First of all, we can definitely mentally grasp 4D. In fact, we can grasp infinity D [EDIT: pause!], that's why this is a topic of discussion in the first place. But what would make you believe that we can grasp such extra dimensions in our tiny little heads but that they wouldn't (be able to) exist in the world at large outside ourselves?The reason it's an analogy is because we cannot mentally grasp 4D.
When I describe the qualities of a 4D model, I relate it to something we can compute. Which is a 2D plane warped in a third dimension.
A map. It has only two dimensions that you need to consider. The deeper truth is that the 2D space of the map actually exists on a larger 3D object. In a normal process of driving from one point on the map to another, the third spatial dimension is not a factor.
We understand that there is a third dimension. Up in this case. Now understand that X y and z have a dimension that that wrap around in also.
I bring the globe in to it to illustate that, just like x and y are "infinite" on the globe (circular), x y AND x are "infinite" in the cosmos (circular). But not truly infinite because there is a certain size for each axis. Just like there is a certain value for the circumference of the spheroid Earth.
Where many physicists go wrong is to assume that anything that we will never be able to have control over can't exist or isn't worth thinking about. It's because science is fundamentally a narcissistic enterprise that is only concerned with exerting control over nature to gain power.
That's why humble people tend to be a lot wiser than scientists and why philosophy will always be more interesting than physics.


