So the paper's
and it's basically a discussion on the author's "grossone" approach that they've been pushing for quite a while now (at least since 2004).
The tl;dr on it is that they're proposing a slightly abstracted data type, kinda like how complex number data types generalize real number data types, except this one's focused on infinities instead of imaginaries.
The author seems to want CPU's to include support for grossone's in their arithmetic logic units (ALU's), e.g. as specified in their patent. The author's website and publications seem to focus on applied use in mathematical optimization; they seem to justify their work in terms of theoretical work as an afterthought.
Some of the harshest criticism seems to come from folks who're knocking it as-though the author were trying to report their work as some new theory, arguing that it's weaker and less sophisticated than prior work, e.g. hyperreal numbers. The weird part about this is that, as far as I can tell, that's what the author wanted to do; they're trying to make an abstract data type to replace floating-point data types for common calculation units, not solve the mysteries of the universe.
In general, I'd expect folks who get the author's intent to receive it with a bit of a yawn. The arguments for modifying mainstream ALU's don't seem too compelling, and at-current there's likely not much demand for custom CPU's that implement it.
The author appears to have implemented grossone's at a software level, kinda like how computers do complex-number operations. Since they're using a non-primitive numeric data type, their calculations'll be slower for it; but if they distribute the library and can demonstrate how it's worth the performance hit in some useful applications, some folks might start using it in those cases.
In short, it looks like the controversial proposal's for an abstract data type that includes infinities for use in common calculations. It looks like applied work rather than theoretical, a misunderstanding which seems to have ruffled some feathers; however, at the end of the day if the authors can show that it's useful in real-life coding scenarios and make it available, then folks might use it.