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Feeling Greek is considerably less important than being truly human, and in any case no one who reflects deeply on the nature of things feels “at home” or “in synch” with any earthly environment, whether ethnic, geographical, or otherwise.
Are you familiar with the following aphorism? One finds it, variously expressed, in more than one tradition:
“He for whom only his own country is home is but a raw beginner; he for whom every country is home has made some progress in the way; he for whom no country is home has reached the Goal.”
By all means make the trip to Athos, Patmos, and Samos, but do so in order to deepen your spiritual life and draw nearer to God, not to feel more like a Greek.