Here are some of the posts from last week's writing day. Due to the participants writing 34 posts in less than 24 hours (!), I'm re-airing them to let people have a proper chance to read (and comment) on them, in roughly chronological order.
1) Markets are Universal for Logical Induction by John Swentworth
A discussion and proof of the following.
We want to show that any possible logical inductor can be represented by a market of traders - i.e. there is some market of traders which produces exactly the same prices.
2) Intentional Bucket Errors by Scott Garrabrant
Bucket errors are normally thought of as a bad thing. It has "errors" right in the name. However, I want to argue that bucket errors can sometimes be useful, and you might want to consider having some bucket errors on purpose.
3) Logical Counterfactuals and Proposition graphs, Part 1 by Donald Hobson
Within this sequence of posts I will outline a procedure for logical counterfactuals based on something similar to proof length. In this post I present a reimagining of propositional logic in which proving a theorem is taking a walk around a graph of equivalent proposititions.
The post shows two intuitive models of proving propositional logic:
...a directed acyclic graph, as shown above. Under this interpretation, all we have to do is test node identity.
and another graph where:
...all statements that are provably equivalent in propositional logic will be within the same connected component of the graph. All statements that can't be proved equivalent are in different components, with no path between them.
Finding a mathematical proof becomes an exercise in navigating an infinite maze.
In this question I ask a rambling question about why the difference between peak and median human performance is so much larger than for other species, and Vaniver and Carl Shulman give some fascinating answers.