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≥4 there are no nontrivial integer solutions to the equation xn+yn=−zn.
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 n2≥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.
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.
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.)
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.
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.
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.
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.
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!)
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.)
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).
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!