Since Europe has never been a distinctly defined, stable concept anyway, making it include Turkey shouldn't be a problem. All it takes is the political will to do so.
And I think making up hard and fast geographical-cum-cultural rules about what is Europe and who is European is a really BAD idea. The indistinctness of what and who is European is a major asset. Start drawing lines in the sand like this, and the EU project begins limiting itself for no reason or gain.
Me, provided Turkey wants in and complies fully with the agenda, I say the are very welcome. All these Central Asian republics look too Russia, China and Turkey. With Turkey in the EU, that's our door to influencing this place to move in the same direction as the EU itself. Admitting a large historically Muslim country would also be a huge asset in dealing with North African and the Middle East, which is of vital interest for the EU.
In general, the EU project needs to move ahead and it needs to be ambitious, if possible visionary. Closing the borders of a new "Festung Europa" is hardly the way to do it. That said, it doesn't have to mean the indefinate extension of the EU itself, but an extension of some form of matured EU-model.
Either Europeans in general come around and decide that the principles this union is based on are universally applicable, worth protecting and run with the premise, or we can call quits on the whole thing sooner rather than later.
Of course, first we have to agree on what we are about. All this talk about geography, history and culture I take as an indication that the EU for the time being has a large deficit as to what "core values" it stands for. Defining that would seem more pressing than drawing borders on maps.