You should look into soft linear logics, which are logics that express functions limited to certain complexity classes based on the properties of said logic (particularly, the behaviour of the exponential modalities, limiting duplication of variables/data), that function as justification for ultrafinitistic mathematics, just like constructive mathematics is justified by (long story short) computers.
As Girard also puts it, finitism and ultrafinitism have to be distinguished, as finitism will always fail in its rejecting infinitary methods, as it needs them to justify itself, i.e. you cannot have "all" of finitary mathematics without implicitly accepting infinity.
Ultrafinitism on the other hand, encompassing only "some" of all finitary mathematics, is to be viewed at from a different angle, more in line with constructivism: once I'm sold on the matter of the constructibility of my functions, I should care about constructing them in feasible time as well.
If constructivity is about exposing the algorithmics of proofs, linear logic and ultrafinitism should be about exposing their algorithmic complexity: like constructive logics can embed classical logic in itself and give it some constructive meaning, linear logic can embed both into itself and expose the inner complexity of both, by encoding them with the exponentials and linear implication.
I'm not an ultrafinitist, nor do I pend from Girard's mouth, but I recognize that given the existence of linear logic, ultrafinitism has a place in some mathematical system worth studying.
These that I listed are all mathematical relations between mathematical systems, and should be studied as mathematics.
That some sheltered math enthusiast like
>>12385778 has read a wikipedia article about it once and needs something to point at to fill the "schizo contrarian" boogieman role he has heard from others before him is not mathematics.
Simple as.
