Sam Marks

• Sam Marks 15 Jul 2020 22:38 UTC

  5 points

  on: The Gold­bach con­jec­ture is prob­a­bly cor­rect; so was Fer­mat’s last theorem

  So Andrew Wiles’s genius was in showing there were no unexpected obstructions for the “likely” outcome to be true. That’s why the proof is so hard: he was trying to prove something very “likely”, and show an absence of structure, rather than a presence, without knowing what that structure could be.

  This is a poor description of Wiles’s proof; in fact, I would call it diametrically wrong. Wiles proved the presence of a very rigid structure—not the absence—and the presence of this structure implied FLT via the work of other mathematicians.

  I don’t have a great understanding of the broader point you are making, so I don’t know how big an issue this mistake presents. However, be aware that the paradigm you’ve extracted from the ideas in this post has lead to at least one incorrect prediction.

  I’ll try to explain how Wiles’s proof diverges from your model of it by way of analogy. Suppose that we instead wanted to prove Fermat’s first theorem:

  Fermat’s first theorem: For every even integer $n\ge 4$ there are no nontrivial integer solutions to the equation $x^n+y^n=-z^n$.

  Further suppose that in our world, mathematicians know about the notion of positivity and absolute values, but the proof of the following fact has long evaded them.

  Positivity conjecture: For every integer $n$, we have $n^2\ge 0$.

  The positivity conjecture is a very important structural fact about the integers. And it immediately implies Fermat’s first theorem (since the left-hand side must be positive, but the right-hand side must be negative unless x,y,z are all 0). So Fermat’s first theorem follows from an important structural fact.

  However, in our supposed world, mathematicians don’t have access to the positivity conjecture. They might perform the exact same analysis in your post (it goes through verbatim!), and conclude that if you check Fermat’s first theorem for enough n, then it is probable to be true. However, it is not true that the proof of FFT via the positivity conjecture is “proving an absence of structure”—quite the opposite!

  The analogue of the positivity conjecture in the real world is the Modularity theorem. This is what Wiles actually proved, and it was already known that the Modularity theorem implies FLT. And as with the positivity conjecture, the Modularity theorem is a very powerful structural result. To give a slogan, it says that every elliptic curve over $Q$ is “modular,” meaning that it relates in an appropriate way to an object called a modular form.

• Sam Marks 13 Oct 2020 2:02 UTC

  10 points

  on: Why Bos­ton?

  Boston resident here, so I thought I’d add some more points and further emphasize some things.

  • The bike infrastructure is really good, and rapidly improving. In fact, there’s so much bike infrastructure that I want to make the converse warning: if you are a nervous driver, driving around here can be terrifying because of the bikers.

  • The winters can be quite brutal (though they seem to be getting milder). And since Boston is way too far east for its timezone, this means that the winter sun sets very early (think ~4:30pm).

  • New England in general, and Boston in particular, is very lovely. If you like the European town aesthetic, this is probably the closest you can get in the U.S.

  • The food scene is pretty bad—food which is both good and cheap basically doesn’t exist.

  • People here are very young, especially when all the students are in town. Whenever I leave Boston, I’m shocked at how old the people are.

  • Marijuana is legal here. However, the dispensaries can be inconvenient to get to: none have opened yet in Boston or Cambridge.

  • I really love living here, and almost everyone I know also likes living here. The exceptions tend to be Californians, though. Did I mention how brutal the winter is?

  • Whatever those Intangible Qualities of a Happy Place are, Boston has them. I’m not sure what gives Boston this feel; I think it’s some mixture of excellent green space, good walkability, a sense of history, small-town aesthetic blended with big-city resources, and generally well-educated and competent populace. Think of the anxious feeling you get when you feel like the world is falling apart and there are a million little things coming apart at the seams—the felt sense of Boston (at least for me) is the polar opposite of that.

• Sam Marks 13 Oct 2020 2:04 UTC

  6 points

  in reply to: Adam Zerner’s comment on: Why Bos­ton?

  The SSC meet-up during Scott’s meetups everywhere tour drew over 140 people. So there’s a bunch of rationalists, but not any hubs. (Though there is a group house in Cambridge that runs LW meetups.)

• Sam Marks 4 Dec 2020 17:28 UTC

  1 point

  on: Covid 12/​3: Land of Confusion

  Minor correction to the “How Many Undiagnosed Cases” section (I think): the CDC calls the parameter they’re estimating “Mean ratio of estimated infections to reported case counts,” which seems to imply it’s the ratio [estimated actual cases (including reported ones)]:[reported cases]. They say their best guess is 11 with a range of (6,24), meaning that you’ve added 1 to all of these numbers. That would be correct if the CDC’s parameter was meant to be [estimated actual cases (not including reported ones)]:[reported cases], but that doesn’t seem to be the case here.

• Sam Marks 22 Apr 2021 17:33 UTC

  13 points

  on: 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.

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

• Sam Marks 25 Apr 2021 21:07 UTC

  1 point

  in reply to: TurnTrout’s comment on: 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.

• Sam Marks 29 Apr 2021 16:08 UTC

  15 points

  on: 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!)

• Sam Marks 29 Apr 2021 16:54 UTC

  3 points

  in reply to: jimmy’s comment on: 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.)

• Sam Marks 25 May 2021 18:08 UTC

  4 points

  in reply to: cata’s comment on: 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).

• Sam Marks 25 May 2021 21:27 UTC

  4 points

  in reply to: tjleing’s comment on: 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!

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

• Sam Marks 30 Sep 2021 16:25 UTC

  6 points

  in reply to: rsaarelm’s comment on: [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:

  1. “This model I’m presenting is literally how the brain works/​how future AIs will work.”

  2. “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.

• Sam Marks 30 Sep 2021 23:25 UTC

  9 points

  in reply to: Bojadła’s comment on: [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

  1. a statement which is true in every interpretation but unprovable? or

  2. 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).

• Sam Marks 4 Oct 2021 21:39 UTC

  3 points

  in reply to: Viliam’s comment on: [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.

• Sam Marks 17 Oct 2021 18:39 UTC

  8 points

  on: Book Re­view Re­view (end of the bounty pro­gram)

  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:

  • Detailed walkthroughs, i.e. step-by-step instructions on how to do a thing

    • Examples: signing up for cryonics, cultivating shoulder advisors, starting on Zettlekasten, running meetups

    • These posts seem ridiculously valuable to me, and also pretty hard to write, so I’d think they’d deserve a pretty large bounty

  • Overviews of a contemporary research field, i.e. “if you were writing a textbook on the state-of-the-art for subject X, what would the intro chapter look like”

    • Example: state-of-the-art for aging research

    • Examples I would like to see: what does technical biosafety research look like? What are the key research directions and challenges in studying C. elegans’s brain?

      • [disclaimer: I’m planning on writing something on this second topic, so I can’t guarantee my suggestion of a bounty on this is entirely self-disinterested]

  • Life advice, i.e. “here’s an easy thing you could try that might make your life better”

    • Lots of examples, but I’m mostly imagining something like Give it a Google

    • So many people have advice to give that probably it would be better to give a payout only to the top-N best advice posts during the bounty period.

    • What exactly constitutes life advice? Does any rationality technique count here?

    • Note also that this runs the risk of flooding LessWrong with billions of advice posts (because these are easy to write)

    • Probably a bad idea, oops

  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.

Movie re­view: Don’t Look Up

• Sam Marks 20 Feb 2022 23:21 UTC

  LW: 3 AF: 2

  AF

  on: Im­pli­ca­tions of au­to­mated on­tol­ogy identification

  It seems to me that the meaning of the set $C$ of cases drifts significantly from when it is first introduced and the “Implications” section. It further seems to me that clarifying what exactly $C$ is supposed to be resolves the claimed tension between the existence of iterably improvable ontology identifiers and difficulty of learning human concept boundaries.

  Initially, $C$ is taken to be a set of cases such that the question $Q$ has an objective, unambiguous answer. Cases where the meaning of $Q$ are ambiguous are meant to be discarded. For example, if $Q$ is the question “Is the diamond in the vault?” then, on my understanding, $C$ ought to exclude cases where something happens which renders the concepts “the diamond” and “the vault” ambiguous, e.g. cases where the diamond is ground into dust.

  In contrast, in the section “Implications,” the existence of iterably improvably ontology identifiers is taken to imply that the resulting ontology identifier would be able to answer the question $Q$ posed in a much larger set of cases $C^{\prime }$ in which the very meaning of $Q$ relies on unspecified facts about the state of the world and how they interact with human values.

  (For example, it seems to me that the authors think it implausible that an ontology identifier be able to answer a question like “Is the diamond in the vault?” in a case where the notion of “the vault” is ambiguous; the ontology identifier would need to first understand that what the human really wants to know is “Will I be able to spend my diamond?”, reinterpret the former question in light of the latter, and then answer. I agree that an ontology identifier shouldn’t be able to answer ambiguous and context-dependent questions like these, but it would seem to me that such cases should have been excluded from the set $C$.)

  To dig into where specifically I think the formal argument breaks down, let me write out (my interpretation) of the central claim on iterability in more detail. The claim is:

  Claim: Suppose there exists an initial easy set $E_0\subseteq C$ such that for any $E_0\subset E⊊C$, we can find a predictor that does useful computation with respect to $E$. Then we can find a reporter that answers all cases in $C$ correctly.

  This seems right to me (modulo more assumptions on “we can find,” not-too-largeness of the sets, etc.). But crucially, since the hypothesis quantifies over all sets $E$ such that $E_0\subseteq E⊊C$, this hypothesis becomes stronger the larger $C$ is. In particular, if $C$ were taken to include cases where the meaning of $Q$ were fraught or context-dependent, then we should already have strong reason to doubt that this hypothesis is true (and therefore not be surprised when assuming the hypothesis produces counterintuitive results).

  (Note that the ELK document is sensitive to concerns about questions being philosophically fraught, and only considers narrow ELK for cases where questions have unambiguous answers. It also seems important that part of the set-up of ELK is that the reporter must “know” the right answers and “understand” the meanings of the questions posed in natural language (for some values of “know” and “understand”) in order for us to talk about eliciting its knowledge at all.)

2022 ACX pre­dic­tions: mar­ket prices

• Sam Marks 9 Mar 2022 1:09 UTC

  2 points

  in reply to: SimonM’s comment on: 2022 ACX pre­dic­tions: mar­ket prices

  Thanks for this feedback!

  Re 17: You are right to be skeptical, because my methodology for this one was silly and ad hoc. I somewhat arbitrarily turned a 92% chance that unemployment never goes above 6% into a 80% chance that unemployment isn’t above 5% in December. This is completely unprincipled, but I didn’t have any better ideas, and the alternative was to ignore the Kalshi market completely and defer entirely to the 5 betters on Manifold, which seemed worse. If you have a more reasonable way of getting a number here, I’ll happily defer to it.

  Re 15: Thanks! I’ll edit that number in and point to your comment.

  Thanks also for the work you put into doing this last year! That post (along with Zvi’s re-predictions) led to me running a small prediction contest with a handful of friends. That went well, was a lot of fun, and straightforwardly grew into me asking Scott if he wanted me and Eric to run the same thing for the ACX community. So, making up some numbers and hoping I can use Shapley values correctly, I estimate that you get 40% of the credit for this year’s prediction contest happening.