Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn’t?
Excellent question, and as you yourself allude to, it’s a question of bounds. If you can establish and upper and lower bound on a quantity and make them approach eachother, you can measure it.
On a finite 2d surface you can make absolute lower and upper bounds on any area - lower is zero, upper is the full surface. All areas are measurable. But on the same surface you can make a line infinitely squiggly and detailed, essentially drawing a fractal. So the upper bound on the length of a line is infinite. Which means not all lines have a measurable length. And that comparing two line lengths might become the same problem as comparing to infinities of the same type, which is not well defined.
This extends naturally to higher dimensions - in a finite 3d space, volumes must be finite, but both lines and areas can be fractally complex and infinite. And so on.
Excellent question, and as you yourself allude to, it’s a question of bounds. If you can establish and upper and lower bound on a quantity and make them approach eachother, you can measure it.
On a finite 2d surface you can make absolute lower and upper bounds on any area - lower is zero, upper is the full surface. All areas are measurable. But on the same surface you can make a line infinitely squiggly and detailed, essentially drawing a fractal. So the upper bound on the length of a line is infinite. Which means not all lines have a measurable length. And that comparing two line lengths might become the same problem as comparing to infinities of the same type, which is not well defined.
This extends naturally to higher dimensions - in a finite 3d space, volumes must be finite, but both lines and areas can be fractally complex and infinite. And so on.