A very large mesh network *used* to be possible. Not so much anymore.
> There is nothing in the rules of mathematics or laws of physics that prevents such a system.
In fact there the laws of physics DO put some serious limitations on it, especially a true mesh network. In a nutshell, the frequencies that carry over distance and through walls have limited bandwidth, which must be shared by *everyone* who wants to use any kind of wireless communication. Frequencies above 10 Ghz have a lot of bandwidth, but don't go through drywall. Also of course high frequency waves have high energy - think microwave oven.
Mesh networks are horribly inefficient in how they use the limited bandwidth available in desirable frequency bands. You can do much, much better if you have local transmitters around 1 Ghz communicating with local towers which form a backbone connected via high power dishes, or better yet fiber optics. There is a lot more usable bandwidth to go around using the backbone topology rather than wasting most of the bandwidth by using a mesh. That brings up the issue of who owns and controls the backbones.
Given the physics of it all, back in 1990 you could have built a mesh network to replace the wired connections of the day - 48Kbps max bandwidth, with each person using it an hour or two per day, on average. On a new network built today, you'd want 100,000 to 10,000,000 Kbps, with each person using it ten hours per day. So roughly 40,000 times as much total bandwidth. Not going to happen. Not with the physics we know in this century.
There *is* a way we can 40,000 times as much bandwidth as we had in the the 1990s, though. We actually have such a system working in much of Texas. It involves setting the greedy corporate ISPs up in a situation where to make money, they have to compete with other greedy corporate ISPs. Customers choose the best one, so an ISP can't make money if they suck. It's not a perfect system, but it beats the hell out of what I hear people on the coasts complaining about - a single monopoly ISP protected by a government franchise, an ISP that sucks but they don't care because nobody is allowed to offer competing service.