For the first belt musing see here.
When it comes right down to it, a belt is a piece of fabric that holds your pants up, or your jacket together. In the case of a karate gi, it usually doesn't even do that, since the jackets and pants really don't need it. The function of a karate belt is symbolic. It functions as a visible sign of where you are in your karate journey.
So, given that, what does a black belt mean?
For some people it means the end of the journey. The first year as a shodan has a higher drop-out rate than practically any other part of a karate student's time - excepting the first few tenuous months. For those people, a black belt is their graduation certificate, the symbol of their having "learned" karate. If that is how one sees the black belt, then it's not at all surprising that people would drop out - after all, why would one keep going to school after graduating, unless you're one of the few inspired to become a teacher?
Even of those who stay, some feel the black belt symbolizes a graduation, the shift from student to teacher. A mark of "I know what I'm doing," in some essential way.
This isn't to say these people are wrong. A black belt often is a teacher - indeed I would think it difficult to become a black belt without doing at least some teaching, if only for the extraordinary amount of learning that occurs when one sets out to teach something you're sure you know - only to discover that the student will teach you far more than you teach them.
Yet that isn't how I feel about a black belt. Maybe it's because I've done at least a little teaching from fairly early in my karate career. Maybe it's because I know down to my bones that I will never be done learning about karate. Maybe it's because I've become aware of just how little I will know, when I become a black belt, standing there with my bucket beside the ocean of knowledge. But to me a black belt doesn't feel like a graduation certificate.
It feels like being handed my library card.
If the black belt certifies something about me, it certifies that I've learned my ABC's. I can write them, pronounce them, even sound out words with them. I'm making that phenomenal leap, from letters on a page to words, sentences, meaning.
The black belt means that I've mastered the fundamentals of my karate craft. That I have my understanding of stance, strikes, blocks, focus - all the vital minutiae - sufficiently for me to start seeing the meaning behind all this stuff. The katas become less about the individual steps and strikes and more about flow and bunkai (explanations of the purpose of the moves, for the non-karate readers). Fights become less about finding the target and hitting it - but not too hard! - and more about strategy and tactics.
It's a world away from "toes forward, heels out, knees bent," and yet it incorporates all that - just as someone writing a letter must still form their 'H's correctly, lest a shirt become a skirt. The letters must be correct, yet the letters become words, and the words become sentences that one could never imagine when still struggling to understand the individual letters.
And that is what a black belt means to me - the opening up of meaning from my letters of kicks and punches to the stories of karate.