At this point it's either wrong or meaningless to call LLMs "next token predictors". Yes, pretraining involves predicting the next token. But after that you're in the world of RLHF, RLVR, and other post-training.
A model is _sort of_ designed to predict the next token in a string of text. But it's equally accurate to say it's designed to edit code until it passes a suite of unit tests.
Yes, they generate sequentially, so "next token predictor" isn't technically incorrect (which is why I chose the word "meaningless"). But it's pretty clear there's a mechanism inside them that allows them to "plan" what they generate. I might as well call you a next word predictor.
> A model is _sort of_ designed to predict the next token in a string of text. But it's equally accurate to say it's designed to edit code until it passes a suite of unit tests.
Training is not design. Design is the mechanism, namely a statistical method of learning distributions, and of executing those distributions computationally as an IO function.
The model is not designed to write unit tests. It is trained to do so, i.e. the distribution learned and reinforced is one such that it maps the I distribution to the O distribution that is what human beings typically do or want when writing unit tests. The design is the suite of algorithms used for this (which is ultimately global next token prediction)
I think you are conflating design (the mechanics) with their cultural or user level function.
You are basically saying "my bike isn't just rotating wheels on a frame because it takes me places". Hopefully you can see how both parts of that statement are true and why contrasting them as if the first clause is somehow false or inaccurate is incorrect.
Both these are true statements. They operate at different levels. People are constantly conflating these levels in the "is it a next token predictor" (the answer is yes) debate.
Yeah, fine. I was using the word design loosely. I'd also happily say something like "humans are designed to like salt and fat" and feel like that was a pretty reasonable statement, even though humans aren't designed either.
Its a stateless function that takes in x amount of words and calculates logits. It is in fact a next token predictor. Its literally nothing more than that.
Isn't most of our current understanding of how life on earth came to be (eg: primordial soup, hydrothermal vents, etc), natural evolution, all basically "just statistics"? They all require the correct "setup" (time, location, etc), but in a large enough universe, it's _statistically_ likely (mediocrity principle) and "just statistics" again, right?
Well yeah, that's tautologous because statistics is a mathematical abstraction. Everything, viewed a certain way, is "statistics". Just like everything, viewed a certain way, is "physics". And everything, viewed a certain way, is "narrative".
Mathematics models the world, it does not have substance. It is a useful way of looking and a useful way of manipulating, but many models are equally useful. Capital T Truth, beyond models is very difficult to get at. We can show that things are true for certain models well enough. Showing that something is true in all models (a universal truth) is far more difficult. You face incommensurability.
I see a lot of people in technology fall into this extremely basic logical fallacy called Reification constantly.
Do you believe in a soul? It’s fine if you do, but I don’t, and I don’t understand how people can reject the notion of soul and still say things like this.
I mean, when you really get down to it, either the particles and such operate on very specific rules, and the rules govern how all our brain and body work. Followed to the conclusion, the universe is computable and we can simulate life.
That sounds like a next token generator to me, which I think is more accurate. The word "predict" is doing a lot of heavy lifting in your formulation. What exactly is being predicted?
"next token predictor" has always referred to inference, not training. Training steps produce and bend weights, inference predicts tokens based on those weights.
The 'predictor' part referred to the fact that base models would effectively answer the question "what token is most likely next". Training away from that default behavior doesn't escape the technical meaning of those words- answering "what token is most likely next in a solution to Navier Stokes" is still predicting a next token- but at that point the label just means it outputs text serially.
> "next token predictor" has always referred to inference
Going back to first principles, wikipedia mentions [0] inference as:
> "Inferences are steps in logical reasoning, moving from premises to logical consequences. Inference is traditionally divided into deduction and induction"
Is that exactly what next token predictor means, or is it correlated?
My understanding is that the underlying part is a next token predictor and that's what is "actually" editing the code but the part that decides what to do (to achieve a successful unit test result for example) is a mix of all the things including post training.
I think what breaks people's minds is just how far one gets with probabilistic approaches.
Which is interesting, given that life itself is merely that. It's just that humans decided that that ain't good enough and invented something alien to nature - logic.
Yes, I didn’t mean that literally. Since “attention is all you need” it’s more like “predict the full output in full for this blob of input text”. That doesn’t change the “wtf is going on with the weight understanding bit” in what I wrote.
> it’s more like “predict the full output in full for this blob of input text"
Hm. I would actually go so far as to say "predict" is just the wrong word to use in 2026, as it smuggles in an out-of-date picture of what LLMs are that's from the pre RLVR era, and which is frequently used to suggest they don't model anything real in the world but rather the statistics of language, and thus have all sorts of limitations that they've now surpassed ("an LLM will never solve a major open mathematics problem, they're just predicting tokens"). These days I tend to use the more neutral phrase "generate the next token"; I'd only say "predict" in the technical context of (say) the pre-training part of the training run where the object is indeed to minimize the cross-entropy loss function by predicting next tokens in existing text. Generating a token with a modern LLM is better described as something like, "given a context and the weights from the totality of its training, including not just next token prediction with pre-training but also solving vast numbers of math and coding problems with RLVR, the LLM generates tokens one by one." (Consider that an LLM trained purely to optimize for next token prediction with pre-training is terrible at programming and math tasks, as we all remember from 2024. In that sense the amazing results we've seen in the past two years are due to the departure from pure "next token prediction" that RLVR made possible.)
Put another way: would it be useful to describe the AlphaZero chess engine as "just a next move predictor"? (It, too, was a large neural network trained with RL.) I would rather say: "it's playing chess by generating the next move given the state of the board." Which is what humans do too, though we do it with different hardware.
Anyhow. I'm not sure the weight understanding is all that strange. (It's impressive, but so are many thing LLMs do these days.) LLMs are very very good at coding and math problems, of which this is one, and in particular, their RLVR training presumably includes many tasks involving neural networks, doing tasks with PyTorch, etc. It's amazing and impressive, I agree, but I'd say less amazing than proving major open theorems in mathematics, which we already know they can do.
It's still prediction executed by a statistical model. Even RL models are just predicting what the training gauntlet taught them the trainers want to see, in terms of typical behavior.
This doesn't make it any less fascinating that by combining this basic "do as I say" training with massive amounts of all of humanity's data, you can solve major problems—that is still incredible. But that doesn't fundamentally change what is going on, mechanistically.
I don't understand these pseudo-religious desires to read into models something "more" than what we know. We built them. We don't go around making wild claims about how cars are actually more than just a mechanical system, how they aren't just taking us places—we do, but usually such points are cultural not about the mechanica capabilities of cars. People in the LLM space seem to be constantly conflating these two views. It's entirely accurate to say, on the one hand, next token predictors combined with RL, and massive troves of existing human work, can solve culturally interesting problems, and to hold, on the other hand, that mechanistically what is happening is statistical next token predictions over global linguistic space and more or less "monkey see monkey do".
I know how LLMs work. I understand the output projection and logits and sampling just fine. I don't think we should call what is happening prediction because it is a misleading term with a lot of baggage associated with claims that LLMs are just statistically modeling language and "guessing" tokens without any internal representation of meaning. Since your reply is very short I can't tell whether you mean it in that sense or not.
And yes, I understand "prediction" is a standard term in the field. I just think it is a bad one and we should stop using it.
unless you are doing something where you stay in latent space without having a random number generator pick a word, the next token predictor part still matters. without making the tokes concrete in between, that plan doesnt exist
if it was designed to edit code, id expect the first translation from latent space out to tokens to be the final code, not be assembling the code by random number generator.
its not really designed to edit code until the tests succeed, we try to manipulate the distribution of tokens to predict by prompt engineering with the test results.
It is not clear that there is a mechanism that allows them to plan anything. It is clear there is nothing of the sort.
You can demonstrate definitively that there nothing other than "next token prediction" by running the same inference engine/model/prompt/settings multiple times in a row on the same hardware at temp 0 with a seed set.
If you turn the temp up, then you might let the "mechanism" select pseudorandomly from a probability distribution of k-likely tokens with minimum p-cumulative probability. Even then, if the next token has a very high probability, deviation will be uncommon.
I just performed this experiment right now on a few dozen runs of a dense model at thousands of tokens of output at temp 0, and all were identical. Then I did it with a MOE model, which might get some benefit of picking different experts each time. Nope. Outputs were again identical.
> You can demonstrate definitively that there nothing other than "next token prediction" by running
Describing LLMs as next token predictors isn’t incorrect, it’s meaningless. It’s the equivalent of describing brains as “a bunch of atoms”.
The problem with these descriptions is that they say nothing about high-level capabilities. If you can predict the next token in a proof of the Riemann hypothesis, then you can prove the Riemann hypothesis.
In a similar vein, if the string of "next tokens" based on 300k input tokens is: "run this bash command and paste the results into the token stream", followed by another call that includes the output of that bash output giving "Based on the output from that bash command, now run head -n 30 /foo/bar/baz.txt and paste it here", finally followed by "Based on that, here's the problem with your systemd configuration: [...]"
The part that really bakes my noodle: `/foo/bar/baz.txt` can be a filename that no one has ever typed before, the error inside the file can be a typo that no one has ever typed before, and it will still correctly "predict the next token" to solve your problem.
I don't think that's true and I don't think your comparison is accurate.
In the case of the brain, it'd be more akin to the reduction of the brain to just the neuronal dynamics (obviously experience phenomenally is much more than that to us!)
But both views can be useful, depending on what you are trying to understand.
The problem with these arguments is that people pick and choose based on their persons preferences. People who are in love with LLMs will, on the LLM side, argue against reductionism "they are way more than next token predictors!" then they'll turn around and equally try to argue "the substrate doesn't matter! The brain is "just" neurons and digital activity so of course we can replicate it!" (this is also incorrect, btw, the brain is also an analog system, even in its electrical connections and not just in its chemistry)
The on the other side the anti LLM crew will do the opposite (launch into reductionism when it comes to LLMs, resist it when it comes to humans). I'm a human being and I like humans and I think people have adopted a strange view of LLMs, which are a tool and should be treated as such, so I tend to side with the humans. The stalwart defenders of LLM honor that would balk at any attempt to reduce them in stature to what they fundamentally are, mechanistically, are... an odd bunch. I'm confused why people are so passionate about it. If someone told me "well your IDE is just a fancy AST walker" yeah I think that is a dumb take but I wouldn't feel like I had to argue against such a take either saying "no! no! it's so much more!"
I think the point was meant to be more simple and broad:
A system is not just its little bits.
Where "system" and "little bits" can be just about anything. But people endlessly repeat this mistake, and it has been a go-to misunderstanding/misrepresentation for a lot of people on this forum.
Right, Banks' Culture seems fairly mainstream and because it doesn't need to make episodic adventures it often gets to actually do utopian stuff Trek wouldn't even if only as a backdrop to the exciting story. The repeated instances in which Culture children play happily, insulated from whatever terrifying stuff we're reading about is an example.
Or if Comic Books are in scope, obviously "Miracleman: The Golden Age" is a utopia. Explicitly so. It is about the ways people are still people in a utopia.
Less mainstream, Greg Egan's Amalgam is also clearly a utopia. The couple in "Riding the Crocodile" choose to die at the end of that story, more out of some sense that stories need endings than because of any outside force.
Probably and the model had access to all those results. The question is whether 500 person-years of effort would have been enough to get there from the same point that the model started. And also not any 500 person-years, not 100 mathematicians independently working on it in parallel for 5 years, but 500 tightly cooperating person-years, 20 mathematicians in the same room for 25 years.
Is that true? I'm actually asking because all the ones that I have seen mentioned from the arena were generated with Claude and I just checked a few random ones and they all have commits by Claude.
I don't know how much was created by LLMs but can you still call these independent implementations?
Yes? Very little about Claude is deterministic. If given the same sufficiently complicated problem, but via slightly different input prompts, it can generate entirely different code. It would be shocking if they all had a single shared bug.
If it's a complicated but narrow problem then regardless of the prompt its possible that Claude will always converge to the same solution type from its training data. So it will generate different version of the same code that have the different implementations of the same procedure that 'works' but is incorrect in some subtle way.
This is true of literally any proof of sufficient complexity, human-authored or otherwise.
It takes time to convey the core ideas. It takes time for other mathematicians to come up to speed on the new techniques and approaches or the novel combinations of existing ones.
The actual experts in this field have asked for the argument to be explained to them, and are now at the point where they believe this is plausible. Another person in this thread has produced a Lean proof and presumably is not trying to pull a fast one on us by exploiting a bug in the solver.
Not sure I agree - they dropped hundreds of papers, most of which were so badly written that they were incomprehensible. If it wasn’t for the lean certificates on 30% of them, the mathematical community would have told them to go and whistle.
I think the consensus here is that most of the papers are effectively correct. But it would take years of study just to understand most of the problem statement - so we have to wait for the mathematical community to digest the borderline-nonsense that OpenAI spewed out.
Just tell the (AI) authors to go back and write clear explanations for why they matter and what the specific claim is.
I gotta say I think mathematicians brought this roblem on themselves, most math papers are absolutely incomprehensible to anyone who is not an expert in the field. Mathematicians spend so much effort trying to make their argument to other mathematicians that they've gradually abandoned making their field or findings comprehensible to anyone else and left that to science journalists. In turn, those journalists churn out very formulaic stories written to a template: 50% the human story of the mathematicians social/academic wanderings in pursuit of their goal, 50% an explainer with some nice graphics on the subfield they operate in and a mention of what sort of problem their insight solves, perhaps with a few colorful metaphors thrown in.
The last mathematical paper I can think of (but I'm not an expert, I'm sure there are some others) that was both mathematically rigorous and accessible enough to be readable was Mitchell Feigenbaum's Quantitative Universality for a Class of
Nonlinear Transformations. This is accessible int he same way as a textbook - you have to work at it, but hte author is explaining his thought process and drawing the reader's attention toward things that matter or stand out as unusual.
It seems like, once the IPO is done, this flood of papers will dry right up: the inherent value of the papers themselves is unlikely to justify the cost to churn them out. It makes as much sense as OpenAI churning out hundreds of slopped programs no-one wants to run, or hundreds of novellas no-one wants to read: it's just showing they can.
These papers are just first steps. They are going to keep churning out papers, in part because the generated papers are going to get better. And better.
But up until now, the mathematics community has valued the "prove" part much more highly than the "deliver an insight" part. Mostly because with a little work they went hand in hand.
And going from zero proofs to one proof (even a sloppy one) is a big deal regardless of whether it was written by AI or a human.
Getting recognition is a fundamental human motivation to work on hard problems. Being first is a really important part of getting recognition, measuring by the last few hundred years.
You might as well say "the obsession with sex has always held humanity back". Maybe, but it's complicated...
Probably true for the dedicated problem solvers (of which Tao is one IMO). But I doubt there's ever been a better time to be a theory builder (more like Peter Scholze, or Grothendieck).
Some up with an idea and leave the system to check it 15 different ways, and see whether you can simplify an existing body of theory. It'd be like having an army of lightning-fast grad students.
I'd even say Scholze is not a great example here. Most of the work he's known for is progression towards the Langlands program - which is very much a problem to be solved, and one I would imagine he'd not be thrilled for an AI to one-shot. I agree that it is somewhat 'up the chain', in the same way that software engineering has not immediately disappeared now that performing coding tasks is largely automatable.
But I also take issue with 'never been a better time' - e.g. is this really the greatest time to be a software engineer? Everyone has AI psychosis and feels like they're a couple of breakthroughs away from being unemployable. The same is even more true in math - we've gone from failing IMO problem 6 last year, to solving NS. The rate of change is formidable, it feels like there may not be many places to hide in a few years.
This is coming next. There's nothing particularly special about theory building. Successful theory building is always oriented towards solving a problem, because otherwise even humans can easily spam out a bunch of nonsense. This is a real problem that the math community has experienced on multiple occasions pre-AI. I'd go further and say that theory building is irrelevant if it doesn't help solve problems people care about.
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.
A model is _sort of_ designed to predict the next token in a string of text. But it's equally accurate to say it's designed to edit code until it passes a suite of unit tests.
Yes, they generate sequentially, so "next token predictor" isn't technically incorrect (which is why I chose the word "meaningless"). But it's pretty clear there's a mechanism inside them that allows them to "plan" what they generate. I might as well call you a next word predictor.
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