The unmeasurable set

The diagonal proof is one of the prettiest proofs in mathematics, but there is another proof that I quite like, and it illustrates the use of the fabled axiom of choice.  It's known as the Vitali Theorem, and it proves that not all sets of real numbers are measurable.

What is measure?

But first we need to talk about what it means for a set to be "measurable".  The measure of a set is, on a basic level, the length of the set.  For example the set of all numbers between 0 and 1 has a measure of 1.

However, it is incorrect to say that measure and length are the same thing.  "Length" is an intuitive concept, and it simply doesn't work for all sets of numbers.  For instance, what is the length of the set of all rational numbers between 0 and 1?  Mathematicians need a more rigorous concept so they can have definite answers to these questions.  And so they define measure, which resembles the concept of length, but is not the same thing.

In order for the concept of "measure" to resemble the concept of "length", we will take a few simple axioms:*

1. If we take the set of all the numbers between A and B, then the measure is simply the distance between A and B.
   Example: The set of all numbers between 5 and 7 has measure 2.
   
2. If we take a set of numbers and shift it up or down, then the measure stays the same.
   Example: We can shift the set of all numbers between 5 and 7 upwards by 3.  Now we have the set of all numbers between 8 and 10.  The measure is still 2.
   
3. If we have a countable collection of non-overlapping sets, and we join them together, then the measure of the resulting set is equal to the sum of the measures of each of the smaller sets.
   Example: Consider the set of all numbers between 5 and 7, joined with the set of all numbers between 8 and 10.  The measure of this set is 2 + 2 = 4.

*I am specifically talking about Lebesgue Measure, which is just one kind of measure.  This may not be a complete set of axioms.  For a more rigorous treatment, don't read blogs, read textbooks on real analysis.

With our new concept of "measure", we can answer our earlier question: What is the measure of the set of all rational numbers between 0 and 1?

The number of rational numbers is countable.  That means we can list them one by one, just as we can list all the positive integers one by one:

{0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, ...}

Therefore, we can split this set into a countable collection of sets:

{0}, {1}, {1/2}, {1/3}, {2/3}, {1/4}, {3/4}, ...

Each of these sets has measure zero, because a single point has length zero.  When we put all the points together, we just add all the measures up (by axiom 3), for a total measure of zero.  Therefore, the set of all rational numbers between 0 and 1 has measure 0.

The Vitali Theorem

Vitali's Theorem shows that there exists a set of numbers that is not measurable.  That is, if you try to assign a measure to the set, you will run into contradictions!  We will construct the set as follows:

Take the set of all rational numbers:

{0, 1, -1, 1/2, -1/2, 1/3, 2/3, ...}

Then we shift it upwards by some quantity x, with x being any number between 0 and 1.  Call the resulting set S(x).  So for example,

S(π-3) = {π-3, π-2, π-4, π-5/2, π-7/2, π-8/3, π-7/3, ...}

So far, we have an infinite collection of sets S(x).  But some of these sets will be duplicates of each other.  For example, S(0) = S(1/2), because if you add 1/2 to any rational number, it is still a rational number.  Similarly, S(π-3) = S(π-5/2).  So let's divide the collection of S(x) into many collections, each of which contains an infinite number of duplicates.

The next step is to pick one S(x) from each set of duplicates.  That way, we have a bunch of unique S(x), with no duplicates.  The trouble is, which one do we pick?  Any of them.  It doesn't matter which one we pick!  But how can we pick one from each collection of duplicates?  That would require an infinite number of choices.

And this is where the axiom of choice comes in.  The axiom of choice says that we can make an infinite number of choices, though we can't always write down what all these choices are.  So let's do that.  Now we have a bunch of unique S(x).

The set of all x that we chose is called a Vitali set.  I cannot write down what is and isn't in the set, because I used the axiom of choice to construct it.  However, I will be able to show that the set is unmeasurable.

Suppose that the Vitali set is measurable, and its measure is M.  Now let's shift the Vitali set upwards by a rational number q less than 1.  Take only the fractional part of each number in the set, since we want to keep it between 0 and 1.  Call the resulting set V(q).  By rule 2, V(q) has measure M.

For example, if our original Vitali set was:

{0, π-3, ln(2), e-2, ...}
Then V(1/3) = {1/3, π-8/3, ln(2)-2/3, e-8/3, ...}

The collection of sets V(q) is a countable collection, since the set of rational numbers is countable.  Also, none of the V(q) overlap, because no two elements in the Vitali set have a rational difference.  Lastly, if we join all of the V(q) together, we will get the set of all numbers between 0 and 1.

By rule 3, the measure of the resulting set (ie the set of all numbers between 0 and 1) is equal to the sum of the measures of V(q).  Each V(q) has measure M, and there are an infinite number of V(q).  Therefore:

1 = M + M + M + M + ... ad infinitum

This leads to a contradiction.  If M is 0, then 1 = 0.  If M is not zero, then 1 = infinity.

In conclusion, the Vitali set is not measurable as we first assumed.  If it has length, how did we fit an infinite number of copies into a finite length?  If it has no length, then how did we make a length out of a countable number of copies?

Is the axiom of choice true?

The axiom of choice has been used to derive some strange conclusions, which has led many to ask whether we should really be assuming that the axiom is true.  But that's the wrong question!

Axioms are not true or false.  They simply define a set of ground rules for mathematical analysis.  The best we can hope for is that all the axioms are consistent with each other.  As far as we know, the axiom of choice is consistent with the other axioms of set theory.

But the axiom of choice is not the only axiom that would be consistent.  For example, we could have taken instead taken the axiom of determinacy.  This axiom implies that all sets are measurable, and therefore the axiom of choice is false!  The axiom of choice and the axiom of determinacy are mutually exclusive with each other.  But this does not mean that one axiom is more "correct" than the other, just as flat geometry is not more "correct" than spherical geometry.

The series on infinite sets:
Hilbert's Hotel
Doubling the Sphere
Larger infinities and the Diagonal Proof
Power sets and the Chef Paradox 
The unmeasurable set

What's the deal with this blog?

I know many people were eying this survey kind of strangely.  "Is miller caving to his audience?"  No, that's not what it was about.  I will continue to blog about the same things for as long as those things interest me.  But when I write what I want to write, who am I writing to?  For instance, if I found out that everyone skips my physics posts, I might try to make my physics posts more accessible, and if I found out I had an audience full of physicists, I might make them more in depth.

There is also a particular audience that I am tracking: my asexual readership.

When I first found the asexual blogosphere, frankly I felt it was in shambles.  It now shows signs of picking up (a regular blogging carnival has just been announced), but it still pales in comparison to the lively skeptical blogosphere or the massive atheist blogosphere.  I'm a small fry in the skeptical blogosphere, but in an environment full of dead, dying, and newbie asexual blogs, I'm suddenly notable.

Long story short, I now have a lot of incoming visitors from asexual sites.  I can see this from the site statistics, but site statistics don't give me a good idea of who sticks around and who doesn't.  Thus the survey.

Survey Summary

With my small sample size, uncertainties are around 10-25%.  Therefore, I think it is misleading to give specific numbers, so I will summarize in words.

About half came for asexual/queer topics, half came for physics, half came for atheism/religion, and half came for skepticism/critical thinking.  That means that on average people came for not one but two topics.  That either means I lose the readers that only like one topic, or it means that people have more diverse interests than I give them credit for.  In any case, the tentative conclusion based on terrible statistics is that asexuals do not yet dominate the readership.

The only topic that gets skipped a lot is puzzles.  I could have guessed this before, and it will not affect my blogging.

The homosecular acetheist

It occurs to me that some asexuals might feel uncomfortable with a blog that associates them with atheism?  If that's the way anyone feels all I have to say is, "Tough luck!"  It probably also makes some physics enthusiasts and even some skeptics uncomfortable.  I write what I like and I don't expect everyone to agree.

In fact, I'm sort of evilly rubbing my hands together at the thought of introducing skepticism to a bunch of asexuals and introducing advanced queerness to a bunch of atheists.  Mwahaha, I have finally found a way to do more than preach to the choir!  Which is what people accuse you of doing when they're not busy accusing you of proselytizing.

As for asexuals who feel uncomfortable being associated with LGBTQ, this discomfort is fully intentional.  I am unabashedly opposed to asexual separatists (in a friendly way of course).

Unrelated(?) blog updates

I designed another sidebar graphic to link to the LGBTA category of my blog.

I also wrote an Asexuality 101 page, linked on the side bar.

I am currently debating whether to keep the rainbow flag and asexual flag, which now seem slightly redundant.  I may also decide to make the extended link list more prominent.  Any thoughts?

Bering is still wrong

Following the discussion of whether homophobia is an evolutionary adaptation, Bering responded to his critics with an interview with Gallup, the author of the original study.

There is a lot of incidental wrongness in Bering's response.  Like when Gallup says that men have a monopoly on paraphilias and kinky sex... Really, Gallup, you think so?  Yoder points out even more errors.  But I will have to ignore most of these so I can stick to the main point.

I agree with Bering on a few points.  First, Gallup's theory in no way implies that Gallup is or is not a homophobe.  In fact, that was never the issue.  Second, I agree that minor-attracted people get short shrift in this whole discussion.  If it were true that gay men were more likely to be attracted to minors, then those people deserve to be helped, not ignored.  However, my agreement with Bering on these points only serves to underscore the fact that political incorrectness was never the problem.

The problem is that Gallup's evidence fails to even partially exclude alternative hypotheses.  In other words, it fails to behave as evidence.

Here are some alternative "plausible" hypotheses which are also completely consistent with Gallup's data:

1. Null hypothesis: Culture causes many to believe that gay men will harm their children or turn them gay.
2. Partially null hypothesis 1: Homophobia is adaptive.  However, people's specific fear of homosexual contact with children is culturally learned.
3. Partially null hypothesis 2: Fearing for one's children is adaptive.  Culturally learned homophobia interacts with these fears.
4. Spandrel hypothesis: There is a genetic basis for wanting to protect children for homosexuals.  The same gene also causes some other trait which happens to be adaptive, and that's why the gene prevailed.
5. Alternative just-so story: Homophobia, especially to protect children, is adaptive, but not for the reason Gallup supposes.  Gay men can still have opposite-sex partners and reproduce, but this is less likely if as children they have contact with gay adults.

What new evidence do Bering and Gallup bring to exclude these hypotheses?  None.  None at all.

In response to concerns about "adaptive storytelling", Gallup responds:

We’ve shown that a person's voice is also related to fitness. Just as people with more attractive faces are more symmetrical, the same is true for people with more attractive voices. The sound of a person’s voice conveys information about their gender, age, body configuration, hormonal status, when they lost their virginity, how many sex partners they’ve had, their propensity for infidelity, whether they are on birth control pills, and whether they are in the fertile phase of their menstrual cycle.

Wow, that is... completely irrelevant!  Even if they have enough evidence to show that a person's voice is related to fitness (and PZ suggests that they don't), what does that have to do with the fact that they don't have enough evidence for the adaptiveness of homophobia?

Another new piece of evidence is a replication of the study in Taiwan.  Not only does this fail to prove cultural universality, but cultural universality would fail to prove a genetic basis, which would fail to prove adaptiveness, which would fail to prove the particular just-so story that Gallup thought up.

And the next piece of evidence... oh, that's it.

I have not been sufficiently convinced that this hypothesis even merits further research.  Not all questions are good research questions; in fact, very few of them are.

Adaptive homophobia: politically incorrect?

Speaking of political correctness, Bering in Mind discussed a scientific question he considers politically incorrect: Is homophobia a product of natural selection?

Unfortunately, the discussion that follows is a tiny bit of science followed by fantastic leaps to conclusions.  The tiny bit of science is an old Gallup poll showing that homophobic reactions increase when the homosexual person is in proximity to children.  The fantastic leap (made by Gallup, not Bering) is that this is because homophobia evolved to reduce the likelihood that one's offspring would turn gay from contact with gay people.

Bering has already been criticized by biologist Jeremy Yoder:

Lots of people, including some evolutionary biologists, speculate about the adaptive value of all sorts of traits—but in the absence of solid evidence for heritability or fitness benefits, such speculation tends to get derided as "adaptive storytelling."
...
What this amounts to is arguing that homophobia is an adaptation favored by natural selection because homophobia is a thing that exists.

And here's another biologist, Jon Wilkins:

In fact, it is trivially easy to come up with a plausible-sounding evolutionary argument to describe the origin of almost any trait. More importantly, it is often just as easy to come up with an equally plausible-sounding argument to describe the origin of a hypothetical scenario involving the exact opposite trait.

According to Jon Wilkins, it's fairly common for evolutionary psychologists to claim that they're just being attacked for political incorrectness:

[Evolutionary psychology] deflected criticism by claiming that politically correct academics didn't want them to ask these questions, painting itself as a field of martyrs who were bravely trying to do science, when the actual criticism was that the science was bad.

Obviously, the biologists can say it better than me, so follow those links for more.  However, I would like to comment on Bering's posturing as politically incorrect.

Consider this a warning: the theory I’m about to describe is likely to boil untold liters of blood and prompt mountains of angry fists to clench in revolt. It’s the best—the kindest—of you out there likely to get the most upset, too. I’d like to think of myself as being in that category, at least, and these are the types of visceral, illogical reactions I admittedly experienced in my initial reading of this theory. But that’s just the non-scientist in me flaring up, which, on occasion, it embarrassingly does. Otherwise, I must say upfront, the theory makes a considerable deal of sense to me.

This seemed rather strange to me for several reasons.  For one thing, science doesn't require that people be unemotional, just that they use a method which cannot be affected by such biases.  For another, science isn't about what makes "sense", since what makes sense to people can be about as biased as their emotional reactions.

As for whether this is politically incorrect, I'm not upset at all by the idea that homophobia could be adaptive.  I'm pretty sure that would have no relevance to whether it's morally acceptable.  What's possibly more upsetting is the suggestion that homosexuals are more likely to be pedophiles, and their victims more likely to become homosexual.  Gallup offers some evidence for this claim, but Yoder showed that it doesn't really hold up.  Yoder also commented that the hypothesis originated in the 1980s when it aligned with prevailing attitudes.  Politically incorrect, or just incorrect?

[Aside: This isn't the first time I've seen Bering posture a bad idea as politically incorrect.  He wrote an article on asexuality which was generally positive, but ended with the "unsavory" suggestion that studies should subject asexuals to erotic stimuli.  Actually I don't think that's all that unsavory if it's necessary for study, but I do think that it's an unfruitful direction for research.]

Question: Does emotion play any role in our rejection of Gallup's claim that homophobia is adaptive, or is it just a matter of looking at the evidence?

I think emotion does play a proper role in science: motivation.  Gallup's claim is personally relevant to me and other gay people, so of course we would take a closer look at the evidence!  If the evidence is found wanting, we have motivation to loudly call it out.  But the method for examining the evidence is itself neutral.  In fact, I've criticized evolutionary psychology even when it's made conclusions I liked.

(via Pharyngula)

Take this survey!

It's a really short survey!  Given the wide range of topics on my blog, I want an idea of which ones people come for, and which ones they stay for.

Update: survey closed

Full rejection with an open mind

Some months ago, PZ Myers expressed the opinion that no evidence could convince him of the existence of a god.  But this caused much disagreement, since no critical thinker likes the idea of being closed to any evidence.  And so, this debate has been echoing around the atheosphere, continuing until today.

Put me on PZ's side... sort of.  The thing is, I can imagine a few scenarios where I'd become convinced of a theistic universe.

Like if I stumbled into another universe through a wardrobe, and found myself adventuring with a talking lion-god.  Or if I had a compass which would always answer any question truthfully, including moral questions, and do it with attitude.  On the other hand, lucid dreams are fairly common, so I'd need it to be repeatedly verifiable by multiple people.  Once verified, I'd believe it, at least for the remainder of the dream or hallucination.

Mind you, these scenario would only convince me of a supernatural world, and not convince me of any specific version of supernaturalism.  In particular, they would fail to convince me of any of the major religions.  In fact, nothing could convince me of the major religions, because any such fantastic show of evidence would be out of character for them, and thus argue for something completely different.

Also, that will never happen.  Seriously, just because I can imagine it doesn't mean it's possible.

What my imagination can do is privilege a hypothesis.  The probability of a god (as conceived by religion) is extremely small, and we can't even conceive of numbers that small.  So by merely imagining the god hypothesis, we've tricked our minds into overestimating its likelihood.  And before we know it, we start thinking that evidence like this is anything other than pathetic.

In short, I agree with PZ Myers that no evidence could convince me that gods exist.  But how do I reconcile this with my strong principles of allowing for the possibility that I may be wrong?  Easy.  To convince me that gods exist, you need two things:

1. A strong philosophical argument showing that my current thinking is completely off-base (not just a little off-base).
2. Lots and lots of evidence.

Not only is the god hypothesis lacking evidence, it's philosophically unsound.  The major arguments for gods are not about evidence, they're about explaining why no evidence is necessary.  This is a fundamentally problematic argument, and fundamentally problematic attitude.  No amount of evidence will fix it.

And that doesn't mean I'm totally closed-minded.  That just means that I first need to be convinced that these flaws are not so fundamental after all.

Pascal's swindle

Blaise wanted to borrow some dough
For an investment he said would grow.
"I'll pay back tenfold,
Before fall takes hold!"
But I frowned, and let my doubt show.

To every concern he'd reply,
"How can you ignore stakes so high?"
Then he'd up the reward
Infinite! Absurd!
All I could say was, "Nice try."

Suddenly I think we should turn all arguments for and against God into limericks.  Give them the level of seriousness they deserve.  But I don't like poetry, so someone else would have to do it.

On taking offense and derailing

I don't know how obvious it is, but over the past year, my views have been skewed towards a form of political correctness.  I know that PC is used nearly universally as a pejorative, and thus has come to describe many truly negative things.  This is especially true in skeptical circles, where it is thought to prioritize politeness over truth.  But sometimes PC is appropriate precisely because we value truth.

So that's my extraordinary claim, that critical thinking and political correctness sometimes align.  It is my new long-term blogging project to support this claim, and also attack it so that we may know its limits.  I'm very interested to know where my readers stand on this so I know what direction to go in future posts.

Exhibit A: The "easily offended"

On The Thinker, Jeffrey said, "It is easier for a camel to pass through the eye of a needle than for the 'easily offended' to become good critical thinkers."  Based on his personal experience, he proposed three (non-exhaustive) categories of easily offended people.

First are those who tend to use being offended as a manipulation tool to stifle discourse on a topic, thereby avoiding arguments they don’t wish to face.
...
The second form is related to emotions. Some people are thinkers; others are feelers. Feeler-types tend to become emotionally attached to their opinions.
...
The third category is comprised of people who actively seek opportunities to get offended by anything that appears (or can be made to appear) to run counter to their pet cause.

Now, I can see where Jeffrey is coming from.  I don't think Jeffrey is an atheist, but this is a pretty common response to atheists.  Do something as simple as put up a sign advertising an atheist student group, and it will offend people.

And yet these categories don't sit well with me, because they echo standard derailing arguments used against marginalized groups whenever they complain about said marginalization.  Just to name a few of these arguments:

You're Being Overemotional
You're Just Oversensitive
You Just Enjoy Being Offended
Being Offended Is Great For You
You're Taking Things Too Personally

I asked Jeffrey what he thought the difference was between the arguments he made and the ones appearing on "Derailing for Dummies", and he thought it was a matter of whether there was a legitimate basis for being offended.  Well yes.  People who have a legitimate basis for being offended have a legitimate basis for being offended, and those who don't don't.  But I think this is failing to get at the heart of the issue.

Instead, I would try to draw a distinction based on the question: Who is doing the derailing?  Is it the offended person who says, "I don't want to talk about it, and you ought to be ashamed of yourself for bringing this up in public"?  Or is it the other person who says, "You're taking this way too personally!  Talk to me after you've calmed down."

Guess what?  Both sides can derail the discussion.  It's not just the side that gets emotional about it (with or without a legitimate basis).  It can also be the person who can't see past their opponent's emotion, and decide to make that emotion the focus of the argument.

(For the fallacy geeks: This is a form of the Fallacy Fallacy.  That's when you spot something resembling a fallacy, such as an emotional argument, and proceed to ignore any real arguments they make.)

But let's take it one step further.  Riddle me this: Is it really so counterproductive to derail an argument that would itself be unproductive?  I slam down trolls all the time, saving me time and energy for the discussions that at least have a chance of being productive.  Of course, this depends on me having a realistic view of which discussions would be productive and which would not.  I wouldn't want to fool myself into thinking that all discussions attacking my beliefs are unproductive.

What can we get out of this?  Not much, since we're talking pure generalities.  If someone is getting offended and trying to derail, you should say something to persuade them that the discussion will not be so unproductive as they think.  Start by avoiding cliched arguments.  And if you're going to derail the argument by focusing on your opponent's emotions, at the very least do it with some self-awareness.

Asexuals: Who are they and why should you care?

I facilitated a workshop at the 2011 Western Regional LGBTQIA Conference.  Excuse me while I rattle on about it, because I was pretty happy with the results.

Asexuals: Who are they, and why should you care?

PPT slides
PDF slides
Outline

(Update: More recent slides are hosted on Asexual Awareness Week.  Also e-mail me at skepticsplay at gmail dot com for even more recent stuff.)

The workshop is in lecture format, because I'm not going to have a bunch of non-asexuals have an open discussion of what asexuality means to them.  I've seen that happen and it doesn't work!  So instead, it was mostly information conveyed from me to the audience.  However, I did have Q&A, and I did allow them to discuss possible solutions to asexuals being excluded from the queer community.  And though I didn't assemble a panel, I did tell a few real-life experiences of other asexuals.

Reactions

The workshop was wildly successful.  The room was packed with about 100 people.  People were asking questions, laughing at my jokes, and participating in discussion.  Somebody asked about demisexuality, which is just about the hardest question you can ask, but I think I handled it fine.  Several people came up to talk to me afterwards and said it was really awesome.  A couple wanted a similar presentation elsewhere.

I only found out later, but Maymay, one of the BDSM panelists attended my workshop, and was live-tweeting it like every five minutes.  He has over 1700 followers.  I got to meet Maymay later, and he's a cool sexual freedom advocate with a nerdy bent.  Suddenly I think I should educate myself more about the intersection of asexuality and kink.  No, that's not a joke, there are lots of kinky asexuals.

Another response I really liked actually came out of the practice presentation I did for Oscar Wilde House, an LGBT themed coop.  Someone designed flyers, and they included this comic strip.

Long-time readers know of my love for Dino Comics.

I also got interviewed for Hot Pieces of Ace, an asexual vlogging channel.  It's not up yet.

Why I’ve put this presentation online

Many people asked for the slides, and that is reason enough to put it online. But without me speaking in front of the powerpoint, and without the audience participation, there are probably better educational tools out there.

No, the primary reason to put this presentation online is for other asexual activists. I hope that this inspires others to give presentations or workshops, and gives them ideas on how to do it.

In particular, pay attention to the direction of my presentation. It covers Asexuality basics, yes, but rather than getting bogged down in further details, it moves on to how asexuals interact with society and the LGBTQ community in particular.

This image was taken from Queer Secrets and included in my presentation.

I wished to focus on the elephant in the room. There is a reason that the LGBTQ community needs Asexuality 101, and it’s not because they simply haven’t seen my workshop yet. It’s because asexuals do not participate in the larger queer community, do not feel comfortable there, or are made invisible. This is a problem that can be solved, and many activist-minded queers would be happy to help if only they knew what the deal was. As asexuals become more visible in the larger queer community, people will become better at educating themselves, and asexuality 101 will be needed less and less.

Reactions to Western Regional

Last weekend, I attended the 2011 Western Regional LGBTQIA Conference at UC Berkeley.  I facilitated a workshop, but I'll post on that later.  First, I want to give my reactions to the rest of the conference.

All in all, it was a pretty terrific conference.  There's always the worry that the organizers will do something horribly wrong and inadvertently exclude a whole segment of attendees, but I'm not aware of any significant problems.  They had a really great lineup of workshops, and a great set of keynote speakers, including Victoria Kolakowski, the first transgender person to serve as judge in the US.

Disjointed highlights:

The first keynote speaker Juana Maria Rodriguez, who explained a little about this year's theme: "Unchecked Boxes: The Faces Behind the Labels".  Nobody really fits into all these labels and boxes, and they're all socially constructed.  And yet, those boxes still matter.

There was also a lot of entertainment.  I don't particularly care for drag shows, but they were there.  Noteworthy, an a capella group, performed "straight boyfriend" and "Poker Face".  One of my friends is in the group, and I asked him if they were copying that old YouTube video.  He said it was the same group.  Cool.

I also really liked speaker Rinku Sen, who explained how many people of color issues are queer issues.  She asked, who is the group most affected by Don't Ask Don't Tell?  Turns out the answer is black women, not white men as the media coverage would have you believe.

Rinku also noted that the queer movement currently spends most of its resources on same-sex marriage.  This is a pretty questionable choice of priorities considering how many more pressing issues there are.  The question is, is the queer movement prepared to quickly mobilize to fight for other things once same-sex marriage is legalized?  In fact, why even wait until then?

They had this community project which they called Queer Secret.  Attendees made PostSecrets and put them up for display.  I wonder if the organizers knew that there is a Queer Secrets tumblr.  This is a rather silly secret to highlight, but I really liked this one that had a drawing of a TARDIS, with the words, "My TARDIS looks like shit :(".  I think the submitter might have been asexual.

I also love this t-shirt design.

It's a Rubik's cube!

I didn't actually take any pictures during the conference.  But that's okay because I suck at photography anyway.

Workshops:

The conference had a great selection of workshops, covering about every topic imaginable, but the workshops themselves can be hit or miss.  I already knew that from last year.  A couple of the workshops I attended were simply open discussion, which I don't think always works out so well.  They're interesting, but they lack direction.

But some workshops erred in the other direction.  I attended an Intersex 101 given by Jeanne Nollman, but it was too much lecturing about the details of various intersex conditions.  That's great, but most of that I will forget.  I wanted to hear more about how intersex people interact with society and with the LGBT community.  I'm sure if there were a longer Q&A session, that's what everyone would have asked about.  She did talk a little bit about how doctors tend to hide any medical history about operations they did to "correct" intersex conditions.  And how many parents freak out about intersex conditions.

And then there was the BDSM 101 workshop, facilitated by my best friend.  He had assembled a really great panel, including Maggie Mayhem and Maymay, both prominent bloggers in the kink community.  There was also a small demo where my friend got tied up.  But too much time was wasted on introductions, 101, and resources, and the panel was too short.  There was this odd moment where one panelist implied that her interest in masochism was due to her history of self-harm.  That was... poorly stated.  Another panelist clarified that self-harm and masochism come from completely different mind-sets, the former from self-hate, and the latter from complete trust.

I also suspect many things in the panel were going over my head.  Maymay tweeted that he disagreed three times with the other panelists.  But I only caught one disagreement, which I didn't understand: they disagreed over whether BDSM could be a solitary activity.

Wait, why was I reading Maymay's twitter you ask?  Because he live-tweeted my presentation, causing it to dominate the #WRC11 tag on twitter.  There will be more on this in a separate post.

Logicians with hats

Three logicians play a game with colored hats. (That's always how the trouble begins.)

Each logician wears a hat, and that hat has a 50% chance of being red and a 50% chance of being blue.  The probabilities for each hat are independent of each other.  None of the logicians is allowed to see the color of her own hat.  They may make a game plan before starting, but during the game they may only look at the other hats without communicating.

And yet, the logicians guess the colors of their own hats.

They guess simultaneously, writing their guesses on paper which are only revealed after all guesses have been made.  They may write "red", "blue", or "abstain".  If at least one logician guesses right and none guess wrong, then they win.  Abstaining counts as neither right nor wrong.

What strategy should the logicians use to have the highest probability of victory?

This puzzle was taken from Haidong.  He also asks what happens if there are 2^k - 1 logicians.  The general case is rather difficult, but you should at least try for 7 logicians.

Solution posted

What I'm doing this weekend

I don't think there's any point in saying more until the conference has actually happened.  I'm facilitating a workshop.  That's why I've been too busy to blog much in the last week.

Electricity, magnetism, space, and time

[One of my most popular essays is "Relativity + Electrostatics =  Magnetism", which is about how the electric force, combined with Special Relativity, explain the existence of the magnetic force.  I think this is a topic worth revisiting, because I am a better writer than I used to be, and can make better pictures.]

The magnetic force is weird.  You're all familiar with bar magnets, which have a north and south pole.  Opposite poles attract and like poles repel.  But even though bar magnets are very familiar to us, they're relatively complicated, so we'll start with a more basic example.

A simple model of two wires.  In each wire, the electrons (in blue) are moving to the left, while the atomic nuclei (in red) remain stationary.

Two wires.  Wires conduct electric current, meaning that the negatively charged electrons move to the left or the right.*  The positively charged atomic nuclei remain stationary.  If the two wires have current going in the same direction, then they attract.  If they go in opposite directions, then they repel.

*I am making a simplification, since this is not true for all materials that the wire could be made of.  For reasons that I won't get into, some materials conduct electricity by things other than electrons.

Bar magnets are sort of like circular currents.  If you have a north and south pole together, these currents are going in the same direction, and they attract each other.  If you have a north and a north pole together, these currents are going in opposite directions, and they repel.

On the left, the circular currents (dark red arrows) are going in the same direction, making the magnets attract.  On the right, the currents are going in opposite directions, making the magnets repel.  Alternatively, we can think of the magnets as having north and south poles, where opposite poles attract and like poles repel.

But let's return to the wires, because they're simpler.  The reason the wires attract is specifically because the electrons attract.  One way we know this is the electrons tend to gather towards each other at the closer edges of the wire (this is called the Hall Effect).  The electrons drag the rest of the wire with them, thus causing the wires to attract.

The reason why the electrons attract and the atomic nuclei don't is because that's just how the magnetic force works.  The magnetic force is proportional to the speed of the particle.  The electrons are moving, so they attract each other.  The atomic nuclei are not moving, so they don't attract each other.

But hold on!  Doesn't the speed of a particle depend on which way you look at it?  What if you put these wires on a moving truck, won't all the electrons and nuclei be moving faster than before?  For that matter, what if you put these wires on a moving Earth, doesn't that affect the speed?  Why is it that we can use motors and generators (which both require magnetic fields) without worrying about the earth's motion?

This question was the motivation for Special Relativity Theory.  You may have heard about Einstein's Theory of Relativity as something that radically alters our notions of space and time.  But that wasn't really the point of the theory.  The point was to explain something about magnetic fields.  The radical view of space and time was just an added bonus.

Einstein's Special Relativity is sort of like an expansion of rotations.  If you rotate your head 90 degrees, and look around you, several things change.  Ceilings become walls and walls become floors.  Right-left becomes up-down, and up-down becomes left-right.  Everything that was pointing in one direction (like gravity, which originally pointed down) is now pointing in a different direction.  But physics behaves the same way, just rotated.

The same is true if you're moving at constant velocity.  If I'm on a moving truck, things that were previously stationary are now zipping behind us.  If the air was previously still, it now becomes wind at our faces.  But physics behaves the same way, it's just that directions have changed.  In this case, we don't call it a rotation, we call it a boost.

According to Special Relativity, a boost does not just affect the motion of objects.  It also affects the electric and magnetic fields.  Just as right-left became up-down, and up-down became left-right when we rotated, electric fields can become magnetic fields and magnetic fields can become electric fields.  It's not quite like rotation (the math is harder), but the idea is the same.  A boost causes electric and magnetic fields to mix into each other.

And of course, it's not just electric and magnetic fields that mix into each other, it's space and time too.  But at everyday speeds this is difficult to observe, because they're such small boosts.  It's like rotating your head a fraction of a degree.  What was once horizontal is still mostly horizontal (but with a slight vertical component).  Similarly, at everyday speeds, what was once a distance is still mostly a distance (but with a small time component).

The mixing of electric and magnetic fields is deeply intertwined with the mixing of space and time.  But this is very difficult to demonstrate without getting into the mathematics of boosting.  There is one example where it is easy to demonstrate, and that is the example I started with.  Two wires.

In a real wire, the electrons are not really moving at the same speed.  But let's simplify and imagine that they are.  And then let's boost to a perspective where the electrons are not moving at all.

Two wires, boosted.

After we've boosted, it is no longer the case that the electrons are moving left while the atomic nuclei are stationary.  Now it is the electrons that are stationary while the atomic nuclei are moving right.

Since the magnetic force is proportional to speed, it can no longer affect the motionless electrons.  But this is easy to explain in terms of Relativity: what was once magnetic fields have now changed into electric fields.  The electric force causes the electrons in each wire to attract.  The same electric force also cause the atomic nuclei to repel, but that's okay!  The atomic nuclei have some speed, and are thus attracted by the magnetic force.  The magnetic force and electric force on the nuclei cancel out.

So we really have the same picture as before.  The electrons attract (but now due to the electric force, not the magnetic force), and they drag the entire wire along with them.

Now I will connect this to time and space.

If you look at my diagram, you'll notice I did something sneaky.  After the boost, there are more nuclei than there are electrons.  This is due to the mixing of time and space.  Previously, the protons had some spacing, some distance between them.  But as time and space mix, some of the distance changes into a time component, and the distance becomes shorter.  It's called Lorentz contraction.  When we boost from a frame where nuclei are stationary to a frame where they are moving, they become closer together.

Similarly, when we boost from a frame where electrons are moving to a frame where they are stationary, they become further apart.  It's backwards Lorentz contraction.

The end result is that each wire now has a positive charge because the nuclei are more dense than the electrons.  Electrons are attracted to positive charge by the electric force!

And so, there are two methods of solving the problem.  The first method is to mix the electric and magnetic fields through the mathematics of boosting.  The second method is to mix space and time, and then look at the resulting electric and magnetic forces.  Both of these methods are equivalent.

Hold on!  I know you have some questions.

Does that mean that the magnetic force isn't real?  No.  I'm not sure that it's meaningful to talk about whether these things are "real" or not.  What it does tell you is that the electric force, magnetic force, and Special Relativity are all connected.  Physics wouldn't make sense if you only had two out of the three.

What happened to the missing electrons?  The number of electrons does not change when we boost, so they must have gone somewhere.  The reason it seems they disappeared is because I only drew part of the wires.  For a wire to conduct electricity, you need to have the wire in a loop to complete the circuit.  The electrons will gather in the parts of the loop where they were going the opposite direction. There, they will be doubly Lorentz contracted.

What about two electrons by themselves moving alongside each other?  In the wire example, electric and magnetic fields don't change over time.  In the two electron example, they do change over time.  This significantly complicates the way things work.  But suffice it to say that the faster the electrons move, the stronger the electric force repelling them.  The magnetic force between the electrons is counteracted by this stronger electric force. 

Aren't the electrons all moving at different speeds?  Yes.  But it works out the same.  It makes the math a lot more complicated.  The entire premise of this post is to simplify to an example where I don't have to show that.  If you would like to see it, you should be studying physics academically, not from blogs.

Is Lorentz contraction really that big?  No.  I exaggerated it so you can see it clearly.  In a typical wire, the electrons are moving with an average velocity of 10^-5 meters per second, which corresponds to a Lorentz contraction of 1 part in 10²⁷.  If it seems impressive that such a small Lorentz contraction can create a measurable force, that's because it is.