No particular theme. Quoted text is me, rest is GPT.

(Old notes from 2016 that I stumbled upon).

I just finished a 1 month research program called REMIX (Redwood Research Mechanistic Interpretability Experiment).

For a finite set, one can describe a uniform distribution. There isn't a natural way to do so for a countable set. But for a hyperfinite set, one can describe a uniform distribution through a probability density. So in some ways the countable is "bigger".

What is an action?

The usual setup of a habit is

just like how infinitesimals are useful because they're indiscernible from 0, but have the advantage of being able to be divided by.

Take an interval. Cut it into H pieces, where H is hyperfinite. This serves as the index set of a stochastic process, among many other uses. Imagine that for each of the H steps, you flip a coin to get -1 or +1. Then move an infinitesimal distance left or right based on the sign. This is Brownian motion. Each infinitesimal piece of the timeline is profitably thought of as a Planck time.

Blog power laws

August 2016: Jiuzhaigou, Sichuan Province, China[^maybe it should be timespace instead of spacetime, better to put the time first, what if the place identifier no longer exists? Plus timestamps are orderable.]

Why

Instrumental convergence is an issue I think about a lot because it's so natural. Being rich and powerful almost never works against you.

$$e^x$$ was first defined by a formula for continuous compound interest:

Fundamental theorem of arithmetic¹: every whole number can be split into a product of prime numbers in a unique way.

Through a somewhat messed up example:

I spent the last hour looking and failing to find the paper that made this specific claim: