I just watched the last episode of the most recent season of Doctor Who. I won't reveal exactly what happened, or the climax of the episode, but some spoilers are inevitable, given that they're the jumping-off point for this post.
Is everyone who minds that gone? Good.
There's a scene in which we see the young Amelia Pond sitting in a parent-teacher conference. She's just painted a night sky with the moon and some stars and her teacher is very concerned. She takes her out to look at the sky and we see a crescent moon... and no stars. The adults tell her that everyone knows that stars are only a legend, and when she's left the room they worry about her joining a 'star cult' when she grows up.
Setting aside the sheer impossibility of this for a second (even in the show it's only possible because of a highly unusual set of circumstances) I think the writers fail to explore the implications of this setting. Besides the references to a legend of stars (somehow) and some distinct oddities in a museum, the world in which Amelia Pond lives seems very similar to our own.
But I suspect that without the stars to try to figure out, science and math would have suffered greatly. There wouldn't be as much impetus for research. And cults aside, mainstream religions would have developed very differently. Human cultures would be very different indeed, and the world would probably be largely unrecognizable.
Unfortunately for us science fiction fans, such speculation outstrips the time, and very likely the budget, available for a single hour-long episode. It's an interesting seed for a story, though.