Allow me to clarify that point. The idea is that - conceptually speaking - mathematics cannot be used to tell you empirical axioms about the world in which we r

Allow me to clarify that point. The idea is that - conceptually speaking - mathematics cannot be used to tell you empirical axioms about the world in which we reside. All it can do is tell you what must be true given certain well-defined assumptions. It is a phenomenally powerful tool for proving things about the world from empirical axioms, but which empirical axioms we choose to rely on in engineering is mostly the domain of physics or the sciences. The underlying point here is mostly a philosophical one, but it has some bearing on the matter at hand. In effect, the definition (or lack thereof) for division by zero in fields is of no practical consequence for the real world impact of implementing an operation which admits division by zero. I can define an algebraic structure in which division by zero is sane (it is not in fields, despite what the article states!), or I can define an algebraic structure in which division by zero is insane. Both can be coherent, consistent and genuinely useful. But whether or not I can define something that works has nothing to do with what happens "when the rubber meets the road", so to speak. It was a mistake to open up an argument about programming division by 0 using the field axioms in the first place. There is no "one truth", there are only facts which must follow as consequences from assumptions. This is especially the case for programming, considering that computers are fundamentally incapable of working with real numbers in the first place.