Or let's put this another way. Show of hands - how many of you "spicy autocorrect" / "stochastic parrot" people had "AI will start mass-solving Erdos problems" on your forecast list a couple years back? Huh, none of you? Fascinating!

Take some time to reassess your priors. And while you do so, understand that, yes, they are doing logic / reasoning.

They weren't discovered by an LLM. They were known conjectures that were proven by an automated solving language that was linked to an LLM.

I'll take "Things That Didn't Happen For $200", Alex.

Only a handful of meaningful proofs have ever been done by automated formal theorem solvers (the Four Colour Theorem being the most noteworthy example - but its proof is so long that humans can't verify it). By contrast, AI tools have been solving Erdos problems en masse. The majority of them just bog-standard commercial models. In case you need help, the only ones on that list that were hybrid (AI / non-AI) in the actual solving phase are:

1) AlphaProof / DeepMind Prover Agent / AlphaProof Nexus
2) Aristotle (Harmonic)
3) Seed Prover / Seed Prover 1.5 (ByteDance)
4) AxiomProver (Axiom Math)

In each of the above, LLMs come up with the lemmas / strategies but then use Monte Carlo search ("brute force") or likewise to investigate what they came up with. These are a minority. In the "AI Standalone" category, these "hybrid" tools made up only ~20% of attempts and successful proofs. Hybrid tools actually made more of a contribution in the "AI Alongside Literature" (related literature found afterward) and even more of the "AI Building On Literature" (related literature known beforehand) categories, which is the opposite of what people like you expect.

And even with the hybrid tools, it's still the AI doing the heavy lifting when it comes to strategy. Non-AI theorem solvers, again, don't have a spectacular record for churning out novel proofs to unsolved problems. Tools like Lean are more about mathematical rigour - a passive environment that requires a driver (a human or AI) to feed it actual strategies, lemmas, and proof steps. And no, you cannot brute force "strategy" in the vast majority of cases, which is, again, why automated theorem solvers don't have much of a track record with unsolved mathematical problems.

Let's take a random example: the disproof of the unit distance conjecture. It was solved purely by a general purpose commercial GPT model, not custom-trained to mathematics, with no external tools. Read what the various mathematicians reviewing / commenting on it have to say (sections #3 and onward). Seriously, don't skip reading them, actually read them. This was one of Erdos's favourite problems. He mentioned it commonly in his lectures. Essentially every mathematician working in complex geometry has thought about this problem. The approach that the model came up with was highly novel approach, based on CM-fields and class field towers.

I know you don't want to accept this reality, but it is the reality, so you better improve your ability to accept it,. The field of mathematics is already doing so.

Dang, link didn't post.

Ah, good 'ol "Model Collapse" theory that people have been pounding on for years now, predicting an imminent collapse in model capabilities. How has that been working out for you?

Except what the US is actually facing, at least in the near term, is just the opposite, a worker shortage.

You can't pay royalties to the entire internet. That's not realistic.

What is practical are things like taxes to fund public benefits, or requirements of returning things to the commons (for example, open models).

Windows search used to be kind of OK. Never as good as Everything, but Everything didn't exist for most of history. But they really fucked it up somewhere along the line, which I didn't notice because I was using Everything, and now it's very challenging to construct a complex search without learning a whole new language of keywords.

If they keep the facilities small enough then you might be right, but then they miss out on economies of scale. They will still need some nurses in order to put up the semblance of providing health care in the process, not in the facilities perhaps, but around them.

The top shareholders and the executives are. Hence Luigi.

The insurers are motivated to drive health care costs up by the so-called affordable care act, which caps their profits at a percentage of those costs. Since they're not the ones paying the bills, the insured are (and via APTC, the government is, which means the taxpayers are) they want those costs to go up because they get to collect more profit. You need to not ignore reality if you want it to make sense.

Every nation does this, they mock the failures of their adversaries. We do it constantly relating to China and Russia. So what?

Cool to see you admit to crime in public

You're the one making the claim that modern frontier models still make this mistake (they don't), so how about you go to a frontier model, reproduce what you're claiming happens, and click the "share" button and post the link here?

Don't worry, I'll wait.

Your inability to reproduce the thing that you claim happens is duly noted.

You are speaking irrelevant nonsense. LLMs are trained in words

They are not. They are trained in tokens. Tokens do not align with word boundaries, and an arbitrary word can be tokenized in many different ways.

and they think in words

They do not. They don't even think in tokens. The process is: words are split to tokens, tokens point to an embedding position (latent space) while RoPE encodes a relative position, and all reasoning is done within latent space, which is not at all verbal (concepts are directions in latent space, and math is done on concepts, not words).