It is an old saw at this point, but what an LLM does still cannot be divided into hallucination and non-hallucination. This is literally an anthropomorphism trap.
Layers and layers of application-specific verification can reduce the risks inherent to LLMs, to a really remarkable degree, but nothing about what these tools are suggests that this problem will go away; it will just bubble up again somewhere else.
An old saw unless something that's widely accepted, but sadly it seems that many people don't recognize this, even many people working in the field.
For all that I saw over the last few hundred hours with AI on software engineering, hallucinations are no longer a problem at all.
Not once have I seen a task fail due to what would have been a "hallucination". If they still occur, they can apparently be detected and corrected automatically, or are subtle enough to escape notice with presumably no significant impact on the results.
Why would this not also be the case for mathematics?
"Test suite passed" when it actually errored? Obvious hallucination, unless it ran a command that returned the wrong error code.
But is running a malformed command that does not achieve the expected effect itself a hallucination?
I'd say a hallucination (very misleading word) or confabulation or "making shit up" happens when an LLM uses factual / evidential language purely based on local statistical expectations of the text, instead of it drawing from actual evidence in its context pointing to it.
This is murkier in the case of general knowledge questions, like when and where was some famous person born. It may then be a spectrum from fully making something up based on how the name sounds, all the way to confidently retrieving it from its weights correctly. In between, we can get hallucinations. But newer models are taught to use Web Search when unsure, and it works pretty well, though not perfectly. I don't see any fundamental limit here. It's just not perfect. Trying to solve "the hallucination problem" is basically like saying "our dog vs. cat classifier is pretty good already with its 99% accuracy, now all we need to do is the tiny little task of eliminating the 1% error, and we will be golden". Like, no shit, there is some error yes. People are working to reduce it. It will never be absolutely 100%. It's not an insight to say we should remove hallucinations.
To fold that back in (and ham-it-up a bit) how about:
> There is nothing qualitatively different when Pet Classifier <ironic-quote>maliciously misreports</ironic-quote> your cat as a dog, compared to when it works <ironic-quote>honestly</ironic-quote>.
I agree that hallucination isn't some kind of "different" operation than "normal". It's not like when a train derails and you can point to it. It just operates as normal and sometimes that yields correct factual outputs, sometimes not. You don't have to metaphysically ascribe any kind of intent to it.
I'm not sure how people conceptualize these things who weren't doing classical machine learning before all this. To me, "hallucination" is shorthand, and we know it's not like humans on drugs or something. It was used in the literature also for any kind of generative imputing of missing information from a learned prior. For example in image inpainting a GAN "hallucinates" the missing part of the image. This terminology was already used in the 2010s and probably earlier. Or in image colorization of grayscale photos, the model "hallucinates" the color information.
Then the word escaped into the mainstream and people have weird connotations about it.
One of my bugbears is when people abuse the term "Ponzi Scheme" to refer to literally anything the think is unsustainable. (As opposed to something that, at a minimum, requires someone telling factual-lies about assets.) Kind of like if folks started calling every kind of software error a "Buffer Overflow."
This is quite a specific sense, and isn't equal to just any kind of "unsustainable" system. Things can be unsustainable for reasons other than being a pyramid scheme relying on rapidly growing numbers of people being brought in by the ones already in the system.
1. Actual Fraud
* A Ponzi Scheme requires lying to people about what assets currently exist, or how much of the asset is reserved as "theirs".
* A Pyramid Scheme just asks people to collectively speculate (without guarantee) that tangible profits will arrive.
2. Layers
* A Ponzi Scheme can be one criminal and an indefinite number of unwitting victims. [0]
* A Pyramid scheme (like the structure) requires *growing interior layers* of participants that blur the line between victimizer and victim.
To an arbitrary degree.
Just like all of science. Reduce the error to the desired margin.
Hold up, that's an even bigger assumption in the opposite direction, and I don't see anything to support it.
At least in terms LLMs getting all the "AI" hype these days, there is no structural/mathematical reason to believe they won't continue to have the same problem they've always had of generating plausible text over rational text, and I don't think anybody even has a clear idea how it could eventually be accomplished.
I've seen "then the magic singularity occurs and somehow it solves the problem for itself", but I would classify that more as mysticism than engineering.
Two years ago, hallucinating that the code worked or that a task was accomplished was a common occurrence.
We have seen that now agent swarms across thousands of agents can coordinate to achieve a result.
Clearly hallucinations are no longer the problem they once were, since now we can get working results for long horizon tasks that require massive compute.
Consequently it would seem unwise to assume that current limitations will remain as they are and prevent LLMs from coming up with solutions that they can explain to humans.
It happens in more subtle ways, but it still happens often enough for me to notice. For example I have had hallucinated checksums show up in lock files as recently as yesterday using a SOTA model.
This is not surprising, since the whole basis of LLM training is to produce output that humans will accept _as a proxy for actual training goals_. In a sense, the training process of an LLM “wants” to produce output that is statistically plausible much more than it “wants” to produce correct output. It’s always going to be a struggle to drive that system towards other goals (and we see this bourne out in practice by the amount of effort that is required to be spent on RL).
I think there will be some threshold of correctness (something like 99.999% of the time) that if the model surpasses it, I can stop needing to check it, but I think we’re still at 99% or something which sounds good, but when you are producing a ton of output you hit that 1% frequently.
> Consequently it would seem unwise to assume that current limitations will remain as they are and prevent LLMs from coming up with solutions that they can explain to humans.
I 100% agree with this. In fact explaining things to humans is something LLMs are particularly well suited for.
This is just a fact. I'm sorry if it messes with your narrative.
I work on a device that gets firmware updates over USB-DFU. If a DFU fails, its bootloader restarts, re-enumerates USB, and waits for a new DFU to begin. AI decided there's a risk of bricking the device if an update fails. There's no brick risk, a retry fixes it.
The same device uses CAN bus, with the common bosch_mcan peripheral. Claude decided that calling can_mcan_stop followed by can_mcan_start somehow left the transmit buffers intact, and tried to implement a fix which manually cleared the buffers. They're cleared automatically by the hardware when can_mcan_stop is called, and that's documented in the comments of can_mcan_stop.
Both cases could have resulted in unnecessary changes getting deployed which wouldn't have fixed the actual issues. Since it's embedded that could mean significant delays to getting fixes to customers.
And yeah I get hallucinations all the time still. Maybe it's because I'm working on harder/more niche problems (like a compiler with an unusual type system), but it happens quite a lot. I don't record all of them.
Although the most common one you can find is them misattributing the source of changes from themselves and also other agents (Fable, Opus 5.5, deepseek, whatever). They'll say "your changes" or "you changed" or "your ruling." I didn't decide anything and it's in their own chat log, and yet...
Where does this "will probably not hold in the very near future" come from? People correctly warn about extrapolating current things onto the future, but then just throw some vague "probabilities" without providing any argument why their "probably" is somehow more grounded than others.
Maybe if we start with giving simple AI generated analysis of those clumsy humans with their measly 2700 elo moves
With Waymo and Tesla increasingly doing what they said they would do, and a small number of early adopters happily paying money for their services, that do work.
So what's the critique? That the timelines are not correct? Sure. And how about the timeline of the people who said "research level math, never in my lifetime" and the people inside the ai companies who are apparently increasingly spooked by how quick the progress is? How about the various levels of code/programming jobs that AI was supposedly never going to be able to do, but, in reality, now just does?
We are engaging in some very one-sided discrediting, and I am not sure, why.
https://www.wsj.com/articles/self-driving-cars-dont-do-snow-...
More telling: Waymo just rolled out in Denver (1 month ago or so), apparently fairly confident they got this handled given the upcoming winter.
Progress on the obvious stuff keeps happening (which kind of brings me my to the first comment here).
But also: It does not have to do "snow" to be useful! A lot of cars/people don't drive when it snows heavily, and that's something we have always been okay with (at a societal level, YMMV of course). If was only useful 95% of the year that's still great. A lot of technology works like that.
You just don’t notice that because evolution has given you 99% of what is needed to drive a car before you were even born.
Maybe the really hard thing isn't abstract cognitive capability, but perception+movement.
[1] https://dev.to/natcher/researchers-develop-method-to-train-l...
There is literally not a single shred of evidence to indicate either of your supposed eventualities. The core technology of an LLM is sampling from a distribution so there is literally no way to make it deterministically robust (only probabilistically).
An LLM mostly deterministically (except parallel processing nondeterminism that can be mitigated) produces a probability distribution that can be sampled deterministically: just take the highest probability token or use beam search.
What has been demonstrated is a process that outputs lean proofs based on those probabilities. This happened after decades markov chain producing garbled texts and very shortly after gpt2 producing stories about unicorns.
You might have a point if the goal was to have LLMs that spit out a correct proof without chain of thought or tool use. LLMs + agent harnesses are more than capable of self verification and course correction.
This means when writing documentation, tutorials or commit messages, their output is often a garbled jumble. Assuming shared context, using invented terminology without explaining, leaking conversational states due to improper epistemic boundaries and failing to model the reader. This all usually leads to their freely generated explanations being terrible. Getting good explanations requires chaining questions that force them to line things up properly, which is not easy the less you know. These failures as something LLMs naturally struggle with make sense, given the nature of attention and RL with weak signals from human data.
Math is not merely a collection of proofs, it's a way of understanding. A proof presented in a manner that cannot be incorporated remains useless. It does not make it's way to physics like Riemannian geometry and matrix math did. This is no less true when done by humans too.
Your hallucination conclusion, checking if a proof is one, is exactly the counterproductive cost.
Most of us cannot verify that the claims in the OpenAI lore dump are in fact all correct. It will take tons of work from experts to do this. It took subject expert mathematicians to identify the discrepancy and disconnect in the Navier Stokes proofs, for example. LLMs will struggle to make use of their own proofs or turn them into knowledge that accumulates over time.
The act of proving is often more valuable than the proof itself. Human constraints and limitations force us to invent tools and abstractions that a 100,000 x 1M context swarm can bypass. The tradeoff from that AI swarm advantage is work that doesn't usually lend itself to being built upon. It's like doing all the side quests and reading all the books of an RPG versus min maxing a straight path with a guide. We might try to identify new abstractions, but the fact that we don't get access to CoT and that much of it will be illegible means mining LLM traces for what human mathematicians produce naturally will be a tedious chore.
This is a feature, and a huge step forward.
If you expect AI to do serious work, you can’t have it guessing what you “really meant”. Every sufficiently advanced task depends on very subtle details in the problem statement, and the correct default behavior for advanced AIs is to solve the task exactly as stated, unless a system prompt or other constraint tells it to do otherwise.
Here's a longer article which goes into a bit more of the details: https://terrytao.wordpress.com/2026/10/05/the-future-of-math...
You’re making the following assumptions:
1. the exercise of struggling to find proofs was not productive, but this is precisely how new techniques in math were produced. Brute forcing solutions doesn’t lend itself to the creation of much new mathematics (except maybe the exercise of developing verifiable proofs)
2. the point of doing mathematics is to be “productive” in the first place. This is silly. Many people get into mathematics because of the beauty of understanding, for example.
Are they independently wealthy? Or do they have a deal with their local supermarket that they can take food for free?
If your point is that capitalism fucks up the incentive structure and makes it all about maximizing productivity then I wholeheartedly agree with you.
To slightly deflect your question, what do you think will arise naturally?
I’d put my money on feudalism if SOTA model providers can stay far ahead of open weights models. However if that proves infeasible and everyone ends up having access to local AGI does that serve to level the playing field? What resources are still going to be scarce in the coming decades?
This is a really interesting point. This is almost analogous to a metric in software engineering like lines of code which is a proxy for something we actually want to measure.
In a way, the SOTA model providers are cargo culting what appears to be what mathematics is.
One is that AI will continue hallucinating in a manner that is not easy to verify, second is that AI will not be enhanced to produced more simplified amd robust outputs, and third that a human will be required to do that. What humans in the loop are doing now is verify the process, propose shortcuts and add legitimacy, through the verification process, if that ends up being succesful its highly likely a lot less mathematicians will be required in the future.
The conclusion that this is not productive focuses on the mathematicians, but it is very productive in terms of hundreds of proofs being produced that had previously consumed uncountable hours of the brightest minds. Unless it ends up being the greatest hallucination ever ofcourse