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: 
Life starting from a single point will create a so called [Fairy Ring](http://en.wikipedia.org/wiki/Fairy_ring).
Life will move outwards, expanding the ring seeking for a new resources. Many concentric ever expanding rings may be created as a new life evolves, that is able to use resources that the life in the previous ring was not able to use.
I was actually at a planetarium this weekend and I thought of this question based on a display that was there. It had the travel times to pluto and Proxima Centauri based on a car, a plane, and a rocket ship and then light. What that drove home to me was that I think most of these answers are looking at things from too high of an abstraction perhaps. 

On earth if I get on a plane there is a limit to how far I can travel; on this flat world there isn't, not in any direction. There isn't going to be just the Panama Canal and the Suez Canal but probably some Northwest Canal and a Hyperion Canal and a Southern Canal and that is still just in the same grid as we are in; There will be many, many more before one gets to the distance of the moon (1.3 light seconds away); which going a constant 80 mph 24 hours a day would take one 124 day to reach (or 248 to travel from one end to the other). 

In this setting what would a MAD policy even mean? one could just out fit a fleet of Zeppelins and travel, but then again there is no assurance that one would necessarily reach a place one could live/conquer even if one traveled ones entire life (or many lifetimes). Likewise, the notion of a "UN"; there would always be nations known to at least some of the member nations that are not part of the UN, and eventually it would reach the point where traveling to and from meetings would take longer than meetings themselves would, or even longer than ones life; while still being much shorter than what can be communicated. 

The speed of light being so much faster than anything else, it would be trivial to be able to communicate with people with whom contact is impossible. Mars is 3 light minutes away at closest approach but traveling at Mach 1 would take about 22 years to reach that distance, so quite close enough to communicate with via radio (or on stackexchange using fiber optics or satellites fixed to the celestial spheres, even if first person shooters might be out of the question) but generalized trade wouldn't really happen, not via shipping or rail or plane or even supersonic flight. 

So there would be a limit to everything as we are finite even as the world would not be; nations may be fine with nuking some cities and conquering areas many times larger than our earth but eventually either in war or in peace stability would be reached as in 1984 in regions where to go further from their central regions is not worth the marginal effort of doing so. 

There would also be many more opportunities for Columbian exchanges to happen, where technologies, foods, and illness (which can be devastating) from one area mix with another (giving the Old World the potato, tomato, chili pepper, corn, chocolate, etc.) Which means that even long trading expeditions could end up being very profitable. 

Sure the fragmentation would mean that eventually the people that one is trading/communicating with may not even be people any more (I mean we already dealt with/are still dealing with that idea in our own pale blue dot) but they are really not any stranger or more dangerous than the people who closer at hand but still far enough away to be different than one is oneself, which as already pointed out can be literally next door.