What's with Objectivism?

If there are two topics in the skeptical universe that I don't quite get, the second one is Objectivism.

Objectivism is a philosophy created by Ayn Rand (1905-1982) that encompasses politics, ethics, metaphysics, and epistemology. It was outlined in several lengthy novels that I am unwilling to read, including Atlas Shrugged and The Fountainhead. Most of what I know about Objectivism is garnered from various internet sources, Michael Shermer, and BioShock. The first thing you should know is that Objectivists are libertarian atheist skeptics, though not all libertarian atheist skeptics are Objectivists.

From what I understand, Objectivists and Objectivist sympathizers constitute a small but significant minority of skeptics and atheists. See, there are liberal skeptics and libertarian skeptics. Libertarian skeptics tend to at least sympathize with Objectivists, to various degrees. For instance, Penn of Bullshit!, a libertarian skeptic if there ever was one, seems to like Objectivism. On the other hand, Michael Shermer, also libertarian, agrees with some of it, but thinks that as a whole it's a bit of a cult. Liberal skeptics tend to think Objectivists are as insufferable as hell. This blog is brought to you by a liberal-libertarian skeptic who really doesn't get Objectivism.

It's not the libertarian part that gets me. I get libertarians, at least as well as I get liberals or conservatives. I also get the bit about rational self-interest (aka greed) being a good thing. I disagree on these things somewhat, but I understand why people might see it that way. What I don't get is the Objectivist epistemology. They have this weird sort of deductivist rationalism, and I don't even see why it should be a viable option. Either I have an insufficient grasp of Objectivism, or Objectivism is complete nonsense; I'm beginning to suspect the latter.

The first time I ever encountered Objectivism, I was perusing a group's website, and they had the oddest statement among their fundamental tenets. The axiom of identity: "A is A". No, it's not wrong per se (except I would call it a tautology, not an axiom). No, of course, it's correct, it must be correct. But what of it? This is a tenet? Because the statement is by itself is useless, I had to read between the lines. I didn't like what I saw there. Basically, the axiom of identity is the first indication of Objectivists' overuse of deductivism. The law of identity is meant to assert that, yes, there are some truths that are absolute. We know they absolutely must be true, with absolute certainty, because they are derived through reason.

Objectivist epistemology seems to be entirely built around this idea. We start with a few axioms, and from there, everything else follows. Just as "A is A" is an absolute truth, so, too, is capitalism. Oh, look, I found an Ayn Rand quote to that effect:

I am not primarily an advocate of capitalism, but of egoism; and I am not primarily an advocate of egoism, but of reason. If one recognizes the supremacy of reason and applies it consistently, all the rest follows.

The mistake that Objectivism makes is that they think every idea can be reduced down to something like "A is A". They think every idea can be derived from deductive reasoning, which is reasoning that leaves no doubt about its conclusions. But that's not the case. There is a major field that investigates what we can know through deductive reasoning; we call that field "mathematics". The vast majority of all other knowledge, most especially including politics, requires inductive reasoning, which is reasoning that leaves at least a little doubt about its conclusions. If Ayn Rand thought she could argue for capitalism through deductive reasoning, she was either delusional, or took a lot more axioms than she thought she did.

That's another thing--there aren't very many axioms in Objectivism. There's the axiom of identity ("A is A"), which I think should be a tautology, not an axiom. There's the axiom of existence ("Existence exists"), which sounds like some sort of pun. And then there's the axiom of consciousness ("Consciousness is an irreducible primary"), which I'm not sure I want to understand. I'm utterly confused as to how we get from these axioms to "... ergo capitalism".

According to Michael Shermer (in Why People Believe Weird Things), early Objectivism was a bit of a cult that surrounded Ayn Rand. Everything she said must have been correct, because she was the most perfect human being, and therefore her reasoning must have been perfectly undeniable. If there was any little disagreement that you had with Ayn Rand, you couldn't properly call yourself an Objectivist. I think this cultish behavior can be credited to the Objectivists' overuse of deductivism. If they had a less nonsensical epistemology, they would realize that reason rarely results in absolute certainty, and thus would be logically compelled to tolerate a bit more heresy.

But anyways, as I said from the beginning, I don't really get Objectivism. Any objectivist readers out there who want to set me straight? Or maybe you're an Objectivist-hater who wants to set me straight in some other manner?

A time traveler's anecdote

5:00 PM, the dining hall

Me: "You know what? I just remembered..."

Suddenly, in the past...

9:30 AM, my room

Roommate: "Just so you know, Sarah is coming by later today. She's going to invite us to the group dinner."

Me: "Okay."

Meanwhile, in the future

5:00 PM, the dining hall

Me: "We forgot about Sarah. She was going to come over."

Roommate: "What about Sarah?"

Me: "Remember? You're the one who told me!"

Roommate: "I did?"

Me: "Yeah, we were going to eat dinner, but we forgot about it. And now, here we are, eating dinner."

Roommate: "Dude, it's lunch."

Time skips backwards

11:30 AM, the dining hall

Me: "Oh."

Me: "I have a poor sense of time."

Loosely based on a true story

Carnival of Mathematics #40

Another carnival! This time, it's the Carnival of Mathematics at Staring At Empty Pages. I submitted my two-parter on how use generating functions to factor dice. The carnival has been MIA for a few weeks, so this is a pleasant surprise.

So if you've never heard of the Carnival of Mathematics before, it includes interesting discussions of mathematical concepts, of math education, and sometimes math puzzles. See the full description here.

If I had to pick a favorite from this carnival, I'd go with the series on paradoxes, because I'm a sucker for paradoxes.

Skeptic's Circle #96

The Skeptic's Circle, a blogging carnival for skepticism is up at -endcycle-. If I had to pick my favorite entry, I'd go with The Perky Skeptic's Personal View of the Harm of Astrology.

My entry was False memories in Atonement. By the way, I didn't read the book; I cheated by watching the film. My sources say the film and novel are about the same though.

Oh, and for future reference (because a commenter asked), no, this blog is not fully focused on skepticism. It's about things that interest me, a skeptic. More often than not, what interests me isn't straight-up skepticism. Um... sorry?

Why is Quantum discrete?

Nowadays, when most people think of quantum mechanics, they think about how weird it supposedly is, what with quantum randomness, quantum entanglement, quantum teleportation, and so forth. But historically, the reason quantum mechanics were first formulated was to solve a relatively simple problem in classical physics. It was to explain how energy levels can be discrete.

What is discrete?

What does it mean to be discrete? That means that there is a first energy level, a second energy level, a third, and so forth, but nothing in between. If energy levels were continuous, we would expect there to be infinitely many energy levels in between, just as there are infinitely many fractions and between one and two. Since the energy levels are discrete, they're more like whole numbers--there is no whole number between one and two. The word "quantum" itself means the smallest distance between two different discrete values.

The fact that energy levels are discrete is incredibly important to physics--starting with atoms. Atoms are made up of a tiny, massive, positively charged nucleus, and a light, negatively charged electron. If it weren't for quantum mechanics, the electron would lose all its energy, radiating it out as light, and would spiral into the nucleus. All this in a fraction of a second, and you can kiss chemistry goodbye (or rather you couldn't, because your atoms would have collapsed). However, thanks to quantum mechanics, there is a lowest energy level for the electron. The electron cannot go any further below the lowest energy level, no more than there can be a positive whole number below 1. Next time you see Quantum Mechanics Personified, you should thank him or her for your atoms.

Wavefunctions, reviewed

When inventing a physical theory, it's not enough to just say, "Energy levels are discrete". Sure, that's nice to know, but we also want to know what those precise energy levels are, and why they are. I won't go through the historical details, but the answer to the why is Schrodinger's Equation:
Now, if you were familiar with complex numbers and differential equations, all I would need to tell you is what all those symbols stand for, and you would eventually go, "Ohhhh! It makes sense now." Well, maybe. But I will not assume that my reader even knows how to read the equation.

Schrodinger's equation basically predicts the behavior of what we call the wavefunction. A wavefunction is like a particle, and like a wave, but it is neither a particle nor a wave. It's a wavefunction. What the wavefunction represents is the probability of finding the particle in any particular location. For example:
If this were our wavefunction, we are most likely to find the particle where the hump is. But there is a good chance that the particle will be slightly off to the side, and a very tiny chance that it will be very far away from the hump.

The wavefunction can actually be negative too. Though a negative probability is impossible, I neglected to mention that the probabilities are equal to the wavefunction squared. A negative number squared is positive, and positive probabilities are okay.

But what isn't okay is a positive probability greater than one. It wouldn't make sense, for instance, to have a coin that has a 2/3 chance of getting heads and a 2/3 chance of getting tails--we can only get one result per coin flip. The total probability must add up to exactly 1. What this means is that we can't have a wavefunction that increases indefinitely as we get further from the "hump". A wavefunction that increases indefinitely is called a non-normalizable wavefunction--a wavefunction that cannot possibly have a total probability that adds up to 1. Non-normalizable wavefunctions are impossible, by common-sense probability.

Bound particles

Most particles fit into two categories: free particles and bound particles. For example, a photon emitted by the sun into space is free. A particle that you trap between two walls is bound. An electron that is attracted to a nucleus is also bound. When we say that a particle has discrete energy levels, we actually mean that a bound particle has discrete energy levels.

I will consider the particle trapped between two walls, and describe the appearance of its wavefunction, as determined by the Schrodinger's equation. I am considering this as a 1-dimensional problem, ignoring the possibility that the particle will go above or around the wall.

Outside the walls, the wavefunction is very small, and gets closer and closer to zero as we get further from the center. However, note that the wavefunction isn't quite zero. That means that there's a small possibility of finding the particle on the other side of the wall! This is called "Quantum tunneling". However, note that this is only on a very small scale, with tiny particles and extremely thin and soft walls. Otherwise, the chance is so unlikely you're better off waiting for a miracle.

Inside the walls, the wavefunction is wavy. It goes up and down, alternating between positive and negative. How many times does it go up and down? That depends on how energetic the particle is. But you can be sure that it goes up and down a whole number of times. It can't go up and down one and a half times. If you tried to make it go up and down one and a half times, it would hit the wall midway through a hump. The end result would be a non-normalizable wavefunction which increases indefinitely outside of the walls.

Here's what the first few possible wavefunctions look like:

[I must admit to taking this from the Wolfram Demonstrations Project]

As I said, the amount which the wavefunction goes up and down is related to how much energy there is. The more times the wavefunction goes up and down, the higher the energy. But the wavefunction must go up and down a whole number of times. Therefore, the energy can only have discrete levels! The same is true for any bound particle.

More to it

I admit there are a few more details that I haven't yet mentioned.

For instance, I neglected to mention that there are a limited number of possible wavefunctions for any bound particle. If the particle has too much energy, it will simply escape from the walls. Furthermore, if these are electrons then there can only be one electron per wavefunction. The electrons will form a sort of stack, with one electron in each energy state. That's why a nucleus can only hold a certain number of electrons (this number depends on how many protons are in the nucleus).

Furthermore, the "free" vs "bound" dichotomy isn't so clear cut. If we are studying an electron that escaped from its atom, it may be free from the atom, but it's still trapped in our lab. If we just consider a large enough space, almost any particle can be thought of as "bound". But as the region of space gets larger, the discrete energy levels become closer. Soon they will be so close together that the energy levels would appear to be continuous for all practical purposes.

There's more! Wavefunctions don't just stay motionless the whole time. And then we have "mixed states." Further explanations are upcoming!

Update: I wrote more on mixed states.

False memories in Atonement

Atonement was a novel by Ian McEwan, adapted to film in 2007. Its main character is Briony Tallis, a rather imaginative girl who seems destined to become a writer. But at 13, she does something that she ends up spending her entire life trying to atone for.

It all starts when she witnesses her sister, Cecilia, stripping down for the housekeeper's son, Robbie. The truth is that Cecilia was stripping down so she could get something at the bottom of the fountain, but Briony never finds out until later. Later, Briony sees a sexually explicit letter from Robbie that was never meant to be delivered. And then, later, Briony sees Cecilia and Robbie making love, and assumes that Robbie was assaulting her sister. In short, through a series of misunderstandings, Briony is convinced that Robbie is a sex maniac after her sister.

This could easily be sitcom material, but instead there is a tragic turn. In the dark, Briony encounters someone raping her friend, Lola. The rapist runs away, but Briony imagines that she saw Robbie's face. She gives her testimony, putting Robbie in prison. This separates Robbie and Cecilia, who have fallen in love with each other. Briony comes to deeply regret her actions as a 13-year old, thus the title of the novel.

I am not going to review this movie, because I am not what you call a "good" movie critic. Instead, I intend to comment on the false memories in the story. The story of false memories is one of the more dramatic stories in the world of skepticism. There is an idea in psychology that traces back to Sigmund Freud that the cause of many psychological conditions is a traumatic event during childhood. People generally can't remember any such event because such traumatic memories are usually repressed. I don't have an exact timeline, but it became especially fashionable in the 70s 80s for psychologists to try to recover these memories. And so it was that many people "remembered" being abused by their parents (or other adults) when they were young. As you can imagine, the resulting legal actions were disastrous to many families. It was a modern-day witch hunt.

But the skeptics eventually won! Nowadays, psychologists know about false memories. Studies have shown that it is not only possible to implant false memories in people, but it is very easy to do so. The truth was that psychologists everywhere were inadvertently implanting childhood memories of traumatic events in their patients by the power of suggestion. It is no coincidence that the psychologists found exactly what they expected! Certainly, not every single memory of child abuse was false, but most of them were. Anyways, there is little evidence that most psychological conditions are caused by traumatic childhood incidents. And there is little evidence to suppose that memory recovery would actually help a patient. To top it all off, the very existence of "repressed" memory is now disputed (note that temporarily forgetting something is not necessarily a repression of memory).

In short, psychology has sinned, and sinned greatly. I do not personally know anyone who has been affected by all of this, but I feel their pain. Families destroyed... feelings of anger, betrayal, and regret... I truly feel that this is one of the greatest tragedies of science.

In the world of fiction, it's different. Part of it is that many scientific theories tend to linger around much longer among the liberal arts than they do in the sciences (I, for one, am disgusted that Freudian psychoanalysis is still popular in some liberal arts). But I think that it's mainly because the manipulation of memory is simply a very useful plot device. It allows you to switch around the order of what the audience sees. Or it allows for character development, or the development of relationships. And because few conflicts go unresolved in fiction, most characters will recover from their amnesia. For extra suspense, they could recover through a series of dramatic flashbacks! Whatever the reasons, it is disproportionately common for fictional characters to have amnesia or repressed memories, and then recover from these. Some of these stories are plausible, if unlikely, but most are not realistic at all.

Atonement is refreshing in that it treats memory far more realistically. Rather than treating memory as a mere plot device, Atonement has at its center a real phenomenon: false memories. Briony "recalls" seeing Robbie's face on the rapist. Here, there is no psychologist who is inadvertently implanting memories through suggestion, instead Briony is wrongly biased against Robbie. Perhaps, if she had not been convinced that Robbie was a sex maniac, she would not have been so quick to blame him. She is also a very imaginative and impressionable girl. As an aspiring writer, she tends to play back her memories in her mind, each time becoming more dramatic. I don't think all these things are necessary to create a false memory, but they probably help.

The consequences of Briony's false memory tap into many of the same emotions caused by "recovered" memories in the 70s. Cecilia and Robbie are, of course, very angry at Briony. They think she was simply being overly imaginative, or worse, outright lying. Briony herself is at first sure of herself, but this wears out as she gets older. She comes to understand the gravity of her action. She becomes less sure of her memory. She blames herself. And even if she did retract her eyewitness account, who would believe her second account over the first one? And who would accept that as a sufficient apology?

Knowing what I know about false memories, I could not blame Briony for her action. I see it as more of a "girl against nature" sort of conflict. Through a fatal trick of psychology, she was put in a situation that no one deserves to be in. And since no one understood false memories back then, the consequences were dire.

But there was one other aspect of the story I thought was sad. After Briony becomes convinced that her testimony was unreliable, she "recalls" the rapist to be the man who eventually became Lola's fiance. I suppose that canonically, her new memory is the correct one, but I can't help but think it is just as unreliable as her first memory. The situation is pretty much the same as it was before. Briony is still an imaginative and impressionable girl, if a bit older now. Now instead of having mixed feelings about Cecilia's relationship with Robbie, she is having mixed feelings about Lola's upcoming wedding. Really, we have to be suspicious of any memory that is "recalled", for the first time, years after the fact.

As it plays out, Briony can't summon the courage to tell Lola of her new memory. Frankly, I was relieved.

Z-Rox: A game from flatland

Check out this free online game: Z-Rox.

I have to commend any game that makes use of flatland-type concepts. Letters, symbols, and shapes have been converted to one-dimensional animations--now you just have to name that symbol! It's as if you're a flatlander, watching as shapes pass through your plane.

(via Jay is Games, where there are answers if you need them)

Ten Greedy Pirates

I've just been informed that today is Talk Like a Pirate Day! You will not actually see me talking like a pirate, but I've got a pirate-themed puzzle classic.

Ten greedy pirates need to split 100 gold coins. These being pirates, they do not split it fairly. The oldest pirate decides exactly how to split the gold. But after he's decided, if more than half of the pirates (including the oldest one) are unhappy with their share, a mutiny will occur. They'll kill the oldest pirate, and start the procedure over.

All pirates have the following priorities in this order:
1. Every pirate wants to stay alive.
2. If they will stay alive either way, the pirates would like as much gold as possible.
3. The pirates are not friendly with each other. All other things being equal, the pirates would prefer mutiny.

The pirates' behaviors are entirely predictable; there are no psychological games being played here. How should the oldest pirate split the gold?

Hint! Try solving a simpler case (ie less pirates) first.

For an extra challenge, replace the bold "more than half" with "at least half". For this challenge problem, I have to add an extra priority. If it doesn't matter to himself, the oldest pirate prefers to give gold to the youngest pirates possible. Have fun with that one.

Spoiler! Solutions have been posted

What does our group do?

Frequently asked question: What's the point of a group for atheists, who have nothing in common but a lack of belief? What do you do, anyways?

In the interest of answering this question with real data, I discretely took notes during one of BASS's meetings last year. Now those notes are on the BASS website.

This is my little way of advertising the group, because I know that when I consider joining a group, I worry a lot about the question, "What could they possibly do at meetings anyways?"

Six random things

Oh, look, I've been tagged by Linda of the blog MindBlink. Linda is a friend I met at the Friendly Atheist, so I'll, er, hold my tongue on Karl Jung for the moment.

Here are the rules:
1) Link to the person who tagged you. Please see above.
2) Post the rules on the blog.
3) Write six random things about yourself.
4) Tag six zero people at the end of your post.
5) Let each person know they have been tagged.
6) Let the tagger know when your entry is up.

Uh, I think the meme just got a potentially fatal mutation. Hey, these things just happen by chance sometimes! I'm sure the species as a whole will survive.

So here they are, six random things about me:

1) I like dark chocolate. I deny that this is symbolic of anything. I have this on my Facebook profile, for reals. If you asked, I'd say that my intent was to question the medium. Postmodernism! No one asks.

2) I'm well-traveled but not much of a traveler. My parents were the opposite! They didn't get to travel much as children, so now they enjoy ambitious family vacations. We once spent a month driving to Alaska and back, staying at campgrounds all along the way. More recently, I passed on their Peru vacation.

3) I have half-Chinese, half-German ancestry. However, my Chinese mother lived in the Philippines and I was born in Korea. Fun fact: yes, there are ethnic minorities in other countries!

4) I was first introduced to the blogosphere through the Internet Anagram Server. Did you know that "The Bad Astronomer" is an anagram of "Moon trash debater"? Good times.

5) I am a not-quite-a-member of the IVCF group on campus. That's the Intervarsity Christian Fellowship for those who don't speak acronym. No joke!

6) I am an avid gamer. Just because I don't write about it doesn't mean it's not true. Shout out: Anyone here played Braid? Braid is awesome, and so sad too. Also: the main artist is the same guy from A Lesson is Learned But the Damage is Irreversible, everyone's favorite inactive webcomic. Um, I am an avid webcomic reader too?

That's all for today. If you don't like the thought of this poor mutant meme's lonely death, tag yourself with it.

Post hoc vs post hoc

Fun fact: The phrase "post hoc" has not one but two different meanings in skepticism. Confusing! I wonder how that happened, etymologically speaking.

There's "post hoc" as in "post hoc ergo propter hoc", which is Latin for "after, therefore because of". This logical fallacy is called "post hoc reasoning" or the "post hoc fallacy". It's when you assume a causative link between two things just because they happen around the same time. For example: autism is usually diagnosed in children around the same time that they get certain vaccinations. Therefore vaccinations cause autism. Basically, it's a kind of non sequitur.

And then there's "post hoc" as in a "post hoc hypothesis" or a "post hoc justification" or a "post hoc rationalization". A post hoc hypothesis is an unlikely hypothesis to explain, after the fact, why the evidence didn't fit your theory. For example, we start with the theory that aliens are abducting people in their sleep. We later discover that the people who experience these abductions actually stay in their bed the entire time. And so we create a post hoc hypothesis to explain away the new evidence: obviously the aliens are fooling us with hi-tech holograms.

A post hoc hypothesis is not a straightforward logical fallacy--obviously sometimes you really do need to modify your theory to accomodate new evidence. But a post hoc hypothesis can be a form of confirmation bias. It shows an irrational unwillingness to change one's original theory, instead opting for a unlikely modification. If the modification were so likely, we should have been able to think of it before the evidence came in, rather than after the fact. Post hoc hypotheses can't be dismissed out of hand, but they are often suspect.

Related to the post hoc hypothesis is an ad hoc hypothesis. An ad hoc hypothesis is a modification to a theory that explains away a single piece of evidence, but otherwise has no effects. For example, if experiments, when done properly, always fail to support the existence of ESP, maybe it's because skeptical scientists have a damping effect on ESP. Ad hoc hypotheses are suspect because they too quickly dismiss the possibility that the evidence indicates a larger pattern, rather than an exception. But again, this is not a straightforward fallacy; an ad hoc hypothesis is not necessarily wrong.

This is a little off the wall, but I mentally associate ad hoc hypotheses with a part of a poem from Through the Looking Glass.

But I was thinking of a plan
To dye one's whiskers green,
And always use so large a fan
That they could not be seen.

I blame Martin Gardner for this. While explaining the concept of an ad hoc hypothesis in one of his books, he showed this poem, along with a picture of the White Knight with a large fan in front of his face. The reason why he did this, at the moment, escapes me, but it apparently made an impression.

Redefining God

There is something I have been thinking about as of late, mainly at Hugo's provocation. Hugo has been discussing the distinction between the God of Faith and the God of Philosophers. Excuse the nonstandard language for a moment. The God of Faith is that which gives us religious experience. The God of Philosophers is the god for which we would seek proof. The question is, can we have one without the other? Can we have spirituality (the God of Faith) without having to believe in all that weird stuff about spirits (the God of Philosophy)? Here are my rambling thoughts on the question.

I have two conflicting opinions about Hugo's efforts.

On the one hand, I agree with Hugo. Yes, we can separate out the God of Faith and the God of Philosophers. You can simultaneously be an atheist and be religious. Really, it's one of the conclusions of atheism. God doesn't exist. Religion and religious experiences do. Therefore, religion and religious experiences do not require God. So if religion (or at least part of it) is such a great thing, why not just participate in it, even if you don't have any religious beliefs? You don't really need all the woo-woo about objectively existing souls, divinity, sin and whatever, no more than you need to believe the 9/11 truth movement to be unhappy with the Bush administration.

On the other hand, I share the atheosphere's distrust of people who try to redefine God or spirituality. Why would you even bother unless you were trying to pull some dirty trick on us? First you introduce God as some sort of harmless concept like "nature", "the emergent principle", "love" or "the source of morals". Then you give God a bunch of new attributes, like "good", "fundamental", "existent", "mysterious", or in the worst cases, "conscious". In the final stages, you accuse atheists of not believing in love, and then pray that they will see the light. Well, it doesn't have to happen exactly like that, but it frequently does.

Things get a bit muddled up from here on, because there are various senses in which I can "agree with", "accept" or "respect" Hugo's views. Now, obviously, I respect Hugo in the sense that I respect any human being. I also obviously disagree with him, in the sense that it is wildly unlikely that there is anyone I can always agree with on every detail. Much breath is wasted on these two truisms, when what we really care about is the meaty middle.

I can say that I disagree with Hugo in the sense that I would never bother trying to separate the two "Gods", not for my own benefit. I, personally, do not really care about the God of Faith (ie spirituality). I don't even understand why I'm supposed to strive for that sort of thing. Is it in the same sense that I'm supposed to enjoy giant frat rush parties, because, you know, that's what all normal, well-adjusted college students like? Man, whatever.

But in what sense can I agree with Hugo? To what degree do I accept this view as a viable option in the universe of personal philosophies? One that I might hold myself in an alternate universe?

The central problem here is an attempt to redefine God and other religious terms. Such an effort is made difficult by all sorts of pitfalls. There are all sorts of connotations to the words. This is especially true of "God". As soon as you've said the word, you're implicitly talking about a god that is good, a god that is transcendentally important, a god that objectively exists, a god that is conscious and pays attention to human affairs. Most of these connotations remain in the mind until you explicitly deny them; some still remain afterwards. For instance, take Stuart Kauffman's idea of God as the emergent principle, or the creativity in nature. That's great and all, in a vacuous sort of way, but he's implicitly elevated the importance of the emergent principle, as if it were some sort of fundamental mode of nature. Oh, come now. What's next, will we declare the Higg's Boson to be the God Particle, because particle physics doesn't already get a disproportionate amount of attention? Oh, wait, someone already did that, and we already dislike him for it.

The point is that if you're not very careful, you could end up saying something you don't want to say. Or worse, you could end up saying something that you did want to say. You know, encoding your own assumptions and biases into the definition, so that you can later push them onto other people without realizing it.

In one sense, I disagree with any attempts to redefine God, because it's a difficult task to do right, and I have little motivation to do it. But Hugo has different motivations and priorities--on a certain level, I accept that. If I understand correctly, his primary motivation is to "build bridges".

Mentally, I am comparing Hugo's efforts to redefine God to the things I learned about God in my Catholic high school. In my high school, we were taught that there are many descriptions of god. Sometimes god is a mother, sometimes a father, sometimes a gardener, sometimes love, sometimes the infinite, etc. The point is that these are all different understandings or aspects of the same thing. The various descriptions are at once all correct, and yet incorrect. So it goes with every description of God. Each one gives a new insight into God, but none is complete, as no description of God can ever be complete.

What like about Hugo's attempts to redefine God, as opposed to my Catholic upbringing, is that he has the right attitude about definitions. Namely, he takes a label agnostic approach (I have been meaning to write an essay to the same effect since forever, but this is as close as I've gotten). That means he is fluid with definitions. For instance, it is completely irrelevant whether "Richard Dawkins is a fundamentalist" is a true or false statement. What matters is what sense is it true, and what sense is it false?

He takes the same approach to redefining God. These definitions are not so much a tool to understand God as they are a tool to understand humans and religion. I think that's great. Furthermore, Hugo treats these definitions as a little experiment, something we can let go of if it turns out badly. A definition is a bit like a lens. If we stick to a single lens, our view can be distorted. But if we switch around the lenses occasionally, we can get multiple perspectives of the same reality. Contrast with my Catholic education. Not only is each description of God a new insight into God (rather than ourselves), the insights are considered to be cumulative. It's like we're supposed to put on all the lenses all at once, and declare the result to be the closest possible description of the ineffable. The insights are also cumulative with other definitions of god--the god of the Old Testament, the New Testament, of prayers, of devotion, etc. This is truly a semantic disaster, a bunch of definitions that have been all wrapped up together into a giant conflation of ideas. They cannot let go of the definitions. I mean, really let go of the definitions, to the point that, in some sense, God doesn't really exist, or God isn't really good.

Actually, I rather like this standard. The best redefinition of God (or perhaps any redefinition) is the one we are willing to let go of.

Monochromatic triangles in multi-colored planes

My earlier puzzle "A painted plane II" asked the following question:

Let's say I've painted each point on an infinite, continuous plane. Each point is either painted red or blue. Prove that there must exist three points of the same color which form the corners of an equilateral triangle.

I did not invent this puzzle. I took it from one of the many sources in my memory. However, the second question was my invention. This is one of those puzzles that just begs to be modified and generalized, so it was just a matter of finding a variation that was still reasonably challenging.

The most obvious way to vary the problem is by adding more colors. Is it possible to prove it for three colors? Four? Must there always exist a monochromatic triangle (meaning an equilateral triangle whose corners are all the same color), no matter how many colors we have?

I've long wondered about this variation, but I've found it too difficult. It is tempting to give up. After all, the mathematical ocean is a large place, and even trained professionals cannot do more than find a few pretty shells on the shore. But I did find a solution! It turns out that you can always find a monochromatic triangle, as long as there are a finite number of colors. The proof follows.

The key is to reduce the number of colors, one by one. Given n colors, we prove that there must exist a set of points which only use n-1 colors. Within that set, there is a subset which only uses n-2 colors. Another subset of that only uses n-3 colors. We keep going until finally we're left with a set of three points--an equilateral triangle--which must use exactly one color. That's the monochromatic triangle we were looking for.

To see how we might reduce the number of colors, let's first try reducing the number of colors from 2 to 1.
All we need to do is find three points (in this case, red, but they could be blue too) all in a row such that they are evenly spaced. If we assume there are no monochromatic triangles, we can deduce that the upper three points are not red, and therefore blue. Those three blue points form a monochromatic triangle, which contradicts our original assumption.

Let's generalize a bit more, reducing from n colors to n-1 colors.
All we need to do is find X number of points, all of the same color, all in a row, and all evenly spaced. Here, we found a row of blue points, but we could have used any of the n colors. Here, X is 7, but it could have been any number. Assuming there are no monochromatic triangles, there are a bunch of points (shown in black) that cannot be blue. In other words, there exists triangle-shaped set of points which is made up of no more than n-1 colors. Hopefully we have enough space to further reduce the number of colors to n-2, then n-3, and so on until we have only one color left. This last color will form a monochromatic triangle.

But what if we don't have enough space to reduce down to one color? We simply choose X to be larger!

This proof assumes that we can find X evenly spaced points in a row and of the same color. It assumes that this is true, no matter how large X is. How do we know this is the case?

Lucky for us, mathematicians have already proven that this is the case. It is called van der Waerden's Theorem. Van der Waerden's Theorem states that for any natural numbers r and k, there exists a number n(r,k) for which the following is true: If we paint each of the natural numbers from 1 to n(r,k), using no more than r different colors, then there must exist a monochromatic arithmetic sequence of length k. That's exactly what we need! (For an incomplete proof of the theorem, see Wikipedia.)

If we have two colors, we only have to look at n(2,3) points in a row, and we're guaranteed to find a row of three evenly spaced points all of the same color. If we have two colors, we need to find a monochromatic row that is longer than n(2,3) so that we're guaranteed to have enough room to reduce down to a single color. To find a monochromatic row of n(2,3)+1 points, we need to look at a row of n(3,n(2,3)+1) to be guaranteed to find one. If we have four colors, we need to look at a row of length n(4,n(3,n(2,3)+1)+1).

The caution is that though van der Waerdan's theorem tells us that n(r,k) exists, it doesn't tell us what precisely n(r,k) is. Mathematicians only have an upper bound for n(r,k). The best current upper bound was found by Timothy Gowers.*

[Here, V(r,k) just means the same as n(r,k)]

The actual value of n(2,3) is known to be 9, but the formula above gives us a number so large that I don't even have the tools to calculate it. I don't know how much n(3,n(2,3)+1) is, but it's probably extremely large. But luckily we're on an infinite continuous plane, so we can have as many points in as small a space as we like.

Did I forget something? Oh, yeah: QED

*see "A new proof of Szemerédi's theorem"

Confirmation bias

It's been a while since I've written anything in the fallacies category. Logical fallacies are something of a staple of skepticism. It goes at least as far back as Carl Sagan's Baloney Detection Kit in The Demon-Haunted World. Well, logical fallacies are good to know. Being able to recognize a fallacy helps to keep yourself and everyone around you honest. But it is a mistake to think that fallacy recognition by itself constitutes critical thinking.

An important addition to any skeptical tool kit is the ability to recognize cognitive biases. Cognitive biases are not so much specific kinds of mistakes, as they are the motivating force for an entire pattern of mistakes. They are subtle tendencies of the human mind to diverge from rationality. One example of a cognitive bias is pattern recognition. Humans are good at pattern recognition. Too good. We are more likely to recognize a pattern that isn't there than we are to miss a pattern that really is there. This cognitive bias results in stuff like pareidolia and the post hoc fallacy ("after, therefore because of"). We might trace this cognitive bias to evolution--to our ancestors, missing a pattern was much more dangerous than seeing a pattern that wasn't there.

But the most important cognitive bias of all is confirmation bias. Confirmation bias is the tendency of people to be more receptive to evidence and arguments that confirm their previous views. The common description of confirmation bias is "Remembering the hits, forgetting the misses". But it's not necessarily a matter of forgetting. More often, we simply underplay, undervalue, or wave away opposing evidence, while being especially receptive to confirming evidence.

It is important to understand that this does not always result in anything we might call a "fallacy". Cherry picking evidence is a fallacy in that it is a misunderstanding of the nature of statistics. But what if we have various pieces of evidence, from all different sources, and we are unsure of the relative quality of each? What if we look into it, and reason ourselves to the genuine conclusion that the confirming evidence is better quality than the opposing evidence? Is that truly "irrational"? What if someone else does the same and draws the opposite conclusion? Cognitive biases allow the rational to blend into the irrational. Even as you use reason, it can betray you, perhaps even giving you more conviction in your wrong beliefs.

Confirmation bias is an ever-present force. It is always in effect, for everyone. Some might even go so far to say that confirmation bias accounts for all of human disagreement.

Ending confirmation bias is a difficult or impossible task. In other people, you can only really point out the most obvious errors that result from confirmation bias. You can point out cherry-picking, or any double-standards of evidence, but that's just about it. In oneself, it's a bit easier, since, to some extent, we can control ourselves. To some extent, we can make ourselves into objective observers. But I wouldn't go too far down that path. You might just end up convincing yourself that you must always be right because you are always so darn objective. In your confidence, your own biases may just will catch you off guard.

If my thoughts may wander a moment, I once said that this was the problem with the Realist movement in art. The artists wanted to be more truthful and objective, just like science. Perhaps they succeeded to some extent, but in the end, they only created a better illusion of objectivity. This was based on a misunderstanding of how science should work.

See, science's solution to confirmation bias isn't to make oneself into an objective, emotionless observer. Science's solution is the peer review process. A bunch of humans with knowledge of the facts, they come together with all their different emotions, opinions, and biases, and they produce truth. That is the best solution to confirmation bias, or about as good as we'll ever come up with.

Of course, outside of science, we don't always have the benefit of peer review. Ah, well, we just have to make do.

Puppies and Kittens

It's time for a blogging experiment! I'm not sure whether it will work out, but as a B-blogger, I'm allowed to try this sort of thing.


We're going to play a game called Puppies and Kittens. It's a variation of Nim. These are the rules:

• There is a pet store with a bunch of puppies and kittens.
• Two players take turns buying puppies and/or kittens.
• During your turn, you may either buy as many puppies as you like, as many kittens as you like, or an equal number of puppies and kittens. You must buy at least one of either.
  
• Whoever buys the last puppy/kitten wins.

We'll play in the comments. Here are three starting positions, in order of increasing difficulty:

Game 1: 5 puppies, 6 kittens
Game 2: 10 puppies, 7 kittens
Game 3: 25 puppies, 16 kittens

The first player has the winning strategy in each game, so you, the reader, take the first move.

If you play, please do not use the name "Anonymous". If you wish to remain anonymous, select the "Name/url" option and just make up a name. This will make it less confusing for me. Also, don't bother saying how many puppies and kittens you buy, just say how many are left afterwards. Feel free to play as many times as you like until you've figured out how to win.

Bad argument: God's burrito

I don't mean for this blog to be all math all the time. Therefore, I present God's burrito, aka the unliftable stone.

This argument is a favorite among folks who like to stump their Sunday school teachers:

Can God create a burrito so hot that even he cannot eat it?

Or,

Can God create a rock so big that even he cannot lift it?

I like the burrito version better, because it's kind of funny and cute for its anthropomorphization. Imagine a tiny little kid asking the question.

But to point out the obvious, little kids do not necessarily have any more insight than adults, no matter how cute we imagine them to be. And Sunday school teachers might be stumped by the question, but they shouldn't be.

Here are some responses that the teacher shouldn't give.

"Stop asking questions." Trying to extinguish a child's curiosity? For shame.

"Don't be a smart alec." An honorable sentiment, but misplaced. It could be an honest question.

"Yes, he can create an uneatable burrito. And then he can eat it." Oh, so God contradicts formal logic?

Let me take a moment to more fully address that last response. Now, I imagine that most folks view God as more fundamental than language. God simply is, and our words can only describe him. The god in question is omnipotent, so he can create any burrito at all. And all we can do is describe those burritos. If we imagine all possible burritos, not one of them could we describe as "uneatable and eatable", because that would break the rules of our language. In our language, "uneatable" is defined as the negation of "eatable", so they simply cannot be paired to describe any possible object. The flaw is not in God, but in our language (if you call it a flaw).

Let us formulate the argument more formally.

1. Definition: God is omnipotent.
2. Definition: An omnipotent object can do anything.
3. Conclusion: If God exists, he can create an uneatable burrito.
4. Conclusion: If God exists, he can eat any burrito.
5. Conclusion: If God exists, there can exist a burrito that is both eatable and uneatable.
6. Definition: An object that is uneatable is not eatable.
7. Conclusion: God does not exist.

Yes, this is a valid and rigorous proof, spelled out as thoroughly as I could make it. Did you notice anything about the argument? There are no premises. This is somewhat unusual in a logical proof. It means that there is no way to question the original assumptions of the proof, because there are no original assumptions.

What you can question, however, are the definitions. It's not that the definitions are wrong per se. In logic, one definition is about as good as another. What's wrong is that these definitions do not necessarily match the definitions we use in everyday language. Who's to say that God is omnipotent? Who's to say that omnipotence really means capable of doing everything?

I mean, who really cares if God can't make uneatable burritos? Who cares if he can't make round squares? As long as God can create universes, answer prayers, and manifest himself in human form, I think most people will be satisfied.

And so, any serious thinker would simply modify the definition of God or of omnipotence to get around the burrito argument. That modification is: the definition of "omnipotence" does not allow for any logically impossible action. It's a rather trivial modification, that doesn't change anything else about God in any way whatsoever. And it's not even like this is a new development for theology or anything--it dates at least as far back as St. Agustine in the 5th century.

It's quite common to include a whole host of other exceptions to the definition of omnipotence. God can't sin. God can't contradict his own will. God can't change his mind. God can't cause himself to lose omnipotence (we can add a further exception to this exception so that God can make part of himself human). God can't contradict free will. Etc. Etc. Just about any omnipotence paradox you can think of can be fixed by a trivial exception. No, it's not really special pleading, not anymore than "You can't divide by zero" is special pleading.

With all these exceptions, you might ask, "How can you still call God omnipotent?" Easy. He's still more omnipotent than you are, right? More omnipotent than anything else imaginable? That is sufficient.

I might add that the free will exception allows for an interesting variation on the burrito paradox, to which the answer is a straightforward "yes":

Can God create a being with will so free that even he cannot control it?

But I wouldn't read too much into it.

M24 puzzle solved

Remember that Scientific American article about some new permutation puzzles based on simple sporadic groups? There was the M12 puzzle (for which I gave a few hints), along with the M24 puzzle and dotto puzzle. The last two puzzles I described as "ridiculously complicated", but it seems that someone out there has gone and solved the M24 puzzle. You can congratulate Baumann Eduard.

See his solution here
You will also need this table

That's really amazing.


In other news, it was recently proven that at most, 22 moves are required to solve any Rubik's cube. Positions are known that require 20 moves, so we now know that the optimal solving algorithm requires between 20 and 22 moves.