Question: Instead of discovering that earth was round, we find that we live on a infinite, flat "earth".

What are some cultural implications if we were to live in a world like that (disregarding any physical impossibilities, and that gravity, biological/ecological systems remain roughly the same as reality).

One interesting idea is that cities could be tens of thousands of kilometers from another and that one can only explore so much of the world in their lifetime.

EDIT: I don't mean flat as in geometrically flat, but flat in general. There would still be mountain ranges and oceans and forests and stuff. But there would be different and exotic ecological systems as you travel.

Answer: 
This would actually be really awesome. It wouldn't really be a planet anymore, but it could conveniently be called a plane or plane-et.

**We could find the epicenter of life.**

Presumably, abiogenesis is quite rare. On an infinite flat Earth it will certainly have occurred elsewhere, but not likely very near to us. Theoretically, we would eventually be able to travel fast enough that we could get to the edge of life. This is similar to how we look at the background radiation of the universe to see its birth.

**Aliens on the same plane-et** 

Because life will have certainly occurred elsewhere (a vanishingly small chance in infinite space is a guarantee) we could meet lifeforms that would be totally alien to us, but clearly able to live in similar conditions. They might even be significantly more advanced than us. 

**Geo-location would be difficult**

We couldn't build geosynchronous satellites. No GPS. For something to remain geostationary it would need to constantly thrust. Orbit wouldn't be possible. Getting the satellites into the same impossible space as the Sun (or moon if we've got it) wouldn't help because they wouldn't be good references for geolocation. The best we could do is built beacon towers. But the range would be incredibly limited and line-of-sight requirements would make it fairly ineffective. 

**Infinite resources**

Obviously we wouldn't need to worry about resources so much. Including fresh air and water. We could pollute as much as we wanted because there would always be more resources further away.

**Nomadic Lives**

Most likely we'd live a long term nomadic life. Is it really worth building a massive city when the resources in the area will eventually be used up? Why would a farmer preserve their land if it's so much easier just to keep moving? When something like land becomes infinite, it's hardly precious anymore. We'd use it up and move on. Maybe we'll circle around in 10,000 years when things have replenished a bit.
With infinite living space, population control might not be such a major concern. 

Isaac Asimov's essay "The Relativity of Wrong" can also be a good reference as to why the Earth was/could still be seen as flat... But what if the concave curvature of your living surface was that of the Earth's *orbit* instead? Idea is from a Larry Niven sci-fi novel "Ringworld". 

An oversimplified immersive approach to visualizing said infinitely flat(relatively!) land would be found in a game of Halo.