A 4D object is just hard (not impossible) to visualize

Bullshit. It is impossible to visualize. If you can visualize it then describe it to me.

and pretty much by definition has a 3D surface

Which probably means that there is no such thing as a 4 dimensional object since it depends on an impossibility.

You're just not imagining hard enough. Don't just scoff and dismiss as ridiculous that which you don't care to comprehend.

That's what religious people say about their god. I do care to comprehend it if it is possible for human beings to do so. Help me out. How do you visualize an object which has length, width, height, and let's call the fourth measurement W. This measurement would have to be in a direction perpendicular to all 3 of the first ones. Sounds like nonsense to me. The concept of perpendicularity doesn't even seem to apply in the case of a 3 dimensional object.

There's no evidence for their physical (not least because, as you may have noticed, we live in a 3D universe so such objects are impossible here), but as mathematical constructs they are perfectly cromulent.

As mathematical constructs with no parallels in the real world maybe, but they are still impossible to visualize. They would in no sense represent any sort of object.

Just because you can't imagine a 3D surface of a 4D object, doesn't mean others can't.

I would certainly not claim that the reason no human currently alive is capable of imagining a 4 dimensional object is because I cannot. No. The reason they cannot imagine it is because it is utterly nonsensical. Like 1 dimensional or 2 dimensional objects, 4 dimensional objects are worse than unproven. The concepts themselves are impossible.

Nothing real, nothing that actually exists, can be measured with either more or less than 3 dimensions. Anything with only 2 dimensions doesn't exist by definition and more than 3 dimensions simply doesn't make any sense.

Mathematicians have been "studying" n-dimensional objects for years.

What is your point? Is that supposed to prove that such entities exist or are not nonsensical. Is it supposed to show that they can imagine such things?