The fundamental problem is that you need to teach the computer how to choose which of a number of computations to run depending on two enumerable dimensions, and you need to embed that into your program in a way that covers all the possibilities, is bug free and maintainable.

Like, I need to be able to add shapes to my program, and things I can calculate about shapes to my program, and whenever I do so I need to either add all the new calculations for my new shape, or add my calculation for every existing shape.

That’s the essential complexity at the heart of this problem.

Double dispatch is the problem you then get when, in looking to solve that problem, you want to express the operations you want to run as functions that take every pairwise combination of types, and you are at a point in your code where you have a couple of objects and need to choose which of those functions to run. Like, I have an operation I want to run (could be calculate area, could be calculate perimeter), and a shape I want to run it on (could be a square, could be a circle) and I need the computer to go and run the correct one of four possible bits of code for me right now.

This is a double dispatch problem no matter what facilities your language has to address it. If your language can pattern match on typed tuples you can use pattern matching to dispatch to the right piece of code. If not, maybe you need to use something like the visitor pattern.