A nice introduction to the history of mathematics is Morris Kline's Mathematics: the loss of certainty. He talks about the difficulties in characterising the real numbers. Also of interest might be one of the appendices to Imre Lakatos' Proofs and refutations where he discusses the history of the concept of continuity, and the difficulties it caused.
More philosophically, there is a paper by Douglas Gasking called (I think) Mathematics and the world from the Australasian Journal of Philosophy and a reply by Hector Neri-Castañeda in the same journal called Arithmetic and reality. These papers are not easy to find, however. The basic question at stake in these papers is whether, if we had developed different methods for measuring the world, whether our number system would have been different.
Finally there is a little book by Donald Gillies calle Frege, Dedekind and Peano on the foundations of arithmetic but this is more about integers than the reals.