"Introduction to abstract entropy" by Alex Altair
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https://www.lesswrong.com/posts/REA49tL5jsh69X3aM/introduction-to-abstract-entropy#fnrefpi8b39u5hd7)*This post, and much of the following sequence, was greatly aided by feedback from the following people (among others): *Lawrence Chan), *Joanna Morningstar)*, *John Wentworth)*, *Samira Nedungadi)*, *Aysja Johnson)*, *Cody Wild)*, *Jeremy Gillen)*, *Ryan Kidd)*, Justis Mills) and *Jonathan Mustin)*. Illustrations by Anne Ore.

Introduction & motivation

In the course of researching optimization, I decided that I had to really understand what entropy is.[1]) But there are a lot of other reasons why the concept is worth studying:

• Information theory:

• Entropy tells you about the amount of information in something.

• It tells us how to design optimal communication protocols.

• It helps us understand strategies for (and limits on) file compression.

• Statistical mechanics:

• Entropy tells us how macroscopic physical systems act in practice.

• It gives us the heat equation.

• We can use it to improve engine efficiency.

• It tells us how hot things glow, which led to the discovery of quantum mechanics.

• Epistemics (an important application to me and many others on LessWrong):

• The concept of entropy yields the maximum entropy principle), which is extremely helpful for doing general Bayesian reasoning.

• Entropy tells us how "unlikely" something is and how much we would have to fight against nature to get that outcome (i.e. optimize).

• It can be used to explain the arrow of time).

• It is relevant to the fate of the universe.)

• And it's also a fun puzzle to figure out!

I didn't intend to write a post about entropy when I started trying to understand it. But I found the existing resources (textbooks, Wikipedia, science explainers) so poor that it actually seems important to have a better one as a prerequisite for understanding optimization! One failure mode I was running into was that other resources tended only to be concerned about the application of the concept in their particular sub-domain. Here, I try to take on the task of synthesizing the abstract concept of entropy, to show what's so deep and fundamental about it. In future posts, I'll talk about things like: