I like the way NKS turns some existing ideas about knowables sideways. The relevant distinctions aren’t inside or outside, model or nature, historical or natural, formal-mathematical or empirical etc. Instead it is rule to behavior and simple or complex. Meaning, simplicity is readily knowable whatever domain it occurs in, but complexity is hard to know about, again regardless of the domain. This tracks our experience better than the usual epistemological puzzles, which always prove too much, and pretend we don’t know things we obviously do.
You can’t know the behavior of a simple program doing complicated things until you’ve done an awful lot of irreducible logical work, stepping through its actual behavior one to one and onto. But you can know the behavior of a simple rule doing a simple thing, with a short cut equation, in seconds. Some things are knowable, others are not, in precisely the same domain. Theories of limits of knowledge that depend on domain categories would put rule 30 and rule 250 in the same box – but in fact their knowability is not the same. The same theories put Popper’s clocks and clouds into the same category and get their knowability difference wrong, too, in an exactly parallel manner.
There may still be additional hurdles to line up formal theories to externals, to be sure. But there are obviously knowable pockets of simplicity in both external reality and mathematics – and complex obscurities in both, as well. The real distinction isn’t between the domains, but cuts right through all of them. Simples are knowable and complexities are hard.