[xg15]

Yeah, the "an n-dimensional vector is just a struct with n floats" way of thinking is great - until you actually want to apply geometrical operations in the vector space, such as calculating a distance or performing a rotation. Then you have a problem: You cannot visualise such a space and "pretending" to work in 2D/3D space is convenient but often extremely misleading.

So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?

[alistairSH]

“Just a struct” plus “measuring curvature and shapes” is where my mind goes into “must visualize this” mode. How does a struct have curvature/shape? Or is curvature overloaded here (with a technical math definition that is very different than the layman’s “surface of a sphere” mental model).

[monadINtop]

the technical math definition is a rigourous formulation that encapsulates exactly the same thing as what we mean when we say things are curved, but one that also extends far more generally into contexts where our old intuition fails.

The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.