I happen to be reading Poincare's Science and Hypothesis right now and he introduced a way of defining the square root of two that I found enlightening. In less articulate form: consider two sets, one of which contains all numbers whose square is less than two and one of which contains all numbers whose square is greater than two. The square root of two is then the symbolic name for the element that divides those two sets.

Poincare says it better in the book though.