"division", "zero", and "impossible".
Mathematics vs Physics:
The main difference between mathematics and physics is that in mathematics the "designer" set the rules and axioms, but in physics it seems that the rules are "fixed". Therefore, a statement like "going faster than light is impossible" should make sense in physics. Bout, what is the situation in Mathematics, or more precisely in each of Mathematics.
Basic definition:
In Algebra, in the definition of groups, they "define" the division a/b as the multiplication of a to the "inverse of b" in terms of multiplication. I don't wanna go in the details, but the bottom line is that this is all "definitions", and to have a consistent mathematics, a user of a mathematics should "follow" them, or switch to another mathematics if he thinks the rules and definitions are very tight for him.
Real numbers:
Let's see what the exact situation with the real numbers. Forget the "addition" operator for a moment. For the multiplication operator, for any real number (except 0), we have another real number which is its inverse. For example, 1/3 is inverse of 3. 1/1000 is inverse of 1000, and so on. We can go very close to zero, 1/10^10000, for example. But, in terms of Mathematics, we cannot get to zero. There is no real number which can be inverse of zero.
It is not the end of story:
But, we want to have something that could cancel zero in multiplication. So, the solution is, or actually was, to "generalize" the real numbers to fix this. It wasn't a problem, but we wanted to have division by zero. So, they added a new "number" to the real numbers, the famous "infinity". Actually, they are two: +infinity and -infinity. The "definition" is modified, and now if you divide any number by zero in this generalized mathematics, you we have infinity.
The point is that there is no "possible" or "impossible" in mathematics. Actually, there is no "unique" mathematics. There are a lot of different and probably contradictory mathematics out there. It is the need of a user that determines which of those mathematics is suitable for him.
Engineering:
The last word is that all these contradictions and surprises in Mathematics are less important in Engineering. In Engineering, the difference between zero and 1/1000000 may be nothing because of many variables that play in a system. So, an engineer doesn't bother himself what is the special behavior of zero, all small numbers in an interval will be the same for him. This is the uncertainty of system that force him to stay on top of a specific granularity level.
Now, let's ask another question: Is it possible to divide zero by zero? Let's think about it.
That's it.
