“Because there are geometric principles that are mathematically and logically expressed in up to five dimensions.”
The same principles can be expressed up to any arbitrary, or even an infinite number of dimensions, because they are simply mathematical abstractions, but that doesn’t tell us anything about the real world.
“A hypercube is mathematically true.”
Sure.
“Therefore even though we can’t directly observe it we know it is logical that it should exist...”
That’s called a non sequitur. Your conclusion does not follow from the argument you made. It’s logical that it COULD exist, not that it SHOULD exist.
“... and it is likewise irrational to argue that it can’t exist.”
I don’t think anyone argues that it “can’t” exist, but if you are doing science, you must demonstrate something exists, through experiment, rather than just positing that, since it might exist, we should assume it does.
“Therefore even though we can’t directly observe it we know it is logical that it should exist...”
That’s called a non sequitur.
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Have you ever entered an elevator and gone to the top of a tall building? Do you ever give any thought to the foundation of the building even though you can’t directly observe it? It’s logical that the foundation exists simply by the fact of your presence at the top of the building.
I don’t need to directly observe everything to know that it’s there. And while that may not constitute a scientific proof to you I’ve yet to encounter the tall building that didn’t have a foundation.
Likewise the existence of more than just four dimensions to me is equally rational and obvious.