In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?
Is that even a valid question, or does it just betray my inability to perceive higher dimensions?
This is fascinating and I'm in awe of the people that do this work.
It's not really a metaphor.
An n-sphere is the set of all points that are the same distance away from the same centre, in (n+1)-dimensional space. That generalises perfectly well to any number of dimensions.
In 1 dimension you get 2 points (0-sphere), in 2 dimensions you get a circle (1-sphere), in 3 dimensions you get a sphere (2-sphere), etc.
EDIT: Also, if you slice a plane through a sphere, you get a circle. If you slice a line through a circle, you get 2 points. If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.
That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)?
From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially in much higher dimensions, like 24 (!). They all "visualized" the equations but in ways that gave them much more insight than you or I might imagine just by looking at the text.
This wasn't atypical of her. She would also say that if your house is on fire then you call the firefighters, but if it is not on fire then you set it on fire, thereby reducing the problem to something that you have already solved.