> If this turns out to be true I think my take above is not correct, in the short term people will have to stop sharing updates because otherwise openai will dishonestly race to finish their work.
Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)
"But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed."
But yes; him reaping the benefits of himself having the idea first was part of it too; as far as I am aware.
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Which is still something completely different than some anonymous organisation keeping mathematical research secret because it is better for hype reasons. One is competition between individuals or groups within a field; the other is boring and sometimes borderline nihilistic generating of mathematical knowledge as an marketing asset.
It is a strong argument in this case though, because Terence Taos expertise is directly linked to his ability to not misstate the history of mathematics.
Also note how the quote by Tao is in all likelyhood not meant as an absolute; rather than a statement of a trend - a handfull of counterexamples do I no way change anything about the truth value of Tao's quote.
On the other heand; consider how absurd it would be if "... in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science ..." would indeed be a misstatement; which would imply that far more promising research directions were not shared with the broader community (i.e.: published). I wonder what different reading of that counterfactual there could be other than secret societies that kept their discoveries and research directions to themselves - which we just learned about (since we would otherwise not be refering to the secret societies and their supposed promising research directions).
All pretty straightforward, I would say - both that "misstatement" is hopefully based an overly strict reading of Tao's quote, and that mentioning Tao's background as one of the fields leading practitioners is relevant as well. Again; to make sure: A few counterexamples achieves nothing here. It would need to reach a certain threshold of such counterexamples before we will have to write the history of mathematics; and before Tao actually made a misstatement here.
I mean mathematics has enough history that something can both have been false for centuries of mathematics research and true for centuries. Sometimes in different places simultaneously.
Terrence Tao can do his job perfectly well without being aware of any mathematical history, though I consider it unlikely that he is. I'm not seeing the direct link you're talking about, in fact history is frequently left out of mathematical teaching even when the history would in fact help in the understanding of some concepts.
> in fact history is frequently left out of mathematical teaching even when the history would in fact help in the understanding of some concepts.
That is well-known I assumed and continue to assume.
> I'm not seeing the direct link you're talking about
You are stating that link yourself; indirectly: "though I consider it unlikely that he is [being unaware of any mathematical history]". Why is it unlikely, precisely?
- Maybe because it is unlikely that he recieved the mathematical teaching that frequently does not contain history of mathematics (wild! I wonder which university you have in mind in particular) that you seem to be refering to?
- Or maybe because he is quite the opposite of a person that never ventures outside of their own area; being blind for other fields, or ones own history; as evidence by being famously collaborative across different fields, having a popular blog where he writes about non-mathematical topics too and last; him being one of the main proponents of foundational topics such as formalization of mathematics; or the use of LLMs for mathematical research.
Does all that really make it more likely to you that Tao is not aware of the existence of counterexamples like those the commenter above mentioned - more likely than the commenter simply having missed a nuance or taking something out of context?
If so; I would be genuinely curious why - people work differently, and I am always happy to learn, or close gaps in my own understanding.
Any one of the arguments you make here is already stronger than the original one. The problems with appeals to authority isn't that they are always false, it's that they are not a good argument.
And you say his "expertise is directly linked to his ability to not misstate the history of mathematics". And frankly, I disagree. If he happened to be misguided or even outright wrong about some parts of mathematical history I wouldn't think any less of him, nor do I think it matters much for the work he's actually paid to do. At worst it would result in an online discussion, which is arguably a good outcome not a bad outcome.
It should obviously not come as a surprise that having areas of the brains working differently can explain an awfull lot of things... but to answer your question directly, yes of course of course there are!
Simply look up all the many, well
-supported and -researched known correlations with ADHD first. In the second step, you can construct the set of all possible correlations, and subtract the well-researched ones if it. What is left is the set of correlations that are either not explained by ADHD (the big majority I would assume) or explained by ADHD, but as-of-yet unknowingly so.
This might be a bit anticlimatic, but clinical psychology is pretty straightforward study design and statistics, and set theory is not that new either, so... no big surprises I am afraid.
100% agree on the ad angle. Ads normalised using digital products/services by paying with data and attention.
That is one of the reasons why I would love to see - reasonably priced - paid services, or even have governments offer them (as in: Source is open, hosting is being paid by the government). Given the role social media, the internet and so on plays today, it might make sense to treat this as a common good.
But that side track aside; I think that privacy respecting, ad free products can be offered for cheap, if their goal is only to finance themselves, development and gain a small margin. And true, network effects / buyin is real, but I also think that the market for ad-free digital spaces/services will continue to grow, as hopefully people become more and more aware of how precious and limited their attention really is.
Given your position and the responsibility that comes with it; I sure hope you updated your mental model in another way than simply "they are acting irrational"... It's not clear from your comment that you did, but it sounded a bit like it.
> I suspect that the next big AI breakthrough will result at least in part from constraining LLM decisions with old AI approaches. Frank Coyle presented the idea of ontologies constraining LLM output about a month ago: https://www.youtube.com/watch?v=Sir59K8ZDPU
Interesting, thanks for sharing!
I plan to give it my own shot with vibing my own harness; I plan to use https://glean.software/ as the central database for everything code, then for documentation. After that, it could be used for bookkeeping for reasoning in a style similar to what you describe; after all, Angle, the query language of Glean, is a https://en.wikipedia.org/wiki/Datalog with some extras (I don't know which yet), so stuff like onthologies could be queried quite well, I would guess.
If you mean letting an LLM generate code for a Prolog planner from an English problem description, that has been done successfully already with last year's models eg [1].
Appreciate the link! That is not what I had in mind though; for them, the Prolog Code is the goal, for me, Prolog is an interface language within a system I build. That code could be generated on the fly, but I lean more towards fixed templates that can be instantiated by input some variables by the LLM that is doing agentic code, rather than having that LLM reproducing that query code completely on the flow.
The reasoning behind it is that I want to put as much logic as possible into the harness where it is deterministic, controled and fast - My goal is to make agentic coding usable for my purposes even with small local LLM models. Those might botcher generating Prolog queries sometimes, but they might might be able to used my API/interface/whatever reliably (and then the harness does everything else under the hood, including instantiating the Prolog templates and running the queries)
Hmm, I would say even older than that (which, of course, is in no way intended to be a value statement of any kind, I like the website and the project, cool idea! :D).
If you squint a little, the linked project is basically a https://en.wikipedia.org/wiki/A*_search_algorithm with optimized implementation, heuristics and so on. I also think that A* was associated with AI due to its use in path finding in early robotics - But I am not sure!
> This is extremely unconvincing. Manual review of 10,000 lines of AI generated code is a terrifying prospect.
With https://en.wikipedia.org/wiki/Lean_(proof_assistant) (and other proof assistants), you need to review only the lines that correspond to the theorem that you want to prove and their types (I am not very experienced when it comes to lean, but I would assume that comes down to a few hundred lines of code, at most). The rest is left to typechecking (which, I would expect many in the field to agree, is at as reliable than your average peer review process in professional mathematics, and likely much more). That's the reason why Lean4 is making such a fuzz now.
That itself is not trivial too, but way easier than reviewing every function and definition used to prove that the theorems have indeed the types they claim.
If one accepts the proof of the https://en.wikipedia.org/wiki/Four_color_theorem, then there should not be new reservations these proofs; except from the maybe new additional failure scenario that the authors (still correctly!) proved theorems that don't state what they think they stated.
To sum it up: There is IMO no domain more suited for using LLMs than mathematical proofs that can be formalized using Lean4. The fact the hype-circle started earlier in software than in maths is due to the difference in monetary incentives I would assume. (Or another, rather radical and not really serious phrasing: "When it comes to Lean4 proofs that typechecks, there is no AI slop" - the theorem being proven might be uninteresting, but the proof itself is very very very very likely to be correct)
> “byte-for-byte equal”
The term itself or its association with LLMs? I would get the latter, if its the former: It's an desirable property to have, I always like seeing people going that far (assuming obviously that they indeed did so, and in the places where it matters!)
> you need to review only the lines that correspond to the theorem that you want to prove and their types
This is (unfortunately) not actually the case—just a few weeks ago, someone "proved" the Collatz conjecture via a Lean proof 1) whose theorem statement was correct, 2) typechecked, and 3) was even verified by external tools with their own implementations of the kernel.[0]
The problem was (AFAIK) that the Lean kernel has a lot of fancy features that aren't yet perfectly understood from a type-theoretic perspective (I don't think Lean is unique in this regard; pretty sure Rocq and Agda are in a similar situation). And so when the kernel implements some feature whose soundness isn't guaranteed, the independent verification tools (or at least some of them) follow suit, and now any issues in the former affect the latter as well.
Huh. Thanks for bringing that into my attention, and the link!
Yes, in hindsight it makes sense that a eager LLM would exploit bugs in the kernel itself. I have not read the details yet (but want to!) and assume that it is rather related to the layers directly before or after the type-theory core; i.e. that the AI managed to get a correct typecheck by sidestepping a check during in one of the translation steps somehow, by manipulating the kernels result or by going totally hacker mode and swap implementation/overwrite memory; that is just speculation on my part though, I mainly base if on what I know about Haskell and the https://en.wikipedia.org/wiki/Calculus_of_constructions in general that have not that many constructs, and I would assume the variants used in Rocq and Lean are proven sound.
For experimental features where soundness is not proven, all bets are off IMO, but of course, that does not stop bad actors from engagement knowlingly abusing unsoundness for their own gain. Fortunately, that kind of manipulation is easy to verify when one has access to the codebase; I am would assume that something like lints for experimental features exist, or better even, something like a "sound mode". That does not help against hacking the kernel machinery though.
Concluding notes:
- In this particular instance, the Collatz conjecture was appearently chosen on purpose to demonstrate the Kernel bug, from the same thread: "Not that this changes any of the above, but I am informed that the person posting the proof was actually aware that this was a Lean kernel soundness bug, and it was not intended to be taken seriously as a solution to Collatz's problem."
- IF the code is made public (and it would be highly suspicious if parts of a proof where hidden), then I would assume this kind of hack is 1) easier to spot that other kinds of hack, since the asset-under-attack is really small and 2) there is not so much incentive to use much time/ingenuity/tokens on finding those hacks (the more are found and fixed, the better of course) 3) and they should be easy to defend against, I would assume; my first thought would be the flag I mentioned above that simply forbids all non-sound features, at the cost of limiting expressive power.
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Edit: I checked, and I think it alleviates my worries in the sense that the bug(s) where not in the type theory or its implementation, but rather the machinery around it. Further context below:
The following is from that thread or links from it:
- "Fixes two things: (1) more strict/nuanced handling for structure/proj interactions, and (2) adds methods for enforcing that generated auxiliary data for inductives, constructors, and recursors are more strictly checked against the assertions in the export file." | That is the fix to the non-Lean-kernel that was mentioned. To me, (2) looks firmly like what I meant with "supporting machinery", regarding (1), I do not know enough to have an opinion about it (i.e. how on what layer those interactions happen(ed)) | https://github.com/ammkrn/nanoda_lib/pull/22
- "For example, pipeline wedges (execute this instruction and the core freezes and never executes another instruction) would not be found by these techniques..." | Power and limits of Lean | https://infosec.exchange/@david_chisnall/117003914014196496
- "@mario @shelldozer it very well may be the most formally correct piece of software we've ever produced, but keep in mind it still has to run on a physical computer it's sharing with less-verified software and is also vulnerable to things like Rowhammer-class ram corruption attacks if one wants to intentionally manipulate it. There's no final escape hatch beyond which a computer can be absolutely guaranteed to always compute the correct answer, especially when someone has a vested interest in getting it to output the wrong answer." | Computer-checked proofs run on computers, which brings its own attack vectors, independent of how well the kernel is written | https://infosec.exchange/@0xabad1dea/117002712346315184
personal, cautios takeaway after reading the details: If you use Lean4 to write proofs, or read that a reputable group of mathematicians publizised a Lean4 proof, you are still highly unlikely to be fooled by a bug, and if Fable 5 decides to exploit a 0day in the core Lean4 machinery, that should still be able to be caught quickly.
Which I don't see a reason for Anthropic and "Open"AI not to, given their not so stellar track record with IP of individuals/entities-that-are-not-rich-enough ;)
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