One head is aware that you’re dealing with the deepest roots of European civilization, and appreciates the way that much of how we think about everything from architecture to athletics, politics to poetry, ethics to Eros, derives from those two ancient civilizations. The other head recognizes that some of the foundations upon which those foundational cultures were built represent ideas and institutions that are profoundly alien, not to say repellent, to modern sensibilities. (Slavery and patriarchy, for starters.) Sometimes the two heads engage in fruitful dialogue; sometimes they argue bitterly about the value and relevance of their subject.
