It does not, because it is not. And the “real mathematicians” that he quotes aren’t supporting his case either, they’re just saying that there are cases where it’s convenient to pretend. If you look at the Wikipedia page for division by zero you may find “it is possible to define the result of division by zero in other ways, resulting in different number systems”: in short, if it’s convenient, you can make up your own rules.
Yes.
People find it confusing that there is no simple model that encapsulates arithmetic. Fields do not capture it in its entirety. The models of arithmetic that describe it end up being extremely complex.
Arithmetic is ubiquitous in proofs of other things, and people like the author of this blog cannot get over it.
Reality is weird, inconsistent, and weirdly incomplete.
Get used to it!
We don’t make up arbitrary rules, though. Well…so-called mathematicians who study systems with completely arbitrary rules are just jerking off. The rules that most mathematicians use are based on our intuitions about what can’t be proven but “has to be” true.