I'm not saying that LLMs can't learn about the world - I even mention how they obviously do it, even at the learned embeddings level. I'm saying that they're not compelled by their training objective to learn about the world and in many cases they clearly don't, and I don't see how to characterize the opposite cases in a more useful way than "happy accidents."

I don't really know how they are made "good at math," and I'm not that good at math myself. With code I have a better gut feeling of the limitations. I do think that you could throw them off terribly with unusual math quastions to show that what they learned isn't math, but I'm not the guy to do it; my examples are about chess and programming where I am more qualified to do it. (You could say that my question about the associativity of blending and how caching works sort of shows that it can't use the concept of associativity in novel situations; not sure if this can be called an illustration of its weakness at math)