In other words, as michaelfeathers said, "the map is not the territory."
I'm surprised this way of seeing the world isn't more wide spread. My opinion is that it should be among people who have a deep interest in science.
Perhaps, the biggest philosophical problem with waves are their integration requirements and the potential for such iterations to exceed cosmic speed limits. When the function is continuous a perturbation at one end must effect the other end.
I don't know much about physics, but my take on this is:
First you model reality with waves and particles, then you see that particles can behave like waves and vice versa. The next step is to model reality with something different that can behave like a wave AND a particle and suddendly there is no paradoxon anymore.