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My apologies. I'll back up, if you're still curious.
Think of the function sin(x).
We can take a number, like x=π/3, and plug it into the function, and we get another number, in this case sin(π/3)=√3/2. (here π/3 is in radians, which when we start doing calculus turns out to be more natural than 60°) We can imagine doing that with every real number, and plotting every (x,y) on a plane, and we get a "sine wave" picture like this. That "plane" gets called ℝ×ℝ, or ℝ², because it's defined with 2 real number (ℝ) lines that form a cross intersecting at one point. It's a great picture! I can think about the function inputs as being the length of lines in one direction, outputs as the lengths of lines in another, derivatives as slopes of angled lines, etc.
But ... how about sin(i), where i=√-1? On the one hand, who cares, because it seems like √-1 shouldn't exist: there's no real number whose square is negative, and even when we found such numbers to be useful intermediate results in algebra problems we still decided to call them "imaginary" as opposed to the newly-named "real" numbers; you'd still expect to have a real number in the end. On the other hand, we soon found "complex numbers" (ℂ, all the numbers x+yi you can make by adding a real number x to an imaginary number yi) to also be useful in engineering problems (they represent oscillation a way similar to how positive numbers can represent growth and negative ones decay), and then we found them to be useful in physics problems (where a "quantum wave function" takes complex values), and at some point it's hard to ignore something as not "real" when it's at the foundation of our understanding of reality.
We can plot a collection of complex numbers on the "complex plane": for every complex number z=x+yi you just plot it as (x,y). One complex number can be described with two reals.
But how do we plot a function that takes complex number inputs and gives complex number outputs? We would need to plot it in ℂ×ℂ, two complex planes that form a cross intersecting at one point. "But two planes meet in a line, not a point", you might object, and that's true, in 3D. ℂ×ℂ only fits in 4D. If I wanted to clearly plot part of a real function y=f(x), I can plot each point as (x,y) in a square, but if I want to clearly plot part of a complex function f(x+yi)=u+vi, I need to be able to plot each point as (x,y,u,v) in a hypercube. I don't have any hypercubes lying around! I can't even visualize a hypercube.
So, we plot garbage like this instead. The xy plane there is the complex plane of inputs x+yi, and for each output u+vi=sin(x+yi), the height z of the red surface is u and the height z of the green surface is v. We plot (x,y,u) and (x,y,w) in the same cube and try to picture the true (x,y,u,w) from the result. Those two 2D surfaces twisting through 3D space are really two aspects of a single 2D surface twisting through 4D space. They're easier to understand if you use that web page to rotate them back and forth and turn them translucent, but still I can't picture the single surface in 4D that they represent. If I could actually visualize 4D then the plot of that single surface would fit in my head as naturally as that first "sine wave" plot did.
I think here it depends on what you mean by "in a 4D space".
If my movements were naturally restricted to a 3D manifold (a "surface" is just a 2D manifold) curving through 4D space then you're probably exactly right. Let's back up to 2D. Imagine as an analog a 2D version of me, living on the surface of a globe. Open my eyes, and if light also follows the globe surface then in any unobstructed direction I look I see the back of my own head one globe-circumference away, but if I'm small enough compared to the globe then it feels almost like I'm in good old flat 2D space. Even if the globe is made of taffy and some 3D monster stretches spikes out of it, mushes parts of it together elsewhere to make a torus or worse, whatever. I can still move around any weird surface I'm stuck to so long as it's smooth enough, to any part of it I want to go to so long as it's it's connected. When I'm on the globe, or on any points of "positive curvature" on a more complicated surface, I might feel a little weird (there's more "room" inside a shape than you would expect from its boundary, so it might be like my skin is getting compressed or my innards stretched). Or, on points of "negative curvature" on a more complicated shape, I might feel like my skin was getting stretched or my innards compressed. But either way, if I was small enough compared to the curvature then I'd still be just a slightly squished-around version of me.
Your "bag of holding" example actually is a 3D manifold - locally I can move parts of my body in no more or fewer than the usual 3 dimensions: up/down, left/right, or forward/backward. But those things are only consistent locally - if I stick my arm 10 inches forward into the bag and then reach 10 inches up, it won't be in the same place as if I reach my other arm 10 inches up (outside the bag) and then 10 inches forward. This 3D manifold has geometry that can't exist in 3D space, but only embedded in a space with at least one more dimension.
But with the same one extra dimension, if my movements were unrestricted? Local senses like touch would get weird too. Imagine that 2D me, previously stuck to the globe like a flat sticker (though free to move parallel to the globe surface), suddenly peeled away into the air. I can still wiggle around in my accustomed two directions, but my orientation with respect to that third direction is at the whim of the breeze. On a globe I might be able to look or propel myself north/south vs east/west, but 2D me has no muscles that can turn his limbs up/down. Even if someone took pity and stuck me back on the globe so I could move around its surface again, if they stuck me on backwards then I'd be backwards for the rest of time; clockwise would seem to be counter-clockwise and vice-versa. 3D me in a true 4D space would be in the same boat; my arm has no way to reach hyperup/hyperdown.
I think I understood the latter half of that and it was a fun ride, thanks for writing it.
Thanks! If you haven't already read Flatland, you might enjoy it. It lacks some of the mathematical sophistication (when it was written, general curved manifolds were still a cutting-edge idea) and brevity (though it is only 100 pages, and a fast read) of my ripoff here, but it does retain some attributes I had to drop like "social satire" and "literary quality".
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