While math may be dense and difficult at times, the results it can prove are sometimes beautiful, mind-boggling, or just plain unexpected. At the turn of the 20th century, a lot people were entranced by a new branch of math called Set Theory. The thinking of the time was that anything could be turned into a set: The set of all types of fruit and the set of all US Presidents were both completely valid. Russell decided to get meta about things and described a set that contained all those sets which do not contain themselves. The set of all fruit doesn’t contain itself, so it can be included in Russell’s set, along with many others. Another bit of math you may remember from school is geometry, which is the part of math where doodling in your notes was the point.
A Turing Machine is like a very basic computer: it uses an infinite string of tape and 3 symbols, and then operates given a set of instructions.