Just starting out this page. I want a log of the mathematics I think about on a daily basis, keeping track of the progress I have been making on my mathematical self improvement. On the 31st of December I was on a flight from Edinburgh. I spent that time listening to the original 'A Mathematician's Lament' by Paul Lockhart, and it inspired me to work with some ideas I had swimming in the back of my mind. I ended up playing with the concept of Fuzzy Graphs, and established the basic idea for a metric between vertices of this fuzzy graph. Would love to explain more with drawings, but there is more playing and some background reading to do.
I did some background reading on fuzzy sets, and realised that my formulation is slightly different from the usual one. My formulation of fuzzy logic uses multiplication as conjunction, but I now realise that this isn't standard. There are a lot of systems of fuzzy logic and picking one requires a pickyness I'm too stupid to have.
Also for my formulation of fuzzy graphs, my vertex set is a normal set, while the textbook I read uses fuzzy sets for the vertex and edge set of a graph. This is really weird to me, and I think I might stick to my formulation, adapting theorems to suit my formulation.
I want to think more about motivations for fuzzy graph theory. I think there are already tons of known results adapting classic graph theory results to a fuzzy context, but I believe there is more to this. Fuzzy Ramsey theory seems like an unexplored corner.
Watched a video about the construction of the Hyperreals. From what I understand it is a way to contruct a number system which allows for the existance of infinitisimals. It uses ring theory, quotient groups and an interesting theorem in first-order logic called the transfer principle. I cannot say that I understand it, but I find it fascinating. Will investigate further.