if it is genuine, but it is worth noting that this wasn't a matter of "ChatGPT, solve this unsolved problem. Make no mistakes." and required substantial domain expertise and human research beforehand.
[1]https://arxiv.org/pdf/2607.13335
It is lean-verified, so it can be trusted unless the Lean statement of the hypothesis is not an accurate description of the hypothesis.
Saying “solve this problem” doesn’t get good results most of the time with humans either, it’s entirely underspecified so the person assigned that problem may solve it in a variety of unacceptable ways or not at all or perhaps worse solve the wrong problem because you weren’t clear about its definition. This actually happens all the time. What matters is the ability to communicate clearly and with precision as well as the “harness” which for humans is procedure, training, planning and management.
The subtext of this whole post (or at least a subtext that some might read), is "we don't need mathematicians/programmers anymore" or "we will need much fewer mathematicians/programmers". So the fact that this result required a year of prior research and a 10 page prompt of specialized knowledge goes against that subtext. You still needed the human just as much to get to the result, and the LLM ended up being a tool to find the last bit.
> Saying “solve this problem” doesn’t get good results most of the time with humans either
Sure. That is not even remotely the point I was getting at. Already we see the thread filling up with comments about how human skills are irrelevant, using a mathematics PhD applying his expert skills in a way that the people who are saying that could never have done to justify their inane conclusion.
Digression: You can link directly to a page in a pdf with a url like this: https://arxiv.org/pdf/2607.13335#page=27
> but it is worth noting that this wasn't a matter of "ChatGPT, solve this unsolved problem. Make no mistakes."
It wasn't the case for this, but when OpenAI disproved the Unit Distance Conjecture, it was really done autonomously by an automated AI pipeline with a completely AI-generated prompt. No human expertise required at all in the process (well, except for the final human verification).
Without human verification, an LLM can generate correct or incorrect proofs but it can't tell the difference. A human is necessary to be able to tell one from the other.
Saying that's a solution "done autonomously by an automated AI pipeline" is like saying that a self driving car that can only take you to the nearest train station after which you have to ride the rain to where you're going is "autonomously" driving you to your destination. Which is exaggerating the autonomy of the system, rather.
The automated AI pipeline also had an automatic grading model to try to reduce false positives.
But anyway, my point was that in that case the prompt involved was indeed pretty much "hey, ChatGPT, solve an unsolved problem, thanks."
What I'm feeling is that there's a need to study how to use AI well. I've seen professors using AI, and it was amazing. In that sense, I think AI prompt input will become stratified. In the past, implementation skills were very important, but these days, concepts feel more important this is one of those things.
It's not that AI brings equality, but rather that the output varies depending on how much background knowledge you have. You could call it a stratification of input
I'm starting to feel like there's no place left for programmers like me who focus on quickly churning out MVPs.
You’re at least 18 months out of date claiming that prompting will be the new hot skill. Turns out LLMs are also good at prompting other LLMs.
Calling it prompt engineer is doing it a disservice. With agents we’re well into process engineering, which is a ton more interesting.
The obvious baby’s first process is “plan -> execute” but as we learn about the strengths and weaknesses of LLMs you have to start unpacking that process into planning, prototyping, testing, validation, reviews, and tons of research. If you treat it like an extension of your brain that can automate some thought processes, it becomes a lot more powerful.
One of the key skills of a professor is asking the right questions. Figuring out something worth working on, and then framing it in an appropriate way and asking questions that allow someone with specific tools and skills to make progress in the topic. Usually the tools and skills are those available to a new student, but working with an LLM is similar.
That skill comes with experience. Most people don't have it immediately after PhD.
>That skill comes with experience.
Well it seems more and more that 3 months of 500k GPUs churning through data 24/7 to build high dimensional landscapes also counts as experience.
Ah, but who prompts the prompters?
I find it strange that people sometimes think of knowledge as 'public property for everyone.' The essence may be one, but the mental model of knowledge is individual. For an LLM's knowledge to become mine, I need to digest it to some extent.
And programming, as the programmer who created Eliza once said, is the act of becoming a legislator of your own universe. So even if there are black boxes, if you want to build a program that fits your own worldview, studying is essential.
That doesn't make any sense; you can't have one LLM to read your mind to prompt another LLM.
One might even say that this remarkable facility can be used to “read” the ideas from one entity’s mind, such that after sufficient dialog the second entity obtains a (possibly lossy, but there are mitigations for this) copy of the ideas of the first. You might further be surprised to learn that this sort of idea-transfer business using language has already been happening in our society and species for quite some time indeed.
Made my day XD
This is a lot of words to say that a human can prompt an LLM to tell it what they want.
edit: it reminds me of all that I have to wade through after I've asked an LLM a straightforward question and the answer should have been "yes, you're right."
You mean promoting, right? Did you read the thread?
so, promoting?
I would not characterize it thus.
I'm going to keep on repeating this on HN threads until I'm blue in the face, but:
There are two ways to solve a problem. Either solve the problem, or deem it irrelevant.
The implication here is that, you, the human operator, clearly are just confused. The LLM knows best. You're just a stupid human. The LLM knows objective truth, you do not. You have concerns, questions, the LLM didn't understand your question "properly"? Do not worry, the LLM objectively knows the optimal course of action. It thought through the implications of what you said, took into account all possible data, and came to the objectively correct design for your software, your society, your life.
In some sense, this problem would have been a societal problem within the next several decades anyways, but it's been hyper-accelerated by AI.
Waiting for the next Neuralink announcement...
That's still prompting, just justing a different interface.
Rather than prompt engineering, I think it should be called overall harness engineering. Anyway, that's how I feel these days
I think harness engineering is more broad, including not only the - system - prompt but also tools and skills made available to the LLM.
And yet in this case a human prompted the LLM for this result, not another LLM
I think there's a lot of interesting things to the side of development that don't get the resources they deserve
Debuggers, testing techniques, testing layers
Essentially things that could be used to ground your ai back to reality and work good for humans too
I actually think people who are great at understanding problems, coming up with requirements and designing solutions (all things I would expect someone who is good at churning out MVPs would be good at) are exactly the people most empowered by the current batch of LLMs. Its the people who are only good at working on small chunks of problems that I'm concerned about..
> I'm starting to feel like there's no place left for programmers like me who focus on quickly churning out MVPs.
Of course there is. The same way this was only possible as a result from the professor who prompted it with his specialized 10 page prompt and most importantly his deep knowledge of the problem space, the muscle memory and intuition you've built over the years is what will allow you to get more out of any AI than some guy who says "make a door dash clone" as the entire prompt
So these days I've been writing down my thoughts on my personal homepage. Things I've learned, my background knowledge, and so on.
I've been realizing that there are more books tied to my background knowledge than I expected, but I'm not sure what will happen as AI advances further.
These days, I'm living for the fun of building my own personal wiki on my homepage
Why write it down? LLM crawlers will ingest it in a second.
Sharing knowledge is good, but just because an LLM crawls it doesn't mean it fits my mental model. The act of writing is fundamentally about drawing the shape of my own mental model.
nah, scientist with the name Shakey Onail found this and all creds are given to LLMs is crazy
Do you have a link to Shakey's publication?
could machine-learning even handle a TEN PAGE PROMPT just a year ago?
this is changing my mind, at least about experts using advanced tools like any profession where it's like the magic of watching a lifetime of hard-earned skill at work
> After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy.
> After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check.
Waiting for comments saying that LLMs can't produce anything new and general goalpost moving.
From the post lol
>So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.
If knowledge is a Swiss cheese, LLMs can help fill the holes, but not make the cheese bigger.
Today maybe. I disagree in the long term.
While they’ll never have the same subjective experience as humans, what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture?
They are prediction machines, and so are we in a way. We can give them nearly limitless resources to scale their predictive capabilities. We have billions of years of training baked in. They distill directly from our knowledge and can walk down paths that no human has before.
It’s silly to say they’ll never do anything novel.
At their current capabilities, it sounds like they are already capable of being a specific type is research assistant. What will that look like in 10-20 years?
They also have ability to go deep and wide in a way that humans just can't. We have limits, get tired, distracted and biased where AI does not. I think there a lot of problem where all the information needed to solve them is there, but we just can't put the pieces together. Like no matter how many people you throw at some problems, you hit human limits and more people won't help, but AI will because it is just relentless.
>biased where AI does not.
AI can be totally biased...
The fact that it can spout bullshit all day long to a human who can be tired and would actually act on the said bullshit, is not very comforting...
For example, an LLM could confidently declare something a tired human would take as a fact, but would backfire in a real world.
Not really the kind of biased I meant though. There was a recent article about a AI disproving I think an Erdos conjecture by doing similar things humans have tried, but it was much messier and less "beautiful". I think it is a common bias in science and math that things should be "beautiful" but there is no real reason to think that.
>what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture?
One thing is that an LLM can never assume, or find out, an inconsistency in its training data. Novel ideas often require correction of existing assumptions. As far as I understand, it is impossible, by design, for LLMs to contradict what is in its training data.
For example, an LLM trained on the data from an internet comprised of people who believe in the earth centric hypothesis can never say "Hey, that cannot be correct", or come up with the heliocentric alternative
But maybe it is not applicable to pure Math...
They can, but it's limited to that specific chat context.
They can spot contradictions in the the prompt. But not in their own training data.
> While they’ll never have the same subjective experience as humans
You state this as a fact - are you aware the question is unresolved?
EDIT: I'd love to know why you're downvoting me for stating a known fact.
Ok, I guess never is a real long time and eventually when we merge our consciousness with the machines we may have the same subjective experience.
I can confidently state that GPT-5.6 Sol is not experiencing the same reality as me. They _might_ be "experiencing" and I personally think they are, but their reality and experience is not the same as ours.
Well, sure. I'm not experiencing the same reality as you, either. I guess I assumed you were implying something more - a lesser experience or something. No?
Fear spreads.
Famously, all of maths is axioms and tautologies, so I'm not sure this will assuage any professional mathematicians currently having an existential crisis.
Maths was already infinite, it's still infinite, but who wants to spend all their lives changing rooms inside Hilbert's Hotel?
this is a fairly bleak outlook even when you're trying to make it sound the opposite. Only the cream of the crop talent will have value going on?
Most of us aren't Terence Tao
so it seems like The New Big Question In Math is
How's It Hanging, Brother?
The author explains he's an expert in the domain and that he had worked sporadically on the problem for about a year, also with the help of previous LLMs. So whatever he means by "I wouldn't really say that this result is using or creating some fundamentally new techniques" it doesn't mean that the result was trivial. Also, says it might not make sense to work on low or even medium hanging fruits in the future- and I bet that's by far the largest share of work for most mathematicians.
Sure, it's not a breakthrough that opens new roads in mathematics- is this where the goalpost has moved now?
HEH. Don't know why you're getting downvoted. It's painfully obvious that there is a vicious AI backlash now, where every amazing advancement is met with denial and loathing.
Oh wait, sorry, I do know why you're getting downvoted. Fear.
A lot of people who thought they were special and “better” than mere blue-collar workers are realizing that in fact;l they are the same just working with a different medium.
I see a trend line from anti-AI back to anti-evolution to vitalism all the way to Galileo.
Humans have a deep need to be special magic flowers - and they can't stand it when science eventually shows them they're not.
Humans just want to be able to work and feed their families. Notice how this is something that is never addressed for example in Asimov's writings where robots do everything. How do people do anything to be able to afford these robots? How do Solarians make money to maintain their massively gigantic estates? It is never explained, but we always jump straight to the "robots do everything, everything is free, nobody works". How about the in between?
Oh yes, I agree entirely. I look at The Jetsons or Star Trek, and I see the fruits of technology shared by all of humankind.
But if the current trends continue, Elon will own all of the robots and the rest of us will be at his mercy.
How we change that - I don't know. But I do know we don't change it by putting our heads in the sand. AI is here and it is real. We MUST take it seriously. Shunning is not a viable option.
We're going to have to (collectively at some point) decide that we shouldn't tie survival to working.
Whether that's UBI, or Fully Automated Luxury Gay Space Communism, or UBS, or some AI managed economy, or some other such thing, we need to decide that "hey, we don't want to tied survival to output under our current system."
That's really the Rubicon we have to cross, and there are a LOT of people who can't really come up with a better way in their own heads yet.
Look, we all saw how things can be during the pandemic. That was a dark time, but it also gives us a model, we can just... do things. We just have to decide to do them.
Oh! Damn! Stupid me! "We have to collectively decide this", when is the meeting to vote on that again? When is the last time we all decided something collectively?
WTF does the pandemic have to do with this? "dark time", oh yeah, people stuck at home watching Netflix, what an incommensurable suffering!
Buddy, I can admire the naive childish optimism to a degree, but come on. "We just have to all decide". Do you live in a Disney movie?
> WTF does the pandemic have to do with this?
I'm not sure, but if I may guess, I suspect he's talking about the fact that the entire US shut down for a year and people survived because the US Government printed money and gave it away free to businesses so they wouldn't collapse. (PPP loans.)
If we can do something like that for the pandemic, we can do something similar when push really comes to shove.
Thank you! We can decide to just... you know... do things.
Like, am I crazy here? The rules that we live by are largely made up and the points don't matter - we can decide to live in a better world if we want to. It's hard, and there are obstacles, for sure, but this appeal to doom for the sake of doom just... why? We have made unfathomable progress over the last century and if we keep trying we can make progress like that over the next century too!
> We can decide to just... you know... do things.
OK, this is Twitter-level conversation here, not interested. But hey, stay positive, more power to you.
You’re the one saying that we can’t. Not me.
I mean, a at least something like million people died just in the US with the most conservative estimates that I've seen (and by 2022). It was pretty dark, yeah.
But we also decided during that time to literally pay people to stay home for awhile. That was a really revolutionary thing and it was awesome. We could decide to do that again.
You can ad hominem until the cows come home, but yeah, we literally just kind of have to collectively decide that there are better ways to do things.
There's a fantastic book called "The Last Emperor of Mexico" I read a few years ago that really talks about how the idea of a Republic or Democracy in general was a pretty novel concept in the mid 1800s. People were a lot more skeptical about it than we're lead to believe now. But eventually, the ideas of aristocracy and some "well bred" group of various types of monarchs became silly on it's face. Now the default is that we should have some sort of democratic representation. That would seem utopian AF in 1820.
Well, we're going to have to bridge that sort of gap for getting rid of the need to justify our existence through work too. The transition is going to be weird, but we'll have to come up with something else and run with it.
Dream big buddy, I know it's hard, lord knows I do, but dream big, and work little by little towards those things you want to see in the world.
Oh brother I can tell you didn't read the entire article.
Genuine question: If you still or did think LLMs are just stochastic parrots that just summarize everything and have no form of creativity, what do you think after seeing results like this?
I'm very curious how people reconcile their fear/hatred of AI with actual objective reality. This is actually what interests me most about the whole AI thing. How we tell ourselves what we tell ourselves.
No matter what there will always be people who refuse to believe AI is anything beyond a string of if statements.
I had to create an account to respond to this because I am quite convinced these math problems they are "solving" are pure marketing. Why is it only GPT doing this, why not Claude? Why does Terrance Tao do marketing for OpenAI? I suspect OpenAI has hired math researchers to solve obscure problems and put them in their training set, purely for marketing reasons.
There was a good comment on the Pelican bicycle svg yesterday about how these models aren't getting much better beyond what the companies focus training them on. I think that's what's happening in this case too, they probably put this in the training set.
It's such a weird train of thoughts lol. You're using the fact that
- Claude isn't doing that
as evidence to support the assumption that
- it's a marketing trick
Which is obviously non sequitur, as if it were a marketing trick, Anthropic could do it too. Anthropic isn't known for not spending on marketing.
Honestly, nowadays I question human's reasoning ability more than I question AI's.
I (author of the paper) can't speak for others, but I have no affiliation whatsoever with OpenAI and have not received anything from them (I even pay for my subscription lol). I've thrown this problem at each model with new model releases, and 5.6 was the first one to solve it, so that's my side of that story.
Thanks for responding. I don't personally believe you have affiliation, more that OpenAI saw the problem was being worked on, gathered training data for solving it and training the model on it, then eventually you found it through the model (and now it's a big impressive thing for OpenAI)
Part of me agrees with the other comments that it sounds absurd, especially because of how involved/intricate your prompting was. But given you have been prompting this problem for a year, OpenAI could easily have seen your attempts/progress, and worked towards solving it with human intelligence that the agent was then trained on. Given the lengths these companies go to for training these things (Meta literally reorganizing their engineering department to provide training data for engineering problems), and also how much difficulty these models have with problems outside their dataset, I really have to wonder if it's the model being "intelligent" or if it has been trained on it
> Why is it only GPT doing this, why not Claude?
Because Claude can't do it. Anyone who tells you that Fable is better than GPT 5.6 at pure math is lying to you.
kind of a hilarious conspiracy theory
You are correct that LLMs are trained on existing proofs but hiring researchers to solve unsolved problems is just unrealistic, both in terms of how none of the mathematicians simply came out and took credit for their own discovery or exposed this, and how training sets are not easily memorized (rather, the meta techniques are learned).
OpenAI just has better training methods and techniques for pure math over Anthropic, it’s one of their biggest strengths
Terrence Tao getting paid by openAI is, to you, the most probable conclusion... much more so then the LLM actually being able to come up with math proofs?
Terrance Tao has for a fact appeared in promotional material for OpenAI. Based on my Googling the consensus seems to be he is paid for it, but I cannot confirm that.
I do think it's very likely that OpenAI pays for solutions like these to put in the training set, and then we get material like this Reddit thread. They market themselves as selling "intelligence", and solving these math problems is something people view as highly intelligent. I'm not a mathematician, so I cannot fully judge it, but based on my experience using LLMs for novel problems in other domains, they seem to really struggle with things that aren't common. That leads me to believe they train for specific outcomes like this. Also, there are a lot of jobs out there for data annotation, including software problems (Meta has basically reorganized its entire engineering department to create training data for coding problems).
This comment on the Pelican svg better articulates what I'm getting at: https://news.ycombinator.com/item?id=48950883
You can go through my commenter history and know I'm no fan of LLMs. I don't overstate LLM capabilities and am highly skeptical of them in general. 5.6 Pro is genuinely pretty good at certain kinds of math problems that just require trying out lots and lots of solutions, mostly because it's stubborn and can run a bunch of instance in parallel. It is NOT good at coming up with unique ideas or recognizing when its proof approach is doomed, and if the correct approach isn't in its "bag of tricks" for tackling a specific kind of problem, it is not going to get it without a lot of guidance. That said: I 100% believe that it's solved the problems people are claiming that it solved.
The way you should read this is (IMO) not that LLMs have somehow achieved AGI, but that a lot of mathematical research is more about knowing a huge amount of mathematical background, being stubborn, and getting lucky with an approach than it is about brilliant insight. Many people who don't think of themselves as particularly mathematically gifted could have made progress on these problems if they were given enough time and were interested enough. What's notably different about 5.6 (and born out in benchmark after benchmark) is that it does seem to genuinely "reason" through stuff at all -- without that, persistence is pretty worthless because the LLM just goes wildly off the rails if it's put to work for long enough (5.6 itself will still do this if it can't find an answer in a reasonable amount of time).
Terence Tao also uses Anthropic's models in his work. Oh, you didn't know that? Well, now you can pivot to saying that he's getting paid off by both companies. This is actually one of the hallmarks of both conspiratorial nonsense and military-grade cope. Any fact, regardless of how mundane or extraordinary, gets re-imagined as evidence of the same mad-hatter conclusion.
I hope people are screenshotting this stuff. This really needs to be documented. It's remarkable how wild it's getting.
I hold my stance that LLMs are stochastic parrots.
Making the parrots ever more complex and training on ever more data produced by intelligent, creative beings may make them more useful or convincing but does at no point give rise to intelligence or creativity.
I won't touch creativity, but if this and other results like it do not demonstrate intelligence, what does? How was it able to solve problems that specialist mathematicians have tried and failed to solve for years?
Mathematics is a language. If anything, it's much more well defined and formal than most others. Train on enough examples and statistical autocomplete gets you places. I'm surprised how anyone would even consider this intelligence?
Comical human arrogance...
With such high standards, most HN commenters also do not have intelligence nor creativity. I don’t think we can set the bar that high.
Heh. I see you're being met with screeching and downvotes.
Not much to do about it, I guess, but continue to call it out.
They aren’t really stochastic parrots, they’re next token prediction machines. They aren’t intelligent nor creative imo. However they are quite useful at some tasks. Why would you assume people who don’t kneel at the alter of AI are somehow fearful or hate AI? Most of the hate I’ve seen have been for the people and companies involved with AI not the technology itself.
I'm very curious why people conflate thinking LLMs are stochastic parrots with "fear/hatred" of AI. It seems like you're arguing with people who agree that it works and it helps, but you're trying to insist that this implies that they should kneel down and pray to it.
Is "stochastic parrot" too disrespectful for you? Do you think it is a slur?
edit: and this is a genuine question, also. How do you do stochastic parrot = "just summarize everything" = "no form of creativity" = "fear/hatred" so quickly?
Are summaries not creative? Are Maxwell's equations not summaries? Do people hate and fear parrots?
I have absolutely no problem with people disliking or fearing AI. It's energy consumption, effects on education and potential for displacing good jobs are all quite disturbing. But "stochastic parrot" means that "all it does is randomly repeat things that it has seen before without understanding them." It's infuriating to see this written about an instance of an AI solving an open math probably. Do you think the models are just randomly repeating facts until they accidentally emit a proof? If so, then how do they synthesize that knowledge into something logically coherent?
Alternatively, if you think that even Maxwell was a stochastic parrot, then presumably almost every human who has ever lived was also a stochastic parrot except a few rare examples like Einstein. Not sure what definition you are using but it seems too broad to be useful.
I think it's quite clear the proof here shows that it is not a parrot. It objectively isn't.... that's the only rational conclusion. Yet many people claim that it is, so the main conclusion is fear/hatred is causing people to rationalize their logic to fit the narrative they prefer.
Must be nice knowing you have a clear understanding of "objective reality" that others don't.
Is it not objective reality that the feat performed by the LLM here is much more then parroting or summarizing something?
It's doing math proofs. At this point, it's fully clear that objective reality is that the LLM is not parroting anything here.
It's parroting proofs. This is no different than parroting a story.
https://lean-lang.org
Except solving problem is probably the least (even though it's important) interesting thing in research...
The most interesting thing in research is finding new questions, that we understand and that we know why they are important. And that's something that humans need to do (by definition)
I keep hearing this but lots of maths problems are practically important! We want to know the answer because it will be useful for applied science, or statistics, or engineering. It’s not all just about knowledge for its own sake.
Is this interesting? AI does what we made it to do, news at 8?