What an LLM can't do (at least with the current architecture) is sustain a commitment that it can be held accountable for later. It doesn't have durable "perspective". It can't update its priors. Maybe a new architecture will come along that gives LLMs memory plasticity, at which point they will shift into being individuated quasi-persons, but that's not what we've got now.
I've written some about this here, if you're interested: https://elliotmilco.substack.com/p/deontic-scorekeeping-comm...
I think you are also suggesting Mathematics and Language are the same thing.
I'm not suggesting mathematics and language are the same thing, but I am saying (which shouldn't be controversial) that mathematics as a discipline exists within the medium of language, and that mathematical reasoning, discovery, and proof happen via people proposing, justifying, testing, and objecting to formulations in language. This isn't a metaphysical claim about whether math is real or captures inherent structures of the universe etc.
You ended your first post with: "An LLM encodes the normative regularities of our linguistic/conceptual games into its weights, but it cannot be a real participant in our games because it is incapable of staking durable commitments or changing its dispositions over time."
And so if there is an agent, a thinking loop, that can recursively adjust over time...how does your statement hold up? I'm particularly talking about 'staking durable commitments or changing its dispositions over time'. Recursive self improvement is all about these 2 things, especially when it's the 'outer thinking loop' doing the improvement.
But you're right, Mathematics exists as a medium within Language. But then Language is just a transfer of ideas from one mind to another.
This principle is the heart of what the article is talking about: a transfer of logical mathematical ideas passed from a human to another. And the sense of 'shared understanding' that comes with it. It has nothing to do with keeping score.
It's about getting a 'sense' for why someone thought or rationalized an idea in order to connect some other idea. You get an 'ah-ha' moment, and some of those moments are gone, or a different path is taken to reach the moments now. That is what the 'weeping' is about (not that I agree with the article. I think anyone can spend however they want. Math doesn't have to change, it's that people want it to, apparently, because it's 'easier' to do something different, or for some other reason I don't know about).
Both depend on Logic, which is where "deontic scorekeeping" comes in. You can only establish something (score) is true and believable by 'keeping score' and having some level of understanding about the language or math you are reading.
The article is mostly referring to the connection that comes from shared understanding between humans, particularly mathematicians, as they build new shared understanding, and realize there are limits to what they can know and 'leave behind' logic for others to pick up and carry on.
That connection has been severed, or strained a bit, but I'm sure new ones will be created. It just might be of the same flavor that mathematicians prefer, therefore they weep.
And if I understand "deontic scorekeeping" correctly, these forums are a better representation of the concept. You will lend some amount of belief to what is said/read based on what you know.
I suppose we would see some interaction with AI in that sense, but it'll take place over the course of learn cycles and when AI starts ingesting its own ideas it typically goes wrong somewhere down the line.