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1. Do the Immortal Olympians exist? Do other deities exist (i.e. Hades, Persephone, Hekate, etc.)?
All of the theoi exist.
I would categorize myself under panentheist (gods permeate all of nature, but exist apart from nature as well), animist (I believe some things that are complex enough have a soul, like my car, although it is NOT a god), and a form of “hard” polytheism called henopolytheism (devotion to one pantheon without denying the existence of other gods outside of that set – I believe in, but do not follow, my husband’s path in which he is devoted to the Norse gods of his ancestors). With “hard” polytheism described as believing that there are distinct and literal gods and goddesses (or forces) in multiplicity with unique personalities who aren’t reducible to one force or being. I don’t think They are limited to the Olympians, either.
2. Of those beings that exist, can they interact with the world (e.g. cause real world changes)? Do they?
Yes. Sometimes They come to us in our dreams and influence us there. In-fact, the gods of dreams are said in the Orphic hymns to deliver messages from the gods. Sometimes the theoi take the personage of someone we know, or even a stranger, to test or guide us. Sometimes they take the role of natural forces that change the world in a direct way. However, different gods will interact more or less with the world and with us. Hestia is content to stay by the hearth, while Hypnos comes to us nightly to bless us with sleep, and hopefully, I hope I will not meet Nemesis in my lifetime – but acknowledge that She has every right to touch it should I stray from the just path or succumb to hubris. Her sister, Oizys (Anxiety and Guilt), is one that I’ve contended with since adolescence, and has motivated change in my life. Other gods once interacted with the world, but no more, like many of the Titans.
3. Of the beings that can and do interact with the world, are they concerned with the fate of mortals? Are they concerned with individuals or only groups?
It depends on the god and on his or her generation. With each generation, Their focus is more specific, and they are more approachable and take more direct influence upon specific mortals. Apollon inflicts plague upon many, while people pray to his son Asklepios for cures.
4. Has this knowledge of Hellenic history changed (or enlightened) your religious beliefs? Of the beliefs of the different periods, which do you feel is closest to your own beliefs? In what ways?
Yes. A scholarly approach, and the content gained from it, enrich my beliefs and give them shape and substance. But this is tempered by the practicalities and needs of modern life. I don’t have enough knowledge yet to distinguish very much between the different periods, but I lean towards the pre-Christian and pre-monotheist emphasis that replaced religion with philosophy.