I have a super dumb question.
So, if you allow me to divide by zero, I can derive a contradiction from the basic rules of arithmetic to the effect that any two numbers are equal. But there's a rule that I cannot divide by zero. In any other case, it seems like if I can derive a contradiction from basic operations of a system of, say, logic, then the logician is not allowed to say "Well...don't do that".
So there must be some other reason for the rule, 'don't divide by zero.' What is it?
For the real numbers, the equation a x = b has infinitely many solutions if a = b = 0, no solutions if a = 0 but b ≠ 0, and exactly one solution whenever a ≠ 0. Because there's nearly always exactly one solution, it's convenient to have a symbol for "the one solution to the equation a x = b" and that symbol is b / a; b but you can't write that if a = 0 because then there isn't exactly one solution.
This is true of any field, almost by definition.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.