The next transportation breakthrough will come from computers and communications: drive by wire. You get in the car, punch in the location (or better yet, say the location) and the car takes you there.
Once that system is in place, flying cars will become possible. We've long been able to support a vehicle with counterrotating fans; the problem is that the average, untrained consumer can't be trusted to operate one safely. Eliminate the need for a pilot and the market breaks wide open.
By the way, doesn't the "math" say that a hummingbird can't fly?
Perhaps you're thinking of bumblebees. When people couldn't get the equations to work out, what it was really telling them is that they didn't properly understand how a bumblebee moved its wings. Once they got that right, everthing worked out. It turned out that "the math" was right all along.