Sometimes people ask me what math they should study in order to get into agent foundations. My first answer is that I have found the introductory class in every subfield to be helpful, but I have found the later classes to be much less helpful. My second answer is to learn enough math to understand all fixed point theorems. These two answers are actually very similar. Fixed point theorems span all across mathematics, and are central to (my way of) thinking about agent foundations.
This post is the start of a sequence on fixed point theorems. It will be followed by several posts of exercises that use and prove such theorems. While these exercises aren't directly connected to AI safety, I think they're quite useful for preparing to think about agent foundations research. Afterwards, I will discuss the core ideas in the theorems and where they've shown up in alignment research.
The math involved is not much deeper than a first course in the various subjects (logic, set theory, topology, computability theory, etc). If you don't know the terms, a bit of googling, wikipedia and math.stackexchange should easily get you most of the way. Note that the posts can be tackled in any order.
Here are some ways you can use these exercises:
The first set of exercises is here.
Thanks to Sam Eisenstat for helping develop these exercises, Ben Pace for helping edit the sequence, and many AISFP participants for testing them and noticing errors.
Read the following.
Please use the (new) spoilers feature - the symbol '>' followed by '!' followed by space - in your comments to hide all solutions, partial solutions, and other discussions of the math. The comments will be moderated strictly to cover up spoilers!
I recommend putting all the object level points in spoilers and leaving metadata outside of the spoilers, like so:
Here's my solution / partial solution / confusion for question #5:
And put your idea in here! (reminder: LaTex is cmd-4 / ctrl-4)