Math is a tool (and sometimes abused for pure pleasure, 200 years later applied to make hard crypto work). If your definition doesn't make sense for the application, fix your definition and get over it.
Another example I've recently often bitched about in discussions is modern measure theory and its application to probability calculations. People just don't get the concept of theorytically possible event, but probability 0, i.e. ignore this. But without Lebesgue integration L_p function spaces are not complete and an awful lot of stuff stops to work properly. Among them essentially all of modern physics.
The sane approach is to get over the "this doesn't make intuitive sense" bitchering and just use defintions to derive useful results. And after a few years of playing around with stuff and applying the un-intuitive definition, it's becoming intuitive ;-)
It turns out to be so. If a reply is not a rebuttal, it is usually preceded with something like "To clarify, ..." or "I wanted to add, that ...". I just got confused without it.
>If you want to know what something "really is", consult philosophy or something.
You mean like the "foundations of mathematics"? http://en.wikipedia.org/wiki/Foundations_of_mathematics
"No really". Mathematics just isn't concerned with this stuff. Sometimes infinity it defined as single point making the real number compact, sometimes a "positive infinity" and a "negative infinity" are defined. Sometimes you add points to a given function to make it more tractable and sometimes you don't. But none of this "means" anything. The real number line can be embedded in a number of topological spaces. At least two division rings and various things (the complex numbers are most common). The way you extend a given function (say e^x) is going to vary depending on what space you're looking at as well as what topic you're interested in.
Math works with definition systems and get theorems out of them. If you want to know what something "really is", consult philosophy or something.