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?

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.