Using this reasoning, would you argue that a new proof of a theorem adds no new information that was not present in the axioms, rules of inference and so on?

If so, I'm not sure it's a useful framing.

For novel writing, sure, I would not expect much truly interesting progress from LLMs without human input because fundamentally they are unable to have human experiences, and novels are a shadow or projection of that.

But in math – and a lot of programming – the "world" is chiefly symbolic. The whole game is searching the space for new and useful arrangements. You don’t need to create new information in an information-theoretic sense for that. Even for the non-symbolic side (say diagnosing a network issue) of computing, AIs can interact with things almost as directly as we can by running commands so they are not fundamentally disadvantaged in terms of "closing the loop" with reality or conducting experiments.

This is simply not true.

Modern LLMs are trained by reinforcement learning where they try to solve a coding problem and receive a reward if it succeeds.

Data Processing Inequalities (from your link) aren't relevant: the model is learning from the reinforcement signal, not from human-written code.

Theoretical "proofs" of limitations like this are always unhelpful because they're too broad, and apply just as well to humans as they do to LLMs. The result is true but it doesn't actually apply any limitation that matters.

That's true. But if we take your implied rebuttal then current level AI would be able to learn from current AI as well as it would learn from humans, just like humans learn from other humans. But so far that does not seem to be the case, in fact, AI companies do everything they can to avoid eating their own tail. They'd love eating their own tail if it was worth it.

To me that's proof positive they know their output is mangled inputs, they need that originality otherwise they will sooner or later drown in nonsense and noise. It's essentially a very complex game of Chinese whispers.

I think my track record belies your very low value and frankly cowardly comment. If you have something to say at least do it under your real username instead of a throwaway.

Obviously you accidentally recreated a language from the 70s :P

(I created a template language for JSON and added branching and conditionals and realized I had a whole programming language. Really proud of my originality until i was reading Ted Nelson's Computer Lib/Dream Machines and found out I reinvented TRAC, and to some extent, XSLT. Anyway LLMs are very good at reasoning about it because it can be constrained by a JSON schema. People who think LLMs only regurgitate haven't given it a fair shot)

Languages with reasonable semantics are rather similar and LLMs are good at detecting that and adapting from other languages.

I recently asked an LLM to give me one of the most basic and well-documented algorithms in the world: a blocked matrix multiply. It's essentially a few nested loops and some constants for the block size.

It failed massively, spitting out garbage code, where the comments claimed to use blocking access patterns, but the code did not actually use them at all.

LLMs are, frankly, nearly useless for programming. They may solve a problem every once in a while, but once you look at the code, you notice it's either directly plagiarized or bad quality (or both, I suppose, in the latter case).

> In fragmentary form all of these elements must have been present in the input

Yes, and Shakespeare merely copied the existing 26 letters of the English alphabet. What magical process do you think students are using when they read and re-combine learned examples to solve assignments?

No, it transformed your prompt. Another person giving it the same prompt will get the same result when starting from the same state. f('your prompt here') is a transformation of your prompt based on hidden state.

I'm not asking about averages. I'm asking about any. There is no need to perform an academic research study to prove that humans are capable of writing original code because the existence of our conversation right now is the counter-example to disprove the negation.

Yes, it is true that a lot of humans remix existing code. But not all. It has yet to be proven that any LLM is doing something more than remixing code.

I would submit as evidence to this idea (LLMs are not capable of writing original code) the fact that not a single company using LLM-based AI coding has developed a novel product that has outpaced its competition. In any category. If AI really makes people "10x" more productive, then companies that adopted AI a year ago should be 10 years ahead of their competition. Substitute any value N > 1 you want and you won't see it. Indeed, given the stories we're seeing of the massive amounts of waste that is occurring within AI startups and companies adopting AI, it would suggest that N < 1.