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What is gravity?
(mander.xyz)
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Sounds like we know what it is, we just don't know the reason for its value. (Edit: Unless I'm misunderstanding what you mean)
Wikipedia link
The strangeness of the Fine Structure Constant isn't it's value, it's that we don't know what it is.
Other constants have units that explain what they are doing. Like converting miles to meters we multiply by meters/miles. But this is just a number that is needed. That's so strange I can't think of another example.
Meh, there's pi, it has no units because it's the ratio of one distance to another...
I feel like this might be another example of OPs meme. Feyman called it a magic number we have no understanding of. It's one of the great mysteries of modern physics.