I've always assumed you can't know how many cards and my group plays that you don't, but I did go look it up.
My 2008 RGG rules say:
"Each player stacks his acquired civilization cards face down on the designated space on his player board."
To stack implies squared up so that you can't see how many. While I realize different versions of the rules have slight differences, I can't imagine they changed this, and the space on the player board doesn't allow for fanning out the cards, so the best you could do would be estimate by the size of the stack.
So my vote is no. In my opinion, part of this game involves paying attention to what other people are doing. I'm not really very good at memorizing stuff and "counting cards," but I don't expect rules to change because of something in a game I'm not very good at.
EDIT: But if this got you to pull the game out again, that's at least good.