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.
I've come to understand that the key thing that determines success in math is ability to compress concepts.
When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concept in their minds. For them, taking the next step to multiplication is almost trivial.
When a college math student learns the triangle inequality, >99.99% understand it on a superficial level. But <0.01% compress it and play around with it in their minds, and can subsequently wield it like an elegant tool in surprising contexts. These are the people with "math minds".
I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))
I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.
The foundations for these concepts were laid by Piaget and Brissiaud, but most of their work is in french. In English, "Young children reinvent arithmetic" by Kamii is an excellent and practically oriented book based on Piaget's theories, that you may find useful. Although it is 250 pages.
This approach has become mainstream in maths teaching today, but unfortunately often misunderstood by teachers. The point of using different strategies to arrive at the same answer in arithmetics is NOT that children should memorize different strategies, but that they should be given as many tools as possible to increase the chance that they are able to play around with and compress the concept being learned.
The clearest expression of the concept of compression is maybe in this paper, I don't know if it helps or if it's too academic.
Once again I wanted to thank you for slowing down and taking the time to leave this thoughtful comment, if everyone took 5 minutes to try to understand what the other person is saying to see if they can help, the world would be a considerably better place. Thank you.