I had an extensive discussion about the meaning of the word “space” recently. Having been (somewhat) trained in math, that biases my understanding of the word, I think. But since most of my adult life has been spent programming, I think that biases it more. The ultimate bias, however, is the medium in which our bodies sense and act. This discussion centered around “visualization” and how/why visualizations (particularly of simulations) appeal to us humans, why they help us understand the often cryptic mechanisms inside a computation. I believe they help us understand things because they appeal to the 4D spacetime in which we live. And hence, there is an intuition that’s evoked by a visualization (as well as an audialization or … a hapticalization? … how about an olfactorization? … or even an inertialization?).
Given that there are these 3 basic usage domains for the word “space” (math, compsci, and natural), when someone talks about a visualization, what do they really mean? I tend to think they mean the latter. They mean “render your abstract stuff into something in 4D so my natural senses are stimulated”. That forces me to contrast the three uses of “space” and try to establish the distinctive properties of a natural space. I won’t go through all the hemming and hawing to do that. I’ll just make my assertion. What I think makes a space natural is affine geometry, basically the preservation of parallel lines. That’s what gives us our intuition about the translation and rotation of rigid objects, distance, perspective, etc.
Of course, people are different. So, it’s reasonable to assume that some people think more in terms of graphs (networks), some in terms of sequence, order, number, etc. But I tend to think those are in the minority. I’d enjoy evidence to the contrary.