What I've been doing

For those who haven't been paying attention, there has been a bit of discussion about the Global Warming post I wrote earlier. Pat Frank dropped by and responded. I think this is great. He provided extra information here, answered a few questions, and did it all politely (always a plus). Was I wrong in my essay? Perhaps, perhaps not. I am still trying to work it out (and I wouldn't mind some help).

And I've been busy with other stuff too. Look what I made in Mathematica.
It is an image of a top. My program makes it spin! See, I am trying to model an extremely complicated type of motion called "rotation". It's so complicated, I don't know if I should even try to explain it. It involves vectors, tensors, matrices, and differential equations...

Anyways, I am still trying to figure out the global warming thing. Isn't it so much easier to talk about fallacies and reasoning rather than specific examples thereof?

I've been published!

Good news! For me!

I now have a single-authored research article published in an undergraduate science journal! I am quite happy about this. The paper has something to do with magnetospheric waves that are thought to energize electrons in the Van Allen Radiation belts.

The natural follow-up question is, does this make me a scientist? I like to think that it does.

The Nine-Square Fold

Here's a fun hands-on puzzle.

Take out a sheet of paper, and divide it into nine squares. Label each square with the letters above. It might help to label the back of the squares too. Next, crease the paper along all the lines. Now it's time to do some complicated folding and tucking!

The idea is that you fold the nine squares down to a single square. The top square should be an S. The next square underneath should be a P. The next should be an I, and so forth, until you've spelled the word "SPINELESS". If you do it right, maybe a squid will appear when you unfold it?

If you figure it out, it might be too difficult to describe each step of the fold, so it should suffice to tell me how the E's and S's are ordered.

The "spineless" puzzle is my own creation, but it was inspired by a similar puzzle created by Robert Neale. I only created my own mostly because it was a fun challenge, but also because Neale's puzzle is a tad more difficult.

Using this paper, you must spell the eight pseudonyms of Beelzebub: Bel Zeebub, Bub Blezee, Ube Blezbe, Bub Zelbee, Bub Beelze, Zee Bubble, Buz Lebeeb, Zel Beebub. The final challenge is to spell his real name, Beelzebub. If you do it right, maybe the fallen angel will appear when you unfold it?

See my explanation of how to find the solution (actual solutions not included)

[For further reading, see "The Combinatorics of Paper Folding" in Martin Gardner's Wheels, Life, and other Amusements.]

Innumeracy in Global Warming skepticism

There's an article in the latest issue of Skeptic Magazine called "A Climate of Belief" by Patrick Frank. It says that the case for Global Warming being caused by CO2 is severely hurt by the fact that computer models of the climate are uncertain. At first, I thought it had raised a fairly good objection, at least good enough that I, mostly clueless about climate science, would have no idea how to refute it. But it turns out that the article fails at basic statistics.

The main argument of the article goes like this:

Computer models of climate show error bars in their results, but these error bars only show one kind of error: the variation between multiple runs of the simulation. What the error bars don't show is the "physical uncertainty", the measure of difference between the predicted and actual.

How do we estimate the physical uncertainty? We use the climate model to "retrodict" past climate, and then compare to the actual climate we had during that time. Frank shows that such retrodictions only calculated the total cloud cover with 10% accuracy. Of course, to show this, he uses retrodictions of the 1979-1988 period, and compares them to observations of 1983-1990. I have to wonder if it's good practice to compare different decades.

He goes on to say that 10% cloud cover has a huge impact on global temperature. How big? 1.1°C a year. That means that after a hundred years, the uncertainty is 110°C! See the graph below of the uncertainty as it increases with time.

This graph is what really set my skeptical bells ringing. Yes it's true that if the uncertainty is very large, we can draw no conclusions. But how can the error be so large? Intuitively, it does not make sense. If all your results are accurate within, say, 10°C, but the error bars are 100°C, that either means you've overestimated your error, or you got really, really lucky. Even global warming deniers will grant that the models are accurate within 10°C. Are they feeling lucky?

So where does his estimate of uncertainty go wrong? Frank's problem is pure statistical innumeracy. Unfortunately, statistics is not common knowledge, so this sort of innumeracy can go right over some people's heads. Allow me to explain.

Problem 1: Uncertainties do not add! If you have 1.1°C uncertainty in the first year, and 1.1°C uncertainty in the next year, what is the cumulative uncertainty? You might guess 2.2°C, but this assumes that both uncertainties are always in the same direction. Half of the time, they will be in opposite directions and partly cancel each other out. The result when you work out the math is a total uncertainty of 1.56°C after two years. Sure, it's possible that it will be off by 2.2°C, but error bars are only supposed to cover the most likely data. The uncertainty does not increase in a straight line. It should be proportional to the square-root of time. That is, it will increase more slowly after a little while. I was extremely shocked at such an egregious error. Has Frank never taken a statistics class?

Problem 2: Uncertainties are reduced in a stable system. The environment is a mostly stable system. That is, it doesn't swing wildly in temperature every century. If the temperature is a little higher than average one year, something will push it towards normal temperature. For instance, higher temperature might increase cloud cover, which reflects more of the sun's light away from Earth. Therefore, a temperature uncertainty this year may not survive to the next year. When I said the uncertainty is proportional to the square-root of time, I assumed that the system has no stabilizing mechanisms. In fact, the uncertainty will increase much more slowly than that.

Problem 3: What's the difference between Frank's uncertainty and the already reported error bars? Frank asserts that they are different, but I'm not so sure. Frank bases his uncertainty estimate on the predictions of cloud cover. But is this uncertainty different from the uncertainty between different runs of the simulation? I imagine each time the simulation is run, it gives a slightly different prediction of cloud cover in the same way that it gives a slightly different prediction of temperature. So not only is Frank calculating the uncertainty incorrectly, it may have already been accounted for.

Frank seems incredulous that we can estimate the temperature decades from now when we can't even estimate next year's temperature accurately. But actually, this makes sense. We can't predict the whether next week, but we can predict overall trends between seasons. Large, overall trends are easier to predict than year-to-year fluctuations!

I only spot the statistical errors, because that's the part I know. Given the kinds of errors I see, I wouldn't be surprised if the rest of it were also riddled with flaws.

[This post has been cross-posted at BASS. Visit the site and take a look around!]

Update: Pat Frank responds! See the BASS website for discussion.

The God Delusion reviewed (Part 3)

Back to the beginning of the review

The Ultimate 747

I was rather surprised to see how Dawkins analogizes everything to evolution. No wonder he likes the idea of memetic evolution. Nevermind that meme hybridization and amemogenesis are completely commonplace. It's like Dawkins has spent so much time popularizing evolution that he can't fully shift his focus. Well, maybe I sound the same way with physics, so I'll forgive him. The upside is that Dawkins gives a simple, well-balanced overview of how religion might have evolved.

The downside is his "Ultimate Boeing 747" argument. I've got to admit, though I'm familiar with most atheist arguments out there, this is one that I had never heard (except with reference to The God Delusion). And with good reason. See, here's how it goes, more or less:

According to the proponents of Intelligent Design, complexity is incredibly unlikely to form through random chance. It is as unlikely as a full Boeing 747 being assembled by a hurricane moving through a scrapyard. In fact, they misunderstand evolution, which is not completely random chance (though chance plays a part). Evolution slowly climbs up a complexity ramp. God, by contrast, must be at least as complex as the things he designs. But unless God was created through a complexity ramp, he must be even more unlikely than that Boeing 747.

Dawkins is trying to turn an Intelligent Design argument on its head. But it fails because a bad argument in reverse is still a bad argument. More specifically, Dawkins buys into the Intelligent Design concept of "complexity", which is just as wrong as any other Intelligent Design idea. "Complexity" is ill-defined in evolution; even more so in metaphysics. If you use standard definitions from information theory, complexity is actually very easy to create, and not at all unlikely. And there are just so many other things wrong with this argument. It's a shame that this was Dawkins' primary positive argument against God.

I sort of understand what he's trying to say here. It's a weird variation on Occam's Razor arguments. Because God is complex--the convolution of many unlikely elements--his existence is unlikely. I think Occam's Razor arguments are rather weak, but they would still be an improvement on this.

But I'd like to end on a positive note, so I'll say that Richard Dawkins certainly has succeeded in his main goal, which was to raise consciousness. People are talking. Atheists are sort of a big deal now. They're not to be ignored, hated, or feared anymore. For that, thank you, Richard Dawkins.

Expect cross-blogging

Check out this website: Bruin Alliance of Skeptics and Secularists. This is UCLA's only student skeptical/secularist group. They have a neat website and blog. I am now a contributor to that blog. Expect me to do some cross-posting between the two blogs. Obviously, they have somewhat different focuses, so not everything will be cross-posted.

And yes, for the record, I am a member of BASS. I am a student at UCLA. I know you've all just been dying to know my super-secret location, so there you go.

The God Delusion reviewed (Part 2)

See Part 1

Now it's time for some criticism of Richard Dawkins.

Dawkins' way with words

Richard Dawkins has an irritating habit of using the wrong word, or otherwise saying some very silly things.

Example 1: "Delusion" The number one sign that you're dealing with an uncareful skeptic is when the skeptic chalks everything up to insanity. People believe weird things not because they're clinically insane, but because they're normal. They have normal cognitive biases. Everyone does. Religious beliefs are no different except that they're even more commonplace than other weird beliefs. Calling it all a delusion is simply sloppy.

Dawkins fans will come to his defense, saying that he carefully defines "delusion" as "a false belief or impression", eschewing any psychiatric connotations. But that's not the case. Dawkins is surprisingly ambiguous. He endorses a quote by Robert M. Pirsig: "When one person suffers from a delusion, it is called insanity. When many people suffer from a delusion it is called Religion." It's as if Dawkins wants to satisfy both parties. Well, I am not satisfied, because I see too many people claiming that religion really is a delusion, and Dawkins is at least partly to blame for it.

Example 2: "Child Abuse" The same thing is going on here. When Dawkins says religion can be "child abuse", he inadvertently implies that it should be illegal, because child abuse is illegal. It would be great if Dawkins explicitly denied any such connotation, but he's surprisingly ambiguous.

Example 3: "The Neville Chamberlain School of Evolutionists" Dawkins uses this term to describe people who are interested in trying to reconcile evolution with religion. Does he not realize that he's breaking Godwin's Law? And furthermore, indirectly comparing religion to Hitler? The Hitler Zombie has chomped Dawkins' brains.

Example 4: "Darwinism" Dawkins seems persistently ignorant of the fact that this term is dated in the U.S. It's not even accurate either, since it's not like Darwin had the final word on evolution.

Example 5: "The Jewish Lobby" Dawkins doesn't actually say this phrase, but he refers to how Jews, though a minority, have a powerful lobby. It is much more precise that there is a powerful "Israel lobby" which is not made up of all Jews. In fact, the idea that there is a "Jewish Lobby" is a common anti-semitic notion, and it's not good that Dawkins blindly accepts it.

Example 6: "The Selfish Gene" Ok, wrong book. But it's worth noting because it's an older phrase coined by Dawkins that has caused much confusion among the public. People seem to think the idea refers to the selfishness produced by evolution, when really the whole point is that selfish genes lead to cooperative lifeforms. It's so darn painful when people mix it up. Dawkins clearly regrets this particular example, since he has made every effort to correct the confusion (even mentioning it in The God Delusion).

Example 7: "Brights" Again, not specifically from this book, but it's still another stupid word endorsed by Dawkins. The Brights campaign was meant to replace "atheist" with a more positive label, "bright", just like "gay" was used to give homosexuals a positive label. That's nice, but the word choice was extraordinarily poor. What's the opposite of Bright? Anyone could have predicted that the campaign would flop.

Ok, I don't have any more examples off the top of my head, but Dawkins perpetually seems to say the exact wrong thing. I sometimes wonder if he's doing it on purpose, just for the shock power. Can't he think of a way of shocking people that isn't so careless with the truth?

In part 3, I criticize the "ultimate 747" argument.

Solution to "Flipping coins in parallel universes"

See the original puzzle

This was a tricky one, if I do say so myself! Question A is a puzzle classic (though the Futurama references are my own), while Universe 1 and the Tale of Interest are my own variations.

First, recall that there are four possibilities when two coins are flipped. The first coin has two possibilities (called H and T for Heads and Tails) and the second has two possibilities. 2x2=4. Those possibilities can be represented with the following:

HH
HT
TH
TT

But wait! Aren't both coins flipped at the same time rather than one after the other? Yes. As a result, we cannot tell the difference between HT and TH. However, each of these possibilities was assigned a probability of 1/4, and if we combine them together, it will have a total probability of 1/2. The remaining three possibilities will no longer have equal probabilities, as shown below:

HH 1/4
HT 1/2
TT 1/4

Now let's examine what happened in each universe. Details are key!

In Universe A, Fry asks whether at least one coin is heads. Leela says yes. Therefore, we can eliminate the TT possibility. What's left?

HH 1/4
HT 1/2

But the probabilities don't sum up to 1! The numbers are nonsensical. What do we do?

Consider a simpler case. If I flip one coin, it has a 1/2 probability of being heads. However, if I tell you the actual result was heads, the probability changes from 1/2 to 1. The same concept applies here. After we've eliminated a possibility, we have to multiply all the other probabilities by a number (here, that number is 4/3) such that the total probability is 1 again.

HH 1/3
HT 2/3

Conclusion: There is a 1/3 chance the both coins are heads.

Universe 1 is nearly the same as Universe A, but with a key difference! Fry simply asks what the coins are, but Leela does not simply answer yes or no. She instead says that, "At least one coin is tails." She just as easily could have said, "At least one coin is heads." How did she decide between these two statements? Perhaps you cannot prove this, but the most reasonable way for Leela to decide is to look at the coins and pick one of them.

Now we've got to break up the 3 possibilities into 6 possibilities, since Leela can either pick the first one or the second one. It doesn't matter that Leela doesn't know which is first or second, nor does it matter whether the first and second happen to have the same face showing. Here are the possibilities:

HH; Leela picks H 1/8
HH; Leela picks H 1/8
HT; Leela picks H 1/4
HT; Leela picks T 1/4
TT; Leela picks T 1/8
TT; Leela picks T 1/8

We know that Leela picked tails, so let's reduce the possibilities, and then make sure the probabilities add up to 1.

HT; Leela picks T 1/2
TT; Leela picks T 1/4
TT; Leela picks T 1/4

Conclusion: There is a 1/2 chance that both coins are tails.

In the Tale of Interest, we are given additional information that the two universes are two sides of the same coin, so to speak. So is the probability 1/2 or 1/3? I admit that I wasn't sure myself at first. But we can divide up the possibilities into 6, just like before, and find out. The following list will be slightly more complicated:

Universe 1: TT; Universe A: HH; Leela picks H 1/8
Universe 1: TT; Universe A: HH; Leela picks H 1/8
Universe 1: TH; Universe A: HT; Leela picks H 1/4
Universe 1: TH; Universe A: HT; Leela picks T 1/4
Universe 1: HH; Universe A: TT; Leela picks T 1/8
Universe 1: HH; Universe A: TT; Leela picks T 1/8

Based on our information from Universe 1, we can eliminate the first two possibilities. Based on our information from Universe A, we can eliminate the first three possibilities. Here's what we have left:

Universe 1: TH; Universe A: HT; Leela picks T 1/2
Universe 1: HH; Universe A: TT; Leela picks T 1/4
Universe 1: HH; Universe A: TT; Leela picks T 1/4

Conclusion: There is a 1/2 chance that both coins are heads in Universe 1.

One question I anticipate is, "How can the probability change between Universe A and the Tale of Interest? Aren't they asking the same question?" It's the same question, but there is additional information given in the Tale of Interest. Leela in Universe A has no idea what Leela in Universe 1 is up to. New information can change probabilities. It's just like when I tell you what the actual result of a coin flip is.

Ok, so I've gone through a lot of detail here, but my experience with probability puzzles has taught me that many people will look at this and still think it's intuitively wrong. Oh, the flaws of intuition! Please ask, and I'll attempt to convince your intuition. But you must realize that it's the math that has authority, not intuition.

The God Delusion reviewed (Part 1)

So I managed to get my hands on a copy of The God Delusion (Thanks, Roy!) and I've finally read it. Right now, you are thinking, "miller, are you normally late to parties?" Yes. Yes I am.

Overall, The God Delusion is well written. Richard Dawkins isn't a famous popularizer of science for nothing. However, for me, it wasn't very informative. Dawkins doesn't really say anything I didn't already know. I would recommend this book to anyone who wants a broad overview of the issues that concern the atheist movement (keeping in mind that not everyone agrees with Dawkins).

The Courtier's Reply

One of the major reactions to The God Delusion, echoed in countless reviews, was, "Dawkins focuses too much on fundamentalist views of God, and ignores sophisticated theology." PZ Myers famously called this reaction "The Courtier's Reply". Here's the money quote:

Dawkins arrogantly ignores all these deep philosophical ponderings to crudely accuse the Emperor of nudity.

The Courtier's Reply is disdained by mainstream atheists, though reasons vary wildly. If you've never noticed, PZ doesn't actually refute the Courtier's Reply in his famous post--he just makes an analogy to ridicule it.

But it is deserving of ridicule, because there is something very wrong about it.* Dawkins does not ever mention [insert sophisticated theologian here] because it would not fit in with the rest of the book. You cannot put "The Argument from Beauty" and Godel's modal ontological argument next to each other. That's right, Dawkins has a section on "The Argument from Beauty" ("Without God, how do you account for Shakespeare?"), which I feel is a self-refuting argument. Dawkins' audience isn't a bunch of philosophers--it's a popular audience.

Dawkins' purpose, after all, isn't to make the end-all refutation of God. His purpose is to effect real change, among real people. As such, much of his writing focuses on things that are down to earth. Even his discussion of the ontological argument was filled with anecdotes, at the expense of including any actual refutations (despite how easy it is to refute). If Dawkins ignores sophisticated theology, it's because such theology is not really all that important to the common person. Furthermore, what need is there to attack the best of religion? Isn't it the worst of religion that needs to be changed the most?

In any case, Dawkins does not focus on fundamentalist religion. If you were looking for a critique of biblical literalism, only the bare bones are there. Good thing too, because I find Bible talk to be boring. His primary focus seems to be on ordinary positions in the middle. Apathetics. Agnostics. Liberal and moderate believers. People who don't believe themselves, but think belief is good for others. (It's been a while since I've used a comic strip to illustrate a point, but here goes!)



Whether Dawkins' arguments against all these different positions are valid is another matter. For the most part, he manages to cover all the basic points.

And yes, Dawkins mentions sophisticated theology too, only to say that it's obscurantist. Dawkins' opinion seems to be that it's all just meant to look real advanced and sophisticated when there's not really anything there. Ironically, this is also a good description of the Courtier's Reply itself. Usually, the person who gives the Courtier's Reply will simply drop a name ("[such-and-such] is way more sophisticated than Dawkins") without including anything of substance. That's all you can expect from a short review in the popular press.

Similarly, that's all you can expect from Dawkins in his short, popular book, especially since its range is so broad.

See part two, in which I am more critical of Dawkins.

*Note that most of my argument against the Courtier's Reply is based on the premise that we're talking about Dawkins' book. Some misguided atheists like to declare "Courtier's Reply!" in a variety of situations as if that were a complete and self-evident refutation of all of theology. Not all of theology is always irrelevant!

Bouncing electrons (Part 2)

See Part 1

When we last stopped, I had shown that electrons move along magnetic field lines. But if they simply follow the magnetic field lines, they'll eventually hit the Earth. Some do hit the earth, but others actually bounce back in a process called "magnetic mirroring". Why is this?

Ok, so I'm using the same picture. This time, I want you to notice that near the poles, the lines are very close together. The lines spread out much more when they are far away from the earth. When the lines are closer together, this means the magnetic field is stronger. When they are further apart, it means the magnetic field is weaker. It should come as no surprise that Earth's magnetic field is strongest when you are close to Earth.

This is a bit of an oversimplification, but the basic idea is that stronger magnetic fields repel electrons.* Therefore, if an electron is traveling along a magnetic field line, getting closer to Earth's north pole, it will eventually turn around. And then it will follow the magnetic field line all the way to the south pole. But since the magnetic field is stronger near the south pole, the electron will turn around again. As a result, electrons will bounce back and forth from pole to pole. Each bounce happens in a matter of seconds.

One of the results of this bouncing is that some regions of Earth's magnetic field are like traps for electrons (as well as other charged particles). And so we have the Van Allen Radiation belts, where lots of high energy radiation is trapped. Their shape can be described as "toroidal" or "donut-shaped". There are several other important regions above Earth with similar shapes.

Mmmm... radiation donuts.

Ok, so electrons are doing two things at once. They are gyrating and bouncing from pole to pole. But that's not all!

There is a third type of motion caused by Earth's gravity. Electrons, though very light particles, still fall. Only they don't fall. Remember, they're still trapped on magnetic field lines. If they fall down, they will very quickly circle around back up. So perhaps gravity has no effect at all? But it does have an effect! Unlike the magnetic field, gravity actually slows down and speeds up electrons instead of simply changing their direction. And faster electrons make larger circles! One side of the circle (the one closer to Earth) will be larger while the other side will be smaller. The resulting motion will look something like this.
As weird as it sounds, downward gravity causes the electron to "drift" to the side! Specifically, electrons will drift eastward. It takes a few minutes for them to go all the way around the Earth. Positively charged particles will also drift, but in the westward direction. Negative charges drift east, positive charges drift west, and we've got an electric current! This is called the ring current. Scientists measure the ring current to determine how many particles are in space, which tells us something about how the "space weather" is going.

And so, electrons above Earth have three types of motion. They gyrate, making hundreds of circles every second. They bounce from north pole to south pole in a matter of seconds. They drift eastward, going around the earth in a few minutes.

There's one last detail I want to add (and there are always more details), because it is related what I researched. All of the above types of motion conserve energy. The electron doesn't really change its speed much. However, this assumes that Earth's magnetic field is constant. It isn't. A stream of particles called the solar wind is always coming out from the sun. When these particles hit the Earth's magnetic field, they cause the magnetic field lines to vibrate like harp strings. Now, each of the three types of motion occurs at a different frequency. If the harp strings vibrate at a frequency near one of the types of motion, a resonant interaction will occur! For example, if the magnetic field line fluctuates every few minutes, it will resonate with the drift motion. The electrons might move between field lines, or speed up. We think this is the cause of one of the Van Allen Radiation Belts, but we're not sure. To find out, we must take lots of data in various circumstances to see if the evidence all lines up!

*Electrons don't actually slow down when moving into stronger electric fields, they simply transfer some of their forward motion to their circling motion. The technical description of this is that electrons conserve their "magnetic moment" under ordinary conditions.

Blog stats chat

Dear internet, I am feeling chatty. So let's chat.

Oh, look, it's my blog stats.
It looks like my number of visits (not necessarily unique) increases at a rate of 0.45 people per day per day, or 400 per month per month. I guess that's good?

Blog statistics were one of the things that surprised me the most when I started blogging. I mean, I didn't even know before that webmasters can tell where their visitors come from. Furthermore, I didn't expect that the majority of visitors would come from Google. Or is it just my blog that's like that?

My rate of visits is probably pretty modest. I guess I'll never conquer the world... And I guess my ponderous brooding is sort of a small niche. Anecdotes, human interest stories--I'm not very good at that. Data and information--I'm not particularly good at that either, except within a narrow specialization. I'm mostly good at making abstract rhetoric.

Oh, and puzzles! I can't say I've ever seen a really good puzzle blog (though there plenty of good websites with other formats). Most seem heavily biased towards sudoku and other grid puzzles. Grrrr...

Anyways, part of it is that I'm no good at self-advertisement. I'm too modest to submit to carnivals that often. It's precisely because of the low quality of the Carnival of the Godless that I feel most comfortable submitting to that carnival. And despite skepticism being the central theme of the blog, I don't actually write anything that I would consider submitting the the Skeptic's Circle. My stuff is too speculative and abstract. I wonder if there are other carnivals worth submitting to. But darn it, I don't even keep track of carnivals anymore.

Oh the other route to fame is through commenting on other people's blogs. The problem is that I hardly participate in the blogosphere conversation. Those two outlying points on the above graph are the one time that I linked my null physics post on Pharyngula. It remains my most popular page, despite being one of the more poorly written ones IMO.

But forget fame. Forget the conversation. Blogging has done wonders to my writing ability!


Oh, and in other news, same-sex marriage is made legal in California! Awesome!

Bouncing electrons (Part 1)

An electron is simply a very light particle with a negative charge. Usually, they're paired up with atoms, and they give us the full range of chemical reactions. But when electrons are by themselves, they're not really all that complicated, are they? In fact, above Earth's atmosphere, there are plenty of electrons all by themselves. Of course, by "plenty" I mean a near-perfect vacuum, but it's a lot by empty-space standards. There aren't really enough for them to bump into themselves very often. So what could they possibly do besides float leisurely in space?

It turns out that electrons do a lot, because they interact with Earth's magnetic field.

Magnetic fields, if you didn't know, are different from electric fields. Sometimes people get the electric force and magnetic force mixed up because on the surface they're so similar. The electric force, (which manifests in lightning and static electricity) causes like charges to repel and opposite charges to attract. Similarly, the magnetic force (which manifests in magnets and compasses) causes like poles to repel and opposite poles to attract. But magnetic poles do not attract or repel electric charges. They have a much stranger interaction.

The way that magnets work is by creating a magnetic field. The field consists of invisible lines that go from the north pole to the south pole. Technically, Earth's north magnetic pole is actually in the southern hemisphere, so sometimes "north" and "south" are mixed up (to the dismay of geophysicists).

When electrons, or any charged particles, move through the magnetic field, the magnetic field pushes them in a direction that is perpendicular to their motion and perpendicular to the magnetic field. Since the electron is being pushed neither forward nor backwards, it doesn't speed up or slow down. Instead, it simply changes direction, and travels around in circles. This type of motion is called "gyration".

The arrow V shows which direction that the electron (light blue dot) is going. The arrow F shows which way the magnetic field is pushing the electron. Because F is always perpendicular to V, the electron will constantly change directions and travel in the circular green path. The magnetic field in this example is coming out of the screen towards you.

Now, personally, I think it's just amazing that this weird physical force causes electrons to move around in circles of all things (even more amazing when you find that it is a consequence of Relativity). Another weird consequence is that the electron make circles at the same rate, no matter how fast it's going. If it's going really fast, it will simply make larger circles. In Earth's magnetic field, electrons will make hundreds of circles per second, regardless of their speed. This rate is known as the cyclotron frequency.

The electron cannot get too far away from the magnetic field line because it will simply circle back on itself instead. However, it can travel along the magnetic field lines without any resistance at all. The result is that the electrons are free to move along the magnetic field lines (in helix-shaped paths), but cannot jump from one line to another. It turns out that those invisible magnetic field lines aren't just mathematical curiosities, but they actually tell us where the electrons can move.

So where do the electrons end up? If you follow the magnetic field lines, don't they simply hit the earth? Yes, they do! Some of those particles will hit the upper atmosphere, creating colorful displays of light: the aurora. That's why the aurora is most common near the north and south poles--because that's where most of the particles come down and hit the atmosphere.

But not all of the particles hit the earth. Some of them "bounce" back along the magnetic field line! Find out why in Part 2.

Colbert interviews an Astronaut

Check this video out:


Maybe I'm only saying this because I'm currently studying rigid body rotation, but if I ever went into space, I'd probably spend inordinate amounts of time observing that spinning Wriststrong bracelet.

(Thanks to the Bad Astronomer for the link.)

Induction and the Bayesian

Early in my blog, I distinguished between two types of reasoning, deduction and induction. Maybe you wouldn't know it from my writing, but I am absolutely fanatical about the concept. If the terms were common knowledge, I would talk about them all the time; as it is, I am afraid of confusing everyone.

So to recap: deduction is a type of reasoning that always gives absolute truths, given certain premises. Induction is a type of reasoning whose conclusions only have a certain probability of being true. Both types of reasoning are absolutely necessary, though induction is far more common. Read my post linked above for details and examples.

What I haven't yet explained is that there's actually a mathematical formulation of inductive reasoning. In practice, it's hard to apply rigorous mathematics to life, but I still think this will give us better insight into the inner workings of reason.

Bayes' Disease

To demonstrate, let's first use an example that is explicitly mathematical. Let's say there's a particular disease, "Bayes' disease" that occurs in 10% of the human population. We have a simple way to test for Bayes' disease, but there is a 25% chance (assume independence) that the test will get the wrong results. Let's say that you've just taken the test, and the results say you've got Bayes' disease. What is the probability that you really do have Bayes' disease?

Now, we could make an inductive argument, and say that because the test turned out positive, you're more likely to have Bayes' disease than before. But how much more likely?

To solve this problem, we first find the probability that any random person will test positive. This is equal to 0.25*0.9 + 0.75*0.1 = 0.3. Next, we find the probability that any random person will test positive and actually have Bayes' disease. This is equal to 0.75*0.1 = 0.075. Last, we divide these two numbers: 0.075 / 0.3 = 0.25. Conclusion: there is only a 25% chance that you actually have Bayes' disease.

The way of stating this as a mathematical formula is called Bayes' theorem. See Wikipedia for a short derivation.

In the above formula, P(A) is the "prior" probability of A, and P(B) is the "prior" probability of B. In my example, P(A) is the 10% chance of having Bayes' disease, while P(B) is the 30% chance that the test will come out positive for a random person. P(B|A) is the "conditional" probability of B, given A, while P(A|B) is the "conditional" probability of A, given B. In my example, P(B|A) is the 75% chance that the test will succeed, given a person who is infected with Bayes' disease. P(A|B) is the probability that you actually have Bayes' disease, knowing that you tested positive.

A More Realistic Example

Let's say we're looking for a particle predicted by an advanced physics theory. We use a particle accelerator, and repeat an experiment thousands of times. We notice a pattern in the results, but we're not sure whether it is the particle we're looking for, or if it is a random error due to the Uncertainty Principle. If our theory is correct, then there was a 40% chance of getting this pattern. If our theory is incorrect, there was a 10% chance of getting this pattern.

Here, P(B|A) = 40%. P(B) can be calculated from P(A) and the other numbers. But the problem is that we don't know P(A), the prior probability that our theory is correct. In reality, our theory either has a 100% chance of being correct, or a 0% chance--we simply don't know which. A good estimation is that P(A)=50%, but this is in some ways naive. I can't just create any random theory and declare it to be 50% likely, pending more evidence. For example, I might claim "I have an apple", and then claim "I have an apple and a banana." These two claims can't both be 50% likely.

Some say that we can't assume anything about the prior probability. But I think that's also naive, since it leaves us with practically no way to know the universe. So we will assume P(A) = 50%, just so we can work through the math. Hopefully, P(A|B) will turn out to be so high that it doesn't matter what we picked for P(A). If that happens, nobody can complain!

The probability that our theory is correct given this new evidence turns out to be 80%. And remember, we had to assume a value for P(A). If I had assumed P(A) was much lower, say 10%, the end result would be about 31%. Is it 80% or 31%? We can't say which. This is what we mean when we say science is uncertain! Luckily, most established scientific theories are far more certain than that.

In Real Life

Of course, in most of life, we don't have any numbers at all! In fact, it's generally a bad idea to try to quantify things that are so uncertain. But Bayes' theorem still gives us insight into what makes a good inductive argument.

• P(B|A) should be high: the piece of evidence should have a high chance of occurring, given that our theory is correct.
• P(B) should be low: the piece of evidence should have a low chance of occurring in general
• If you've made your argument correctly, P(A|B) will be higher than P(A), no matter what P(A) is (unless P(A)=0). That is, a good inductive argument makes a claim more likely. Induction works!
  
• If P(A) is low, an inductive argument won't help much. That is, a claim that was extremely unlikely prior to the argument will still be rather unlikely, unless you've got an extraordinarily good inductive argument.

In conclusion, you should frequently mention things like "prior probability" in your arguments in hopes of confusing your opponents! No, but seriously, think about Bayes' theorem when you use inductive arguments. Soon, mathematics will infect your whole mind! Mwahaha!

Question: science and change

It's often said (correctly) that science constantly changes its paradigms and dominant theories. One example I'm thinking of is Mercury. Before 1965, it was thought that the same face of Mercury always faces the sun. After 1965, it became clear that the previous data was flawed, and that Mercury rotates with respect to the sun.

So here's a question for all you out there. Let's say you lived before 1965. Would you be right in thinking that Mercury always has the same side facing the sun? And in what sense of "right"?

Science and Religion: Independent

There are three broad paradigms concerning the relationship between science and religion. In the first paradigm, science and religion are in conflict; in the second, they independent; in the third, they are to be in dialogue, or even joined together. I primarily advocate the independence paradigm, but I disagree with the usual formulation of it. The usual formulation to which I refer is the idea of NOMA (non-overlapping magisteria). NOMA is a concept invented by Stephen Jay Gould that states that science and religion cover two mutually independent realms. More specifically, science occupies the empirical realm of fact and theory, while religion covers ultimate meaning and moral values. The trouble with NOMA, in my view, is that it simplifies both science and religion.

Science simultaneously covers more and less than Gould seemed to think. Science covers the empirical realm, yes, but in principle, it can cover nearly anything ever thought. Science can consider any hypothesis--absolutely any--and ask, "What are the consequences of this being true?" Either there are some consequences, or there are not. If there are consequences, you can, in principle, test the hypothesis. If there are no consequences, well, it's hard to imagine that there are many hypotheses that are important, but have no consequences.

Of course, in practice, science doesn't cover quite so much. In practice, many hypotheses have important consequences, but are too difficult or not worthwhile to test. Furthermore, science in practice, is quite a bit more rigorous than the above description would suggest. You can't really disconnect the scientific process from the scientific establishment and still call it science. If there are no peer-reviewed journal articles on the topic, it's not really science. Maybe, in principle, it's scientific, but it has none of the authority we would accord to what we would normally call science.

As for religion, NOMA also gets its realm wrong. Religion is about far more than just meaning, purpose, and morality. Religious people are wrong if they think religion is all about believing this and believing that. Religion is also about ritual, community, devotion, and the general human experience. Often times, beliefs about ultimate reality are the ground for these practices, but they're not really necessary. You could hypothetically choose to interpret the beliefs semi-symbolically, or try a religion that has completely different beliefs, yet fulfills the same needs.

Of course, many religious people emphasize right belief, and it is not my place, from the outside, to say that this is any more or less true to religion. I can't just look at fundamentalists, who take a literal interpretation of the Bible to be essential to their faith, and declare that they're not doing religion right. But ask any religious person which is more fundamental: believing in miracles, or loving God? The latter is not in itself a belief.

So perhaps my reader has already begun to see the argument I intend to make. Science and religion are independent because, first, scientific practice simply doesn't cover most of what we consider religious beliefs, and second, religious beliefs are not the only component of religion. In a sense, they are NOMA after all, but Gould wrongly emphasized certain aspects just so he could fit them into neat little parallel boxes.

That's not to say you can't criticize religion. Obviously, I think you can. But the sort of criticisms that you can level against religion are not scientific. Maybe they're scientific "in principle", but no matter how you move around the definitions, it will never have the authority of science. Disagreeing with scientists about evolution is plainly ridiculous; disagreeing with scientists on religion is no worse than disagreeing with pundits on politics, or philosophers on philosophy. In fact, I would say the criticisms against religion belong in the category of philosophy. Some even in the philosophy of science. But that's not science.

Flipping coins in parallel universes

It's time for a probability puzzle! Now, with more Futurama references!

Question A:
In Universe A, Leela flips two coins. Fry saw her flip them, but didn't see how they came up. He is morbidly curious, and asks, "Are they both heads?" After Leela refused to answer, Fry asked, "Is at least one coin heads?" Leela reluctantly replied that at least one coin came up heads.

What is the probability that both coins are heads?

Question 1:
Meanwhile, in parallel Universe 1, Leela flips two coins. Fry saw her flip them, but didn't see how they came up. He politely asks, "How did they come up?" Leela, not wanting to give it all away, says, "At least one coin is tails."

What is the probability that both coins are tails?

Bonus question: A Tale of Interest
What if you are an external observer who has one additional piece of information to what I've previously shown? That is, you happen to know that Universe 1 and Universe A are intimately connected. Whenever a coin is flipped in Universe 1, it comes up on the opposite side in Universe A.

Given this new information, what is the probability that both coins are heads in Universe A?

Update: The solution has been posted.

The Cosmological Argument

The Cosmological Argument is one of the classic arguments for the existence of God. There are actually a ton of variations on the cosmological argument. The general form is as follows:

1. I exist.
2. If there is something that exists, then God exists.
3. Therefore, God exists.
   

Simple, right? If you accept the two premises, the conclusion logically follows. Of course, the major part of the argument is within the second premise. There are a variety of ways to try to justify the second premise. Here's the form that most people think of:

1. Every object has a cause.
2. Causal chains cannot be infinite.
3. Therefore, if something exists, then it must be the result of a finite causal chain.
4. The beginning of that chain is God.

The obvious objection here (the one that most people will think of) is that premise 1 applies to God. What caused God? However, this reasoning is specious. It's relatively easy to modify the argument so that no such contradiction appears.

1. Every contingent object implies the existence of another object, namely, its cause.
2. Causal chains cannot be infinite.
3. Therefore, if a contingent object exists, then it must be the result of a finite causal chain, which can only begin with a non-contingent (or "necessary") object.
4. That necessary object is God.

I hasten to add that "contingent" is sort of a technical word in philosophy. Contingent means that it is true in some possible worlds, but not in all of them. Necessary means that it is true in all possible worlds. For example, I am contingent, since I might not have existed, but 1+1=2 is necessary, because it absolutely must be true.

Now that we've gotten to a reasonably good formulation of the cosmological argument, we can question its premises.

The first premise assumes every contingent object has a cause. But is that really true? I'd question the very idea that causation is a truly fundamental concept. For example, it is possible to describe physical laws without the notions of cause and effect. Furthermore, you have to consider the full range of contingent objects. My keyboard is contingent, and has a cause, but what about more abstract things? Is time contingent, and does it have a cause? Is love contingent, and does it have a cause? Maybe you think so, maybe you don't.

The second premise assumes that causal chains cannot be infinite. But why can't they? Plenty of religions see the world as being in an infinite cycle, so what's logically impossible about that? If a particle is moving along in space, you can say that its arrival at any point B was caused by its previous position at point A. And then you could follow its path back indefinitely. Of course, in our universe, it is difficult to actually do this, because the path will eventually be traced back to the Big Bang. However, as a matter of science, I will say that the Big Bang is not certainly the beginning of the universe, and is almost guaranteed to be a mere wall past which our current understanding of physics is inconsistent. Near the singularity, Quantum Mechanics and General Relativity contradict each other, indicating that we need better theories before we can reliably determine that the Big Bang was truly a beginning.

But my above objections, while valid concerns, are not the real killers of the cosmological argument. The unsolvable problem in the cosmological argument is in step 4. This problem remains in every variation of the cosmological argument I've ever seen. Though we might prove the existence of a necessary being, why must it be God? Perhaps you'll say that the necessary being is God by definition. But if you've given God this new definition, there is no guarantee that it will have all the other traits normally attributed to God. Must the necessary object be conscious? Must it be good? Must it be omnipotent? Must it be singular rather than plural? Would it even make sense to worship, respect, or ponder this object? None of these are guaranteed, much less the idea that the necessary object listens to prayers, or sends prophets.

Personally, I think the necessary object is probably the world. Nothing unusual about that. The world is simply the ground of all possibilities. Without the world, things couldn't possibly exist within it. There you go.

Motorcycle physics

Some one I know has a motorcycle, and he told me that it's much more difficult to drive than a car. Furthermore, in order to turn right, you push the right handlebar. Counterintuitive, huh? I was puzzling over this for about a month, trying to figure out how this was physically possible. After all, shouldn't the wheel turn to the left if you push right?

Well, I've finally got my answer! The wheel is precessing!

To visualize this, imagine that you are sitting on a motorcycle. The motorcycle must be moving very quickly, much faster than a bicycle. If it were moving at the same speed as a regular bike, then pushing right would indeed cause the wheel to turn left. But if the wheel is spinning very quickly, there will be too much energy invested in rotation for the wheel to simply turn like a bicycle's wheel.

The wheel of the motorcycle is spinning forward. This corresponds to an angular momentum going to your left. When you push on the right handle bar, you apply torque in the upwards direction. The angular momentum changes in the direction of the torque, tilting upwards. As a result, the entire spinning wheel will lean to the right. Thus, you turn right by pushing right.

Precession: not just for physics demonstrations!