LINK: Study demonstrates politically motivated innumeracy
Original article: Motivated Numeracy and Enlightened Self-Government.
Easy-to-read summary of the results: Science Confirms: Politics Wrecks Your Ability to Do Math.
Easier-to-read summary of the summary: A group including both liberals and conservatives were first tested for numeracy, then given either fictional stats on the efficacy of a rash treatment (how many people get better/worse, with/without the treatment) or quantitatively identical fictional stats on gun regulation and crime (in how many cities crime went up/down, with/without gun control laws). They had to say whether, according to the stats, the treatment was effective or whether the laws reduced crime. (In half of the cases the answer was yes, in half no). For the rash problem, numerate people did better in giving the right answer, irrespectively of their politics. For the gun control problem, numerate people (of both political sides) were much better at giving the right answer when it agreed with their political beliefs; when the conclusion implied by the stats disagreed with their political beliefs, they did no better than innumerate people.
LINK: Infinity, probability and disagreement
I saw this conundrum at Alexander Pruss's blog and I thought LWers might enjoy discussing it:
Consider the following sequence of events:
- You roll a fair die and it rolls out of sight.
- An angel appears to you and informs you that you are one of a countable infinity of almost identical twins who independently rolled a fair die that rolled out of sight, and that similar angels are appearing to them all and telling them all the same thing. The twins all reason by the same principles and their past lives have been practically indistinguishable.
- The angel adds that infinitely many of the twins rolled six and infinitely many didn't.
- The angel then tells you that the angels have worked out a list of pairs of identifiers of you and your twins (you're not exactly alike), such that each twin who rolled six is paired with a twin who didn't roll six.
- The angel then informs you that each pair of paired twins will be transported into a room for themselves. And, poof!, it is so. You are sitting across from someone who looks very much like you, and you each know that you rolled six if and only if the other did not.
Let H be the event that you did not roll six. How does the probability of H evolve?
After step 1, presumably your probability of H is 5/6. But after step 5, it would be very odd if it was still 5/6. For if it is still 5/6 after step 5, then you and your twin know that exactly one of you rolled six, and each of you assigns 5/6 to the probability that it was the other person who rolled six. But you have the same evidence, and being almost identical twins, you have the same principles of judgment. So how could you disagree like this, each thinking the other was probably the one who rolled six?
Thus, it seems that after step 5, you should either assign 1/2 or assign no probability to the hypothesis that you didn't get six. And analogously for your twin.
But at which point does the change from 5/6 to 1/2-or-no-probability happen? Surely merely physically being in the same room with the person one was paired with shouldn't have made a difference once the list was prepared. So a change didn't happen in step 5.
And given 3, that such a list was prepared doesn't seem at all relevant. Infinitely many abstract pairings are possible given 3. So it doesn't seem that a change happened in step 4. (I am not sure about this supplementary argument: If it did happen after step 4, then you could imagine having preferences as to whether the angels should make such a list. For instance, suppose that you get a goodie if you rolled six. Then you should want the angels to make the list as it'll increase the probability of your having got six. But it's absurd that you increase your chances of getting the goodie through the list being made. A similar argument can be made about the preceding step: surely you have no reason to ask the angels to transport you! These supplementary arguments come from a similar argument Hud Hudson offered me in another infinite probability case.)
Maybe a change happened in step 3? But while you did gain genuine information in step 3, it was information that you already had almost certain knowledge of. By the law of large numbers, with probability 1, infinitely many of the rolls will be sixes and infinitely many won't. Simply learning something that has probability 1 shouldn't change the probability from 5/6 to 1/2-or-no-probability. Indeed, if it should make any difference, it should be an infinitesimal difference. If the change happens at step 3, Bayesian update is violated and diachronic Dutch books loom.
So it seems that the change had to happen all at once in step 2. But this has serious repercussions: it undercuts probabilistic reasoning if we live in multiverse with infinitely many near-duplicates. In particular, it shows that any scientific theory that posits such a multiverse is self-defeating, since scientific theories have a probabilistic basis.
I think the main alternative to this conclusion is to think that your probability is still 5/6 after step 5. That could have interesting repercussions for the disagreement literature.
[Link] Offense 101
From Julian Sanchez, a brilliant idea unlikely to be implemented:
American politics sometimes seems like a contest to see which group of partisans can take greater umbrage at the most recent outrageous remark from a member of the opposing tribe. As a mild countermeasure, I offer a modest proposal for American universities. All freshmen should be required to take a course called “Offense 101,” where the readings will consist of arguments from across the political and philosophical spectrum that some substantial proportion of the student body is likely to find offensive. Selections from The Bell Curve. Essays from one of the New Atheists and one of their opponents, and from hardcore pro-lifers and pro-choicers. Ward Churchill’s “little Eichmanns” monograph. Defenses of eugenics, torture, violent revolution, authoritarianism, aggressive censorship, and absolute free speech. Positive reviews of the Star Wars prequels. Assemble your own curriculum—there’s no shortage of material.
For each reading, students will have to make a good faith, unironic effort to reconstruct the offensive argument in its most persuasive form, marshaling additional supporting evidence and amending weak arguments to better support the author’s conclusion. Points deducted if an observer can tell the student doesn’t really agree with the position they’re defending.
Only after this phase is complete will students be allowed to begin rebutting the arguments. Anyone who thinks it’s relevant to point out that the argument is offensive (or bigoted, sexist, unpatriotic, fascistic, communistic, whatever) will receive a patronizing look from the professor that says: “Yes, obviously, did you not read the course title? Let’s move on.” Insofar as these labels are shorthand for an argument that certain categories of views are wrong and can be rejected as a class, the actual argument will have to be presented.
Rationality Quotes August 2012
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