The solution is either to distinguish between countable and enumerable and decidable, or to use intuitionistic logic.

Isn't that a fairly recent perspective, though? My memory is that the concept of math for its own sake (purposefully divorced from "real world" applications) is only a couple hundred years old at least. Whereas for most of its history, from the Egyptians up to Newton, mathematics was developed as a tool to understand the physical world.

So while some people today might consider that mathematics is not intended to describe the real world, they're working on top of a deep system that was.

So now we start to argue about concepts that are not well defined in the context, introducing new terms, even?

Art for art's sake is just one aspect of science and craft. You might argue that mathematical knowledge as it is incorporated in the material world, e.g. in brains, books and soundwaves, has become a real matter which entails all the complexity that makes it so difficult to grasp.

We do, but you have to remember that mathematics is an activity invented by humans largely for their own amusement. This is especially true when it gets down to the foundations of mathematics.