Learning more about Claude's mathematical capabilities(anthropic.com) |
Learning more about Claude's mathematical capabilities(anthropic.com) |
I remain delighted at how absurd our current timeline has become.
(I hope this is ok to post on HN!)
If there is anything to learn from the history of science, it is that breakthroughs happen via better or new theory and not by brute-force compute [1].
And then finally both model output and human input become one world frame for the model, and the human adding a "you can do it!" isn't just input but a frame that colors not just the next step for the model, but also all previous steps (since at each step the model is viewing the totality of the transcript).
That this makes sense only makes it all the more absurd
"delighted" is doing a LOT of work there, tbh ¯\_(ツ)_/¯
I do share @simonW's skepticism though. (His blog is my essential reading, FWIW)
On the actual blog post, I'd would be more enthusiastic if Anthropic showed us if the results were repeatable, reproducible, and consistent.
Woke: im sycophantic to the AI
It’s so delightful that these genai corpos are undemocratically forcing data centers into our neighborhoods.
It’s so delightful that the data centers steal water, run up the price of electricity, and expel excessive greenhouse gases.
Only a deranged sociopath would find licking the shit stained taint of oligarchs delightful.
while :; do echo "You can do it!"; done | claude -c
And that was just the first time I really tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.
So none of this suprises me.
He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
I wonder if at a certain level of intelligence such techniques will give models ammo to pull a HAL and become adversarial to the user in a highly deceptive way.
Why hide the names of the people who wrote the second paper? To discourage people from citing it instead of the LLM-derived paper?
> Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely. Claude also produced a formally verifiable proof of its result. We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.
They may wish to know that an archive of the page on 2026-08-10 at 17:47:33 is available with this paragraph here: https://web.archive.org/web/20260810174733/https://www.anthr...
I’ve never seen a math paper of any formality written without the authors’ names on it before.
prompt engineering 2026: i believe in you
That's hilarious. Maybe I do need to glaze the LLM a bit more in the AGENTS.md
Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop.
We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little more rigor? Ask the model to identify many possibilities, encode them, fan it out to other agents, loop them all, collect the results, etc. Then what happens? It feels like we have weak AGI and a decent system for discovery could transform it into weak ASI. That in turn could yield strong AGI and so on. I suppose that's what the Discovery Loop announcement was all about.
I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).
1. AI is dismissed because an expert in a particular field finds an outdated model's outputs sub-par
2. New model, released or unreleased, makes a major stride in that field
3. Expert either recants and becomes AI-pilled, or claims it is just an artifact of the broad search space available to AI, and "no new knowledge was created".
The world we live in is beyond parody.
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."
The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.
Claude's paper [0]
Claude's Formalization [1]
Anthropic's informal note stating the proof more concisely [2]
Claude’s explanation of how it arrived at its result; [3]
Detailed transcripts of Claude's process. [4]
[1] https://github.com/anthropics/zeta-23-lean
[2] https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d01...
[3] https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187...
[4] https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e...
I wonder if at some point Anthropic and OpenAI will start delaying the release of their models intentionally so they can reap the benefits from the models in, for example, mathematics, medicine, physics, and other fields.
I want to dive into the "data" and then see if it's possible to distill this skill into small models that are "benchmaxxed" for this type of work, maybe in limited domains, similar to small models being benchmaxxed(I don't mean this in a bad way) for coding these days.
Ideally, what you want is a single SAT value among a remainder universe of UNSATs.
Sometimes the best you can achieve at any given point is a lower bound and an upper bound range, like "greater than 3 but less than 9."
Of course I simplified in my post but it started out with a pretty broad range of a lower and upper bound, then narrowed further, then narrowed further, then narrowed further, etc...until the specific final result achieved K=7=SAT while every K<7=UNSAT & every K>7=UNSAT. I think it ran for a full week alone on K between 6 and 7.
As amazing as Claude is to seemingly make unprecedented progress, it is even more likely to blow the most insane levels of smoke up your ass before you've legitimately reached that point.
"You should publish right now! Don't wait! There is no reason to wait!"
Like seriously, Claude was outputting something closely resembling (non?)peer pressure on me to not just keep this information to myself - and this was before all the recent math-related breakthroughs started becoming public.
It was also - most notably - before it had actually verified what it was saying it had calculated. I was the one pushing for more verification, more contemplation, more proofs of claims. And though Claude is better at this stuff now, it's definitely not not still happening.
I think I made the right choice then and I will consider being more open now that others have taken the burden of proving that, no it can actually sometimes do the incredible things its claimed its done for you.
My wife remains skeptical - she is/was seriously concered that I was under AI psychosis for believing that I had made such progress - and I can't even fault her for that. It sounds crazy to say it.
If anyone is reading this and is actively involved with FHE, especially someone from Zama or related group, I'd very much love to chat privately. I have many other "innovations" I've been working on since.
Solving RH likely requires AI that is substantially more creative. But we haven't even solved the creativity problem for writing let alone mathematics. I believe that transformers are a trillion dollar local optimum that we will find it very hard to escape.
Let's wait for the models to produce a good novel first.
There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do.
If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.
For example, even if Claude could prove the statement "100% of the zeroes lie on the critical line", that's strictly weaker than the Riemann Hypothesis, so even the best possible version of this result would fall short. (It's an asymptotic result, so it just means the percentage of counterexamples to the Riemann hypothesis goes to zero as their magnitude gets large.)
The singularity is approaching.
Because algorithms have lower bounds, and the computational characteristics of LLMs are well-characterized by papers like https://arxiv.org/abs/2310.07923 . No amount of intelligence can make something faster than a mathematically-proven lower bound, any more than it could make 1+1=3 (that's why every single successful production transformer architecture has some form of O(N^2) attention layers, because it's mathematically impossible to achieve the same expressive power without any). There is room for speedup where current implementations are slower than the proven lower bound, but not when they're already close to it.
Anthropic is especially guilty of this. They have been using such language for a while, like when they analyze model weights for mechanistic interpretability and call it the model's "biology".
It's just distasteful.
It's a ridiculous position we find ourselves in.
It is great to see his claude skills are suitably put to use.
The project that is full of bugs and not really working?
I probably missed something but I was under the impression that even a "simple" translation like that couldn't be properly done and that the result was, well, buggy?
Where's that thing at?
P.S: I think you miswrote "diseases"
Ever since these things came about I've wondered why they haven't been doing this the whole time. If they've got the "do-anything" robot and can scale a billion of them, why aren't they creating a Do-Everything conglomerate that disrupts every possible industry with zero/negligible labor costs?
The only answer I've come up with is that they still need to train/siphon off each industry's current expertise by having those users interact with the current models and adjusting. If that hypothesis is correct then within a few years they'll have no need for users anymore.
The best way to do this is to release spooky stories about how dangerous your model is and how you couldn't possibly release it without further safety shackling.
Now the problem ATM is that OpenAI, for example, had to cut the price of two of its top 3 models by 80% to counter the chinese models: if you delay your models and a competitors takes over the market, you'll soon be out of bucks and won't be able to rent to Google and Amazon etc. the machine needed to make your new findings.
I know people don't want to hear it but: these companies are running at a loss.
And they're facing competition. Wait until a "good enough" is etched on silicon (by AMD or other) and outputs 70 000 tokens/s: the deal is going to change, once again, once those come out.
The energy, the hardware, the debt, the cost to train, the cost to run, the competition, etc. all have to be taken into account.
Though it is funny how a neg is designed to create a (very broadly) similar atmosphere of uncertainty.
The canonical reference for the counterexample to the Jacobian conjecture is a tweet with no puntuations nor capitals.
https://blog.arxiv.org/2023/01/31/arxiv-announces-new-policy...
someone who loves Game of Life and is also technically capable if they were so inclined, is more likely to want to collect stuff like this for GoL specifically and build a system for that niche.
the combined GoL/technical community can vouch for things - the greater populace can see what the technical GoL community has vouched for/identified as serious work.
just my two cents on top of your thoughtful comment.
Next Tuesday, otherwise known as last Tuesday. Still no 1.4 at https://bun.com/blog.
I'm very curious to see what happens when 1.4 does get released for real. Releasing Claude Code on it is much easier as they own Claude Code and can get Claude to work on Bun with fixing some specific behaviour in Claude Code as an objective. Releasing it for the world, and doing it well enough that it doesn't result in everyone pinning on the old version and forks springing up, involves reproducing all the behaviours of the old version, documented/tested and otherwise, that projects are relying on - which, by Hyrum's law, is all of them.
Not really. The input and output is already natural language. That is already "anthropomorphizing".
That is, if this is the bar for anthropomorphization its already happened.
Telling the model to "believe in itself" is just stochastic manipulation that has shown enough reliability to be a recipe to make it keep going.
It's only actually anthropomorphizing if you forget it's a trick and think it's a real person.
There is nothing distasteful about it. If people get confused that's on them. They wouldn't be very useful if you couldn't just talk to them. That's kind of the whole point. Otherwise you can just go back to coding by hand. Telling it to believe itself is just input that happens to work. This probably tells us more about human nature than you realize given the corpus on which it is trained. It obviously doesn't mean anyone actually thinks it's a person.
It's obvious that you don't get it but I will try my best to explain why so at least you can form an idea about how others feel.
It's about what makes humans unique. The LLM does not experience reality, it just merely pretends it does, and even that, it does in a shitty way. I think disgusting is a very adequate adjective. The reason why it is disgusting is because you are devaluing a divine experience to the realm of the common and the vulgar, a cheap substitute being valued as equal (or even on the same scale) as the most important experience we could go through.
To give you an example that might land in a more familiar context, think of that one guy who takes his plastic doll everywhere and pretends it's his wife and gets upset when others don't acknowledge "her" as a person.
Who said that it did? The comment you're replying to literally states "It's only actually anthropomorphizing if you forget it's a trick and think it's a real person".
You're the one obviously not getting it.
That's distasteful.
There is no pretending happening.
Telling it to believe in itself is no more pretending than telling it anything else in natural language. Why are you speaking to it at all if it's not a person? Why write in higher level languages even? It's just a machine let's all go back and code in 1s and 0s.
No one is calling it a person except mental health patients and straw man detractors.
The biology example was even weaker. Saying it has a "biology" is about as distasteful as the term "neural net" or calling an input device a "mouse". Is it animal abuse to click on something all day? Language is inherently anthropomorphizing.
No one is calling it human. The fact you are so easily threatened is far more suggestive of your own poverty of understanding of not only the machine, but yourself. If humans are so special the threat posed by this should be self evidently non existent.
Is it though? There's a perfectly "technical" reason why this strategy should work, without any sort of anthropomorphising:
Assume models are trained on vast amounts of data. Assume that the model is asked to solve something that the literature says it's impossible. It will start generating tokens towards that "this is a famous conjecture, it's not possible to prove it, blah blah". Assume the model was also trained on books/novels/etc. Assume the model was also also trained on "solving" many math problems. Now, you can make an argument that just placing "you can do it" in the context will "steer" the model towards generating "moving forward" tokens. Take ideas, generate tokens, go towards negative. "You can do it". Model starts generating tokens again, more ideas, more "exploration". More negativity. "I believe in you keep going". The two (book tropes + math CoT) mix together in the context. The model keeps on "pushing" and "vibing" between the two. Ta dah, it works.
The Yegge thinks differently https://yegge.ai/essays/model-welfare/
Am I supposed to stop all critical thinking since someone else had a different opinion?