Even if we grant all of this -- that humans are good "axiom generators", that AIs are not, and that this is important -- it is totally unconvincing because it conflates
current AI tech in practice with
computers in principle. It's always important to remember that humans aren't made of fundamentally different stuff; we're just very strangely-implemented computers, and that if it's possible for us, it would be astounding if it wasn't, at least in principle, possible for an AI. There's no ethereal module for second-order logic bestowed by the universe only on biological creatures. Whether that's a hard thing to build and why isn't a topic given any space in the article; it's just sort of conclusory.
In fact, in the five years since this article was written, many of the specific supporting details have become questionable, or would now at least need to be more precisely stated and argued. Is AI really so obviously bad at metacognitive thinking? Doesn't seem like it: I bet GPT can generate fresh and interesting axioms given the right prompt about axioms, which is how humans do this kind of thing too. Second-order statements are just...statements, so there's no reason to expect AI to be bad at working with them. And here we're arguing that computers are incapable of this kind of thing in principle?
Just feels like copium for dualists.