The whole point of Lean is that you don't need to understand the entire proof to be sure that it's correct. You only need to understand the definition of the theorem being proven, and you need to trust that the relatively small core of Lean is correct.
> In July 2026, a disproof of the Collatz conjecture was verified not only by Lean, but another formal verification system Nanoda. However, investigation quickly revealed that the proof exploited bug(s) in these verifiers.
Why should you trust that the relatively small core of Lean is correct?
The core of Lean got a lot less correct when a well-meaning AI system probed Lean for corner cases (bugs) that would "prove" a false conjecture. Corner cases so arcane that no human exploit in a proof. Basically, humans are too stupid to break human-created Lean, but the AI is not.
Lean does have libraries, but since they are also in lean they are subject to the same rules. It's basically a super strong type checker. If it compiles the proof is valid. Unless there is a bug in the type checker.
They could, for example:
1. Rely on users flagging suspected repos (with human moderators reviewing afterwards).
2. Deploying AI agents to evaluate repos against known AI-generated-code patterns (also with human moderators afterwards). Of course, the irony of the situation - using slop machines to fight slop.
Indeed, but having more data might be able to solve that? The whole problem seems to be that benign conditions sometimes look scary because we're currently not able to predict well enough whether it's something that will eventually cause problems.
Not in the same way, no, because they have not been targeted, while they should have if the same rules applied, according to Anthropic's depiction of the situation.
This is potential tyranny aimed at Anthropic, specifically.
Europe doesn't seem to care so much about erratic and hostile governments when it cozied up to Russian gas for decades, something it still continues to do just hiding behind third party countries.
It's a clear statement that European morals are purely performative
Just like how the EU is hostile towards US companies, but very light to the touch when it comes to corruption with HSBC, FIFA or VW. With such hostile and erratic allies, who needs enemies?
Let's not even get into Orban. You can never trust the EU again since who knows if they're capable of electing someone like that in the future? Trust is broken forever
Great, great point. That's why people who say NATO can't be relied on anymore aren't making sense - NATO never elected Trump, so his involvement or opinion on it doesn't matter.
Verifying that every step in a (potentially long) proof is sound can of course be much, much harder than verifying that a definition is correct. That's kind of the whole point.
That's not what the parent comment meant. They meant checking the Lean-language definitions actually match the mathematical English ones, and that the Lean theorems match the ones in the paper. If that's true then you don't actually need to check the proofs. But you absolutely need to check the definitions, and you can't really do that without sufficient mathematical maturity.
Yes, and the child comment’s point is that formalizing the problem is likely easier than having the LLM verify that each step of a long deduction is correct, which is why Lean might be helpful.
But both of you are ignoring the parent comment! Actually you're ignoring the context of the thread.
Originally someone said "I wish I was math smart to know if [this vibe-mathematics proof] worked or not." They did NOT say "I'd like to check but I am too lazy." Suggesting "ask it to formalize it in Lean" is useless if you're not mathematically mature enough to understand the proof, since that means you're not mathematically mature enough to understand how to formalize the problem.
Then "likely easier" is a moot point. A Lean program you're not knowledgeable enough to sanity-check is precisely as useless as a math proof you're not knowledgeable enough to read.
It’s not useless, because you can, for example, ask multiple frontier models to do the formalization and see if they agree. And if they have surface-level differences in formalization, you can also ask them whether apparently-different definitions are equivalent.
This isn’t perfect of course - perhaps every single model is wrong. But you are too quick to declare that something isn’t useful for arriving at an answer. Reducing the surface area of what needs to be checked is good regardless.
By "online wallet" they were likely referring to the Bybit website being the wallet of those customers that held their coins there rather than keeping them in their own private wallets, and not whether the hack involved a hot wallet or a cold wallet. Calling it a custodial wallet would have been more accurate.
> It would be similar to if I claimed that an LLM is an expert doctor, but in my data I've filtered out all of the times it gave incorrect medical advice.
Computationally it's trivial to detect illegal moves, so it's nothing like filtering out incorrect medical advice.
> Computationally it's trivial to detect illegal moves
You're strictly correct, but the rules for chess are infamously hard to implement (as anyone who's tried to write a chess program will know), leading to minor bugs in a lot of chess programs.
If you move beyond legal positions into who wins when one side flags, the rules state that the other side should be awarded a victory if checkmate was possible with any legal sequence of moves.
This is so hard to check that no chess program tries to implement it, instead using simpler rules to achieve a very similar but slightly more conservative result.
That link was new too me, thanks! However: I wrote some chess-program myself (nothing big, hobby level) and I would not call it hard to implement. Just harder than what someone might assume initially. But in the end, it is one of the simpler simulations/algorithms I did. It is just the state of the board, the state of the game (how many turns, castle rights, past positions for the repetition rule, ...) and picking one rule set if one really wants to be exact.
(thinking about which rule set is correct would not be meaningful in my opinion - chess is a social construct, with only parts of it being well defined. I would not bother about the rest, at least not when implementing it)
By the way: I read "Computationally it's trivial" as more along the lines of "it has been done before, it is efficient to compute, one just has to do it" versus "this is new territory, one needs to come up with how to wire up the LLM output with an SMT solver, and we do not even know if/how it will work."
> You're strictly correct, but the rules for chess are infamously hard to implement
Come on. Yeah they're not trivial but they've been done numerous times. There's been chess programs for almost as long as there have been computers. Checking legal moves is a _solved problem_.
Detecting valid medical advice is not. The two are not even remotely comparable.
As I wrote in another comment - you can write scripts that correct bad math, too. But we don't use that to claim that LLMs have a good understanding of math.
I'd say that's because we don't understand what we mean by "understand".
Hardware that accurately performs maths faster than all of humanity combined is so cheap as to be disposable, but I've yet to see anyone claim that a Pi Zero has "understanding" of anything.
An LLM can display the viva voce approach that Turing suggested[0], and do it well. Ironically for all those now talking about "stochastic parrots", the passage reads:
"""… The game (with the player B omitted) is frequently used in practice under the name of viva voce to discover whether some one really understands something or has ‘learnt it parrot fashion’. …"
Showing that not much has changed on the philosophy of this topic since it was invented.
> I'd say that's because we don't understand what we mean by "understand".
I'll have a stab at it. The idea of LLMs 'understanding' maths is that, once having been trained on a set of maths-related material, the LLM will be able to generalise to solve other maths problems that it hasn't encountered before. If an LLM sees all the multiplication tables up to 10x10, and then is correctly able to compute 23x31, we might surmise that it 'understands' multiplication - i.e. that it has built some generalised internal representation of what multiplication is, rather than just memorising all possible answers. Obviously we don't expect generalisation from a Pi Zero without specifically being coded for it, because it's a fixed function piece of hardware.
Personally I think this is highly unlikely given that maths and natural language are very different things, and being good at the latter does not bear any relationship to being good at the former (just ask anyone who struggles with maths - plenty of people do!). Not to mention that it's also much easier to test for understanding of maths because there is (usually!) a single correct answer regardless of how convoluted the problem - compared to natural language where imitation and understanding are much more difficult to tell apart.
I don't know. I have talked to a few math professors, and they think LLMs are as good as a lot of their peers when it comes hallucinations and being able to discuss ideas on very niche topics, as long as the context is fed in. If Tao is calling some models "a mediocre, but not completely incompetent [...] graduate student", then they seem to understand math to some degree to me.
Tao said that about a model brainstorming ideas that might be useful, not explaining complex ideas or generating new ideas or selecting a correct idea from a list of brainstormed ideas. Not replacing a human.
Obviously not, but that is tangential to this discussion, I think. A hammer might be a useful tool in certain situations, and surely it does not replace a human (but it might make a human in those situations more productive, compared to a human without a hammer).
> generating new ideas
Is brainstorming not an instance of generating new ideas? I would strongly argue so. And whether the LLM does "understand" (or whatever ill-defined, ill-measurable concept one wants to use here) anything about the ideas if produces, and how they might be novel - that is not important either.
If we assume that Tao is adequately assessing the situation and truthfully reporting his findings, then LLMs can, at the current state, at least occasionally be useful in generating new ideas, at least in mathematics.
> in the absence of other celestial bodies the satellite would be in a stable orbit
Presumably entering such an orbit is only possible due to forces from other celestial bodies in the first place, since otherwise if you reversed time it would spontaneously leave its orbit. In other words, the act of the earth "capturing" the object is ultimately performed by external forces?
