>LLMs are neither smart nor stupid.

by that reasoning then neither are there smart or stupid designs, questions, answers, or any of the millions of things that were described as smart or stupid, that did not possess any brain to actually be smart or stupid long before LLMs showed up.

The analogical process implied in many common English usages means that describing an LLM as smart or stupid is perfectly reasonable.

By this logic a human is only $130-$160 worth of Oxygen, Carbon, Nitrogen and some trace elements. Perhaps structure sometimes makes things that are more valuable than their inputs?

That said, this is also inaccurate at a technical level.LLM's are very capable of doing math and they ARE calculating internally. Most of what they do is calculation, not storage. It's just not done in a way that it's trivial to explain here.

It's described in some detail below, though it's a bit dense.

https://www.lesswrong.com/posts/E7z89FKLsHk5DkmDL/language-m...

During conversation, we are statistical token generators whose results are dependent upon our training set. Seriously, write that definition out rigorously. It encompasses virtually everything. It is totally meaningless. So to say "nothing more" is effectively also a tautology.

This argument was asinine in 2024. It is insane to be saying these things in 2026. Where have you been? What have you been looking at? How many articles explaining why the "statistical parrot" analogy fails have you missed? How much mental gymnastics do you have to do to explain how a modern LLM can solve novel math problems that fall really far outside of its training set?

It absolutely understands how to do math, by whatever reasonable definition you want to provide to the word "understand". For example, the identification of the addition expression is understanding, and no, it does not do tool calling for basic arithmetic any more than humans might. Isolation of individual concepts in intermediate layers can already be demonstrated, or else transfer learning wouldn't possibly work. Nobody is saying that LLMs are humans. But we need labels for some of the things that we observe and dismissing them because "statistical" is laughable.

Look at the proof of this: https://github.com/anthropics/formal-math/blob/795efb86f1917... . Forget the Lean, look at the underlying argument construction. At the very least, this is continuing from an argument that was hinted at in the literature in 2024, but these proceedings were difficult enough that humans were not able to do them within two years. Do you attribute this to the harness alone? If so, that's a pretty sophisticated bit of engineering, I would say! Probabilities are far too small to argue infinite monkey theorem.

If there was even a shred of a reasonable argument that LLMs were incapable of concept extraction and manipulation, I and my colleagues would be all over it. We would relish in it. It would bring us comfort. It is unbelievable that people think they can spew whatever basic garbage they think of as a gotcha, and think that minds all over the world haven't already considered that. This is like climate denial at this point.

I'm not anthropomorphizing anything, I literally said that the training data for the formulas and equations is baked into it. It only "knows" things because a crawler and scraper acquired the information from an existing written source. In just about the same way that information is baked into a printed encyclopedia.

Reductionistic fallacy, among other errors. "understanding" is best defined operationally. (Note that teachers and educational institutions test understanding operationally.)

> They are statistical token generators whose results are dependent upon their training data set and involve a degree of randomness.

You haven't demonstrated why this matters.

> Nothing more.

Are you contending that complex systems cannot be more than the sum of their parts?

A market is nothing more than offers and counter offers.

A ant colony is nothing more than scent trails.

All life on earth is nothing more than reproduction with variation.

> This still falls under the purvey of statistical token generation.

Stating the mechanism does nothing to provide insight into capability. For instance: a nuclear power plant boils water by using fuel rods for heat. What does that tell us about the capability of nuclear power?

> This is not "doing" or "understanding" math.

Asserting something purely by stating it does not prove anything but that you intuitively believe it to be true.

Right, it was really more as a test of how much was contained in the training data set. For my purposes Vincenty is quite accurate enough. This isn't for millimeter level precision land surveying or measurements, but for distance in meters between microwave or millimeter wave band radio sites, point to point links. Even a distance difference of 4 meters plus or minus on a 12 km, 11 GHz band link is going to have no appreciable difference on link budget/reliability calculations, it can be that crude. But not so crude that I just want to throw Haversine at it when Vincenty exists and is not computationally expensive.

As this was for a test of "what happens if..." I also watched to see if it did any web searches or external data retrieval to build the test script, and it didn't.

I intentionally didn't give the LLM a direct copy of the software or a link to it, to see what it would do. In my case it was a randomly chosen example I could come up with in 10 seconds of imagination to see "hey what if I ask it to do this...". It also implemented a perfectly usable parabolic millimeter wave antenna gain efficiency calculator based on variable surface smoothness parameters, which is a lot more basic math.

I would really love to hear your reasoning on this!

of course they can do math, how is this even controversial in 2026?

GPT-6 can do math directly just fine. Fable apparently can't because they broke its self-estimate of thinking effort.

https://x.com/maksym_andr/status/2100364212207837560

I don't know for you, but it would take me more than 30s to find and translate the open source code implementing the formulae/algo into small usable program. The more hesoteric the optimisation in the original code, the more time I need.

So maybe it is more of a smart completion engine than a SQL answer.

> they found bits of code that are associated with "math" and the supplied arguments

How is this different from a human using an algorithm they have memorized, or reading it from a reference site written by a human and then writing the same formula into a custom one off piece of python code?

I could have gone and spent a couple of days teaching myself the math behind Karney and reading its reference implementation (very possibly just copy/pasting big chunks of it to save time) and writing a wrapper around it. It would have produced the same result.

Maybe it’s just a different and in some ways better way of doing mathematics? Maybe how we think and process mathematics of physics is just but one way to do it? I’m not suggesting an LLM will prove 2+2=6 because of course that’s nonsense but maybe it can invent a new calculus?

> The LLM is not suited to giving deterministic answers to math problems.

Less so with formal mathematics proofs maybe but I think in general humans don’t provide deterministic answers to math problems or questions either. Humans get it wrong all the time and when you ask a human to solve a problem they may solve it in a different way than before.

reasoning models can trivially do math (open up astra and ask it some undergraduate problems), but eventually break down (similar to how humans start to lose track if asked to do math without any assistance)

It would be more accurate to say they can do math instantaneously without even thinking, at a level far beyond what humans can do. (I assume you're talking about doing arithmetic.)

  TLDR: Astra has 8.6x better odds of doing a reasoning task without CoT than the
  next best model (Fable 5.1), and can do 7.2 serial arithmetic steps in a forward
  pass vs 4.1 for the next best model (Gemini 3.8 Flash/Fable 5.1)

Reasoning models can do math on their own without external tools.

No it isn't. Even without any harness at all, modern LLMs are better at maths than the majority of undergraduate students in mathematics. Seriously, we need to face facts, not just comforting ourselves with what they were like a year ago.

LLMs can in fact do arithmetic, just not reliably owing to how numbers are represented probabilistically: https://arxiv.org/abs/2410.21272

I just asked ChatGPT to multiply two 4-digit numbers, and two 7-digit numbers without external help. It got both right. I'm sure it wouldn't have a 100% success rate, but saying it can't do arithmetic is just false.

You're talking about doing arithmetic; GP was obviously pointing out that "do math" can refer to other things.

LLMs are bizarrely good at non-tool-assisted math these days. They can multiply multiple digit numbers without reasoning! I can’t do that. I’d love to understand better how the LLMs do this.