Gram Stone

Incorrect hypotheses point to correct observations

Succubi/incubi and the alien abduction phenomenon point to hypnagogia, and evo-psych explanations of anthropomorphic cognition are often washed down with arguments that anthropomorphism causes good enough decisions while being technically completely false; there's an old comment by JenniferRM talking about how surprisingly useful albeit wrong it would be to model pathogens as evil spirits.

Topological Fixed Point Exercises

An attempt at problem #1; seems like there must be a shorter proof.

The proof idea is "If I flip a light switch an even number of times, then it must be in the same state that I found it in when I'm finished switching."

**Theorem. L**et e a path graph on ertices with a vertex oloring uch that if hen Let s bichromatic Then s odd.

*Proof. B*y the definition of a path graph, there exists a sequence ndexing An edge s bi**chromatic i**ff A subgraph f s a st**ate i**ff its terminal vertices are each incident with exactly one bichromatic edge or equal to a terminal vertex of The co**lor o**f a state is the color of its vertices. There exists a subsequence of ontaining the least term of each state; the in**dex of** a state is equal to the index of its least term in this subsequence.

Note that none of the states with even indexes are the same color as any of the states with odd indexes; hence all of the states with even indexes are the same color, and all of the states with odd indexes are the same color.

For each state there exists a subsequence of orresponding to the vertices of and the least term of each subsequence is either r some hat is the greatest term in a bichromatic edge. Thus the number of states in

By contradiction, suppose that s even. Then the number of states is odd, and the first and last states are the same color, so the terminal vertices of re the same color, contrary to our assumption that they are different colors. Thus ust be odd.

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What To Do If Nuclear War Seems Imminent

I see that New Zealand is also a major wood exporter. In case of an energy crisis, wood gas could serve as a renewable alternative to other energy sources. Wood gas can be used to power unmodified cars and generators. Empirically this worked during the Second World War and works today in North Korea. Also, FEMA once released some plans for building an emergency wood gasifier.

Making a Difference Tempore: Insights from 'Reinforcement Learning: An Introduction'

The Lahav and Mioduser link in section 14 is broken for me. Maybe it's just paywalled?

Anthropics made easy?

Just taking the question at face value, I would like to choose to lift weights for policy selection reasons. If I eat chocolate, the non-Boltzmann brain versions will eat it too, and I personally care a lot more about non-Boltzmann brain versions of me. Not sure how to square that mathematically with infinite versions of me existing and all, but I was already confused about that.

The theme here seems similar to Stuart's past writing claiming that a lot of anthropic problems implicitly turn on preference. Seems like the answer to your decision problem easily depends on how much you care about Boltzmann brain versions of yourself.

Does Thinking Hard Hurt Your Brain?

The closest thing to this I've seen in the literature is processing fluency, but to my knowledge that research doesn't really address the willpower depletion-like features that you've highlighted here.

Learn Bayes Nets!

It's also a useful analogy for aspects of group epistemics, like avoiding double counting as messages pass through the social network.

Fake Causality contains an intuitive explanation of double-counting of evidence.

Set Up for Success: Insights from 'Naïve Set Theory'

Re: proof calibration; there are a couple textbooks on proofwriting. I personally used Velleman's *How to Prove It,* but another option is Hammack's *Book of Proof*, which I haven't read but appears to cover the same material at approximately equal length. For comparison, Halmos introduces first-order logic on pages 6 and 7 of *Naive Set Theory*, whereas Velleman spends about 60 pages on the same material.

It doesn't fit my model of how mathematics works technically or socially that you can really get very confident but wrong about your math knowledge without a lot of self-deception. Exercises provide instant feedback. And according to Terence Tao's model, students don't spend most of their education learning whether or not a proof is valid *at all*, so much as learning how to evaluate longer proofs more quickly without as much conscious thought.

Part of that process is understanding formal things, part of it is understanding how mathematicians' specialized natural language are shorthand for formal things. E.g. my friend was confused when he read an exercise telling him to prove that a set was "the smallest set" with this property (and perhaps obviously the author didn't unpack this). What this means formally when expanded is "Prove that this set is a subset of every set with this property." AFAICT, there's no way to figure out what this means formally without someone telling you, or (this is unlikely) inventing the formal version yourself because you need it and realizing that 'smallest set' is good shorthand and this is probably what was meant. Textbooks are good for fixing this because the authors know that textbooks are where most students will learn how to talk like a mathematician without spelling everything out. I find ProofWiki very useful for having everything spelled out the way I would like it and consistently when I don't know what the author is trying to say.

Finally, I have a rationalist/adjacent friend who tutored me enough to get to the point where I could verify my own proofs; I haven't talked to them in a while, but I could try to get in touch and see if they would be interested in checking your proofs. Last time I talked to them, they expressed that the main bottleneck on the number of students they had was students' willingness to study.

For those wondering about the literature, although Kahneman and Tversky coined no term for it, Kahneman & Tversky (1981) describes counterfactual-closeness and some of its affective consequences. This paper appears to be the origin of the missed flight example. Roese (1997) is a good early review on counterfactual thinking with a section on contrast effects, of which closeness effects are arguably an instance.