I very much appreciate the way faith and religion are handled in this game.
To be sure, the Final Fantasy series has usually taken a sceptical — if not outright cynical — approach towards religion, which is almost always portrayed as a power-hungry organisation used by the clergy to manipulate the masses. This was especially the case in say, Final Fantasy XII,Final Fantasy X and Final Fantasy Tactics.
The matters of gods, primals and eikons in this game, on the other hand, underscore the deeper reasons for why faith and religion continue to persist not just in Eorzea, but also in the real world. Our world.
All people need to believe in something. In our own, modern times, belief in abstract ideology has come to dominate much of the political dialogue on a whole range of socio-cultural issues. But, even so, religion continues to hold tremendous moral sway.
It's become a reflex action among many in the West to have a low opinion of religion — not unlike how Garleans regard gods-worshipping peoples as backward savages — but, the truth is, it's not necessarily regressive to believe in gods and the divine.
Rather, it's about understanding that it's different strokes for different people. For some people, abstract ideas are enough to motivate them. But for many others, they need something more "concrete" they can relate to on a personal level, and who can give them an affirmative direction to follow.
Without being too overly flippant, consider how masses of people become besotted with celebrity singers, actors and athletes, and unwittingly take these individuals as role models, no matter how flawed their behaviour and personal lives. You'll then better understand what I mean by the need for people to have "concrete" idols they can readily emulate.
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To bring this all back to the topic in question: Do the Twelve exist? Well, they haven't manifested themselves the way primals and eikons have, and yet we generally accept that primals and eikons, despite being powerful entities, are all false gods.
So, what are the Twelve? They're the embodiment of the 12 universal virtues and ideas that all Eorzeans seem to hold dear, and in common across the realm.
In that sense, then, the Twelve exist. Not as physical realities like the primals, but certainly as ideas that very much rule the lives of men and women across Eorzea.