Religion is not delusion

In recent news, there is the story of Javon Thompson, a sixteen month old boy who was murdered by his mother and other members of 1Mind Ministries. From the Washington Post:

Answering to a leader called Queen Antoinette, they denied a 16-month-old boy food and water because he did not say "Amen" at mealtimes. After he died, they prayed over his body for days, expecting a resurrection, then packed it into a suitcase with mothballs. They left it in a shed in Philadelphia, where it remained for a year before detectives found it last spring.

This was all discovered last year, but it's in the news again because the mother and four other members of 1Mind Ministries are currently being tried for murder. Which raises the question: could they plead insanity?

Psychiatrists who evaluated Ramkissoon [the mother] at the request of a judge concluded that she was not criminally insane. Her attorney, Steven Silverman, said the doctors found that her beliefs were indistinguishable from religious beliefs, in part because they were shared by those around her.

"She wasn't delusional, because she was following a religion," Silverman said, describing the findings of the doctors' psychiatric evaluation.

"At times there can be an overlap between extreme religious conviction and delusion," said Robert Jay Lifton, a cult expert and psychiatrist who lectures at Harvard Medical School. "It's a difficult area for psychiatry and the legal system."

When this story appeared on Friendly Atheist and Pharyngula, many people were quick to comment on the close relationship between religion and delusion. For example:

Believing in things despite all evidence to the contrary is delusion. Talking to those things, as in prayer, is psychosis.

I think these people are far too enthusiastic to echo the title of Richard Dawkins' book, The God Delusion, and have missed the point. Let us ponder for a moment the implications of calling religion a delusion, particularly with regards to this story. If religion is a delusion in this context, then people who take actions based on their religious beliefs are legally insane while doing so. Therefore, they can plead "not guilty by reason of insanity". That's quite an unfortunate implication! I think religious people should be held responsible for their religiously motivated actions, don't you?

If we want to avoid this implication, we cannot consider all religion to be delusional, at least not in the legal sense. Of course there may exist some religions which are delusional in the legal sense. For instance, in the case of 1Mind Ministries, it looks like there are a lot of cult-like practices (though "cult" is an imprecise term to use). However, I will leave it to the courts to decide, since obviously I don't have all the relevant information. In any case, it would be hard to argue that 1Mind Ministries is representative of religion, since it has no more than a dozen adult members.

I contend that the statement "Religion is a delusion" is incorrect not only in the legal sense, but also in the psychiatric sense. If religion were a delusion in the psychiatric sense, then that would also have many unfortunate implications. For one thing, why should it be delusional in the psychiatric sense but not the legal sense? For another, wouldn't that imply that religion is better treated through psychiatric methods rather than, say, a cultural movement led by atheists? Doesn't it imply that religious beliefs, however common, are the result of abnormal mental processes (as opposed to normal processes which have gone wrong)? What are we trying to say here?

Of course, most atheists aren't trying to say any of those things. See, it starts like this:

Just look at the definition of delusion. "A mistaken or unfounded opinion or idea". Religious beliefs are mistaken and unfounded, therefore they are delusions. (Yes, in the same sense that I'm delusional when I think I have my pencil in my pocket when I actually left it on my desk.)

A delusion is "an erroneous belief that is held in the face of evidence to the contrary". Religious people are constantly ignoring the contrary evidence right in front of them, therefore they are delusional. (But we are assuming that they indeed see lots of contrary evidence and understand it properly. As a student and skeptic, I would never trivialize the process of proper understanding by saying it's easy.)

Religious people see and talk to God. What is that but hallucination? (How do we know that they aren't simply interpreting much more mundane mental processes as communication with or to God? That's what I did when I was Catholic.)

It is not a single person who says all these things. One person starts with a harmless comparison between religion and delusion. Another person sees the idea and takes it one step further. At every step, the claim is elevated and elevated, becoming more over the top. Richard Dawkins himself stops short of calling religion a psychiatric delusion. However, through innuendo, he has planted the concept in everyone's mind, and many atheists do not stop where Dawkins did.

I prefer to not call religion a delusion at all, except in the very weakest sense. It is important to recognize the true source of mistaken beliefs. They are not the result of some abnormal mental process. They are most often the result of a normal mental process which has gone wrong. Critical thinking is not a trivial task, and is not always successful even when executed perfectly. And then there is a large social component to religion. By calling religion a delusion we are at best being sloppy about the cause of religion.

Calling religion a delusion is more shocking than accurate. But just look at the case of Javon Thompson. The reality is shocking enough, without enhancement!

The purpose of ritual

A lot of irreligious people don't really understand the purpose of traditional rituals and ceremonies. But then, neither do a lot of religious people. Some might think the purpose is simply because God wills it. But even if that were true, the explanation of God sure isn't instructive. It's really hard to say why God wills it. And since you don't know why, you know very little about the exceptions, when ritual and tradition are no longer worthwhile.

So what is the purpose of ritual? Sometimes, there is a functional purpose. For instance, every time I practice my flute, I warm up first. I play long tones, scales, vibrato exercises, etc. I don't really know how or why, but I believe that these warm-ups improve my subsequent practice session, and have long term benefits to my technique. But I certainly don't have any scientific papers to establish these facts; as far as I'm concerned, it is simply folk wisdom taught by my teacher. And that's fine. The purpose of this ritual is to apply wisdom which has been accumulated over however-many generations, even if I don't fully understand the benefits.

But if that seemed like a strained example of a ritual, that's because it is. After all, it doesn't really matter how exactly I warm-up. It doesn't really matter whether if I play the scales in chromatic order, or in order of fifths or fourths, or if I decide to play harmonic minor scales instead. I could skip it altogether, and all that happens is I sound like I haven't warmed up. What matters is the functionality of the warm-up, rather than the warm-up itself.

But the essence of a ritual is not in its functionality, whatever that may be. The essence of a ritual is in the order it imposes in our chaotic lives. For instance if you want to marry someone, you might be confused about how to propose. I know I would be nervous. I would worry about all the details, about any little thing that might go wrong. Luckily, there is a standard format for a proposal which everyone knows about. You drop down on one knee, and ask, "Will you marry me?" Above and beyond that, you may add whatever you like.

The same goes for the marriage itself. If marriage ceremonies didn't exist, I would feel kind of weird about just saying, "Okay, I guess we're married from tomorrow forward. We'll be married... after midnight?" It would feel as if nothing important had happened. Like we could just go back to how it was in an instant. But wasn't it important? Shouldn't it have felt more special?

I could go on and talk about other ceremonies marking important moments of our lives, such as baptism, confirmation, or funerals. But there are also lots of rituals which are performed on a more regular basis, such as a weekly church service.* The reasons for these regular rituals is much the same. But rather than providing a structure and format for your entire life, they provide a structure and format for your daily life. If, for example, you're a church-going Catholic, you go to church because it gives you some comfort in your chaotic life. In that brief service, you know who you are and what you are doing--you are a Catholic, and you are participating in a ritual with a prescribed role. You also know what to feel--perhaps it is a sense of devotion, or a feeling of "I'm going to go out there and do good this week!"

*Please note that weekly services are not entirely ritualistic, but also serve several functional purposes.

Or maybe you feel none of this. Maybe you feel even more confused, or maybe you find the whole thing positively boring. I daresay that many religious people practice their religion not because they are getting anything out of the rituals, but because they simply feel obligated to do it. Well, here's what I think. Regular rituals are not for everyone. Nor are ceremonies. If a conscious and benevolent god existed, it would not want everyone to be forced into rituals, not even the subset of people who believe in him.

105 ways to promote skeptical activism

Daniel Loxton of the Skeptic's Society has recently released What Do I Do Next?, which includes 105 practical ways to promote skeptical activism. There's a quick reference guide, a sixty-some page pdf, and a podcast interview with Daniel Loxton.

Well, this is great! While I am pretty enthusiastic about skepticism, I've always felt a little defunct as an activist. Well, I suppose I have this niche-within-a-niche blog called--you may have heard of it--Skeptic's Play. Item 86 of the guide suggests starting a blog which focuses on a very specialized part of skepticism. The problem is that though this blog is a niche-within-a-niche, it is not, and never will be focused or specialized. I am skeptical about the effectiveness of my blog as a tool for skeptical activism.

I'm also the secretary of the campus' skeptical/atheist club, BASS, and I've made the meeting minutes public online. I also use my written rhetoric to carry out my sinister plans in the background (you know, picking out discussion topics and stuff). But I wield no real power. I note that our group breaks one of the rules stated in item number 26: "Clubs should have a clear mandate for either skepticism or atheism but not both." Daniel says that groups which advocate both skepticism and atheism frequently go into conflict and "flame out." For what it's worth, our group strikes a good balance, and is fairly successful.

But if you ignore everything that I do, you'll find that I don't do anything to promote skepticism. How can we fix that?

One of the easiest things on the list is to write letters to editors, reporters, and politicians either approving or disapproving of their stories and actions. I write a whole lot anyways, so why not write letters, where people might actually read what I say? But what to write? Next time something comes along, I promise to write some sort of letter. Not because a single letter will change the world, but because all patterns and habits need to start somewhere.

The internet better hold me to my promise, or I'll never follow through.

There was one suggestion that I found vaguely intriguing, but also decidedly vague:

98. If you are a student, use your technological networking talents for skeptical activism — but get credit for it!

What's that?! How? I do not understand.

Online activity has the reputation of detracting from schoolwork.

Well, luckily, I have the advantage that I already know everything and therefore do not need to study.

Rather than pouring effort into undirected work at less-coordinated “social” blogging networks with no mission statements, some students find ways to reap academic rewards for their work online.

Students who are interested in this type of work should get in touch with science youth groups, education support, and monitored message boards so they are not only safe but also reaching an audience that needs to be reached.

I'm not really sure what that means, but it sounds like I would have to work or something? For some reason, my initial reaction is, "I don't have time for that!" And yet, here I am, wasting all my time blogging. Curse my internal double standards!

Appeal to future evidence

It's logical fallacy time!

"My theory may sound pretty crazy now, but soon everyone will come around. The scientists will find evidence validating my theory, and then there will be the biggest scientific revolution since Einstein! If you buy my book, you can be at the front of that revolution, and tell your grandchildren all about it."

This should be an obvious one. Evidence is useless unless it is present. As in, the evidence exists right now, at this moment. Evidence from the future doesn't mean a thing. 'Cause if the evidence doesn't exist yet, how do you know that it will ever exist? What evidence do you have to support such a prediction?

Future evidence is never a certain thing. Evidence is essentially a piece of information or an observation which makes a claim more likely than we previously thought. If we knew for sure that we were going to find evidence in the future, than this piece of knowledge counts as present evidence.

Science is not about going out and finding evidence for your ideas. It is about determining whether there is evidence for your ideas, or evidence against them.

Anyways, there is no shame in being wrong when the current evidence is incomplete, thus pointing you in the wrong direction. If, for instance, you believed in evolution prior to the 19th century, you would be correct in the sense that evolution was correct. But you would be incorrect in that you believed a scientific claim without any evidence to support it. If you believe lots of things which are unsupported by current scientific evidence, you might occasionally be correct. But far more often you'll be incorrect, because you've basically abandoned the scientific method in favor of the guessing method.

On Lent

I've discovered that a lot of atheists have at least a passing interest in the traditions of Lent. What's up with Lent? Why do Catholics abstain from meat on fridays, but are okay with fish? Isn't fish a meat? Well... yes...

But as an ex-Catholic, all I can say is this: there is nothing inherently objectionable about Lent. It's just a collection of customs and traditions in the forty days* preceding Easter. The forty days represent the time Jesus spent in the desert being tempted by the Devil. Easter, of course, celebrates the most important event of Christianity (as in, should be more important than Christmas), the resurrection.

*Actually, I just found out that it lasts forty-six days, because they don't count the Sundays as part of Lent. I guess I never really bothered to count the days before.

I'm sure there are a lot of Lenten traditions which have disappeared or changed over the ages, but there were a few that even I used to practice. I guess most importantly, there are some special church services. For instance, on Ash Wednesday, the first day of lent, there is a service which ends with everyone with an ash cross smudged on their foreheads. On the fifth sunday, Palm Sunday, we had what I always considered the most exciting service, because instead of being bored as usual, I had a palm leaf to play origami with. Ah... that was a long time ago.

And of course there's the well known tradition of giving up something for lent. I never really bothered with this myself, but I don't consider it any different from the practice of making new years resolutions. If anything, it's more reasonable; many people successfully keep up their sacrifices for forty days, but nobody actually keeps their new years resolution for the entire year.

And then there's the meat. Why doesn't fish count as meat? Because the tradition's motivation has nothing to do with vegetarianism. I imagine the tradition is meant as another form of giving up luxuries. Back then, other meats were more expensive than fish, though nowadays it is reversed (I quite openly prefer tuna sandwiches to turkey sandwiches myself). But honestly, I have no idea. I also have no idea why on Chinese New Year, we receive red packets with money and say "Gung Hey Fat Choi" (it's probably a pun--the Chinese take their puns very seriously). Whatever the motivation was, the current motivation is that it's tradition.

I never really took these traditions very seriously myself. I recognized them for what they were, and assigned them the appropriate priority. It doesn't really matter that much, and no one cares if you skip it. Okay, I guess a few people care. In particular, I bet the Church cares. I think it might have used to be a mortal sin to break some of the dietary restrictions. But that's silly.

To hear a mockingbird

I think it was many years ago that I first heard the call of the mockingbird. I was lying in bed, trying to sleep, when I suddenly realized there was a strange noise coming from outside. I wasn't sure if it had just started, or if it had been going for hours. For all I knew, I could have been hearing that same sound every night for my whole life, and never noticed it. But for whatever reason, I noticed it this night.

At first, I had no idea what it was. Why is there a bird singing in the middle of the night? Do birds do that? And its call was very distinctive. It would tweet one way three or four times in a row, then it would tweet another way three or four times in a row. And then it would tweet another way, and another way, never repeating itself. So I suddenly realized, it's one of those mockingbirds I've heard so much about!



I recall that in To Kill a Mockingbird, there was a single reference to mockingbirds:

"Mockingbirds don't do one thing but make music for us to enjoy. They don't eat up people's gardens, don't nest in corncribs, they don't do one thing but sing their hearts out for us. That's why it's a sin to kill a mockingbird."

But Miss Maudie! There's a mockingbird nearby that's always singing its heart out at 2:00 am. It's pretty loud too. Maybe if its call weren't so interesting to listen to, I would mentally block it out like I do with the freeway noises, but as is, I think it's cutting into my sleep! (Not that I would ever shoot at it...)

The three nihilists of Watchmen

One of my favorite aspects of Watchmen (I'm mostly talking about the graphic novel here, not the movie), was the rich characterization all the heroes. In particular, I found it fascinating that at least three of the characters, the Comedian, Rorschach, and Dr. Manhattan, could be described as nihilists--though each has his own distinct flavor of it. I thought it might be enlightening to compare and contrast these characters.

[I will not be revealing any major spoilers in this essay. Just characterization.]

The book begins with the murder of the Comedian, one of the old costumed heroes. But as we look into the Comedian's past, we see that the comedian is hardly heroic, and hardly funny either. Instead the Comedian gives us the raping and pillaging which we might expect when we hear the word "nihilist". As a costumed "hero", he interacted with the worst of humanity, and found it overwhelming. What is the point of morality in a world where nukes might start flying any moment? What is the point of morality when so many people simply ignore it? The only way he could find to deal with the cruelty of humanity was to become a parody of it.

The Comedian saw little point in fighting it. When the second generation of costumed heros attempted to form a group called "The Crimebusters", the Comedian literally burned up their plans. To him, there was no point in fighting crime, because there are far greater evils which can never be solved by heroics. Of course, it seems that he did continue to fight crime. He also went on to help the government, fighting in the Vietnam war. But this is all just a front for the Comedian to do whatever he likes, to exercise his sadism and depravity. He's not trying to stop the joke, however bad it might be, he's just trying to play along. The worst part is that he seems to get away with it, even with the government's approval.

One of the few characters that actually admires the Comedian is Rorschach. I found this rather ironic, because Rorschach really hates criminals--and he thinks everyone is a criminal. He views the city as populated by drug dealers, child pornographers, whores, and other scum. Even when costumed heroes are outlawed, Rorschach continues to terrorize the streets, often torturing people in bars in order to get his information. He has a strong, uncompromising sense of justice, one which would grate on the liberal-minded.

And yet he seems to like the Comedian. Perhaps it's because Rorschach and Comedian both have similar insight into the evil of human nature. Rorschach knows that the Comedian "gets it", so he's blind to the Comedian's sadistic actions (not to mention Rorschach's own actions). But even though they have similar insight, they react in very different ways. Where the Comedian reacted by joining in the violence, Rorschach reacted by fighting it (...with violence). As Rorschach said, "It is not God who kills the children. Not fate that butchers them or destiny that feeds them to the dogs. It’s us. Only us." It is our fault, and our fault alone, therefore it is our sole responsibility to fight it, however futile it may be.

At this point I should mention a fourth nihilist in the novel, a minor character (if you only saw the movie, you missed this one). There was a chapter about how Rorschach basically brought his psychologist, Dr. Malcolm Long, over to the dark side. After that's done, Dr. Long says in his notebook, "I looked at the Rorschach blot. [...] The horror is this: In the end, it is simply a picture of empty meaningless blackness. We are alone. There is nothing else." I thought this slightly ironic considering that the Rorschach ink blot test is mostly discredited nowadays (it really is meaningless). Dr. Long's reaction may sound bleak on the surface, but there's more to it than that. After his experience, he began to have marriage troubles. Why? Because he became too concerned with helping others, and his wife was used to him being a lot happier.

Of the three (or four) nihilists in the story, Dr. Manhattan is the odd one out. Of course, he would be the odd one out of any group of humans, since Dr. Manhattan is hardly human himself. Once a physicist, he died in the terrible accident, only to reappear as an omnipotent blue being. He's basically a god. He can see his own past and future as if they were all occurring right now. And yet he occasionally changes his mind, sort of like how God does in the Old Testament. Characters find this frustrating, since it's almost as if he simply doesn't care to exercise his free will to change the future. Dr. Manhattan would say he simply doesn't have free will.

But it's also true that he just doesn't care. At first, he wants to, but why would he? To him, humans are about as interesting as termites. He can simply point at them and disassemble them. This would horrify most people, but to Dr. Manhattan, it's just a trivial rearrangement of particles. While the nihilism of the Comedian and Rorschach was motivated by the dissonance of human cruelty, Dr. Manhattan's nihilism is motivated by something more philosophical. What is so special about humans that he should assign them any more meaning than, say, the geological processes on Mars? What, don't the geological processes on Mars interest you?

Eventually, he does get his answer. I felt his answer echoes (and predates) this famous quote from Richard Dawkins: "We are going to die, and that makes us the lucky ones. Most people are never going to die because they are never going to be born. The potential people who could have been here in my place but who will in fact never see the light of day outnumber the sand grains of Arabia." Each life is extraordinarily improbable and extraordinarily specific. There are so many vastly different people that could have been here instead, but aren't. Dr. Manhattan compares it to a statistical fluctuation which reduces entropy. (Of course, the more I think about it, the poorer this seems as a basis for ethics. As long as there's no nuclear war, I doubt Dr. Manhattan cares about the quality of human life.)

The Comedian, Rorschach, and Dr. Manhattan together display the complexity of nihilism. Most people, when they think of nihilism, probably first think of someone like the Comedian. After seeing to much evil in the world, he decided that there was no meaning in it all. So he joined in on the evil, literally raping and pillaging. But that's not necessarily the inevitable conclusion. Where the Comedian simply played along with a meaningless world, Rorschach forcefully imposed his own meaning onto it. Rorschach responded to nihilism with a single-minded pursuit of justice and retribution. Dr. Malcolm Long shows a similar type of nihilism as Rorschach, albeit with less of the crazy. And there is a final form of nihilism shown, not motivated by a cynical view of human nature, but by a philosophical failure to find value in life. And yet, as Dr. Manhattan finds, humans are very valuable even by standards which are completely alien to ours. If you're even a little more human than Dr. Manhattan, then the value of life is definitely there, somewhere.

Talk like a physicist

Today is March 14. It's best known as Pi day, because the date 3.14 is equal to the decimal expansion of pi to the hundredth's digit. Lesser known is that it is also Albert Einstein's birthday. And even more obscurely, it is Talk Like a Physicist Day, an occasion that probably only physics bloggers are aware of...

Therefore, I think now is a good time to muse about those crazy physicists and their crazy naming conventions. Especially particle physicists. This SMBC comic is pretty accurate...

[Pro Tip: Hovering your mouse over that red circle on the SMBC website gives you extra funny]

Sadly, I must inform you that there is no such thing as a "splork" in particle physics... yet. However, I assure you that squarks are a serious possibility.

Let's break it down. These are all honest-to-goodness particle physics names!

• There are basically two types of particles: Fermions and Bosons. Fermions are "matter-building" particles, while Bosons are associated with forces.
  
• Bosons consist of photons, gravitons, gluons, and the W and Z bosons, which are associated with the electromagnetic, gravitational, strong, and weak forces respectively.
  
• The Fermions consist of Leptons, and Quarks.
• There are six flavors of leptons: electron, muon, tau and three associated neutrinos. Yes, "flavor" is a technical term.
  
• There are six flavors of quarks: up, down, charm, strange, top, and bottom
• I'm told that the top and bottom are sometimes called truth and beauty. Ironic, considering how elusive these particles are.
  
• Each quark has an associated quantum number. There's upness, downness, charmness, strangeness, and bottomness. Usually, no one cares about the top's quantum number, but I've concluded that it should be called "truthiness".
• The quarks always come in groups called hadrons. A quark-anti-quark pair is a meson, and a three-quark group is a baryon. Protons and neutrons are two kinds of baryons.
  
• Every fermion has an associated anti-particle. Not only do you have ups, downs, and electrons, but you also have anti-ups, anti-downs, and positrons, and so forth.
  
• You also got theoretical particles which have not yet been observed.
• The Higgs boson!
• Tachyons! They travel faster than light, and can go backwards in time.
• Super-symmetric particles! According to supersymmetry theory (abbreviated SUSY), every fermion should have a supersymmetric bosonic partner, and every boson should have a supersymmetric fermionic partner.
• The naming convention for supersymmetric partners of fermions is usually to add an "s" at the front of the name. So you got squarks and sleptons. Sleptons include the selectron, smuon, stau, and sneutrino. Squarks include sup, sdown, scharm, sstrange, stop, and sbottom.
• The naming convention for supersymmetric partners of bosons is usually to add an "ino" at the end, pronounced "ee-know". So we got photinos, gluinos, gravitinos, higgsinos, winos and zinos.
• Some mixture of photino, zino, and higgsino gives you the neutralino, which is the leading WIMP candidate for dark matter.

I'd say "I can't make this stuff up", but I think the problem is precisely that it seems I can make this stuff up.

Anyways, I think it's great that we have these names which are accessible and humorous. And perhaps the names give you a small window into how physicists view physics. We don't need some really arcane terminology, because the basic concepts which we describe are really not that arcane.

I watch the Watchmen

I saw Watchmen last week.

Watchmen takes place in 1985, in an alternate history where Nixon has been elected for his fifth term, and the US won the Vietnam war. In this alternate history, there existed people who, inspired by superhero comics, decided to become costumed heroes themselves. These people tended to be a little unbalanced, and many had fascist tendencies, but for a while it was all well and good. But costumed heroes were eventually outlawed when it became clear that standard heroics were very poor solutions to the world's problems. In particular, nuclear war with Russia is a looming threat, and has thus far only been prevented by the existence of a single superhero, Dr. Manhattan. Dr. Manhattan is practically omnipotent, but has lost almost all humanity he once had.

In this setting, we begin with the murder of one of the old costumed heroes, the Comedian. As the plot slowly unravels, we meet many of the other costumed heroes, all of whom are deeply complex characters.

I thought the movie was great! I don't think I've ever seen a movie which so faithfully reproduces the novel it is based upon, but that's good because I absolutely loved the graphic novel too. The novel had a very cinematic flow to it, so at times it seemed the movie copied the novel right down to the camera angles. If you didn't read the book, you must see the movie, and if you don't see the movie, you must read the book. Ideally, you should do both.

So after that brief excursion into movie-reviewing (a form of writing in which I do not excel), you should see this video which delightfully subverts nearly everything that Watchmen stands for.

Hmmm... I think I might write something more about Watchmen in the near future... something suitably philosophical.

Two more measuring problems

See the previous two measuring problems

You have two lengths of rope. Each one takes exactly one hour to burn. However, they are not necessarily identical in length or size. In fact, neither of them is uniform. That is, after half an hour of burning, there might not be exactly half of the original length remaining. Therefore, you cannot simply wait for the fire to reach the half-way point and assume that exactly 30 minutes have passed.

Use these two ropes to measure 45 minutes.

And now for the challenge problem! It's an old one that I had written, and now it's yours.

You have three unmarked, asymmetrical containers, and no others. The first holds 6 cups, the second holds 10 cups, and the third holds 15 cups. The 10-cup pitcher is filled with lemonade, but it's too strong. You need to dilute it to half of its current concentration. You have an unlimited source and sink for water, but please don't throw out any of the lemonade! Remember, all the lemonade must be diluted to half-concentration.

Solutions have been posted, or you can just read the comments below.

Extraordinary plotting

Previously, I mathematically modeled the impressiveness of a claim as a function of the extraordinariness of the claim. Basically, the more extraordinary a claim is, the less likely it is to be true, and thus the less impressive it is. However, an extraordinary claim with extraordinary evidence to support it can be very impressive indeed.

DeralterChemiker asked me an excellent question: where would I place a bunch of claims on the graph? Here I have a mathematical model, why don't I try applying it?
Legend:
A. Big Bang Theory
B. String Theory
C. Evolution
D. Life on Mars (microscopic)
E. Anthropogenic Global Warming
F. Homeopathy
G. Orbital theory (ie gravity)
H. The existence of God (falls somewhere on the line depending on who's claiming it)

Of course, the whole thing is rather subjective, since there isn't any obvious way to quantify the extraordinariness and impressiveness of claims. Nor is it obvious how I should scale it (can gravity and homeopathy really be on the same graph without distorting the scale?). Really, it's the general method and general patterns which are important, not the conclusions.

However, I'd be interested to see some of my readers' conclusions. Where would you place some of the claims you've heard? For instance, Deralterchemiker also suggested plotting Obama's economic policy. But you can also try such favorites as "Brocolli is healthy" or "Human nature is essentially good". Give the coordinates: (extraordinariness of claim, amount of evidence)

When making this graph, it became clear to me that I needed to define terms. For my purposes here, the "extraordinariness" of a claim is basically a measure of how initially unlikely it is. A claim is extraordinary if it initially goes against common sense or common observations. A claim is extraordinary if it is very complex and specific, going against Occam's Razor. It is also extraordinary if it contradicts other well-established claims.

The other that became clear while making this graph, is that it's really difficult to judge where to place claims. I'm not happy with how the scaling turned out. And sometimes a claim is extraordinary precisely because it has evidence against it. How do you separate out extraordinariness and evidence anyways? I guess you gotta remember that "Extraordinary claims require extraordinary evidence" is simply a rule of thumb, not to be quantified unless you're feeling particularly artistic.

Bell's Theorem explained

It may help to see my previous post for background information on Bell's Theorem. However, I hope that this is understandable, even if you didn't read the background.

Quantum mechanics is well known for saying that particles can be in many locations at once, smeared out in a probability wave. Why can't we simply say that particles have definite locations, and that only our knowledge is complete? Because as Bell's Theorem shows, we will run into problems.

The set-up is that we have a source which emits pairs of entangled electrons. Each electron will go into a separate detector. These entangled electrons exhibit strange correlations in their behavior. If you measure the vertical spin of each electron, then both electrons are guaranteed to give the same result. However, we need not always measure vertical spin. We can also measure the horizontal spin, or any angle inbetween the horizontal and vertical. Let's tinker around with it.

We're going to use three different settings for the two detectors. Setting A will ask the question, "Is its spin going up?" Setting B will ask, "Is its spin going to the right?" Setting C will ask, "Is its spin going at a 45 degree angle between 'up' and 'right'?" If both detectors are on setting A, then they will both give the same answer for any given entangled pair of electrons. Likewise, they will give the same answer if they are both on setting B, or both on setting C.

Let us assume that each electron has predetermined answers to each of these three questions. We just don't know the answers yet! We can describe each electron as a set of three answers to the questions posed by detectors A, B, and C. For example, we could use (Y,Y,N) to denote an electron which will answer "yes" to A and B, but "no" to C. Similarly, (Y,N,Y) would denote an electron which will answer "no" to B, but "yes" to A and C.

Recall that if both detectors are on the same setting, then both electrons will give the same answer. This means that both electrons in a pair can be described with the same set of three answers. Each electron pair can be placed somewhere in this Venn diagram:

There are eight possible locations on the Venn diagram, but there's no reason to assume that each possibility is equally likely. Perhaps we could determine the probability of each possibility? But it's rather tricky. The problem is that we can only determine two of the answers for any given pair of electrons. Once each electron has been measured, we've changed their original quantum state and can no longer investigate it. Nevertheless, we can be mathematical detectives and figure it all out, right?

So let's say that we set the first detector to setting A and the second one to setting C. How often is the answer to A different from the answer to C? That is, how often do the electron pairs fall into the shaded region shown in figure (a)? The answer is about 15% of the time, or (1-cos(45°))/2. This can be predicted by quantum theory, but just as importantly, it can be directly observed with experiments.

What happens if we set the first detector to B, and the second detector to C, and we ask how often the two detectors give different answers? That is, how often do the electron pairs fall into the shaded region in figure (b)? Again, the answer is 15%.

How often do electrons fall into the shaded region in figure (c)? Figure (c) is the "union" of the two sets (a) and (b). All electrons which fall into the shaded regions in (a) and/or (b) will also fall into the shaded region of (c). This implies that electrons will fall into shaded region (c) no more than 30% of the time.

To summarize:
(a) Detectors A and C give different answers: 15%
(b) Detectors B and C give different answers: 15%
(c) The union of (a) and (b): At most 30%

Now, what happens if we set the first detector to A and the second detector to B? How often will the two detectors give different answers? If they give different answers, then the electron pair must fall somewhere in the shaded region of (c). Therefore, it couldn't happen more than 30%, right? Wrong! Both quantum theory and experiment show that the answers to A and B are different 50% of the time. We've run into a mathematical contradiction!

What we usually do when we find a mathematical contradiction, is we back up. We look around for assumptions. One of our assumptions must have been wrong! At the top, I said, "Let us assume that each electron has predetermined answers to each of these three questions." Under the mainstream interpretations of quantum mechanics, this assumption is wrong. Electrons do not have predetermined answers to every possible measurement. Instead, we must describe their states with probabilities.

I should also mention that there are ways around this conclusion. Perhaps I made other implicit assumptions, and maybe those assumptions are the wrong ones. There are several other ones, which have names like "realism", "locality", etc. But I'm not going to go into that. I'll just leave Bell's theorem as is, a proof that something is going intuitively wrong.