The use of $$(a,b)$$ as an abbreviation for $$\{x\in\Bbb R\mid a<x<b\}$$ and $$[a,b]$$ as an abbreviation for $$\{x\in\Bbb R\mid a≤x≤b\}$$ is so widespread and so entrenched that I was surprised when I realized it was essentially arbitrary.
A student asked me what it was that way and I got halfway through mumbling something like “It's because closed sets have sharp edges and open sets have fuzzy ones” before I realized that was nonsense: Under this theory $(a,b)$ for a closed interval makes as much sense as $[a,b]$, because closed discs and open rectangles are just as common as open discs and closed rectangles. If the two types of brackets had been switched, back at the beginning of time, I don't think anyone would find have found it less intuitive, if this were the explanation.
I think what I was getting at perhaps is that one can imagine that the $($ curves toward the endpoint, and then away from it again without quite getting there, whereas the $[$ goes directly to the endpoint and stays there for a while. I don't know if that was the intended intuition. Or, indeed, if there was any intended intuition. It's quite possible that whoever invented this notation needed two kinds of brackets and arbitrarily assigned one to each type of interval. Or perhaps one of the notations was already well-established, and much later someone else wanted an analogous notation for the other kind of interval, and simply used a different kind of bracket.
Do we know anything about this? I did not find anything with a Math SE or MO search, and I also looked at Jeff Miller's site, but did not find anything. I have not checked Cajori, but it does not really seem like the kind of thing he covers.