I am very happy because today I arrived to finish a thread of work concerning computing and space. In future posts, with great pleasure I’ll take the time to explain that, with this post serving as an impresionistic introduction.
Eight years ago I was obsessing about approximate symmetric spaces. I had one tool, emergent algebras, but not the right computation side of it. I was making a lot of drawings, using links, claiming that there has to be a computational content which is deeply hidden not in the topology of space, but in the infinitesimal calculus. I reproduce here one of the drawings, made during a time spent at the IHES, in April 2010. I was talking with people who asked me to explain with words, not drawings, I was starting to feel that category theory does not give me the right tools, I was browsing Kauffman’s “Knots and Physics” in search for hints. (Later we collaborated about chemlambda, but knots are not quite enough, too.)
This a link which describes what I thought that is a good definition for an approximate symmetric space. It uses conventions later explained in Computing with space, but the reason I reproduce it here is that, at the moment I thought it is totally crazy. It is organic, like if it’s alive, looked like a creature to me.
There was an article about approximate symmetric spaces later, but not with such figures. Completely unexpected, these days I had to check something from my notes back then and I found the drawing. After the experience with chemlambda I totally understand the organic feeling, and also why it does resemble to the “ouroboros” molecules, which are related to Church encoding of numbers and to the predecessor.
Because, akin to the Church encoding in lambda calculus, there is an encoding in emergent algebras, which makes these indeed universal, so that a group (to simplify) encodes numbers.
And it is also related to the Gleason-Yamabe theorem discussed here previously. That’s a bonus!