But Somebody Would Have Noticed

Would anyone in 1980 have believed that there was an unrecognized allotrope of carbon, stable at room temperature and pressure?

Yes, in 1966 and 1970.

I happen to work with someone who was working on his PhD thesis at MIT and found this gigantic peak in his mass spec where C-60 was, but didn't pursue it because he didn't have time.

Sure thing - http://research.microsoft.com/en-us/um/people/jiansun/papers/dehaze_cvpr2009.pdf

The other thing that happens is that those who notice something that goes against the orthodox view are dismissed out of hand. As in Alicorn's point 2, soothsayers are ignored. They often are outsiders, untrained/unconditioned by the accepted view, so their arguments are frequently inadequate. 

Nowhere is this more apparent than with the abusively named "Cantor-cranks". They have noticed something fishy about Georg Cantor's 3 proofs that the real numbers are uncountable, but because Cantor did such a good job of diverting attention so completely onto the reals, the cranks tend to fall into the same trap.  Yet all along the cause of their dislike of Cantor's proofs lies with his treatment of the natural numbers as a finite quantity.

Generally, the experts dismiss all objections as boring, or as pseudo-maths, and if the crank can argue against one proof, then the "experts" move on to the other proofs, as did Cantor, further reinforcing the original misdirection.

That's not what the English word purple means. *rolls eyes*

Well of course it doesn't work. People are irrational. :-)

A few problems with that. First of all, anyone actually paying attention enough to think about the problem of determining primality in polynomial time thought that it was doable. Before Agrawal's work, there were multiple algorithms believed but not proven to run in polynomial time. Both the elliptic curve method and the deterministic Miller-Rabin test were thought to run in polynomial time (and the second can be shown to run in polynomial time assuming some widely believed properties about the zeros of certain L-functions). What was shocking was how simple Agrawal et al.'s algorithm was. But even then, far fewer people were working on this problem than people who worked on proving FLT. And although Agrawal's algorithm was comparatively simple, the proof that it ran in P-time required deep results.

Second, even factoring is not believed to be NP-hard. More likely, factoring lies in NP but is not NP-hard. Factoring being NP-hard with P != NP would lead to strange stuff including partial collapse of the complexity hierarchy (Edit: to be more explicit it would imply that NP= co-NP. The claim that P != NP but NP = co-NP would be extremely weird.) I'm not aware of any computer scientist who would be so sloppy as to make the statements you assert are often heard.

Overall, Agrawal doesn't compare well to the use of the heuristic here because Agrawal's method (a generalized version of Euler's congruence for polynomials) was an original method.

That said, I agree that such a heuristic can lead people seriously astray if it is applied too often. As with any heuristic it can be wrong. Using any single heuristic by itself is rarely a good approach.

Noticing doesn't necessarily mean they actually saw something.

Which is a large part of why I didn't predicate AA's argument thereon.

If someone has this sort of thought in their head there are likely serious fundamental misunderstandings. They probably won't be solved simply by trying to explain dimensional analysis.

I think it was on This American Life that I heard the guy's story. They even contacted a physicist to look at his "theory", who tried to explain to him that the units didn't work out. The guy's response was "OK, but besides that …"

He really seemed to think that this was just a minor nitpick that scientists were using as an excuse to dismiss him.

Not right away. I'd want explanations for why they had never come forward before, explanations for why no one else had come forward. Other witnesses who would confirm their story or a good explanation of why such witnesses don't exist. I'd like an MRI to confirm they're describing events from memory. I'd like documents confirming the story. Some combination of these things could raise my probability estimation to belief-level. Frankly, it's such a complex conspiracy that a detailed account of how exactly it went down would put it on my radar.

Extraordinary claims require extraordinary evidence.

If they had it, yes. Not otherwise. This evidence would have to cover both the immediate claim (that they were working at NASA at that time) and the larger one (that the moon landing was faked). Evidence explaining why no one else ever came forward would be appreciated but not required if the other two things are present.

If "belief" equals greater than fifty percent, no, I wouldn't believe them. But it would raise my probability estimate. By the tenth such (credible) person, it would raise my probability estimate a lot. So by conservation of expected evidence, the lack of such people can validly lower my probability estimate.

It's most obvious when conspiracy theorists claim things about academia, because I know about academia.

This sounds like Wednesday:

"It's most obvious when conspiracy theorists claim things about the LDS church, because I know about the LDS church. Specifically, I know that it is full of loving, caring, thoughtful and intelligent people. If there was a conspiracy, not only would someone know, I would know."

Wednesday in the story, on the other hand, does have a relatively sheltered life

How do we measure sheltered-ness? How can I be confident that my life is less sheltered than Wednesday's, and seek to correct for that?

It seems to me that posts are pretty much automatically promoted once they reach 20 or so; some posts are promoted before then, leading one to infer that the editor thinks especially highly of them. (Others, by contrast, seem to be promoted only with considerable reluctance; although it might just mean the editor wasn't paying attention.)

Couldn't have said it better myself.

Maybe we should do something like: require promotion to penalize the user 50 karma if the post doesn't get at least 20 net upvotes? (I'm guessing this one of mine would have gotten more than 5 additional net upvotes if it had been promoted...)

I did post a criticism of the idea behind the post, long before I made this one, which got to 6. So did several others, all of whom got to 10+. Significantly fewer comments are being voted up for defending the broad attack on the heuristic. This is inconsistent with the post's rating, and a problem with this post only.

I see no reason to justify having done anything different. Maybe if I didn't mention the name "Alicorn", perhaps, but I strongly suspect someone else would have done it for me if I didn't.

Any other suggestions? That I haven't already taken?

As I understand it, the meta-thread is for meta-level discussion of the site in general: new feature ideas, what norms to encourage, how we can be more welcoming etc. I think you're the first person to suggest moving all meta-level excursions to the meta-thread. This is an interesting proposal (you can discuss it on the meta-thread!) but it isn't yet what users are expected to do. We have meta-level discussions all the time in the comments to top-level posts when the meta discussion deals in particular with our discussion of that top-level post. Sometimes those discussions involve principles than could apply to a broader range of discussions but that doesn't mean we need to move the conversation.

I can't speak for you because I don't know what your values are, but if I knew that the U.S. government was secretly mass-murdering its citizens, I would decide that the best thing I could do would be to reform or overthrow that government. I'm sure if I thought for five minutes I could come up with a way to do this. If there is a 9/11 conspiracy then I really want to know that there is a 9/11 conspiracy.

A better reason for making the 9/11 conspiracy theory harder to notice would involve its sheer implausibility.

9/11 truthers should be proposing wagers on the discovery of robust evidence, etc.

There has been quite some robust evidence.

But, I'll accept a wager. There is someone on LW who has openly bluffed regarding 9/11. This will be a simple issue to settle, all I have to do is to ask one simple question(specifically related to this bluff) to one person(edit: to the person who bluffed) and I wager that he won't be able to give a convincing answer. Deal?

Speaking of which, what's taking so long? You've been around for at least several months, but if you're as intelligent as you say you are, and if your values are as you describe them, then why haven't you undergone an intelligence explosion (or created an AI capable of doing so if you aren't one), deployed molecular nanotechnology, and converted all matter on Earth into paperclips yet? Surely our efforts to refine the art of human rationality can't be of much long-term use to you...

You can't just get out of evidence by appreciating the other person's perspective. The alleged abductee is in a special position to evaluate whether or not she is joking or lying with a confidence others cannot share. But the weight of the evidence still suggests a psychotic episode or hallucination and the alleged abductee does not have privileged evidence regarding that proposition (she might have some reasons to doubt it but not enough to counter the fact that it is the dominant explanation).

All the examples you give are valid examples of bad reasoning. But they if anything underscore Jack's point that engagement in the "someone would have noticed" heuristic seems pretty rare. None of these people said "If God didn't exist wouldn't someone have noticed?" which would be the roughly equivalent argument.

That sounds related to the "Big Lie" trick, actually.

It only fails in cases where you wouldn't notice if somebody else had noticed. In a school full of terrified children, each of whom incurs a huge risk in speaking up unilaterally / going to the media about the evil headmistress, it's easy to believe that no one would have said anything. If it happened today, in the real world, I'd check www.ratemyteachers.com, where the incentives to rat on the headmistress are totally different.

The dominating principle (pun totally intended) is:

P(you heard about someone noticing|it's true) = P(you would have heard someone noticed|someone noticed) * P(someone noticed|it's true)

From there you can subtract from one to find the probability that you haven't heard about anyone noticing given that it's true, and then use Bayes' Rule to find the chance that it's true, given that you haven't heard about anyone noticing...

...I think; I don't trust my brain with any math problem longer than two steps, and I probably wrote several of those probabilities wrong. But the point is, you can do math to it, and the higher the probability that someone would have noticed if it wasn't true, and the higher the probability that you would have heard about it if someone noticed, the higher the probability that, given you haven't heard of anyone noticing it's true, it's not true.

For you to justify the rule in this post, you'd have to prove that people either systematically overestimate the chance that they'd hear of it if someone noticed, or the probability that someone would notice it if it were true.

This might count-- it's the story of a flamboyantly abusive boss who got away with it for a long time. It seems to be partly that he was very good at working the system, and partly that the complaints about him seemed so weird that they were discounted.

One that I didn't want to include in the post because I felt it would make it too inflammatory is this reaction to a particular conspiracy theory.

I assumed you had that exchange in mind. And it makes sense to avoid the inflammatory issue. But "someone would have noticed" was not what I was saying and that makes me wonder how often people actually do say "someone would have noticed".

I actually will site this using Matilda's wording of "Never do anything by halves if you want to get away with it" as Trunchbell's Law, both in terms of conventional actions and when taking Refuse in Audacity, but in my experience once the momentary shock wears off thecurve of people using this heuristic doesn't goes up fast enough to make up for the massive amount of noticing.

Are you out of your mind?! Obviously, 9/11 was an inside job designed to cover up evidence of vaccine deaths, in turn a plot by scientifically connected Illuminati Christian Natinalists such as George W. Bush. You would know this if you even attempted to look at any evidence. Clearly, you are just another sheeple.

Respectfully, the idiosyncracy of Semmelweis's personality isn't directly the point. Semmelweis had established beyond doubt early in his career that hand-washing with chlorinated water before deliveries dramatically drove down the maternal mortality rate. This was a huge finding. Incredibly to most of us now, at one time childbirth was a leading cause of death. The gut prejudice of his peers prevailed, however, and it was to be another 60 years later that the introduction of sulfa drugs and antibiotics again began to drive down maternal mortality. The point relates to pluralistic ignorance and the role of social proof. Social proof roughly means that the greater number of persons who find an idea correct, the greater it will be correct. In situations of uncertainty , everyone looks at everyone else to see what they are doing. One answer to Alicorn's query at the end of her post is to bear in mind the phenomenon of social proof, and the tendency toward pluralistic ignorance. Therefore, look beyond what the plurality of people are doing or saying.

In fairness, Semmelweis didn't handle things very well. He drank heavily, and he engaged in personal attacks on doctors who disagreed with him. ... He wasn't ostracized until his various problems with interacting with people had already started. Before that, many people listened to what he had to say, and many just listened and then didn't change their mind. If he had handled things better, more people would likely have listened.

So he was a 19th century version of me that liked alcohol? ;-)

as long as you have a set of all sets and the ability to take subsets specified by properties, you get Russell's paradox.

Yes, that's true. What I have in mind is restricting the latter ability a bit, by the minimum amount required to get rid of paradoxes. Except if you squint at it the right way, it won't even look like a restriction :-)

I will use the words "set" and "predicate" interchangeably. A predicate is a function that returns True or False. (Of course it doesn't have to be Turing-computable or anything.) It's pretty clear that some predicates exist, e.g. the predicate that always returns False (the empty set) or the one that always returns True (the set of all sets). This seems like a tiny change of terminology, but to me it seems enough to banish Russell's paradox!

Let's see how it works. We try to define the Russell predicate R thusly:

R(X) = not(X(X))

...and fail. This definition is incomplete. The value of R isn't defined on all predicates, because we haven't specified R(R) and can't compute it from the definition. If we additionally specify R(R) to be True or False, the paradox goes away.

To make this a little more precise: I think naive set theory can be made to work by disallowing predicates, like the Russell predicate, that are "incompletely defined" in the above sense. In this new theory we will have "AFA-like" non-well-founded sets (e.g. the Quine atom Q={Q}), and so we will need to define equality through bisimilarity. And that's pretty much all.

As you can see, this is really basic stuff. There's got to be some big idiotic mistake in my thinking - some kind of contradiction in this new notion of "set" - but I haven't found it yet.

EDITED on May 13 2010: I've found a contradiction. You can safely disregard my theory.

A: t=>bool
P(A): (t=>bool)=>bool

t[i] = t[i-1]=>bool + T
A[i]: t[i]=>bool
P(A[i]) = A[i+1]

t = lim i->inf t[i]
A = lim i->inf A[i]
P(A) = lim i->inf P(A[i])

To be clear - Aczel's ZFA and similar systems do satisfy extensionality; they'd hardly be set theories if they didn't. It's just that when you have sets A and B such that A={A} and B={B}, you're going to need stronger tools than extensionality to determine whether they are equal.

Wait, how is NBG set theory relevant to this? NBG is just a conservative extension of ZFC, and only allows something resembling a set of all sets by insisting that this collection is not, in fact, a set. Which after all, it has to in order to avoid Russell's paradox.

Every set is also a subset of itself, by definition.

From Wikipedia "By definition a set z  is a subset  of a set x, if and only if every element of z is also an element of x"

Insufficient data to answer your question :-) See my reply to Sniffnoy.

By the way, quick history of Russell's paradox and related matters, with possible application to the original topic. :)

Russell first pointed out his namesake paradox in Frege's attempt to axiomatize set theory. So yes, it was a mathematical statement, and, really, it's pretty simple to state it. Why nobody noticed before this paradox before then, I have no idea, but it does seem to be worth noting that nobody noticed it until someone actually attempted to sit down and actually formalize set theory.

However, Russell was not the first to notice a paradox in naïve set theory. (Not sure to what extent you can talk about paradoxes in something that hasn't been formalized, but it's pretty clear what's meant, I think.) Cesare Burali-Forti noticed earlier that considering the set of all ordinals leads to a paradox. And yet, despite this, people still continued using naïve set theory until Russell! Part of this may have been that, IIRC, Burali-Forti was convinced that what he found could not actually be a paradox, even though, well, in math, a paradox is always a paradox unless you can knock out one of the suppositions. I have to wonder if perhaps his reaction on finding this may have been along the lines of "...but somebody would have noticed". :)

I don't count a statement of the form "x such that x is not a member of x" as mathematical, because my intuition doesn't want me to talk about sets being members of themselves unless I have a mathematical formalism for sets and set membership for which that works.

The problem is, how do you exclude it from working? If you're just working in first-order logic and you've got a "membership" predicate, of course it's a valid sentence. Russell and Whitehead tried to exclude it from working with a theory of types, but, I hear that has philosophical problems. (I admit to not having read their actual axioms or justification for such. I imagine the problem is basically as follows - it's easy enough to be clear about what you mean for finite types, but how do you specify transfinite types in a way that isn't circular?) The modern solution with ZFC is not to bar such statements; with the axiom of foundation, such statements are perfectly sensible, they're just false. Replace it with an antifoundational axiom and they won't always be false. However, in either case - or without picking a case at all - Russell's paradox still holds; allowing there to be sets that are members of themselves, does not allow there to be a set of all such sets. That is always paradoxical.

It's also not happy about the quantification of x in that sentence; it's a recursive quantification. Let's put it this way: Any computer program I have ever written to handle quantification would crash or loop forever if you tried to give it such a statement.

It's not recursive unless you're already working from a framework where there are objects and sets of objects and sets of etc. In the framework of first-order logic, there are just sets, period. Quantification is over the entire set-theoretic universe. No recursion, just universality. In ZFC sets can indeed be classified into this hierarchy, but that's a consequence of the axiom of foundation, not a feature of the logic.

Russell's paradox, as usually stated, doesn't actually prove anything, because it's usually given as a statement in English about set theory.

It is presented that way to make a point that naive set theory isn't workable.

I don't know whether Russell originally stated it in mathematical terms, in which case it would prove something. I've read numerous accounts of it, yet never seen > a mathematical presentation. Google fails me at present

It is presented rigorously in most intro set theory text books. In ZFC or any other standard set of set theory axioms, Russell's paradox ceases to be a paradox and the logic is instead a theorem of the form "For any set A, there exists a set B such B is not an element of A."

I don't count a statement of the form "x such that x is not a member of x" as mathematical, because my intuition doesn't want me to talk about sets being members of themselves unless I have a mathematical formalism for sets and set > membership for which that works. It's also not happy about the quantification of x in that sentence; it's a recursive quantification. Let's put it this way: Any computer program I have ever written to handle quantification would crash or loop forever if you tried to give it such a statement.

Well, a standard formalism (again such as ZFC) is perfectly happy talking about sets that recur on themselves this way. Indeed, it is actually difficult to make a system of set theory that doesn't allow you to at least talk about sets like this.

I'm curious, do you consider Cantor's diagnolization argument to be too recursive? What about Turing's Halting Theorem?

Let's put it this way: Any computer program I have ever written to handle quantification would crash or loop forever if you tried to give it such a statement.

What encoding scheme would you use to encode arbitrary, possibly infinite, sets in a computer?