Sam Marks

[Book review] Gödel, Escher, Bach: an in-depth explainer

Your original question (about the focus on first-order logic) was a very good one. I asked a logician friend about it, and his answer was the same as the one that you and Samuel settled on here -- first-order logic is likely the best that humans brains (or machines) are able to access. So yeah, basically a version of the Church-Turing thesis.

They referred me to Lindström's theorem as a concrete result in that direction. I don't know enough to understand the statement of the theorem, but it seems to say something along the lines of "first-order logic is the strongest logic which still has [some properties that computable systems ought to have?]."

Fun fact I learned in the conversation: it is a theorem of first-order ZFC that the models of second-order ZFC are constrained enough to single out a canonical model of the integers.

[Book review] Gödel, Escher, Bach: an in-depth explainer

*[Epistemic status: I think everything I write here is true, but I would feel better if a passing logician would confirm.]*

Great question! The important thing to keep in mind is that truth only makes sense *relative to an interpretation*. That is, given a statement *S* in your system, you can ask whether it's true in a particular interpretation, but it might be true in some interpretations and false in others.

Here's a fun fact: *S* is true in every interpretation of the system *if and only if *S* *is provable. (Proof: clearly if *S *is a theorem then it is true in every interpretation. Conversely, Godel's *completeness theorem* says that if *S* is true in every interpretation then it is a theorem.)

So when you ask for "naturally occurring" true-but-unprovable statements, do you want

- a statement which is true in
*every*interpretation but unprovable? or - a statement which is true in
*some*interpretation but unprovable?

If it's (1), then the fun fact above says that no such statements exist. If you want (2), then by taking the contrapositive of the fun fact, you'll see that this is the very same as asking for a statement *S* such that neither *S *nor 'not *S*' is a theorem! In other words, you're looking for a statement which is *independent* of your system. And you probably already know of oodles of these: the axiom of choice is independent of ZF set theory, the continuum hypothesis is independent of ZFC, large cardinal axioms, etc.

--

I have a feeling that this purely mathematical explanation may have done nothing to cure your core confusion, so let me write something else here which I think strikes more at the heart of the question. We often like to think that the integers *exist*; that there's a God-given copy of the integers sitting out there in metaphysical space, not a formal model, but the *real thing* which the axioms are trying to capture. By the discussion above, we know that no matter what axioms we impose, we'll never be able to fully capture the integers (because any formal system will always have unproveable truths, and therefore multiple interpretations, and therefore allow for models of the integers with are different than the *real *ones).

So when you ask for unproveable truths, maybe you're not asking for an unproveable statement which is true in *some* interpreation, maybe you're asking for an unproveable statement which is true in the *one true interpretation. *That is, you want a statement which is true about the *actual* integers, the real out-in-the-world number system that we're trying to model, but unproveable. I don't know of any "naturally occurring" statements like that off the top of my head (though some people think that the Goldbach conjecture might be one).

But if you replace "integers" everywhere in the last paragraph with "universe of sets" you can say very much the same thing. That is, we might intuitively believe that there is an *actual* universe of sets which the ZFC axioms are merely trying to model. And then you can take a statement independent of ZFC -- the continuum hypothesis say -- and ask "is this is true in the real, actual universe of sets?" If yes, then that's your unproveable truth. If no, then its negation is.

So yes, naturally occurring unproveable truths abound, at least in set theory (but depending on what you think the real universe of sets looks like, you might disagree with others about what they are).

[Book review] Gödel, Escher, Bach: an in-depth explainer

Totally agree that a major goal of GEB was to help people build intuition for things like "How could our minds be built out of things that follow purely mechanical rules?" that are now more well-accepted. With that in mind, there are two ways to interpret a lot of the stuff he writes about minds and artificial intelligence:

- "This model I'm presenting is literally how the brain works/how future AIs will work."
- "This model is a plausible idea about how the brain might work/how future AIs might work, presented here to build intuition, not necessarily because I think it's right."

I spent a long time trying to figure out which of these interpretations was intended, and my best guess was that it was different for different claims, but usually somewhere in the middle. The stuff about grandmother modules was more like (1) -- he was really trying to argue from first principles that grandmother modules *must *exist. A lot of the stuff about AI was more like (2), I think (but still with a little bit of (1), which is why I think he still ought to be a little surprised by modern ML).

I'm actually very curious about to what extent GEB helped put ideas like "the mind is a machine" and "it's possible to create a thinking computer" into the water supply. Hofstadter's arguments for these things felt a little different than the standard arguments, so it never occurred to me that he could be partly responsible for the widespread acceptance of these ideas. Maybe GEB convinced a bunch of people, who eventually came up with better arguments? Or maybe GEB had nothing to do it, I honestly have no idea.

lsusr's Shortform

Ahh, I had forgotten that "not stolen" shareholders can also take actions that make their desired outcome more likely. If you erroneously assume that only someone's desire to steal the rack -- and not their desire to defend the rack from theft -- can be affected by the market, then of course you'll find that the market asymmetrically incentivizes only rack-stealing behavior. Thanks for setting me straight on that!

lsusr's Shortform

This is pretty interesting: it implies that making a market on the rack theft increases the probability of the theft, and making more shares increases the probability more.

One way to think about this is that the money the market-maker puts into creating the shares is subsidizing the theft. In a world with no market, a thief will only steal the rack if they value it at more than $1,000. But in a world with the market, a thief will only steal the rack if they value the rack + [the money they can make off of buying "rack stolen" shares] more than $1000.

I still feel confused about something, though: this situation seems unnaturally asymmetric. That is, why does making more shares subsidize the theft outcome but not the non-theft outcome?

An observation possibly related to this confusion: suppose you value the rack at a little below $1000, and you also know that you are the person who values the rack most highly (so if anyone is going to steal the rack, you will). Then you can make money either off of buying "rack stolen" and stealing the rack, *or* by buying "rack not stolen" and not stealing the rack. So it sort of seems like the market is subsidizing both your theft and non-theft of the rack, and which one wins out depends on exactly how much you value the rack and the market's belief about how much you value the rack (which determines the share prices).

For mRNA vaccines, is (short-term) efficacy really higher after the second dose?

Yes cases show up in the data on the day that they first report symptoms, not when they were first exposed. As you say, this means that if the data show some efficacy on a given day, you should actually expect to be protected at that level a few days before.

On top of that, people in the waiting room were talking about how you can tell if you're getting the real vaccine by looking at the syringe. And top of

that, the doctor who gave me the injection basically told me that I got the real thing ("Keep wearing your mask, we don't know yet if these work"), and said something equally revealing to at least one other person I know who did the trial.

Wow. I know that because of side-effects these things can never be fully blinded, but this is just horrifying.

(Technical point: the phase 3's still were randomized controlled trials, they just weren't double-blind. But double-blind is the relevant characteristic when asking whether the different results are due to partying Israelis, so that's fine.)

Covid 4/29: Vaccination Slowdown

It's mentioned in the screenshotted White House statement, but bears emphasizing: the U.S. is also sending vaccine ingredients to India for them to produce vaccines with, effective immediately. This is important because India apparently has good production capacity, but lacks raw materials. Depending on how much raw material we have and how long it takes to turn raw materials into shots in arms, this might be more impactful than the decision to share stockpiled doses over the coming months.

(I'd also like to complain that in the "India" section, all the object-level information about what the U.S. is actually doing is relegated to links. Without clicking through to them, we're only treated to Zvi's commentary on what's being done. Zvi's commentary is great, that's a big part of why I'm here. But after reading that section, I felt like I had no idea what the U.S. was actually doing -- only that whatever it was, Zvi thought it wasn't enough.

In any case, keep up the good work, Zvi!)

For mRNA vaccines, is (short-term) efficacy really higher after the second dose?

Umm ... that's weird. I'll paste in the picture again and maybe that'll fix whatever bug is going on? Let me know if it loads now.

Covid 4/22: Crisis in India

I wouldn't trust the vaccine hesitancy data at the sub-state level. From the methodology here, the state level data come from the Household Pulse Survey (HPS), and the local estimates are produced by adjusting these data using sociodemographic factors:

Our statistical analysis occurred in two steps. First, using the HPS, we used a logistic regression to analyze predictors of vaccine hesitancy using the following sociodemographic and geographic information: age, gender, race/ethnicity, education, marital status, health insurance status, household income, state of residence, and interaction terms between race/ethnicity and having a college degree.

Second, we applied the regression coefficients from the HPS analysis to thedata from the ACS [a survey with local demographic information] to predict hesitancy rates for each ACS respondent ages 18 and older. We then averaged the predicted values by the appropriate unit of geography, using the ACS survey weights, to develop area-specific estimates of hesitancy rates.

Note in particular that "state of residence" is one of the variables in the regression.

More info can be found here.

Some genres of thing you might consider offering a bounty on in the future, ordered by how good of an idea I think they are:

C. elegans's brain?Another idea is to -- instead of putting a bounty on a specific genre of post -- offer to generally pay out to the top N posts submitted each week/month, to generally incentivize the creation of high-quality material of any type.