What do you think Twilight's personality would be like if she wasn't Celestia's student, didn't live in Ponyville and didn't believe in the magic of friendship?

Personally I'd just see a clone of Moondancer.

To be honest, I'm not sure. She'd probably be a loner just going through life.

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Yeah, that sounds about right. Always thought that Moondancer was shown mostly in S5 to show how Twilight could have become without going to Ponyvillie

Well, then I'd dispute the actual extent to which she could be called Twilight Sparkle, but for argument's sake, I'd say she'd probably be exactly like Moondancer. Studious, well-read, introspective and brooding to the ostensible nth degree: more or less what you'd entirely expect from a pony who aspires to great intellectual heights and very specifically chooses to denounce nearly all forms of sociable contact.

Notable exceptions: Spike and Shining Armor, if Spike were even still around if, say, Twilight were to fail her entrance examination. If so, and maybe Celestia simply had Her hooves full or was looking the wrong way just then, then yeah, Spike would inevitably soften her a fair bit. To about the extent she was at the very beginning of the series, obviously.

Shining Armor, as well, is somepony Twilight held very dear even before she had her Harmonic awakening, which means a great deal more than is otherwise appreciated. With Twilight having literally no friends besides him and Spike- which is funny since they're both kind of her brothers- she would evidently have to fulfill all of her regular social needs on just them alone.

That's all sort of implied and explicit in Twilight's life for even the time before she moved to Ponyville, but it's something else to imagine her relationships without her being Celestia's student. I... can't imagine she'd change all that much. If you can picture nerdy Twilight during her tenure as the Princess' personal student, likely spending her days researching and writing astounding theorems on magical phenomena, then you can probably picture the same thing without the presence of Celestia.

She might have been a professor of some kind or something? She seemingly had fantasies of being a teacher by season 8, so she probably would have trended towards passing her knowledge along. In any case, she probably would have found renown in the intellectual world regardless because she is a genius regardless. Really, the basic shape of pre-friendship Twilight is pretty well understood. There's certainly a lot to consider about the minutia of her potential other life, but it's pretty damn hard to change who purplesmart tends to be.

*points to twilight from eqg as she was before meeting the others*

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Oh hey, a clear and concise way to say what I said!

Mhm, definitely just Sci-Twi. I wanna change my answer to Sci-Twi if she was a pony and was intensely connected to Spike and Shining Armor.

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She would similar to that of moondancer but worst in social moments and interaction. She would most likely be a magic researcher or a chemist.

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She either would be loner with no life beyond the books or she would be season finale boss who replacement protagonist needs to overcome.