Comment by xanderlewis

I'm sure you don't have the brain of a gnat, and, even if you did, it probably wouldn't prevent you from understanding this.

As for whether these definitions have a clear meaning that one can relate to 'the world': I think so. To take just one example (I could do more), finite-dimensional means exactly what you think it means, and you certainly understand what I mean when I say our world is finite (or three, or four, or n) dimensional.

Commutative also means something very down to earth: if you understand why a*b = b*a or why putting your socks on and then your shoes and putting your shoes on and then your socks lead to different outcomes, you understand what it means for some set of actions to be commutative.

And so on.

These notions, like all others, have their origin in common sense and everyday intuition. They're not cooked up in a vacuum by some group of pretentious mathematicians, as much as that may seem to be the case.