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.

But isn’t the issue that coastlines have a fractal nature? That depending on your resolution, you could have a finite or infinite length of a coastline? In which case measurement is hard to define.

Talking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two.

Sure, the length of the intervals is easily compared. But saying

there are twice as many elements in the total than there are in half the range

is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.

This is the whole crux of the coastline problem, isn’t it?

Isn’t it a bit like saying “there’s obviously more real numbers between 0 and 2 than between 0 and 1”? Which, to my knowledge, is a false statement.

I mean, you could argue that wise women doing traditional medicine isn’t based on the scientific method, while most of modern pharmacology (by men or women) arguably is.

Doesn’t mean that one works and the other doesn’t, it just means they’re getting their results with different methods.

Sigh… Source?

Why is the picture including the equation of Newtonian gravitation?

The trick is to redirect the conversation into something you’re happy to rant about for hours. “Why?” “Because mitochondria are the powerhouse of the cell, much like Horus was the powerhouse of the campaign to reunify the galaxy.” “Why?” “So in ancient Anatolia, an immortal man was born…”

Completely fair - do you have a counterargument? I’d be interested in hearing the other side.