Suppose I want to prove that negative numbers exist. Well, I could easily do that using a mathematical proof. However, all I would be doing is adding another logical object to a list of known logical objects, none of which appear outside in the real world. It's a bit like proving that you lost in a game of chess by invoking the rules.
Using logic or mathematics to prove things does not relate to the real world directly. You cannot prove objects exist in the real world by using logic because no matter how cunning you are, it still might be the case that the objects do not exist. It is possible that no physical objects exist, but that would not affect your logic.
This explains why all attempts to prove God exists fail. You can't do it just by using logic. You could demonstrate that God exists, but nobody has managed to do that under laboratory conditions. All we have are stories that contradict each other.
So my question is: Can we logically prove that anything exists?