The main argument I've seen against division by zero is because:
"Then 2 + 2 could equal 7!"
If you go through the examples of how they derive that, you can actually fix it quite easily by assigning each 0/0 pair to different variables. This prevents the underhand commutative rule violation.
Then all that you are left with is linear algebra. Whether that equation is solvable depends upon how many constraints you have. But it at least doesn't cause math to violate itself.
That doesn't work. You're changing the setting of the problem from scalars to vectors in order to avoid breaking the uniqueness properties of fields and rings. The solution set for a system of linear equations is not unique (if it is at all) in the same sense that e.g. additive and multiplicative identity elements are unique.
Taking the author's argument that a/b = the number x, for which x * b = a, it's easy to draw this out to it's logical conclusion:
For numerators (that is: a) other than 0, you get impossible equations. That is: 1 / 0 = "the number which, when multiplied by 0 is 1." In equation form: x * 0 = 1 Well no such number exists. Therefore x can be said to be "no number."
But for the case where a and b are both zero, the equation becomes: x * 0 = 0. And in this case, the answer is any number, since any number multiplied by zero = 0.
So we can think of 0 / 0 as "any number" and non-0 / 0 as "no number".
That means assigning each instance of 0/0 to a variable is a very good solution. Because we use a letter to represent an unknown number in basic algebra, but in linear algebra we also use a letter to represent a value which could be any number. As in the equation for a line: y=mx+b, where x and y can both take on any value.
"Then 2 + 2 could equal 7!"
If you go through the examples of how they derive that, you can actually fix it quite easily by assigning each 0/0 pair to different variables. This prevents the underhand commutative rule violation.
Then all that you are left with is linear algebra. Whether that equation is solvable depends upon how many constraints you have. But it at least doesn't cause math to violate itself.