I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. They like to extend common operators like “/“ to things that normally it doesn’t apply to. A computer language will sometimes return a value of int or some other type when the integer 5 is divided by 3. Depending on how the language designer wanted things to work. This isn’t division in a mathematical sense though.
I am not at all concerned with what is or isn't possible in a computer for the purposes of this discussion. My only point with the link is that dealing with inverses in particular situations (i.e. where multiplication has or doesn't have certain properties) frequently requires particular considerations, and the properties of division defined as multiplication by the inverse will have different properties as a result.
To be clear, do you disagree that it is commonplace in complex analysis to extend the complex plane by {infinity} and define 1/0 = infinity, 1/infinity = 0? I find it hard to imagine that you can't have encountered that given how much you seem to know about abstract algebra. Or do you just think that it is a bad idea, despite being commonplace? In either case, to say that mathematicians would not call that operation division as a result is contradictory to my experience, even if those two special cases don't fit the category of multiplication by the inverse.
Also to be clear, I know of no counterexamples in abstract algebra and it would make sense to me that in that context division would mean something very particular, in order to be able to talk about it with any generality. But as it happens, abstract algebra isn't all of math.
This is getting very far from where the original question came from. When talking to a layman one would say division is always multiplication by the inverse. There are nuances involved that a lay person simply can’t appreciate or understand. Had I known you knew about the extended complex numbers I would have answered differently. The extended complex numbers are not a ring, not a group, not an algebra, and so…is it really division then?
In math often times the answer we give depends on the knowledge of the person asking the question. For instance we tell calculus 1 students 1/x is not continuous as a function from R-{0} to R. Of course in the standard induced topology it is a continuous function but explaining this to calculus 1 students would be very difficult.
Giving the answer that is relevant to the situation is very sensible. Saying that "this is the only thing division can mean in mathematics" is evidently false, though (I take it that you agree with me on this now?), and false in a way that is very relevant to the original question, which did not specify a context of abstract algebra and seemed to me to be very interested in expanding the mathematical horizons of the questioner, not restricting them.
The extended complex plane is a great example in my opinion, because it shows that yes there are reasons to extend the numbers in various ways, that can give useful structure, but you may have to give up something else in order for that make sense. In my opinion that is a much more complete answer to the deeper question. (Similarly for the reals mod 1, which do have the property that x + 1 = x).
The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though.
The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would break consistency and that is of paramount importance when doing new things in mathematics. There are agreed upon usages of terms and symbols in mathematics. Why call something division in the true sense of the word when it breaks the conventional usage of what that term means? But, also, why invent a new symbol to denote what is analogous to division? So we abuse notation. This is done all the time. So on the one had we’ll say to calculus 1 students 1/infinity is 0 but also say infinity is not a number. Things are done for convenience but when asked, “Is this really division?” the answer is no.
Of course you can redefine all terms you desire and say things like: A circle can be squared, I just mean something different when I say circle than when you say it. But why do that? All of this is my opinion. You disagree and that is ok.
I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).
Mathematics is a vast subject and I can’t keep track of all developments. In 2010 there was a paper on meadows. I’ve never heard the term before. In that paper it is written:
As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as is demonstrated in [3].
So, as I’ve stated many times, I talked about convention and indicated you can use whatever terms you want. In the paper quoted above they acknowledge what the convention is. That is that division is multiplication by the inverse. They are arguing that it is worthwhile in this new algebraic object to change the usual notion a bit. If people agree to a new usage of the word division then definitions will change accordingly. None of this is pertinent to the spirit of the original question given the context under which it was asked. All of this is highly technical.
Definitions and notions change as new mathematics is created (discovered?). This happens all the time. All you have to do is convince other mathematicians to go along with it.
EDIT: Regarding what you wrote in your other comment: The analogy is not apt in my opinion. It’s hard to say zero can’t exist because the nonzero…. The moment you say nonzero means it does exist. I think a better way to look at the situation is:
I have an object that is a group under a binary operation f. There is another natural binary operation on that object that operates with f in a consistent way. That operation doesn’t form a group but if I add a symbol to my set and give these rules then both operations interact in a consistent, natural way. I get a group under the new symbol with the second operation while preserving the group under the first operation minus the new symbol.
With extended complex numbers you don’t quite preserve the structures or properties that one normally wants so I’d say it isn’t true division. It is division like.
I'm happy to agree to disagree about where the line between "division" and "division like" should be placed. As you say, it is a question of convention and not really a question of math. But I don't agree that a student with the curiosity to ask about extending the numbers in various ways would not find something "division like" with the properties they're interested in to be relevant to the question (even if it is missing some other properties that most mathematicians consider to be essential to the notion of division).
Honestly saying you can't have a number 1/0 because it breaks the ring axioms seems exactly analogous to saying 0 can't exist because it breaks the group axioms for multiplication on the non-zero reals. Is ring multiplication "not really multiplication" because it doesn't satisfy group axioms? That doesn't seem consistent with normal usage to me, but you could imagine a pedantic student coming out of their first group theory course and trying to make that argument.
