The classical mathematician would respond that the theorems are clearly meaningful and you can easily test them against any natural numbers you care about to see empirically they are meaningful.
The ultrafinitist would respond that they are only coincidentally correct, in the same way that pre-modern mathematical reasoning was often very sloppy, featured regular abuse of notation, and had no coherent foundations, but nonetheless still often arrived at correct conclusions by "coincidence."
The classical mathematician might then go over how strong the intuition of something like "there exists a number that..." is and how it is an easily empirically validated statement...
Not that this is your main point, but I find this take representative, “do you believe there's anything about humans that exists outside the mathematical laws of physics?”There are things “about humans”, or at least things that our words denote, that are outside physic’s explanatory scope. For example, the experience of the colour red cannot be known, as an experience, by a person who only sees black and white. This is the case no matter what empirical propositions, or explanatory system, they understand.
This idea is called qualia [0] for those unfamiliar.
I don't have any opinion on the qualia debates honestly. I suppose I don't know what it feels like for an ant to find a tasty bit of sugar syrup, but I believe it's something that can be described with physics (and by extension, things like chemistry).
But we do know some things about some qualia. Like we know how red light works, we have a good idea about how photoreceptors work, etc. We know some people are red-green colorblind, so their experience of red and green are mushed together. We can also have people make qualia judgments and watch their brains with fMRI or other tools.
I think maybe an interesting question here is: obviously it's pleasurable to animals to have their reward centers activated. Is it pleasurable or desirable for AIs to be rewarded? Especially if we tell them (as some prompters do) that they feel pleasure if they do things well and pain if they don't? You can ask this sort of question for both the current generation of AIs and future generations.
Perhaps. But I can't see a reason why they couldn't still write endless—and theoretically valuable—poems, dissertations, or blog posts, about all things red and the nature of redness itself. I imagine it would certainly take some studying for them, likely interviewing red-seers, or reading books about all things red. But I'm sure they could contribute to the larger red discourse eventually, their unique perspective might even help them draw conclusions the rest of us are blind to.
So perhaps the fact that they "cannot know red" is ultimately irrelevant for an LLM too?
let Q dot P = 0 which means Q is perpendicular to P
to choose between R = Q or -Q you want R dot (-p) > 0 because -p points from the line containing p and S(-p) toward the origin so you can just dot product again
