What if Mozart's Parents Were Deaf?

I’m not sure if this is the correct forum for this or not. I don’t see any way there could be a definate answer, but at the same time, it seems likely to me that there might be some evidence which would point to one direction or another. So I’ve placed my question here, and I do apologize to all if it’s the wrong spot.

I was listening to a program today that was talking about child prodigies and the whole thing was rather light hearted. They listed a few famous child prodigies, interviewed a few, discussed why they only seem to appear in a few areas of ability and not others (the literary world has no Mozarts for example), and so forth.

They said, (or at least I think they said, radio reception is terrible up here) that when he was a small child, Mozart was given a toy violin that he rejected almost immediately saying, “It doesn’t sing.” Now, Mozart’s father being a composer, he could understand what Mozart meant. But suppose Mozart’s parents had been killed in some kind of bizarre piano tuning accident before his talent had made itself known to anyone, and that Mozart was then raised by someone who was deaf, and had been deaf since birth. Would Mozart have been able to become the composer we know him as today, or would his talent gone into something else, like math? Or would have gone totally mad?