0/0 = ?
If a is NOT equal to zero, we say a/0 is undefined because it has no real solutions. No real number multiplied by 0 gives a non-zero product. Hence we disallow division by zero.
If a =0, then a/0 which is the same as 0/0 is also considered "meaningless" or "undefined", yet it seems to me that 0/0 has an infinite number of solutions, namely the set of real numbers. Any real number multiplied by 0 gives the product zero.
It's true enough that 0/0 doesn't indicate or demarcate any particular real number, but so what? Other operations can have more than one solution, (When X "squared" = 4, then X = +2 or -2), and that's considered perfectly acceptable. [Maybe mathematicians want to set things up so that "fundamental" operations (like +, multiplication, of course, is just repeated addition) have only one solution.] So, this operation, 0/0, has an infinite number of solutions. I guess because 0/0 has an infinite number of solutions it's characterized as "meaningless".
Maybe it's all just a matter of pragmatics in mathematics, insofar as since 0/0 is equal to any real number, it wouldn't do any interesting work.
Update: The above was just written off the top of my head. (Obviously.) As a commenter correctly points out, in some areas of mathematics, division by zero is accomplished. Albeit with different mathematical rules. I should have made it explicit in my posting above that I was referring to the rules in ordinary arithmetic and algebra. See http://en.wikipedia.org/wiki/Division_by_zero for more information. (Note: Wikipedia's reliability cannot be vouched for.)
Actually, this buttresses my point that pragmatics is involved with whether mathematics accepts division by zero. Different rules can be used to achieve different purposes.