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?

If you want to take a serious look at 'Math without Reals', you could do worse than to read Errett Bishop's book, 'Foundations of Constructive Analysis'. Over the history of modern math, there have been several people who have set out to extirpate the reals, as it were: Kronecker, Brouwer, etc. Bishop is known for delivering the maximum of useful results with the minimum of philosophical decoration. 
