Do we really need reals? It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.
Since almost all real numbers are not computable, it seems that real numbers are a set too big and most of them are unnecessary.
So why do mathematicians love this plethoric set of numbers so much?

Simply put, because they are the most powerful natural descriptor of quantity that we have.
To elaborate only briefly, we can easily describe certain real numbers that we encounter regularly ($\sqrt{2}$ or $\pi$). Throw in the fact that most of our theorems are more powerful and general when proven for the Reals (and often simpler) and you have reasons to include them and none to remove them.
Indeed, Reals are a very natural thing to conceive of - that alone would suffice. Go outside and tell me that the world is merely made up of Rationals.
