Making a mistake has nothing to do with using an accurate system. You can have the most precise ruler in the world and yet use it incorrectly.
My point is that perfection is not required. So, calling something a proof does not mean it must be accurate.
Of course it is. When you see a mistake in a reasoning, it's not called a proof anymore. It's kinda the definition.
The problem is it was called a proof beforehand.
So, their is cases of demonstrated lag between publishing and problems showing up. Therefore in the strictest possible terms, every published result is in some way suspects as they simply have yet to be disproved.
I am in no way suggesting this is some kind of reasonable standard. Just that math is slightly more subjective than often portrayed.
PS: This line of reasoning was simply in response to the suggesting that proof somehow inherently implied accuracy.
Just look at all the retracted papers that have been published. Erratum are surprisingly widespread for such a formal system.