OK, Dr. Stochastic. You've hit upon an age-old scientific debate in quantum mechanics (see Roger Penrose's books, as an example for a good discussion of this). Some people take wave functions to be purely mathematical devices created by a theory which found them to be useful (like your concept of numbers). Stephen Hawking takes this approach. Others (like Penrose) believe they represent an underlying reality. But for now, it's a moot point. If they are an underlying reality, we still can't directly measure them, or changes in them (because measurement causes them to 'collapse' into reality - such as in real particles). What can be definitively said is that wave functions themselves are unmeasurable, and that one can't directly prove that they exist. You can INFER that they do, since they give rise to statistical (though NOT certain) outcomes in experiments. Whether just imaginary number functions, or some underlying reality, we can't touch, see, feel, or hear them. Yet we believe that all existence derives from them. That's how many people feel about God.
Roger Penrose => raving nutter alert. He did some interesting math things in his day, but much of the stuff he has written discredits his sanity and is righly scoffed at by his peers. In other words, you don't want to reference this guy as an intellectual "big gun".