2019-02-15
I continued learning about causal inference.
I returned to learning about causal inference, mostly from Pearl’s book but also this overview paper.
I added a bit to the do operator page.
Not much math; I spent the day relaxing, reading things, socializing, and thinking about some things.
I made some additions to the List of important distinctions in mathematical logic page.
I again didn’t do much math.
I didn’t do much math; I wrote some scribbles about the page List of important distinctions in mathematical logic.
I spent the day reading random things (I was curious about top LessWrong posts I had missed, so I spent some time looking through highly upvoted posts, e.g. the top list for 2018).
I started writing List of important distinctions in mathematical logic.
I think I read more of Peter Smith’s Gödel book.
I thought a bit more about the “two flavors” of incompleteness.
I thought about the Löb’s theorem cartoon and puzzle. My take on the cartoon is that it didn’t help me understand the theorem any better, even if it is a cute cartoon; I found myself having to translate the cartoon proof back into logic, which seems like evidence against the native architecture idea that led to the cartoon (or at least, evidence against this specific execution having succeeded). I started working on a page about the puzzle.
I reviewed the “two flavors” of incompleteness theorems and the easy incompleteness proofs in Peter Smith’s book (chapters 6 and 7).
I started thinking about undecidability in logic, and wrote the page for the Entscheidungsproblem. I made some embarrassing realizations, like finally understanding that all the common undecidability statements (for first-order logic) are equivalent, and also realizing that undecidability for a logic is different from undecidability for a theory. It seems totally insane to me that this isn’t emphasized in the resources I have been consulting.
I continued thinking about the diagonal lemma (I mostly used Peter Smith’s book, some online postings, SEP, and Gaifman’s paper). I think it was on this day that I realized that the proofs of the Rogers fixed point theorem and the diagonal lemma are basically the same, and that the theorems themselves are basically the same (I found a couple of papers and some online postings stating this connection, but I wish more textbooks talked about this).
I wrote the diagonalization lemma page.
I continued trying to understand Gödel’s first incompleteness theorem, diagonalization lemma, etc. Mostly via Leary & Kristiansen’s book, but also Peter Smith’s book. Also this page on SEP.
I started the expresses versus captures page.
I spent some time reading about heavy/fat/long-tailed distributions and the distinctions between these terms. I’m frustrated that people seem to use two different visualizations (the quantity vs individual visualization, and the frequency/density vs quantity visualization) but usually don’t mix the visualizations and don’t prove the equivalence between them either.
I started the Multiplicative process page.
I also started reading more about Gödel’s first incompleteness theorem (Leary and Kristiansen’s book).
I re-read the proof of existence of a computable infinite binary tree without any computable paths, in Stillwell’s Reverse Mathematics. I was able to follow the proof. If you remember, I was having trouble with this proof a while back. One of the errors I had made earlier is to assume that the tree nodes were integers, rather than finite binary sequences. Stillwell actually clarifies this in like the first sentence of the proof, but I still had the misconception! I must have been tired… A picture would have helped, and Stillwell does include one, but in an earlier section (that he references).
I did some searching on sets/relations versus sets/relations. They seem to be equivalent under some definitions, but I wasn’t able to figure out when they are vs when they aren’t.
Started on the function versus algorithm page.
I worked through the proof of completeness of first-order logic in Goldrei and Leary & Kristiansen. I think I was able to resolve one of the questions I had about the proof (namely, why it is that we iterate the process of adding constants rather than just adding constants once). I then started working through some of the exercises in Leary & Kristiansen for that section.
I spent the day reading random things and not doing math.
I did various first-day-of-the-month bureaucracies and didn’t do math.
I went through more of Goldrei’s logic book (e.g. different formulations of soundness and completeness).
I read more of Goldblatt’s Topoi but decided to not continue.
I worked on the page Intended interpretation versus all interpretations.
I spent some time reviewing some things in linear algebra (I had forgotten the justifications of some results).
I continued a bit with logic (Goldrei’s book).
I went through some questions I had written up, and Ankified some of the ones I was able to answer.
I started the page Intuitiveness and simplicity tradeoff.
I started reading Goldblatt’s Topoi: The Categorial Analysis of Logic. I suspect I will want to learn category theory at some point, even though I don’t have an immediate use for it.
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