"Mathematics does not give us truths, it gives us consequences."I'd like to point out that, as I stated in another comment in this thread, the author's supposed refutation of the inconsistency inherent in division by zero within fields is incorrect. Their refutation is as follows:
The problem is in step (3): our division theorem is only valid for c ≠ 0, so you can’t go from 1/0 0 to 1 * 0/0. The “denominator is nonzero” clause prevents us from taking our definition and reaching this contradiction.*
This is a strawman, and it's not the way a formal proof that division by zero in fields is undefined would proceed. First and foremost, the conceptual purpose of the proof is to demonstrate that you cannot define division by zero while still retaining the algebraic structure of a field. Trying to refute this point by stating that the proof cannot make use of division by 0 is begging the question. The whole point is that the proof shows you any way you define division by 0 is going to compromise your definition of a field, or it's going to make fields with nonzero elements impossible.
What the author provided is a very contrived strawman for refutation that belies the actual point. You can define division by zero if you'd like, by using wheels, or extended number systems based on fields (i.e. positive and negativie infinities in the extended Real number system), but you cannot do it using fields. In fact, the modern, axiomatic definition of a field explicitly excludes the unit 0 from the otherwise sane rules of multiplicative inverses.
More generally, I'd like to make a couple of observations from both a philosophical and a practical standpoint. First, mathematics does not give us true statements about the world, it gives us consequences that must follow if we accept various axioms or definitions. You can define division by 0 if you'd like, and you can even do so in a sane and useful way. But you will not have a field. But much more importantly, it's conceptually unsound to base an argument about the practical, programmatic behavior of an undefined operation based on imperfect arguments about abstract mathematics. Technically speaking, computers don't even deal with real numbers. If you find yourself mounting a defense of your programming language's behavior by running through the first lecture of a real analysis or linear algebra course, something has gone very wrong with your enterprise.