strong agree, I feel like it poisons the fabric of society somehow when everything you interact with is fake or even just has a good chance of being fake, regardless of the also-shitty fact that it is also often trying to influence you.

I guess my stance (which is not very well-developed or anything) is that you try to learn to live with the gaps: define everything in terms of only what you can measure and it no longer matters whether a number is rational or irrational, or infinitesimal vs small-but-finite, because you can't tell. Instead of saying "it vanishes" as an absolute statement you say "it appears to vanish from my perspective".

That's absurd. First because nobody uses the word 'sociopath' to mean 'mentally ill', they are using it as a moral judgment / to describe a type of amoral person. Second because the reason one negatively characterized the mechanisms of capitalism is because they are not, like, immutable laws of the universe, but rather things that society has quite a bit of control over (whether or not it currently knows how to exercise that control).

(incidentally I disagree with the sociopath characterization anyway; I'm just contesting the weird use of the word ableist)

any particular reference to what you're thinking of? I am aware of some writings on finitist or constructivist mathematics but they have not quite seemed to get at what I want (in particular doing away with explicit infinities does not require doing away with excluded middle at all, which is what most of that literature seems to be concerned with).

I think it's just a perspective shift. The main idea is that you can't ever measure a real number, only an approximation to one, so if two values differ by less than the resolution of your measurement they are effectively the same. For example consider the derivative f(x+dx) = f(x) + f'(x) dx + O(dx^2). The analysis version of the derivative says that in the limit dx -> 0 the O(dx^2) part vanishes and so the limit [f(x+dx)-f(x)]/dx = f'(x). The 'finitist' version would be something like: for a sufficiently small dx, the third term is of order dx^2, so pick a value of dx small enough that dx^2 is below your 'resolution', and then the derivative f'(x) is indistinguishable from [f(x+dx)-f(x)]/dx, without a reference to the concept of a limit.