Schrodinger's Cat

Schrodinger's Cat is a famous thought experiment. It goes like this:

A cat is kept in an impenetrable box, and no one can look inside. There is also a single radioactive atom. After exactly one half-life, there would be a 50% chance that the atom has decayed, and 50% chance that it hasn't decayed. But since we can't observe the atom, we must describe it in a quantum mechanical mixed state, 50% decayed and 50% not decayed. Within the box, there is a detector which determines whether it has decayed. If it has decayed, then the cat is killed. If not, then the cat is left alive.

However, since we have not opened the box, we must describe the cat in a quantum mechanical mixed state, 50% dead, and 50% alive. So the cat is dead and alive at the same time. When we open the box, the quantum state collapses, and we either find a completely dead cat or a completely alive cat.

Before I get into explaining it in more detail, I'd like to ask, what's your favorite joke involving Schrodinger's cat? Mine is this Saturday Morning Breakfast Cereal comic.

Moving on!

Some people think that the takeaway message of the Schrodinger's Cat thought experiment is that it's completely absurd. "A cat which is dead and alive? Absurd!" But there's more to it than that. It's that the explanation seems unnecessarily absurd. Couldn't we just say that there's a 50% chance that the cat's dead and 50% chance that it's alive? Why do we need to add in this whole idea of being dead and alive simultaneously if no one ever sees the cat that way anyways? It's like we're saying that something really weird is going on, but only when we're not looking at it. It's as preposterous as painting my moustache green, only to hide it with a huge fan whenever anyone looks my way.

The thing is, the weirdness of quantum mechanics may seem unnecessary for the cat, but it is entirely necessary for smaller systems, such as single particles or photons. There's even a proof, known as Bell's Theorem, that at least some weirdness is absolutely required.

Consider single photon. If we build a detector to observe its polarity, we will always see it polarized vertically or horizontally.

Since we only see it one way or another, it would be quite absurd to say that it's simultaneously horizontal and vertical. We only see it one way or another, so why would we even consider that it could be both at once? But in fact, there is a quantum state which is a mix of horizontal plus vertical. There's also another quantum state which is horizontal minus vertical. And we can see both of these states if we just rotate our detector 45 degrees. Now we have the two diagonal polarizations.

If you try to measure either diagonal polarization without rotating the detector, then we will have a 50% chance of seeing horizontal and 50% chance of seeing vertical. But before we observed them, they were both vertical and horizontal simultaneously, so to speak.

But how is the cat anything like those photons? No matter how you look at the cat, it will always be either completely dead or completely alive. Rotating your head won't help. In principle, you can build a detector which can see the cat in a simultaneously dead and alive state. But in practice, such a detector is utterly impossible. You'd probably have to look at every single particle in the cat all at once. And you'd have to isolate the cat, making sure that no stray observers destroy the mixed state by looking at it.

Cat illustrations taken from Griffith's Introduction to Quantum Mechanics

But I could be wrong. As I said, the detector is utterly impossible to build. So if I can't build it, how would I prove that it's possible at least in principle? Perhaps there really is something fundamentally different happening to the cat which is not happening to the photons. It depends on which interpretation of quantum mechanics we take.

The most common interpretation is the Copenhagen interpretation. The Copenhagen interpretation says that at some point, the state of the system "collapses" into a definite state. So when we open the box and look at the cat, the cat "collapses" into a completely alive or a completely dead state. Or perhaps the collapsing happens earlier? Perhaps the cat observes its own survival/death, collapsing itself into a state that is either completely alive or completely dead. Or perhaps the collapse occurs even earlier than that? There is a detector inside the box which determines whether the radioactive atom decayed or not. Perhaps this detector collapsed the atom by observing it.

Might we go even further, and say that the atom collapses itself? Perhaps. But an atom is a relatively small set of protons, neutrons, and electrons. Perhaps we will eventually build a detector which can measure the atom in a simultaneously decayed and not decayed state. So we probably don't want to go as far as the atom.

We can go in the opposite direction too. Perhaps the quantum collapse happens even later than when you open the box. Perhaps you, too, are in a mixed state, simultaneously happy that the cat lived, and sad that the cat died. "Ridiculous," you say. "When I look at the cat, I am 100% sure that it is either dead or alive." That doesn't prove anything at all. You could be in a mixed state, simultaneously being absolutely sure the cat is alive, and absolutely sure that the cat is dead. Maybe you exist in this mixed state until you tell a friend about it, allowing your friend to "collapse" you.

Or maybe you never collapse at all, and nothing ever collapses. Perhaps the entire universe is in a mixed state of having simultaneously N living cats and N+1 living cats. This interpretation is known as the Many Worlds interpretation, because it implies that the world is simultaneously in many different states. It's like we're all together in this huge box which will never be opened. (FYI, this is the interpretation I advocate, so I might be biased in my presentation.)

So that's the basic idea of Schrodinger's Cat. I hope this raised all sorts of new questions... you can always ask.

Blessing Theseus's Ship

I took a philosophy of mind class some time ago, and one of the things they discussed was Theseus's Paradox.

Let's say we have a wooden ship. We replace each plank on the ship, one by one, until none of the original planks are part of the ship. Is it the same ship now? If not, when did it stop being the same ship? Now we take all those extra planks and put them together in the exact same way they were before. Are either of these ships the same ship as before, and which one?

This was the sort of question which made me hate philosophy.

However, Zeno Ferox just gave me an idea to solve this paradox once and for all. Supposedly, you only need to bless a rosary once--further blessings do not increase its blessedness. And if you replace, say, 20% of the beads, you don't really need to bless it again, because the blessing is integral to the rosary itself. Aha, so we know that it's still the same rosary even when 20% is replaced!

So, here's the experiment:

1. Bless a rosary.
2. Replace one bead.
   
3. Ask the priest whether it is still blessed.
4. Repeat steps 2 and 3 until all beads are replaced.
5. Use the original beads to make another rosary.
6. Ask the priest if the rosary is blessed.

There are a few difficulties in this experiment. First of all, it must be double blinded. The priests can't be given any indication how many beads were replaced. Second of all, I'm assuming that the priests can tell whether an object is blessed or not. Uh, they can tell, right? They don't need to be perfect, of course. We'll just repeat the experiment with a hundred different rosaries and average out the results. And we might as well toss in a control group of unblessed rosaries while we're at it.

I wouldn't have thought Theseus's Paradox to be a scientific question, but clearly I was wrong.

Is Karma true?

On Friendly Atheist, someone asked whether karma is a reasonable thing to believe in.

To be perfectly cliché, I like to believe that “things happen for a reason,” that there might be an order to the universe (can an order be secular?), and that sometimes good things can even happen to good people. These ideas deal more in the vein of positive events; I do not subscribe to inane drivel, religious or secular, that scapegoats minorities for natural disasters, the breakdown of family values, etc.

My response to karma will be similar to my response to faith. That is, it depends what kind of karma we're talking about here.

There are, after all, at least a few concepts of karma which I consider to be nonsense. For instance, in some traditions, karma carries over through reincarnation, from life to life. Therefore, someone who was good and holy and practiced the right rituals at the right times would in his next life be reborn in one of the upper castes, while someone who was evil and greedy and disrespectful to the gods would be reborn in one of the lower castes, if any caste at all. And that's why poor people deserve their poverty.

Besides being premised on the belief in reincarnation, and lacking any sort of evidence (how do we know it isn't the other way around, with good people being born into lower castes?), it also seems exceptionally unjust. I don't think I should be held responsible for actions in past lives just because those past lives share some unfathomable soulstuff with my own. It seems to me the whole motivation for karma is the idea of justice (rather than the idea of evidence), and this particular kind of karma can't even get justice right.

Another example of karma which I think rather silly is the Jainist concept that karma comes in small particles. These karmic particles stick to your soul, and have complex (but apparently understood to very fine detail) interactions which influence how you think, how you feel, and what happens to you. This belief seems far more specific and extraordinary than we have evidence to justify.

But of course, people are free to pick and choose the kinds of karma which are not so egregiously wrong. The questioner on Friendly Atheist, for instance, believes in karma, meaning "good things can even happen to good people". I note that he left out the part about bad things happening to bad people, which seems equally reasonable, but let us not dwell on it. This kind of karma, of course, is completely correct. Good things can happen to good people. I've seen it with my own eyes! I might go even so far as to suggest that, sometimes, good things happen to good people because they are good. Isn't that quite amazing and wonderful?

However--and don't let this subtract from any earlier feeling of amazingness and wonderfulness--I'd say that this occurs by completely natural mechanisms which you are probably already familiar with. For example, when you are nice to people, they tend to be nice back. When you break the law, that increases your chance of getting caught from zero to nonzero. If you work hard on your homework, you may occasionally learn something which will help you later. And if you pray to the gods, then you might... then you might...

Okay, so it doesn't always work. We can't always find a mechanism which links good actions to good consequences and bad actions to bad consequences. It might be, of course, that we were mistaken about which actions and consequences were good and bad, but then it might be the case that karma is simply incorrect in these cases. If we can't think of some reasonable mechanism, perhaps that's because there isn't one.

You might ask, what if there's a mechanism and I'm simply not aware of it? That is always a possibility. But it seems equally possible that there's a mechanism which links good deeds with bad consequences and vice versa. Perhaps, in these particular circumstances, the world operates under the principle of "no good deed goes unpunished" rather than the principle of karma. And since we don't have any clue as to the mechanism, how do we know what these "particular circumstances" are? Maybe it only works on Tuesdays, and backfires on Thursdays? Maybe it only works if your name begins with a B? Or maybe it depends on whether a particular butterfly in Australia flapped its wings on May 3rd, at exactly quarter to four in the afternoon. And so the argument from "you don't know either" fails again.

But perhaps I dwell too much on the incorrect concepts of karma (since they are the most deserving of criticism). Let's not forget that sometimes karma is correct, that sometimes good things happen even to good people. Let's not forget how amazing and wonderful that is.

I suggest that it would still be wonderful and amazing even if we didn't call it karma. The word karma suggests some single mechanism, when really there are a collection of mechanisms which causally link good deeds to good results. The word karma also has the unfortunate association with the caste society in India. It's worth thinking about, but don't let that stop you from using words as you see fit.

Classifying Exciting and Boring

My summer research job is all about classifying things. We got this big pile of events, and I want to classify them as Exciting and Boring. So what I do is I go through the events, one by one, inspect them, and then either toss them into the Exciting pile or the Boring pile. Okay, so it's not really that simple. I fiddle around a lot with Receiver Operating Characteristic Curves and Operating Points, and False Alarm Rates and Efficiency, and a whole bunch of other Concepts Which I Capitalize Ironically. (The CWICI will only get worse as I go on.)

Which is all to say, my thinking as of late has been colored by the science of classification. The other day, I classified some pens into the Out Of Ink or Just Needs To Be Shaken categories. Then I classified the different vegetables in my vegetable soup as Too Much or Not Enough. I took a list of books I wanted and classified them as In The Bookstore or Needs To Be Ordered Online.

Yeah, so I'm not actually doing all this ('twas a joke), but you get my point get that I have a point.

One thing you realize in the science of classification is that you will always get some of them wrong. Some of those events which I classified as Exciting will turn out to be false alarms. Some events which I classified as Boring are in fact Exciting. In fact, it's sort of arbitrary. I can always choose to be pickier, so that there will be less false alarms, but I'll miss more of the Exciting events. Or I can be less picky, accepting more false alarms, but being more sure that I'll catch all the Exciting events.

The tricky thing is, we don't really know how many events are actually Exciting. That's what we're trying to figure out! Maybe none of them are Exciting, and all I'm looking at are false alarms. How would I know? What's the False Alarm Rate when nothing is truly Exciting?

In my research, we have a very complex and precise way of estimating the False Alarm Rate.* But sometimes, we are not so lucky to have such a method available to us. So we make do with slightly less precise arguments.

*It probably involves time travel and robots.

For instance! UFO sightings: how many of them are Exciting Aliens, and how many are Overexcited Earthlings? UFOlogists assert that the False Alarm Rate is low, and that at least some of them are Excited Aliens. Skeptics assert that the sightings are well within the False Alarm Rate. How do we know which it is? Well, astronomers tend to inspect the sky much more often and more carefully than lay people. So astronomers would tend to have a smaller False Alarm Rate, while at the same time catching much more of the Exciting Aliens if they indeed exist.

There are far, far fewer UFO sightings among astronomers than among lay people. This indicates that UFO sightings are well within the False Alarm Rate, and there may be no Exciting Aliens to speak of. But hey, all these Overexcited Earthlings are pretty cool and interesting in their own way.

Of course, I stole this argument from Phil Plait, the Bad Astronomer. And then I made it a whole lot more arcane and technical. Fun times.

Science meets Religion

Three Paradigms, Three Questions

Back in high school, I took a class called Science and Religion. In retrospect, this is pretty cool. How many high school students get to take a class that introduces them to the discourse between science and religion? We read a book called When Science Meets Religion: Enemies, Strangers, or Partners? In it, Ian G. Barbour described four paradigms of the interaction between science and religion. Those paradigms are: conflict, independence, dialogue, and integration.

Personally, I never liked how Barbour divided up the last two paradigms, dialogue and integration. Both of these describe the view that science and religion are not just compatible, but are actually friends which can contribute to each other. I'm pretty sure that Barbour advocates the integration paradigm, and that's why he divided it up that way. But if he had advocated some particular sort of independence paradigm, would he instead have split the independence category into two? Surely, the independence and conflict paradigms are complex and diverse enough that they could also be subdivided.

But never mind Barbour. Honestly, I don't remember the book too well. I just remember the three kinds of paradigms: conflict, independence, and partnership.

So the question is, which of these three paradigms to I ascribe to? But I've come to realize that this question is unclear. When we considered the question in high school, the question was specifically about our religion, Catholicism. If someone came up and said that Young Earth Creationism contradicts science, we would have responded, yeah, but Young Earth Creationism is bad religion. And with that, we would have quickly muddled up the issue with normative concepts like "good" and "bad".

Therefore, let us separate out what could be, what is, and what ought to be. For each of the three questions, I will briefly explain my position. You'll see that I give a different response for each one.

What could be

If we want to know the relationship between science and religion, we first have to define what we mean by science and religion. Science is relatively well defined. We may have lots of discussion about what is and isn't science (the demarcation problem), but that only occurs on the fringes. For most things, we can definitively say it's clearly science or it's clearly not science.

Religion is much more broadly defined. In the category of religion, we must include such disparate things such as Christianity, Islam, Hinduism, Chinese traditional religion, Buddhism, primal-indigenous religions--and that's just going down the list of the most common kinds. Even in the individual groups, you will find much diversity. Clearly, not every religion is all about moral guidelines, though perhaps some are. Not every religion makes statements about how the universe was created, though many do.

Therefore, I think that all the paradigms are, in principle, possible, depending on what kind of religion we're talking here. If the religion is purely about moral attitudes and ritual, then obviously that religion will be independent of science. Science cannot say anything about what should be, because the universe would be the same, unaffected by what we think should be. On the other hand, if the religion requires that we believe that the world is a flat disk resting on the backs of four elephants on the back of a giant turtle, then this religion will obviously be in conflict with science. Because we've looked around, and we're pretty sure the giant elephants aren't there.

Now this is a harder example to imagine, but I believe that the partnership paradigm is also possible. Let's say we have a religion which takes the conclusions of science, puts them through some inscrutable black box of theology, and derives religious rituals from them. Science is contributing to religion, ergo they are partners in a strict sense. This might not be the likeliest of religions, but it seems possible.

One thing I do not think is possible, is for it to go the other way around, for religion to contribute to science. Science is more than just a set of conclusions, it is a set of methods which can be used to justify those conclusions. Religion cannot simply offer some conclusion to science. Then it's not science at all, since it does not have scientific justification. We might say that religion occasionally inspires scientists for whatever obscure reason, but that's pretty much the extent of it.

What is

Basically, everything which is possible is probably true of some religion out there, or at least of part of some religion. But if you asked me which paradigm is dominant, I would say the conflict paradigm is dominant, followed by the independence paradigm.

The most obvious real-life example of science-religion conflict is Biblical Creationism. Biblical Creationism holds that Genesis is an accurate account of the creation of the world. Without going into details, this is complete nonsense from a scientific perspective. What is this if not conflict? A common response is that Creationism is not true religion, or is not good religion, or is on the fringe. This is also nonsense. According to the 2008 Gallup poll, about 44% of the US believes that God created humans in their present form in the last ten thousand years. It would be quite reckless to dismiss such a large segment of the population just because you think they ought not to exist.

But even for the non-Creationists, there are more subtle conflicts going on. For example, many people believe prayer is effective. There's really no reason to think so, no more than there's reason to think that homeopathic medicine could be more effective than water. In the few conditions in which prayer has been scientifically tested with proper protocols, it has proved ineffective. So here we got another conflict, perhaps not as big a deal as Creationism, but nonetheless present and widespread.

And then we could get into fuzzier and more philosophical examples. The virgin birth and resurrection, for example. I would never say that miracles (the physical-law breaking kind) are, in principle, impossible. But they are, by their very nature, extremely unlikely, and there's no way a bit of historical evidence could possibly overcome that. Thus there is at least a weak conflict between science and religion here. And on the even fuzzier end, I could talk about "ways of thinking"... but I won't.

Note that this conflict can exist even when the religions themselves deny it, or when they emphasize other non-conflicting aspects. Most people who believe in the virgin birth don't think it has anything to do with science. But then, a lot of Creationists I've met are totally gung-ho about science too.

That's not to say that the virgin birth and Creationism are completely equivalent. There are different levels of conflict, ranging from harmless to subtly malicious to outrageously absurd. I'd say it goes from "mostly harmless" to "terrible" when religion hampers people's study of science. This puts Francis Collins on the wrong side, since he apparently believes that altruism could not have evolved. But he might make a good NIH president regardless, I don't know.

What ought to be

This question is the easiest for me. There should be no conflict with science at all! Because science is knowledge. It's a good thing. Everyone ought to be on board with science. Therefore, if religion exists, then it ought to be on board with science too.

Note that I am ignoring the scenario in which religion doesn't exist. That would render the whole question of religion's relationship with science moot!

As for whether science and religion should be independent or partners, I suppose I would slightly prefer partnership. Because that would mean that religion is even more on board with science. But there are some major caveats. First of all, this should only be accomplished by a change on the religion side, not on the science side. It may require unrealistic amounts of change beyond recognition. And second of all, science may contribute to religion, but religion may never contribute to science. This is because, as I said before, it is impossible for religion to contribute to science, so we might as well not fool ourselves.

Given these caveats, and the risk of accidental conflict, it would be much more prudent to advocate independence. And they must be actually independent; just pretending doesn't count.

...Of course, I have very little power to change the nature of religion from the outside, so maybe this particular question is moot.

So that sums up my position. I hope that even if my readers disagree on the answers (there's a lot to disagree with), they realize the necessity of splitting it into at least three questions.

What's so wrong about PZ?

In the blogosphere, there's yet another ... ehem ... discussion ... going on about "new atheists" and "accomodationism".

The setting: Chris Mooney and Sheril Kirshenbaum wrote a book called Unscientific America: How Scientific Illiteracy Threatens our Future. For whatever reason, an entire chapter or two of the book was devoted to criticizing "new atheists" in general, and PZ Myers and Richard Dawkins in particular. As should have been expected, PZ Myers wrote at least four or five posts about it, and so did Chris and Sheril. And the rest of the blogosphere is chattering about it.

What's especially irritating to me is how much the arguments mostly just swish right by each other. I find that I have to keep track of not two, but four different sides: 1) what Mooney and Kirshenbaum say, 2) what PZ says, 3) what Mooney and Kirshenbaum say PZ says, and 4) what PZ says Mooney and Kirshenbaum say. I have to keep track of all this he said she said junk when I'd rather be keeping track of the issues.

One question I'd like answered is, what exactly do Mooney and Kirshenbaum think is wrong with PZ Myers? As in, which part specifically? Is it PZ's position that science and religion conflict? Or is it PZ's harsh internet persona, and his swarming horde of followers? Or something else?

See, Mooney and Kirshenbaum just hammer on and on about PZ Myers (on the internet, if not in the book). But the problem with focusing on a specific example is that you lose applicability to other people! I mean, I can only care so much about PZ Myers. Because I am not PZ Myers. I share PZ's philosophical position, more or less, but his demeanor not so much. And to show that I'm not alone, there was a recent post on Cosmic Variance in which Sean Carroll argued that science can answer questions which are frequently categorized as religious. And he does it calmly, politely.

And arguably, Dawkins is in this category as well. Dawkins argues a lot about how God is in principle a scientific claim, but Dawkins himself is quite polite. Dawkins doesn't need to swear or be rude in order to shock people; he lets his clearly stated beliefs and positions do the shocking.

So do Mooney and Kirshenbaum have a problem with that, or not?

And if they do have a problem with that, what do they expect me to do about it? Surely they could not be arguing that I should change my philosophical position just because it's politically inconvenient. Do they argue that I should keep it quiet because it's politically inconvenient? Forget that! I happen to like having a bit of a voice, thank you very much. If Mooney and Kirshenbaum wish to convince me, they would need some really good direct evidence that speaking up hurts the cause, not just in the short run, but in the long run. And it has to be my cause, not theirs.

And what if their problem is not with PZ's philosophical position, but with his approach? This implies that they would also have a problem with someone who cusses and swears and pulls shocking stunts as he argues that science and religion are completely compatible. But I'm not seeing it. I can't imagine it.

But assuming that they're really just disagreeing with PZ's approach, and not his philosophical position, some of the same questions appear. Where is the good, direct evidence that PZ's approach backfires, even in the long run? And how do they expect to change it?

My line of thinking is, "That's just the internet for you." People can be mean. Different websites tend to have completely different accepted levels of politeness. Internet culture has far too much inertia to change it easily. Look at it this way. No matter what you do, no matter what you argue, you will never change me into a PZ Myers. I don't have the right temperament for it. So how would we change someone like PZ Myers into someone like me? And how are you going to do it to his tens of thousands of readers?

Sometimes, you have to accept the social landscape as is, working around it if you have to. And who knows (since I have yet to see any direct evidence otherwise), maybe a variety of approaches is a good thing after all. You know, don't put all your eggs in one basket. And recognize that our audience is diverse, our goals are diverse.

Feynman Lectures on Tuva

Bill Gates bought the rights to a series of lectures by Richard Feynman called "The Character of Physical Law". And now they're available for free online!

I've heard, of course, of the legend that is Feynman, but this is the first time I've actually cared to look into it. Now that I've seen him, he's a great lecturer for sure. Though the lectures are on the basic, general level, they say things that even I've never thought about.

So far, my favorite lecture is number 2, "The Relation of Mathematics and Physics". He even goes so far as to refer to Lagrangian Mechanics as a mathematical alternative to Newtonian Mechanics or field mechanics. That makes anyone awesome in my book.

Dividing a plane

Let's say we have a plane. Draw N straight lines on the plane, any way you wish. Try to divide the plane into as many different regions as possible. How many regions is that? For example, if we draw 1 line on the plane, we can divide it into two regions. If we draw 2 lines, we can divide it into four regions.

Followup questions: Draw N perfect circles on a plane, of any size, anywhere you want. Into how many regions can you divide the plane? Next, draw N perfect ellipses on another plane. Into how many regions can you divide the plane?

This is, by the way, what's called a combinatorics puzzle. Combinatorics is the branch of mathematics which is all about counting. Fabricated statistics say that 50% of my readers will run away at the first mention of math, but surely no one's afraid of a little counting?

Solution has been posted

Sexual orientation and love vs sex

I was reading a paper called "What does Sexual Orientation Orient?" by Lisa Diamond, and it blew my mind. In it, Diamond presents a model of romantic love and sexual desire. She argues that the two are functionally independent, and that people can have one without the other.

Love and sex, after all, evolved for different reasons at different times. Sexual desire primarily evolved for the purpose of reproduction (not to say that sexual behavior didn't further evolve for other purposes). Romantic love, on the other hand, evolved so that parents stay together long enough to care for their offspring. There are two separate mechanisms, so perhaps they don't always come together.

And in fact they don't always come together. People generally accept the idea of sexual desire without romantic love, but it goes the other way too. A high number of people report experiencing infatuation without the sexual attraction. But of course, many other people do experience the two together. This is because, most of the time, sexual desire for a person is soon followed by romantic feelings for the person. Diamond argues that this link can go in the other direction too. In other words, romantic feelings can lead to sexual desire. Interestingly, the link between romantic love and sexual desire appears much stronger among women than among men.

There are, by the way, two stages of love. Diamond states this not as a new idea, but as one which comes from the scientific literature (not that I had ever heard of it before). The first stage, known as passionate love, or infatuation, is characterized by a desire to be close to the person, a fascination with their appearance and behavior, and so forth. The second stage, known as companionate love is characterized by "feelings of calm, security, mutual comfort seeking, and deep affection." Note that both stages of love can possibly without sexual desire.

Diamond suggested one rationale for these two stages, which I thought rather enlightening. Companionate love requires sustained proximity between the two people. Passionate love motivates people to be close together, allowing for the development of companionate love.

Diamond goes on to explain a surprising evolutionary path for love (Warning: evolutionary psychology ahead!). Before the evolution of adult romantic love, there was the love between the infant and the caregiver. Rather than creating a whole new mechanism of love from scratch, evolution took the mechanism for infant-caregiver love and made what is now romantic love between adults. You might compare it to how feathers originally evolved as insulation, but were later adapted for flight.

I'm not sure I buy this evolutionary mechanism for love. With evolutionary psychology, it can be hard to tell the difference between a well-supported theory and a just-so story. It's especially difficult for me, because Diamond talks about about all this oxytocin and stuff and I don't understand any of it. It seems to be well-supported, so I will accept it for the moment.

Diamond goes on to say that infant-caregiver love does not have an orientation! Infants don't prefer one gender over the other, that would be maladaptive. She uses this fact to argue that romantic love is not intrinsically oriented towards either gender either. And so Diamond's model comes together. Heterosexual people can fall in love with the same gender, and homosexual people can fall in love with the opposite gender. And since romantic love can lead to sexual desire (especially among women), people can end up having sexual desire for a gender which is contrary to their orientation. This goes a long way to explaining why heterosexuals sometimes unexpectedly fall in love with a single specific person of the same gender, and why homosexuals sometimes unexpectedly fall in love with a single specific person of the opposite gender.

However, I must voice a small disagreement I have with Lisa Diamond's model. If romantic love had no intrinsic orientation towards either gender, then why do the majority of people fall in love only with the opposite gender? Diamond argues that this is because of other influences, such as cultural influences. For example, people tend to hang out more with people of the opposite gender, so they fall in love with the opposite gender more often. But... I'm not sure that's really true in all cultures.

I also find the evidence for this assertion to be rather weak. Even if I accept that romantic love evolved from infant-caregiver love, that doesn't mean that they are identical in every respect. Perhaps romantic love later evolved to be oriented towards one gender or the other. Outside of the evolutionary argument, Diamond admits that there are only a few pieces of direct evidence for her claim. Her main piece of direct evidence revolves around prairie voles, and simply isn't convincing to me.

It seems to me more reasonable to say that there are two orientations: sexual orientation and romantic orientation. For most people, these two orientations are the same. But occasionally they differ, and who knows, biromantics might be more common than bisexuals, or maybe everyone is weakly biromantic. This adds another intriguing layer of complexity to the idea of sexual orientation.

Also see "Emerging Perspectives on Distinctions Between Romantic Love and Sexual Desire", a shorter article with the same thrust. Or see Sexual fluidity, a whole book by Lisa Diamond.

Minesweeper solution

See the original puzzle

There are five different mine arrangements which are consistent with the given minesweeper board. See here: Spoiler alert!

Each of five arrangements is equally likely.* So if we want to maximize our chances of winning, we should first try the square which is least likely to contain a mine. The upper left unknown has only a 1/5 chance of containing a mine, so we might pick that. If it turns out to be a 3 or a 5, then we can figure out the rest of the mines from there. If it turns out to be a 4, then we'll have to guess one more square. Let's just guess the square below it. In total, we have a 3/5 chance of surviving.

*The reason for this is that we are assuming that all arrangements are equally likely at the beginning of the game. As we click more squares, we prune the possibilities, but never do we make any of the remaining possibilities more likely than others.

We know there is no way to improve on this survival rate, because if we picked any other square first, we would already have at least a 2/5 chance of dying.

However, there is one alternative strategy which matches the 3/5 survival rate. If we pick the lower left right unknown, there is a 2/5 chance of dying. If we survive and it is a 3, then we can deduce the rest of the mines. If we survive and it is a 2, then we know that the upper left unknown is clear, and we can use the number there to deduce the rest of the mines.

Some crazy LIGO

Everyone is asking me, "Hey mr. miller, what crazy things are you doing this summer?" Well, as I've already let slip, I'm working on LIGO, the Laser Interferometer Gravitational Wave Observatory. We're looking for gravitational waves. Specifically, I'm in the group which looks for gravitational waves which come from compact binary coalescences (CBCs). That basically means when two black holes smash together.

Have we found any gravitational waves yet? Well, the other day, I was at Chandler Cafe, and I found one in my noodles. It looked sort of like this:

Graphic made using a Mathematica Demonstration

Unfortunately, I don't really own a camera, and I was hungry. So I slurped it up, and it made a sound like this: voooooooooooooooooooouP (also available as mp3, from "Gravitational Wave Sounds"). I guess we'll just have to find another one now, huh?

A slightly more serious answer: I couldn't tell you even if we had seen anything. It is "privileged" information. Exciting! But let me say this: We have a fairly good idea of the density of binary neutron stars and black holes, and how "loud" they would be when they merge. So we can calculate the expected rate of detection. By one estimate (see arxiv), the expected detection rate of neutron star mergers is once per two hundred years of observation (probably even smaller for black hole mergers). Basically, we don't expect to see anything. The real excitement will occur when "advanced LIGO" starts in 2013, increasing the observation rate to about 20 per year.

Of course, there are other sources of gravitational waves--Gravitational Wave Pulsars, Big Explodey Things, etc.--so maybe we'll see some of those. I think these other sources aren't as well understood, so we don't have such precise estimates on their expected detection rates. So who knows, we may be lucky.

And if it turns out we're lucky, you probably wouldn't notice right away. LIGO data is littered with what we call "non-Gaussian noise", meaning that every so often, there's a data glitch, causing the measurement to jump up by some really high number. These glitches look like gravitational waves; the computer has trouble telling the difference. And there are so many of them. We toss the glitches through every statistical filter we can think of, and we're still overflooded with them.

But we still have some tricks up our sleeves. I'm working on one of those tricks. What I do, is give the computer a bunch of false signals and a bunch of "real" signals (which are inserted artificially). Then the computer uses these to learn the difference between the two. It's basically Skynet, except it's not even remotely like Skynet.

Instead, I would analogize it to a tree. You throw a bunch of apples and oranges at the tree, and then the tree tries to tell a supercomputer what the difference is between a fruit and a black hole. (I am joking! Don't take my analogy too seriously. It's not really like a tree at all; if anything, it's a forest.)

So basically, if you want to know what I'm doing this summer, you can visualize me tossing a bunch of black holes at trees in hopes of finding delicious spaghetti. That's more or less the right idea.

54th Carnival of Math

The Carnival of Mathematics is a collection of links to the best mathematical blogging in the last few weeks or so. The posts range from technical to popular. One of my posts has been submitted to the 54th Carnival of Mathematics, the one about Godel's Modal Ontological Argument.

Oh geez, it's been a while since I've participated or read a blog carnival. I haven't looked at all the entries yet, but one entry I would really like to read is this series on Polya's enumeration theory. I remember back in my high school puzzling days, I had a very brief encounter with Polya's enumeration theory. You can use it to quickly calculate the number of different ways to paint the faces of a cube. I thought it was mathematical black magic.

What is faith?

Ask an agnostic or atheist what they think faith is. Faith is believing without evidence, they'll say.

It should be clear from this definition (I think so anyway) that faith is not at all a virtue, and may in fact be a vice. If our beliefs are not constrained by evidence, then we could, in principle, believe anything, anything at all. Only a few of those possible beliefs would be correct, only a few would be good to believe in. And since there is no evidence, we don't know which ones are the correct beliefs. So why are we playing this guessing game? And why should it be a virtue to play the guessing game?

In practice, though, this is a rather ineffective argument against Christianity. A lot of people will simply say, "You just don't get it, do you? That's not what faith is at all." Which is potentially a good point. I wouldn't want to be constructing some sort of straw man of faith. So let's talk about faith as Christians use and define it, not how atheists and agnostics define it.

The thing is, Christians have a rather ... confused (for lack of a better word) understanding of what faith is. I suppose a more positive way of putting it is "complex" or "diverse". I think it's a fairly common sermon topic to explain yet another point of view on what faith really means. The number of ways to understand faith is perhaps as large as the number of Christians, possibly larger.

Allow me to explain what faith meant to me a long time ago, back when I was still Christian. When I was Christian, I was taught that Christianity does not require a leap of faith. There are, after all, many arguments we could use in favor of Christianity. None of these arguments are particularly effective by themselves, and absolute proof can never be achieved. They only allow us to get closer and closer to God by degree, never fully reaching him. Therefore, if we ever want to reach God, we have to cross this little gap. Faith is that which gets us across that gap.

A lot of you probably think this concept of faith is really unusual or bizarre. Which is the point. Everyone seems to have a different idea of what faith is.* Faith is our gift to God. Faith is a gift from God. Faith is a relationship. Faith comes from a feeling deep within. Faith is supernatural assent. Faith is an experience. Faith is trust. Faith is confidence. Faith is an attitude. Faith is the will to believe. Faith is believing in things not seen. Faith is what gives us certainty. And yet, faith can coexist with doubt. Faith can coexist with evidence.**

*It would be hilarious if another "agnostic free mason" told me off because I can't appreciate the monolithic nature of true Christianity.

**Most of these ideas come from a list of definitions collected by Greta Christina. Greta took the definitions from religious sources.

Well, how could I possibly attack this huge amorphous blob of "faith"? By chopping into smaller pieces, of course. Allow me to define two kinds of "faith".

Faith-1 is any kind of faith which involves arriving at beliefs by circumventing the proper routes to knowledge. For example, believing without evidence is faith-1. Jumping the gap from uncertainty to certainty is also faith-1. Subtly "pushing" oneself in one direction towards belief is faith-1.

Faith-2 is everything else. For example, trusting someone you know (assuming it really is someone you know) is faith-2. Having a positive or optimistic outlook on life is faith-2. Believing in something which you have evidence for is faith-2.

This is my position: Faith-2 is perhaps justifiable, but faith-1 is not justifiable at all. Most religious believers have both faith-1 and faith-2, and may not necessarily see them as distinct. In fact, that's a major problem, that they do not usually distinguish between the two. They can argue for faith-2, which sounds very reasonable, and then later switch it around for faith-1. It would be very difficult to defend faith-1 directly, so people often cheat.

I myself could say that I have faith-2, but I would strongly prefer not to call it faith at all. I do not want to be complicit in this confusion of faith, this common equivocation. I have trust in my friends (the ones who are reliable anyways), not faith. I believe, or accept most established scientific theories because the evidence is inconsistent with the alternatives.

The saddest thing

It's occasionally thought that skeptics are in the business of taking away hope. This is sometimes true. Not always true, of course. Sometimes skeptics are in the business of delivering good news. There is no conspiracy to take over the world! Cell-phones are pretty unlikely to give you cancer! Demons only exist in your imagination! You can save your money because all that alternative medicine is useless!

But other times, we're in the business of delivering bad news. You cannot speak with your dead wife! You will never be psychic, no matter how hard you try! Your personal experiences are often inaccurate and worthless as evidence! All that alternative medicine is useless!

Really, we're in the business of delivering news which we think to be accurate and true. Whether it's good or bad news is sort of besides the point. But since people are more likely to have misconceptions they like rather than misconceptions they don't like, our news will be, more often than not, bad news. Tough.

So in a sense, we are in the business of taking away hope. But we're doing it in the interest of truth and accuracy. So we're really in the business of taking away false hope.

And why should false hope be desirable? Well, our emotions and desires can be rather irrational. But in my own experience, false hope is not even emotionally desirable. False hope is not happy. It's the saddest thing in the world.

I've had false hopes before (I will remain unspecific). Guess what happens when they go unfulfilled. It kills me, everytime. I become more bitter and more cynical. I trust people less, I trust myself less. It is the experience of disillusionment. And yet, though I feel disillusionment, I am not instantly disillusioned. Hopes are often resilient enough that they can withstand several blows of reality. Things didn't work out because maybe I did something wrong. This is all my fault. Or maybe it's all your fault. If I just keep on hoping, it will come. So let's try again. And the loop repeats...

There's an easier way than this cycle of quasi-disillusionment and blame. Have a realistic picture from the beginning. Then we can get an early start accepting the things we cannot change, and changing the things we can.