Different language models learn similar number representations
arxiv.org109 points by Anon84 a day ago
109 points by Anon84 a day ago
Different models, similar number representations. Different models for different languages, similar concept representations. They have to learn all of this from human text input, so they're not divining it themselves. It all makes a strong case for universal grammar, IMO.
> It all makes a strong case for universal grammar, IMO.
What about through the lens of the Norvig-Chomsky debate?
This would be a sort of convergence? They were both right in part (Chomsky that there was structure there, Norvig that it could be sussed out using brute force statistics). As is often the case, when two smart people who have thought a lot about something complicated disagree, the truth comes out when their unstated assumptions are finally exposed to the light.
In this case, Chomsky's LAD almost certainly relies on Baldwin-effect structure to get around the paucity of stimuli, and the LLMs are just getting to "the same place" through sheer masses of data.
Surely the "universal grammar" is "every country adopting Western Arabic numerals, largely for commercial reasons, but also acknowledging that their indigenous systems kind of sucked in comparison." The fact that there are different languages truly means nothing, Arabic numerals spread much further than the Latin alphabet.
I really don't think this is evidence for "universal grammar" in any sense. It is evidence that we are all using the same very specific grammar for very specific cultural reasons.
Title is editorialized and needs to be fixed; the paper does not say what this title implies, nor is that the title of the paper.
HN automatically removes the word "How" from the beginning of titles. I suspect this title is one instance of that
Unfortunate if so, but I'm finding plenty of counterexamples in the past day alone: https://hn.algolia.com/?dateRange=last24h&page=0&prefix=true...
The exact phrase appears in the title. There is a title length limit. In this case, I don't think that it is wrong to pick the most interesting piece of that title that fits in the limit.
The eigenvalue distribution looks somewhat similar to Benford's Law - isn't that expected for a human-curated corpus?
I would expect that for any sampling of data that has a roughly similar distribution over many scales.
Which will be true of many human curated corpuses. But it will also be similar to, for natural data as well. Such as the lengths of random rivers, or the brightness of random stars.
The law was first discovered because logarithm books tended to wear out at the front first. That turned out to because most numbers had a small leading digit, and therefore the pages at the front were being looked up more often.
(Pardon the self promotion) Libraries like turnstyle are taking advantage of shared representation across models. Neurosymbolic programming : https://github.com/jdonaldson/turnstyle
What exactly is the Platonic Representation Hypothesis?
You just don't "learn reality" by getting good at representations. You can learn a data set. You can learn a statistical regularity in things such as human languages. You can analyze the concept spaces of LLM's and compare them numerically. I agree with that.
What the hell does "learning an objective shared reality" mean?
This reminds me of EY saying that a solomonoff inductor would learn all of physics in a few days of a 1920x1080 data stream. Either it's false (because it needs to do empirical testing itself), or it's true, but only if you presuppose the idea that it has a perfect model of all the interactions of the world and can decide between all theories a priori... so then why are we even asking if it's a "perfect learner"? It already has a model for all possible interactions already, there's nothing out of distribution. You might argue, "Well, which model is the correct one?" That's the wrong question already - empirical data is often about learning what you didn't know that you didn't know, not just learning about in-distribution unknowns.
I just get an ick because I associate people talking about this hypothesis as if "LLM's converge on shared objective reality => they are super smart and objective, unlike humans". LLM's can be smart. They can even be smarter than humans. It's also true that empiricism is king.
It's going to turn out that emergent states that are the same or similar in different learning systems fed roughly the same training data will be very common. Also predict it will explain much of what people today call "instinct" in animals (and the related behaviors in humans).
Evolution is an optimization process. So if platonic representation hypothesis holds well enough, there might be some convergence between ML neural networks and evolved circuits and biases in biological neural networks.
I'm partial to the "evolved low k-complexity priors are nature's own pre-training" hypothesis of where the sample efficiency in biological brains comes from.
The "platonic representation hypothesis" crowd can't stop winning.
Potentially useful for things like innate mathematical operation primitives. A major part of what makes it hard to imbue LLMs with better circuits is that we don't know how to connect them to the model internally, in a way that the model can learn to leverage.
Having an "in" on broadly compatible representations might make things like this easier to pull off.
You seem to be going off the title which is plainly incorrect and not what the paper says. The paper demonstrates HOW different models can learn similar representations due to "data, architecture, optimizer, and tokenizer".
"How Different Language Models Learn Similar Number Representations" (actual title) is distinctly different from "Different Language Models Learn Similar Number Representations" - the latter implying some immutable law of the universe.
> latter implying some immutable law of the universe
I think the implications is slightly weaker -- it implies some immutable law of training datasets?
I don't understand your argument
"How X happens" still implies that X happens, just adds additional explanation on top
"using periodic features with dominant periods at T=2, 5, 10" seems inconsistent with "platonic representation" and more consistent with "specific patterns noticed in commonly-used human symbolic representations of numbers."
Edit: to be clear I think these patterns are real and meaningful, but only loosely connected to a platonic representation of the number concept.
Is it an actual counterargument?
The "platonic representation" argument is "different models converge on similar representations because they are exposed to the same reality", and "how humans represent things" is a significant part of reality they're exposed to.
You should see my reply to convolvatron below.
I don't think this is a correct formulation of the platonic representation argument:
different models converge on similar representations because they are exposed to the same reality
"Not actually capable of understanding" is worthless unfalsifiable garbage, in my eyes. Philosophy at its absolute worst rather than science.
Trying to drag an operational definition of "actual understanding" out of anyone doing this song and dance might as well be pulling teeth. People were trying to make the case for decades, and there's still no ActualUnderstandingBench to actually measure things with.
No, it is partially falsifiable. LLMs clearly don't understand the concept of quantity. They fail at tests designed to assess number understanding in dogs and pigeons; in fact they are quite likely to fail these tests, because they are wildly out of distribution.
We don't know how to demonstrate actual understanding, but we sure can demonstrate a lack of it. When it comes to abstract concepts like "three" or even "more," LLMs have a clear lack of understanding. Birds and mammals do not.
Which "tests", exactly? Do tell. Tests where LLMs don't beat a human baseline is genuinely hard to come by nowadays.
you're right, its just that 'platonic' is an argument that numbers exist in the universe as objects in and of themselves, completely independent of human reality. if we don't assume this, that numbers are a system that humans created (formalism), then sure, we can be happy that llms are picking common representations that map well into our subjective notions of what numbers are.
FWIW it's objectively false that numbers are a system humans created. That's almost certainly true for symbolic numbers and therefore large numbers ( > 20). But pretty much every bird and mammal is capable of quantitative reasoning; a classic experiment is training a rat to press a lever X times when it hears X tones, or training a pigeon to always pick the pile with fewer rocks even if the rocks are much larger (i.e. ruling out the possibility of simpler geometric heuristics). Even bees seem to understand counting: an experiment set up 5 identical human-created (clearly artificial) landmarks pointing to a big vat of yummy sugar water. When the experimenters moved the landmarks closer together, the bees undershot the vat, and likewise overshot when the landmarks were moved further apart.
And of course similar findings have been reproduced etc etc. The important thing to note is how strange and artificial these experiments must seem for the animals involved - maybe not the bees - so e.g. it seems unlikely that a rat evolved to push a lever X times, it is much more plausible that in some sense the rat figured it out. At least in birds and mammals there seems to be a very specific center of the brain responsible for coordinating quantitative sensory information with quantitative motor output, handling the 1-1 mapping fundamental to counting. More broadly, it seems quite plausible that animals which have to raise an indeterminate number of live young would need a robust sense of small-number quantitative reasoning.
It is an interesting question as to whether this is some cognitive trick that evolved 200m years ago and humans are just utterly beholden to it. But I think it requires jumping through less hoops to conclude that the human theory of numbers is pointing to a real law of the universe. It's a consequence of conservation of mass/energy: if you have 5 apples and 5 oranges, you can match each apple to a unique orange and vice versa. If you're not able to do that, someone destroyed an apple or added an orange, etc. It is this naive intuitive sense of numbers that we think of as the "platonic concept" and we share it with animals. It seems to be inconsistent and flaky in SOTA reasoning LLMs. I don't think it's true that LLMs have stumbled into a meaningful platonic representation of numbers. Like an artificial neural network, they've just found a bunch of suggestive and interesting correlations. This research shows the correlations are real! But let's not overinflate them.
Regardless of whether the convergence is superficial or not, I am interested especially in what this could mean for future compression of weights. Quantization of models is currently very dumb (per my limited understanding). Could exploitable patterns make it smarter?
Same with images maybe?
Saw similar study comparing brain scans of person looking at image, to neural network capturing an image. And were very 'similar'. Similar enough to make you go 'hmmmm, those look a lot a like, could a Neural Net have a subjective experience?'
"Subjective experience" is "subjective" enough to be basically a useless term for any practical purpose. Can't measure it really, so we're stuck doing philosophy rather than science. And that's an awful place to be in.
That particular landmine aside, there are some works showing that neural networks and human brain might converge to vaguely compatible representations. Visual cortex is a common culprit, partially explained by ANN heritage perhaps - a lot of early ANN work was trying to emulate what was gleaned from the visual cortex. But it doesn't stop there. CNNs with their strong locality bias are cortex-alike, but pure ViTs also converge to similar representations to CNNs. There are also similarities found between audio transformers and auditory cortex, and a lot more findings like it.
We don't know how deep the representational similarity between ANNs and BNNs runs, but we see glimpses of it every once in a while. The overlap is certainly not zero.
Platonic representation hypothesis might go very far, in practice.
As someone actively researching in the neuroscience field these ideas are increasingly questionable. They do do a decent job of job of predicting neural data depending on your definition and if you compare them to hand built sets of features but we’re actually not even sure that will stay true. Especially in vision we already know that as models have scaled up they actually diverge more from humans and use quite different strategies. If you want them to act like humans or better reflect neural data you have to actively shape the training process to make that happen. There’s less we know about the language side of things currently though as that part of the field hasn’t yet really figured out exactly what they’re looking at yet because we generally know less about language in the brain vs vision. I think most vision scientists are on board with the idea that these things have really been diverging and have to be coerced to be useful. Language it’s more up in the air but there’s a growing wave of papers lately that seem to call the human LLM alignment idea into question. Personally I think the platonic representation idea is just a function of the convergence of training methods, data, and architectures all of these different labs are using. If you look at biological brains across species and even individuals within a species you see an incredible variety of strategies and representations that it seems ridiculous to me that anyone would suggest that there’s some base way to represent reality that is shared across everyone and every species. Here’s some articles that may be of interest if you’re curious:
[1] https://arxiv.org/pdf/2211.04533 [2] https://www.nature.com/articles/s41586-025-09631-6 [3] https://www.biorxiv.org/content/10.1101/2025.03.09.642245v1
Guess I was extrapolating from words and images.
If you could brain scan a human, and identify a shape of the network that corresponds to an emotion, and then could identify that in the ANN, could we say the ANN is experiencing an emotion.
I think its loosely referred to as "neural correlate".
I'm assuming what you are talking about with Convergence, would be these "neural correlate". And no reason we couldn't move beyond images to 'feelings'.
Curious if this similarity comes more from the training data or the model architecture itself. Did they look into that?
They describe that both are important, and researched in the paper, within the opening paragraph.
> Language models trained on natural text learn to represent numbers using periodic features with dominant periods at T=2,5,10.
This proves a decimal system is correct. Base twelve numeral systems are clearly unnatural and inefficient.
This is just a result of base 10 being dominant in our natural languages. I assume if we really used base 12, things would be different.
What would using base 12 in our natural language mean? Number names needed to be based on 12, not 10. Thirteen, twenty-seven, our numbers have base 10 embedded in their naming.
Historically, quite a few languages were (or are) vigesimal. Perhaps decimal is also unnatural.
It's been at least eight score since vigesimal usage was common in English.
It's still used for numbers between 70-99 in French, which is maddening when trying to copy down a phone number as a non-native speaker.
Yeah, I think Swiss French had more? (It's been about four decades since I took French, and high school classes are not very effective.)
French speaker here. (My native language is English but I learned French in France years ago, and can still speak it with near-native fluency). You're correct. French French (that is, French spoken in France, it's slightly confusing that the language's name and the country's adjective are the same) goes "cinquante, soixante, soixante-dix, quatre-vingts, quatre-vingt-dix, cent". In English, that would be "fifty, sixty, sixty-ten, fourscore, fourscore-ten, a hundred". But Swiss French goes "cinquante, soixante, septante, octante, nonante, cent", which would translate to English as "fifty, sixty, seventy, eighty, ninety, a hundred".