River-crossing solutions

See the original puzzle

1. The fox, chicken, and lettuce. Someone pointed out to me that it's usually stated as fox, chicken and corn. Well, sure, if you like your puzzles to make sense. :-)

Bring across the Chicken.
Come back with nothing.
Bring the fox across.
Take back the chicken.
Bring the lettuce across.
Bring the chicken across one last time.

2. Three missionaries and three cannibals. This one was solved by Secret Squïrrel. For those who didn't know, I give free linkage to the first solver of any puzzles. I appreciate it when other people write up the solution so I don't have to.

mmm ccc || nobody
mmm c || cc (2c ->)
mmm cc || c (<- 1c) mmm || ccc (2c ->)
mmm c || cc (<- 1c) m c || mm cc (2m ->)
mm cc || m c (<- 1m1c) cc || mmm c (2m ->)
ccc || mmm (<- 1c) Now the c come across in 3 trips.

I leave it as an exercise to the reader to find out how many missionaries and cannibals you can safely bring across if you have a boat that holds three people.

Why "pseudoskeptic" is a worthless label

A few weeks ago, the local skeptical group decided it would be a good idea to have "pseudoskepticism" as a topic. The idea is that we get to discuss what sort of mistakes denialists make, and in turn elucidate the definition of a "skeptic". It turned out that it wasn't a good topic after all. People really seemed to dislike the word "pseudoskeptic", though it's hard to properly convey what's so wrong about it.

More recently, Matt Nisbet, Scienceblog's notoriously uncharismatic proponent of "framing", attacked the word "denialism" on account of it being inflammatory. I don't know about that. It seems to me that if "denialism" is negative, it's not because of the word itself, but because of the content of its meaning (if that makes any sense). The content of its meaning isn't going to go away, no matter what we call it. But forget Nisbet. (Forget, I say!) My main point relates not to Nisbet but to this Respectfully Insolent response. As an afterthought, Orac says this in his response:

Of course, if you're less pugnacious than Mike, Mark, or me, in my benevolence, I'll suggest an alternative term other than "denier" or "denialist." Lately, I've started to like the term "pseudoskeptic." It captures the essence of what denialists do almost as much as the term "denialist." Remember, a true skeptic is always open to changing his or her mind if the evidence and science demand it.

No! Don't use "pseudoskeptic". It's bad!

The first thing that strikes me about the word "pseudoskeptic" is that it gets us into the "fake vs true" mode. It's the perfect setup for the No True Scottsman Fallacy. Every time I hear the word "pseudoskeptic", it practically begs to be replaced with the phrase, "not a True Skeptic (TM)". After all, if you've never heard the word, that's exactly what the conjunction of "pseudo" and "skeptic" will mean to you. If it came into popular use, I can just imagine the devolution of discussions. "You're a pseudoskeptic!" "No you're a pseudoskeptic!" I hate the word, I hate it!

If my worries seem farfetched, we only need look at the history of the word. (Skeptical history!) It's common knowledge (read: on Wikipedia) that the word was coined/popularized by one of the founders of CSICOP, Marcello Truzzi. CSICOP, the Committee for the Scientific Investigation of Claims of the Paranormal (now simply called CSI), was founded in 1976 as one of the first skeptical organizations to ever exist. Marcello Truzzi was the editor to CSICOP's official journal, The Zetetic. Unfortunately, Truzzi had some sort of falling out with CSICOP because he wanted to include pro-paranormal stuff in the journal. Truzzi left CSICOP, and founded his own journal, The Zetetic Scholar, which included arguments both for and against the paranormal. It was around this time that Truzzi said that the so-called skeptics were becoming pseudoskeptics.

The temptation to simply say, "No, you're a pseudoskeptic" is strong. In my mind, Truzzi was completely wrong, and CSICOP completely right. Truzzi's position was that of pyrrhonic skepticism, the position that we cannot know anything for sure. Pyrrhonic skepticism is wrong because we can know things, not with complete certainty, but with sufficient certainty. I can't know absolutely for sure that psychics don't exist, but I can be sufficiently sure, given the vast amounts of evidence, that they are vastly unlikely. There is a lot of merit to drawing tentative conclusions even when you're not completely sure. Truzzi's position is that of "fair and balanced" journalism, which simply portrays both sides of every issue equally, regardless of the relative merits of either side.

Truzzi, perhaps sensing that the word "pseudoskeptic" would simply bounce back at himself, decided to take another label for himself: "zetetic". A wise move, I say.

But doesn't "pseudoskeptic" have its uses? It seems like it would be useful against factions that call themselves skeptical, the epitome being the "Global Warming skeptics". Now, personally, I don't really think of them as fake skeptics. No, they are true skeptics, albeit under a different definition of skeptical. Skepticism means many different things, after all. When I use it, I refer to the method of determining the veracity of claims through rational and scientific thought. Other times, skepticism means something like Truzzi's zeteticism. In the case of Global Warming skeptics, it simply refers to a position of doubt. It's a moot point whether that doubt has been achieved through proper use of rational and scientific thought.

We don't have a monopoly on the meaning of "skepticism". But of course, we'd like to. Therefore, it's to our advantage if we disassociate the skeptical method from the position of Global Warming skepticism. To do that, we need to give them a different name. But "pseudoskepticism"? Please don't. In my benevolence, I'll suggest an alternative term other than "pseudoskeptic". Use "denier" or "denialist". The meaning of the term is obvious, even to someone who has never heard it before: "One who denies". It may have a slightly inflammatory connotation due to its connection to Holocaust denial, but that's pretty weak. Global Warming denial is distinct from Holocaust denial--I should have thought that obvious.

What the bleep do we know!? reviewed

Some of you may remember hearing about the film What the Bleep Do We Know!?, which came out in 2004. It’s a documentary that explains a rather new-agey interpretation of Quantum Mechanics, and how this might apply to consciousness and daily life. I know that this is old news, but when it comes to something as nearly mainstream as What the Bleep, it doesn’t hurt to revisit it one more time.

The Science

As the title of the movie suggests, one of the central messages of the movie is about how little we know. This is perhaps an honorable sentiment, discouraging arrogance and encouraging open-mindedness. It is in fact true that Quantum Mechanics has several different interpretations which are “up for grabs”. However, it’s not completely up for grabs; you do need to know a bit about the science in order to make an informed judgment on the interpretations. What the Bleep, betraying its supposed open-mindedness, gives you a very spotty and distorted view of the science.

Much of the Quantum Mechanics explanation occurs on "Duke Reginald's court of unending possibilities”, a basketball court governed by quantum mechanics. The film uses fancy effects to drive home the strangeness of quantum mechanics. When you aren’t looking, that basketball is in many places at once. When you look at the basketball, the possibilities collapse into one. What the film fails to mention, is that this is only really true on small scales. Every particle is in many places simultaneously, but most of those places are in a tiny region smaller than a nanometer.

That single omission, by itself, would be forgivable. I've seen worse. However, they go on to deliver a very wild interpretation of quantum mechanics: we choose the results of our observations. Really? So if we wanted, and believed, with every fiber of our being, we could choose to observe a basketball one meter to the left? Wouldn't that contradict quantum mechanics, which states that there is only a small probability of it appearing one meter to the left? I must have missed the "human will" term in Schrodinger's equation, because last I checked, the probabilities are completely independent of the observer's state of mind.

Furthermore, they've completely jumbled their interpretations of Quantum Mechanics. Presumably, they're going by the Copenhagen Interpretation because that's the only one with "observers" in it. Under the Copenhagen interpretation, when we observe objects, they "collapse" into a single possibility. However, "observer" is a technical term that has been abused here. The "observer" need not be human. The observer could simply be a measurement device. The device could be measuring millions of particles, but it will still collapse all of them, regardless of whether anyone is paying attention to the specific results. This is an undeniable experimental fact that must be explained by all viable interpretations of Quantum Mechanics.

You could say that the measurement device itself splits into many different possibilities. In one possibility, the device sees the particle here, and in the other possibility, the device sees the particle there. And then, when we look at the device, it collapses into only one of those possibilities. But then, if you like, you could also say that humans split into many different possibilities, and that no collapse is occurring at all. This is called the Many-Worlds Interpretation.

I'm glossing over all the details here, but this is still way better than what appeared in the film, which didn't even bother to name a single quantum interpretation.

The Copenhagen Interpretation and Many-Worlds Interpretation are the two most popular interpretations of quantum mechanics. In neither of them is the human mind so completely special that it's the only thing in the universe that can collapse quantum possibilities. Why should we think that consciousness plays any role if a mere measuring device gives the same experimental results?

Their flawed interpretation goes completely awry when interviewee Candace Pert goes on to say that human cells can collapse quantum possibilities, and therefore they are the smallest units of consciousness. This led into some rather boring scenes with anthropomorphic cells. So, let me get this straight. Because only human observers can cause quantum collapse (false!), consciousness must play a role. And then, because other things also cause quantum collapse, we reason that those other things must be conscious too. Garbage in, garbage out.

The Pseudoscience

You might think, so far, that the movie simply gets a bunch of details wrong about quantum mechanics. Yes, I'm a pretty harsh judge when it comes to quantum mechanics. However, they're not just wrong on obscure science details, they're wrong on common sense. These are some strange claims that were scattered throughout the movie:

1. Interviewee Joe Dispenza says that when people see a picture and when they imagine a picture, the same areas of the brain light up on scans. He goes on to claim that this means the brain doesn't know the difference between what it sees and what it remembers. I would have thought a guy like Dispenza would realize that a brain scan doesn't tell you absolutely everything about the state of your brain. Just because two brain scans look the same doesn't mean our brains can't tell the difference.
   
   
2. Interviewee Candace Pert claims that Native Americans initially could not see Columbus' ships approaching, because they had no concept of what a ship is. I'd say this is a good example (if it's true) of how we are likely to see the things we want to see, and miss things that we don't want to see. But the film reenacts the scene in the most literal way imaginable. The shaman cannot see the boat, but he can see the ripples. Only after much watching does the boat appear. I've got to wonder, even if the story were true, how would we know it was true?
   
   
3. Interviewee John Hagelin says that in a 1993 experiment, he got 4000 people to meditate in order to reduce crimes in Washington D.C. He claims the experiment successfully reduced crime rates by 25%. But 25% reduction compared to what? It was 25% reduction, as compared to the predictions of a model of their own construction. Great.
   
   
4. The film uses as a prominent example the water experiments of Masaru Emoto. Emoto attached words to glasses of water, and allowed them to form crystals. He then used microscopic photography to capture the resulting crystals. Of course, when the experimenters know exactly which word is attached to which glass, they'll easily be able to pick out whatever crystal best matches. It's perhaps an interesting photography project, but it totally fails as science.
   
   
5. Interviewee Micael Ledwith claims that if you accept with every "rudiment of your being", you can walk on water. What if I say with every rudiment of my being that you can't? Does that cancel it out or something?
   
   
6. Joe Dispenza says some pretty weird things about he "creates his day" through quantum mechanics. He claims lots of things happen that are unexplainable any other way. No examples, though. If you ask me, this is a much better example of people seeing only what they want to see than that Columbus thing.
   
   
7. The punchline: One of the interviewees is Ramtha, a 35,000 year old spirit-warrior from Atlantis. Ramtha is being channelled by a woman named JZ Knight. She says the funniest things through out the movie, but only in the credits does it say who she really is (I, of course, knew ahead of time). If you do some digging, you find that the entire movie was put out by the Ramtha School of Enlightenment. Of course, I'm just an amateur critic, not an investigative reporter, so you'll want to refer to Salon for the digging.
   

The Fable

One common response is, "What's so bad about it if it helps people?" After all, fables are false stories, but they still convey a moral message.

But if we say it's acceptable to use bad arguments when they come to the right conclusions, what's to stop people from using bad arguments to come to the wrong conclusions? What's to stop people from convincing themselves, for instance, that they really are listening to a 35,000 Atlantean spirit? In any case, if the "moral" of the film is so great, then it stands to reason that you should be able to find a good argument in favor of it. One that doesn't involve distortions of quantum mechanics or pseudoscience about transcendental meditation.

But I must say, it actually struck me how negative the message of the film was.

The most striking example came out of Emoto's water crystals. After showing the photos, a mysterious bystander says to the protagonist, "If thoughts will do that to water, imagine what our thoughts can do to us." That would have been a sorta maybe positive thing to say, if it weren't for the one water crystal photo that said "You make me sick. I will kill you." Apparently, if you think negative thoughts, you'll become ugly? The film also says that if you can't control your emotions, you must be addicted to them. Later in the film, this is shown to happen to the protagonist. I understand negative thoughts being bad and all, but should we really instill into people a fear of negativity? Fear of thought is very negative indeed.

On the more philosophical end, I find their use of quantum mechanics shockingly cynical. They make it very clear that without quantum mechanics, there is no free will. Several of the interviewees seem to think free will is vital to being able to truly live. I study physics, and I think it's really important and everything, but if it turns out that quantum mechanics is wrong, or if it turns out that it doesn't have a significant impact on free will, I think I'd still find the will to live. (I've got a fuller discussion of how quantum mechanics relates to free will here.)

In another example of questionable morals, interviewee Jeffrey Satinover states that some psychological problems are not really psychological problems, but the result of bad choices. That might seem like a positive thing to say to people who are trying to overcome their psychological problems, but you're also saying it to the people who have tried and failed. I guess now it's their own fault for failing. Later, the protagonist chucks her anxiety medication while a voiceover says "Try it out yourself". That may work for a few people, but just imagine the pain it would cause to everyone else.

Of course, not every "moral" in the movie was bad, and I am being a little nitpicky. But this is the sort of thing that happens when you accept bad arguments just because they give you good morals. You'll end up getting not-so-good morals too. All at the price of distorting science. Was it worth it?

Extra links:

Skeptico: Excellent review, with a much more complete list of links

Salon: exposes the Ramtha cult behind the film

Blogging Heads: physicist David Albert talks about how the film misrepresented him through editting

On generalizations

"Generalizations are always evil"

"Generally speaking, anyways"

Are generalizations intrinsically good or evil? Neither, obviously.

As an amateur critic, I talk all the time about people and their beliefs. I could either use an anecdote, implicitly generalizing my experiences to a larger whole, or I could skip that step and talk about groups of people as if they were abstract entities. Or I could stop being a critic altogether. Obviously, I have no business unconditionally condemning generalizations. They're all I have.

However, generalizations can obviously be abused. On the extreme end is racism and other bigotry. One could argue that the evil is in the falsity of such generalizations. Women and ethnic minorities are not actually inferior to white males. Homosexuals are not actually perversions created by the Devil. But perhaps a bit of the problem is in the generalization itself? How might we correct for this?

I think a little intuitive statistics goes a long way. Simply put, generalizations apply to groups, and not to individuals.

Let's say you've scientifically proven that a certain group of people usually exhibits a certain trait X. If you take any subset of the group, and count how many people have trait X, you're going to have a certain amount of theoretical uncertainty, just due to random sampling. That uncertainty is roughly sqrt(N) people, where N is the size of the sample. So if you have only a single individual, the uncertainty is roughly one person.

I'm sure somewhere in there is the message, "You can choose", just like they told us in Minority Report.

And that only goes for scientifically proven generalizations. I can assure you that any generalizations I make are not scientific at all. If I were to estimate, I'd say 0% of the statistics and generalizations on this website are backed up by scientific evidence. They should be treated with the appropriate skepticism.

This is why I take personality tests with a grain of salt. Even if they are indeed scientifically tested (which is more than we can say of many personality tests), they're still based on the idea that I should take statistical results and apply them to myself. The uncertainty is one person!

On the flip side, if generalizations can't be applied to individuals, this leaves a rationale for people hold onto their preconceived prejudices despite the clear existence of counterexamples. Hrrrm...

Random Links

I'm awfully busy this week, and there's so much good stuff out there right now. So enjoy!

Catholics for Choice - They published a long document criticizing the Catholic League. The Catholic League deserves every bit of it.

SkepticBlog - Skeptic - That Name Thing again - An enlightening view on the meaning of skepticism

The Thinker - Critical thinking and Skepticism. They're overlapping, but distinct.

Uncertain Principles - There is no moral content in Many Worlds Theory or Multiverse Cosmology (which are two completely different things btw). I'm 100% behind Chad here.

Cuddly Atheism - Precocious - I love the "thank you" letter from the Catholic School.

River-crossing classics

For this week's puzzle, we will have two simple classics. Remember: in the world of puzzles, simple is not necessarily bad, and challenging is not necessarily good.

1. A man must cross the river with a fox, a chicken, and some cabbage. There is only one boat, and only the man can row it, and he can only take one companion at a time. For obvious reasons, he cannot leave the fox alone with the chicken or leave the chicken alone with the cabbage. How can he do it?

2. Three missionaries and three cannibals must cross a river (geez, whoever wrote this one had a dark sense of humor). They have a single boat that can only carry two people, and there must be at least one person rowing it. If at any time, there are more cannibals than missionaries on either side of the river, the cannibals will eat the missionaries. How can everyone get across alive?

Update: Solution has been posted

Adventures in atheist advertising

In recent news, the American Humanist Association (AHA) paid for a bus ad campaign in Washington D.C. This is what the ad says:

If you're interested in the response to the ad, Friendly Atheist has lots of links and an interview with an AHA representative.

I think it's rather interesting to watch how the atheists react to the ad. I see the same pattern of reactions occurring all the time. Some of us enthusiastically support the cause, while the rest of us are left wondering if it was really that good of an idea. You can count me in the latter group, at least with respect to the bus campaign, but allow me to feign some objectivity here.

Oh, the usual questions bounce back and forth. Are we targeting atheists, theists, or the middle? Is it intended to be an attack on religion, a call to freethinkers, or just a call for discussion? If religious people think it's an attack, is that acceptable or not? If not, is it our fault or their fault? Who are we trying to convince, if anyone, and of what? Are we imitating religion, and is that bad? How many people are we "reaching" and how many people are we pushing away? And to sum up all previous questions, was the ad campaign good, or bad?

Sometimes, I despair of answering such questions. I'll stick to physics, thank you.

Perhaps, if I try to explain it, I will help myself understand?

The plain message itself is not too hard to understand. "Why believe in a god? Just be good for goodness' sake." It is basically asserting the secular humanist* position: acting morally is worth it for its own sake, and that believing in a god or religion is not required. It's actually a rather mild message, I think. No, seriously, everyone should agree that goodness is its own reward. I would have agreed when I was Catholic (but apparently that's only because I wasn't Bill Donahue). For those who don't agree, obviously the message wasn't intended for them. The ad only states a position; there certainly wasn't enough space to argue the point. It is not intended to convince anyone, but rather to promote awareness. If you're a nontheist, you aren't alone, and if you're a theist, there are happy people with different beliefs.

*I find it really odd that the American Humanist Association advocates exclusively secular humanism. What about religious humanists? Aren't they important too?

You may also have noticed that the ad uses a slogan nearly like one of the lines in "Santa Claus is Coming to Town". This is simply intended to be a cutesy nod to the season. It's worth noting that many nontheists celebrate Christmas and enjoy the season, at least as much as everyone else does. Some don't celebrate, and there's nothing wrong with that. The so-called "War on Christmas" is a complete fabrication*, and as far as I can tell, there is absolutely no motivation to start such a war. Thus, the Christmasy tone of the ad is not meant to be mocking or subversive. It's simply because Christmas songs are a common cultural touching point, even among nontheists.

*Okay, I guess there's the "happy holidays" vs "merry Christmas" thing, but that's just tremendously silly.

But intentions aside, some people will react badly. The primary problem is that the question "Why believe in a god?" comes off as a challenge. If you ask atheist supporters about this, half of them say religious people are simply reading too much into it, while the other half say, yes it's a challenge, and what's wrong with that? What's wrong is that no one's going to be convinced by a one-liner on a bus, duh. And it builds on the impression that atheists have nothing to do with their time but challenge religion.

And yes, maybe religious people are reading too much into it. It's not really meant to be a challenge. It's just hard to navigate all the little pitfalls and convey a message without offending a bunch of people. But we have to navigate it. If you try to make a subtle point about how hard it is to avoid offending people, it's not going to come accross in a bus ad. So deal.

Another reason people react negatively is because it's similar to those religious billboards. You know, like those godspeaks boards that say things like, "We need to talk. -God" I don't propose to know what their motivation is, but the website says it's to, "create a spiritual climate and get people to think about a daily relationship with a loving and relevant God." Oh, so they can feel good about themselves as they think about all the people who've been inspired by a one-liner on a billboard. They're totally tacky, and I'm glad they don't appear where I live. Doesn't the AHA ad serve the same purpose, to make people feel better about themselves? No, I'd argue that the ultimate purpose is to increase visibility of secular humanism. But it's still totally tacky, at least according to my gut reaction.

Of course there will be negative reactions, but how could we possibly do any better? I dunno. I sort of like this billboard campaign better. It says, "Don't believe in God? You are not alone." It's straight to the point: we nontheists exist. It doesn't challenge religion, it doesn't need to. If we so wanted to challenge religion, advertisement isn't the proper route. Of course, even when the billboard clearly doesn't try to challenge religion, it still considered "controversial". So why bother trying to be "nice" if it'll be controversial either way? I'm not sure sometimes... Because... it's a matter of degree, and we only need generate as much controversy as is worthwhile.

Pill Puzzle solution

See the original puzzle

This was solved by Yoo of Stochastic Scribbles:

...cut each of the indistinguishable pills in half, being very careful not to mix them up, cut the last remaining pill A into half, too, and take a half from each pill each day.

If that wasn't clear enough, you basically add an extra pill A to your pile, and then eat half of each pill. Save the left over half-pills for tomorrow.

This is one of my favorite puzzles. It's simple and elegant, and requires a bit of lateral, but logical thought. I don't know about anyone else, but when I see a puzzle I like, I try to think up a small variation that would produce another cool puzzle. I can't think of one for the Pill Puzzle yet, but maybe one day.

Request a Fractal!

In a previous post, I explained how to make fractals using Newton's method. These fractals can be generated by a java program I wrote for a high school project. The main input is a mathematical function.

So you give me a mathematical function, and I will make a fractal out of it!

You do not need to understand how Newton's Method works to make a request.

The rules:

• Be creative! A simple function (like f(x) = x²) might not produce anything interesting. More complicated functions may not produce anything interesting either, but I can adjust them to make them interesting.
  
• You may use any of the following in your function:
• Any arithmetic: addition, subtraction, multiplication, division, in absolutely any combination.
  
• Imaginary numbers (represented by "i")
• Trigonometric functions: sin, cos, tan, etc.
• Inverse trigonometric functions: arcsin, arccos, arctan, etc.
• Exponentials: e^x, x^(-1/2), x^x, x^i, and so on.
• Logarithms
• Any, absolutely any combination of the above. Even something like x*(cos(tan(x))^(i/x))-log(i+arctan(x)) is possible. But please don't suggest anything that crazy.

More complicated functions tend to generate more complicated fractals; simpler functions tend to generate simpler (more elegant?) fractals. For examples refer to the bottom of my previous post.

Oh, and here's one more example to throw out there. This Mandelbrot look-alike was generated with the function (x+1)*sqrt(x).

I will post a bunch of these at a later date.

Fractals from Newton's Method

Today, I will explain how I created this:

This is a fractal. A fractal is a pattern that contains smaller versions of itself. But it's not just any fractal. It's a fractal I created from something called Newton's method.

Newton's Method

Let's say we have a mathematical function called f(x). I chose one specifically for this demonstration. Here is a graph:

A very common math problem is to find the "roots" of f(x). That means you're trying to find what numbers "x" can you use to make f(x) equal to zero. In a graph, that means that it touches the horizontal axis. In the picture above, the roots are all shown with red dots. You can see that one root is zero, and the others are near 2 and -2.

It's easy for me to make an instant estimate of the roots, but that's because I had a computer graph it for me. What if I were, say, Isaac Newton, and I had no computers? What if I wanted a really accurate estimate of the roots? I would invent a new mathematical method and name it after myself, of course. And that's what Newton did.

Newton's method relies on the fact that most functions are more or less straight. The graph of f(x) sure doesn't look straight--it curves all over the place. But if we zoomed on just one part of the graph, it would be almost straight. An almost-straight line is almost like a straight line. So it stands to reason that an almost-straight line has almost the same root as a straight line.

In the above graph, I started by "guessing" the location of the root at -2. Using this guess, I drew a "tangent line" to f(x). This tangent line is a straight line that just barely touches f(x) at the blue point. Finding a tangent line is a standard method from calculus. If we just find the root of the tangent line, we know it must be fairly close to the root of f(x).

Notice that we started with an initial guess of -2, and we got a much better guess. That means we can take any guess and make it into a better guess! There's no reason to stop there. All we need to do is repeat the process, starting with a better guess each time. You can get a very good estimate of the root of f(x) very quickly.

When you guess badly...

The trouble with Newton's method is that functions aren't really straight. They can curve all over the place! Let's see what happens when I try a different initial guess of -1.

After only one iterations, it looks like we're getting a very accurate approximation of the root near 2. But wait, didn't we initially guess -1? Even though our initial guess is between the first two roots, we end up finding the third root. Newton would probably consider this a bad guess, because we didn't find the root we wanted to. However, we have a different idea in mind.

We want to answer the question: Given any initial guess, which root will we eventually find?

Though the original method was invented in the time of Newton, this is a question that they never could have answered. What if a single guess bounces around for a while, before finding a root? You really need to use a computer to test all the possibilities. So that's what I did.


(Click for a bigger picture.) When you guess badly, you get a fractal!

Allow me to explain the meaning of the fractal. Each color corresponds to a different root. The darkness of the color corresponds to the number of iterations required to get the root. Of course, you never quite reach the root exactly. But once it's within a certain distance, the computer decides that it's close enough. Some guesses are so bad that they don't ever find any root (at least as far as my computer has tried). Those guesses are indicated in white.

It's actually not too surprising that this method would result in fractals. First you have the large regions which correspond to good guesses. Then you have small regions of bad guesses. These "bad guess" regions map to the rest of the number line. And so, the "bad guess" regions will end up looking like smaller versions of the entire fractal.

More complex, More fractal

So far, I've only explained how to make a 1-dimensional fractal. The one-dimensional fractal maps the number line to different colors. But at the top, I showed you a 2-dimensional fractal. The 2-dimensional fractal maps the complex plane to different colors.

The complex plane is a sort of extension of the number line into two dimensions. It includes the "real" numbers, like -1, pi, and sqrt(2). It also includes "imaginary" numbers, like "i", the square root of negative one. And then there are complex numbers, which are in the form a+b*i. The number "a" is called the real part, and "b" is called the imaginary part. The real part is represented by the horizontal position on the complex plane, while the imaginary part is represented by the vertical position.

Otherwise, the method is exactly the same. Only now, it's prettier. And there might be new roots that were previously hidden.

The fractal at the top was generated using the function f(x) = x^3-1

But I have tried much more complicated functions as well. Some of you might recognize this one, because I use it as my avatar in certain internet locales.

This one was generated by the function f(x) = x*cos(x)^i. This function has only one root. The black regions correspond to guesses that never lead to the root.

This was generated by the function f(x) = log(x) + x. I also made a nice desktop-sized version, 'cause it's so awesome.

This is the function f(x) = e^x - x. This function has an infinite number of roots, only two of which are being shown.

This is f(x) = log(x²). I had posted this on my blog last Christmas. The blue "ornaments" are actually an exploit in my computer program; they wouldn't normally be there.

These are all generated using a Java program that I made for a high school project. I have found it very fun to experiment with this math-to-art device. I want you all to have a taste of that. So... later, I will be taking requests for mathematical functions!

Away with All Gods! debate

This is just a pointer to my latest contribution to the BASS website. I attended a debate called "Away With All Gods! Possibility or Fantasy?"

Because I believe in the power of a good teaser quote...

Bartchy played the “Stalin” card. Normally, I would groan at this cliche, but I think it is a fair point against Communism. Sunsara’s response?

Read the entire thing here.

And while I'm at it, I might as well toss some other links around too.

Skepticblog: Tao of Traditional Medicine - Skepticblog is exceeding expectations so far

Greta Christina: Proud -- and Bitterly Disappointed - post-election commentary, way better than the prop 8 post I had too-quickly assembled.

Belief, acceptance, etc.

Something that ever slightly irritates me: skeptical word aversions! I'm not allowed to "know" anything because that would imply that I'm dogmatic. I'm not allowed to "believe" anything, because science isn't about belief. I'm not allowed to "prove" anything because science can never prove anything. Seriously, relax!

I think all of us here realize that scientific proof is not absolute. I think we all realize that very little is truly certain. Does that mean that we have to constrain our language?

I find I have a habit of inserting lots of qualifiers into my language, such as "I think", "it seems that", or "nearly". I try to cut down on the qualifiers, because I want to have more language variety. It should go without saying that everything I ever write is only something that I think. Explicitly saying "I think" serves only to emphasize. It's there to emphasize uncertainty, or show unwillingness to speak for other people, or simply for style. I might have similar reasons for wanting to use a word like "know" or "believe". I don't see why these word-options should be off-limits.

There is a certain breed of essay that draws fine distinctions between words. Many draw a distinction between belief and acceptance. See, belief means dogmatically accepting things despite evidence to the contrary. Acceptance means tentatively believing things because they're currently backed up by evidence. The fundamental flaw with this type of analysis is that I have no reason to agree with the definitions. Maybe when some people say they believe, they mean taking as an article of faith, but that's not what I mean. When I say I believe in science, I mean that I believe in all the established scientific theories because they have been established by sound practice of the scientific method.

Colloquially, the difference between synonyms like "accept" and "believe" is just a matter of connotation and context. "Belief" is indeed used a lot in the context of religion. That doesn't mean religion is its only appropriate context. It does, however, mean that it carries a tiny bit of religious connotation wherever it goes. So I understand wanting to avoid the word most of the time. I understand wanting to draw a distinction between religious and scientific knowledge at every mention. But there's no reason to be anal about word distinctions when they don't even concretely exist. If you can explain the difference between "acceptance" and "belief", then you can also explain the difference between "scientific belief" and "religious belief".

Unhappy about prop 8

It's the day after election day. Are we allowed to talk about stuff that isn't election yet? No? Well, okay...

I'll drop the act--obviously I'm writing of my own free will, and therefore must enjoy doing so.

Obama wins in landslide. I assure you that I am excited about this, even if I'm not much interested in conveying my excitement.

What I do wish to convey is my displeasure about prop 8, the gay marriage ban. It passed. Why, California, oh why must you be so regressive?

As I understand it, there are basically three arguments against gay marriage. There's the religious conservative argument, the secular conservative argument, and the libertarian argument.

The religious conservative argument gets the most contempt, and is most deserved of contempt. Religious arguments can only justify that which you already believed. So it's not a justification at all, but rather, a sign of underlying prejudice. This "justification" deserves no place in politics.

The libertarian argument is that marriage should not be recognized by the government at all. That would be great, or maybe not, but either way it's beside the point. Right now, marriage is recognized by the government, so we might as well be fair about it. Once we get privatization of marriage on the ballot, maybe then we'll talk about it.

The secular conservative argument goes that marriage serves a specific purpose (ie raising children), and gay marriage lacks something vital to that purpose. Put a different way, marriage is a privilege, not a right. I'm fairly sure that gay couples can adopt and raise children at least as well as single parents can. Moreover, if a straight couple chooses not to have children (or is incapable due to fertility issues), we wouldn't want to ban their marriage. The difficulty of implementing a ban on infertile marriages is beside the point--we wouldn't ban it no matter how easy it would be to do so. And the "privilege" vs "right" thing is also beside the point. Either way, it's unfair to grant this privilege/right to some people and not to others. It's racist by word substitution.

I think the secular arguments against gay marriage are BS. Nobody would find these compelling unless they had an interest in finding them compelling. Basically, I think the supporters of prop 8 are a bunch of authoritarians. Authoritarians only believe these subpar arguments because they're on the side of the "proper" authorities, because they're supported by the Republican party. I wouldn't normally do this kind of armchair psychology, but I'm doing it now 'cause I'm so unhappy about the prop 8 results.

On a lighter note...

I really liked yesterday's Dinosaur Comic.



Oh, I think we're all a little racist by word substitution.

The Pill Puzzle

You are on an unbelievably strict medical prescription. Every morning you have to take exactly one pill A and one pill B. You must keep this up for a month, or there will be dire consequences to your health. You have just enough pills to last you the month, and you can't afford to go to the pharmacist to get more.

And so, you are dismayed when halfway through the month, you make a mistake. Right after putting one of pill A in your hand, you accidentally poured two of pill B in your hand. The pharmacists apparently didn't have the foresight to make pills A and B different colors or anything. Now you have three indistinguishable pills in your hand, and you're not sure what to do with them.

How can you keep up with your prescriptions without having to go to the pharmacist to get more pills?

see the solution

Politics!

How about them politics?

I just realized that I almost forgot to participate in that great blogging tradition of aggressively volunteering my thoughts about politics onto my readers near election. You know, because obviously that's what my readers came for--my ever-frequent comments about politics.

Uh, yeah, so I'm voting for Obama. It seems like a fairly obvious choice. Surely you see how compelling a case I make. That is what makes me a great political commentator.

Also important is voting No on Prop 8. Go LGBT rights! Seriously, I hear it will be close.

You should all vote exactly the way I do. That way, it's effectively like I have more voting power.

More importantly, simply vote! It's a well known fact in statistics that the uncertainty of a random sample is 100/sqrt(N) %, where N is the size of the sample (ie the number of voters). That means that one more random voter increases accuracy by 50/(N^1.5) %. Amazing! Of course, it's not really random, which means my analysis is completely BS, but don't tell that to your emotional brain.