Thursday, May 27, 2010

Doubling the sphere

The Banach-Tarski Paradox says that you can take a sphere, cut it into a finite number of pieces, and then move and rotate these pieces around to make two new spheres just like the original.  The Banach-Tarski paradox is derived from the Axiom of Choice, one of the stranger axioms of set theory.  I will not discuss the full proof of the Banach-Tarski Paradox or the Axiom of Choice, but I wanted to show how this is only a bit more complicated than Hilbert's infinite hotel.

As I said in my post about Hilbert's Hotel, one infinite hotel is just as good as two.  This is because if you shuffle around the guests in both hotels, you can fit them into one hotel.  Therefore, it should not be surprising that if you shuffle around the infinite number of points in a sphere, you can make two spheres out of them.  What's surprising about the Banach-Tarski paradox is that there is a "nice" way of shuffling the points.  Rather than moving each point individually, you can take entire sets of points and simply rotate them rigidly.

Let's say we have a ring of circumference 2π.  I'm going to cut this ring into two pieces, rotate them around, and then make a new ring... with one point missing!

To choose the first piece, first take any point.  Then move along the ring a length one, and take another point.  Then move along the ring another length one (ie rotate by 1 radian), and take another point.  Repeat infinitely.  You know you will never end up back where you started because π is an irrational number.  Instead, you will wrap around the ring an infinite number of times.

On the left, the first ten points are shown.  On the right, the first two hundred are shown with progressively smaller dots.

If we rotate this piece around, we can send point 1 to point 2, point 2 to point 3, point 3 to point 4, and so on.  This is exactly the same scenario as Hilbert's Hotel, where we told each guest to move to the next room over.  We're left with the exact same set of points as before, but with point 1 missing.

So here's how you can remove a single point from the ring.  First, cut the ring into two parts, one part described above, and the other containing all the leftover points.  Then rotate one of the pieces by one radian.  Then put the pieces back together.

Can we use a similar process--cutting the ring into a finite number of pieces, rotating them, and putting them back together--to make two rings out of one?  Sadly, no.  That's impossible.

However, spheres are much more complicated than rings.  When you rotate a sphere, you can rotate it in any number of directions.

The set of points on the ring was "generated" by a point and a rotation.  I just took a point, and then rotated it by a particular angle repeatedly.  We can construct a set of points on the sphere in a very similar way, but using four rotations.  The first two rotations are rotating clockwise and counter-clockwise around the z-axis by one radian.  The other two rotations are the same, but around the y-axis.

The crucial fact about these rotations is that the order matters.  If I rotate a point around the z-axis, and then the y-axis, then this will result in a different point than if I had rotated around the y-axis first, and then the z-axis.

When we constructed the set of points on the ring, I analogized it to an infinite row of rooms.  Here, it would be better to analogize the points to a much more complicated arrangement of rooms, called the Cayley Graph.

 The Cayley graph represents how all the different points are related by rotations.  It does not represent how the points will actually look, because they'll wrapped around a spherical shell an infinite number of times. Image credit

The point at the center of the Cayley graph represents our initial starting point.  From there, the graph splits up into four branches.  The right branch is what you get if the last rotation was rotating clockwise around the z-axis.  The upper branch is what you get if the last rotation was rotating clockwise around the y-axis.  And so on.

The "paradox" occurs when we take the left branch and rotate it clockwise around the z-axis.

By simply rotating the left branch (in red), we get three branches (in blue).  Together with the right branch, this makes the entire set.  Similarly, we can take the upper branch and rotate it counter-clockwise around the y-axis, and get the upper three branches.  Joined with the lower branch, we have a second copy of the set.  Amazing!  By cutting the set into four branches and rotating them around, I've made two copies of the set.

I'm leaving out a few details (like what happens with the center point), but that's relatively easy to fix.

Note that I have not yet doubled the sphere.  I only took an infinite set of points in a spherical shell, and duplicated that set.  We still need to duplicate the rest of the sphere.  But actually, the rest of the proof is relatively easy.  I leave it as an exercise to the reader who knows how to use the Axiom of Choice (and no, I haven't used the Axiom of Choice yet).

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 

Monday, May 24, 2010

Quantum Mechanics for skeptics

I've made a PowerPoint presentation called "Quantum Mechanics for the skeptic: How to recognize quantum nonsense without being a physicist".  The title pretty much explains the target audience and purpose.

The presentation features appearances by: Professor Farnsworth, Professor X, the Great A’Tuin, Richard Feynman, and Deepak Chopra.

A lot of the source material for the presentation comes from my review of What the Bleep!?

Saturday, May 22, 2010

Martin Gardner passed away

I just got word that Martin Gardner passed away.  This is a sad day.  Martin Gardner was something of a hero to me.

Mind you, I never knew much about the man himself.  I was only familiar with Martin Gardner's works.

I've been a puzzle enthusiast for a long time, since high school if not earlier.  You do not get to be a serious puzzle enthusiast without hearing about Martin Gardner.  For decades, Martin Gardner wrote a column in Scientific American called "Mathematical Games".  And though he stopped writing the column before I was born, he made enormous contributions to puzzling and recreational mathematics.

Case in point, the puzzle I posted earlier today is credited to Martin Gardner.  This was a complete coincidence, but it is not a surprising one, because a significant number of the "classic" puzzles were at some point collected and popularized by Martin Gardner.

If that weren't enough, Martin Gardner is also considered one of the earliest skeptical writers, in the sense of the modern skeptical movement.  He wrote a book called Fads and Fallacies in the Name of Science, which sounds like a very typical introduction to scientific skepticism, much like others written by Carl Sagan, James Randi, and Michael Shermer.  But this book was written in 1952!  CSICOP (Committee for the Scientific Investigation of Claims of the Paranormal) didn't even exist until 1976.  In fact, Martin Gardner was a founding member of CSICOP too.

Finally, there is one more way in which Martin Gardner is important to me.  I must admit that I've only read one book by him, but in retrospect it seems significant.  I read Relativity Simply Explained back in high school, before I started majoring in physics.  It was the first sensible description of Relativity I had ever seen.  I would not be surprised if this book was part of what motivated me to go into physics.

So perhaps now you've figured my blog out.  I talk about so many different things, but the unifying theme seems to be, "Areas of discussion that are indebted to Martin Gardner."  Thank you, Martin Gardner.

Remove the squares

There are forty matches in a grid pattern as shown above.  Try to remove matches so that no squares remain.  That means no 1x1 squares, no 2x2 squares, no 3x3 squares, and 4x4 squares.

Here's one solution: remove all vertical matches.  No squares are left.  I needed to remove twenty matches.  Can you do better?

I labeled each match with a number so that you can describe your solution easily.

See the solution

Polyomino equation solutions

See the original puzzle

See solution 1
See solution 2

You can ask a more general question: given two kinds of polyominoes, is there some figure you can tile with either one?  In other words, are the two kinds of polyominoes "compatible"?

I know of two websites which try to address this question.  It turns out that some polyomino pairs are compatible, but require large numbers of pieces.  Other polyomino pairs are incompatible.  For many others, it's not known whether they're compatible.

Thursday, May 20, 2010

I drew Muhammad's feet

Today is Draw Muhammad day!

I'm rubbish at drawing people though, so I just drew his feet.  Uh, I'm not sure if that's enough to really offend Muslims, but it's all I could manage.

This is probably one of the great skeptical moments in movie history.
Incidentally, the Wizard of Oz was much nicer as a person than as a giant disembodied head.

While I'm deprecating my own drawing skills, I should also mention that the calligraphy is probably horrible.  I just copied a picture I found on the internet, with no awareness of which details are important and which aren't.

Friendly Atheist has a bunch of other Muhammad drawings.

Last time I talked about Muhammad drawings, I took the angle that the act is intended to promote free speech.

However, the Barefoot Bum offered an alternative perspective that I somehow missed:  The drawings themselves are a substantive criticism of Islam.  It simply isn't sensible to have a prohibition against drawings of your prophet.  Or, if it is sensible, it isn't sensible to apply the prohibition to the rest of the world.  It's just an unfair ploy to get ahead the marketplace of ideas.

Just last week, Austin Dacey offered yet another angle: prohibiting drawings not only infringes on freedom of speech, but also ultimately infringes on freedom of religion.

There are a lot of different perspectives to sort through here.  And different drawings come from different perspectives.  For example, I intentionally drew something very different from the stick figure drawings, though I can't decide if that's better or worse.

Monday, May 17, 2010

Hilbert's Hotel

As it says on the margins, I'm a physics student.  But I have a small confession to make: I'm taking all math classes this quarter.  I have been thinking about writing a series of posts explaining infinite sets at a popular level, one I will likely never finish.

The problem with infinite sets is that they're counter-intuitive. This isn't a problem with the math, but a problem with our intuition.  If you find these things strange, you just have to get used to it.

Let's start with Hilbert's Hotel, the hotel with an infinite number of rooms.  More specifically, there is exactly one room for each positive integer (1, 2, 3, 4, ...).  Hilbert's Hotel has the interesting property that you can always make room for more guests, even if all the rooms are filled.

You simply tell each guest to move to the next room over.  The person in room 1 moves to room 2, and the person in room moves to room 3.  The person in room 10100 moves to the room 10100 + 1.  Every single guest has somewhere to go, because for every positive integer you can add 1 and get another positive integer.  At the end of this shuffle, room 1 is empty.  Now Hilbert can accommodate one new guest.

Some people might see this as an illustration of the statement, "Infinity plus one equals infinity."  But that is an inaccurate way of describing it.  Infinity is not the same kind of number that we usually work with.  We're used to working with "real" numbers, which can be approximated with decimal expansions.  For instance, we can approximate π within 1/100th by 3.14.  You cannot approximate infinity within 1/100th, because every decimal number will still be infinitely far away.  Since infinity isn't a real number, you can't add it to real numbers and get a sensible answer.

But there are other number systems besides the real numbers.  With Hilbert's Hotel, we're working with cardinal numbers, which are a way of characterizing the size of a set.  For example, we could take the set of all rooms in Hilbert's Hotel and give it a cardinal number.  Similarly, we can take the set of all guests in the hotel and give them another cardinal number.

If the guests exactly fill the rooms, then we say that the two cardinal numbers are equal.  For example, if there are three guests and three rooms, then the guests can exactly fill the rooms, and the two sets have the same cardinal number.  This is true, even if the guests decide to all occupy the same room, leaving the other two empty.  The point is that the guests could exactly occupy the rooms if they were shuffled around the right way.

With this in mind, let's return to the original example, where there are an infinite number of rooms exactly filled.  Because they're exactly filled, we can say that the set of rooms and set of guests are of equal size.  If one more guest needs a room, then we can shuffle around the guests to exactly fill the rooms again.  Therefore, the set of guests remains the same size before and after we add a new guest to the guest list.

Suppose that Hilbert has not one but two hotels, all exactly filled up.  But he needs to close down the second hotel and kick out all the guests.  Luckily, the set of guests and set of rooms is still equal. If we shuffle around the guests we can still make room for all of them.  Just tell all the guests in the first hotel to take their room number and double it.  All the odd-numbered rooms are now empty, and Hilbert can accommodate the rest of his guests.

Hey, remember back when Chuck Norris jokes were cool?  One of the jokes was, "Chuck Norris counted to infinity... twice!"  When you understand why two infinite hotels are just as good as one, then you will understand why counting to infinity twice is no harder than counting to infinity once.  I was far more impressed the time Chuck Norris counted the real numbers.

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 

Wednesday, May 12, 2010

Asexuals in fiction

From my gay friends, I've heard a lot of complaints, some obvious, some subtle, about media portrayals of LGBT folk.  I have just one simple complaint about media portrayals of asexuals: there aren't any, none you've heard of.

But it's not that simple.

Even if a writer is unaware of asexuality as an orientation (you know, the kind that affects real people), the writer can still write "asexual" characters, in the sense that they seem uninterested in sex or relationships.  There are some pretty obvious reasons to write such a character.  Maybe it's just too hard to write sexuality well.  Maybe it wouldn't fit into the story or mood.  Maybe it's just to serious a topic to get into.

Or maybe the lack of sexual interest is intentionally meant as a character quirk.  Here are a few well-known examples.

Sherlock Holmes
Sherlock Holmes is portrayed as a cold calculating deductive machine.  Where does an emotion such as love fit in?  According to Sir Arthur Conan Doyle (and many other writers) it doesn't.  In the recent film with Robert Downey Jr., I think he was rather asexy too.  Okay, so there was a fair bit of innuendo between him and Watson, but I think that was more for laughs than for anything else.

Doctor Who
The Doctor always takes with him a companion, usually female, on his time-traveling adventures.  But he's never shown any romantic interest in any of them (apparently this is somewhat questionable in the newer series).  Personally I think the writers just like to leave the shippers in a state of permanent speculation.  But it also makes sense because the Doctor is an alien.  Do aliens count as asexuals?  For that matter, do robots?  Holograms?  Zombies?  Sponges?  These sorts of questions are absolutely important and/or completely pointless!

Dexter, the serial killer with his own TV series.  Not interested in sex, at least not in the books.  This is what I've been told, anyways.  I've never actually read the books or watched the show for that matter.

Sheldon Cooper is theoretical physicist at Caltech, and the most popular character of a sitcom called The Big Bang Theory.  He's portrayed as utterly oblivious to any sexual or romantic cues.  Social cues too.  In fact, he seems to be autistic.  I wonder if the autistic community is happy with his portrayal.

The following dialogue is basically the closest thing to identifying Sheldon as asexual.
Penny: I know this is none of my business, but I just... I have to ask — what's Sheldon's deal?
Leonard: What do you mean, "deal"?
Penny: You know, like, what's his deal? Is it girls...? Guys...? Sock puppets...?
Leonard: Honestly, we've been operating under the assumption that he has no deal.
But since it's a TV show with many writers, you can never guarantee that Sheldon's character will be static.

Have you noticed? Sheldon is an asexual physicist, and so am I!  In theory, I should closely relate to him.  Except... I'm not really autistic at all.  And more importantly, he's part of a sitcom that I think is boring.

There are all sorts of cynical things I can say about the above portrayals.  I mean, asexuals are either aliens, psychopaths, or untouchable geniuses.  On the other hand, the only people who insisted on labeling them as asexual were me and parts of the asexual community.  Obviously, we must derive some enjoyment from the speculation.  Perhaps we think that a stereotype is better than no portrayal at all.

Are there any self-identified asexual characters in major media?  Yes, there are!  I mean, is.  There's one.  I give you Gerald Tippett.

Gerald Tippett is a character in a New Zealand soap opera called Shortland Street.  In 2008, there was a whole storyline where he came out as asexual.  All the relevant clips have been collected on YouTube.

Once you have a self-identified asexual character, there are several very obvious storylines which pretty much write themselves (at least if you're familiar with the subject).  Gerald cycles through basically all of them.  He starts out as a twenty-something whom everyone suspects is gay.  He discovers asexuality while in the middle of a relationship.  He doesn't accept it at first, asks for a cure.  His girlfriend doesn't accept it at first, tries to interpret every quirk as a possible cause.  His sex-positive mother discovers that there is one form of sexual deviancy that she just can't accept (awkward dinner follows).  The coworkers find out and become busybodies.  Gerald tries a meetup group.  Gerald tries an open relationship.  Oh, and there's a perpetual love triangle too.

In some possible future, these stories are absolutely trite.  But right now, they're absolutely unique and wonderful.  I loved it.  Maybe it's just me because I'm asexual?  Is it a complete bore for everyone else?

If we compare Gerald to any of the other characters I mentioned, there's a vast difference.  He's an ordinary character, not an extraordinary one.  He's heteroromantic (or biromantic), not aromantic.  His asexuality is not just another personality quirk, but brings up many difficult issues like identity, social acceptance, asexual/sexual relationships, and so forth.

After a while, the asexuality storyline dropped to the background, and it turned into some ridiculously soapy story about Morgan being a surrogate mother or something like that.  I couldn't watch it after that.

Friday, May 7, 2010

Chalk Muhammad

The Atheists, Humanists, & Agnostics (AHA!) at the University of Wisconsin-Madison are chalking pictures of Muhammad all over campus.  Look at his smile!  Who could feel offended by such a positive depiction?

Many Muslims, of course, are offended.  According to certain Hadith, depictions of Muhammad are discouraged or forbidden.  I believe the underlying motivation is the fear that the visual representation would become more important than the man himself, thus causing idolatry.  But leave it to religion to apply a rule blindly, without any regard to whether its original intention applies.

AHA! chalked over a hundred Muhammads in order to support free speech.  The Muslim Student Association (MSA) had neat response.  Well, first they tried to get the Dean of Students to stop the event, that wasn't so neat.  But their next action was to follow around the chalkers and draw boxing gloves on the stick figures.  Now it's Mohammad Ali!

The MSA is free to draw boxing gloves if they want.  It's free speech.*  I think they'd even be allowed to erase the drawings if they wanted.  Indeed some of names were later erased out, though no one knows that it was the MSA.  But just because they have the right to erase doesn't mean they should.  Though no rules are broken, erasing the drawings would clearly be anti-free-speech in spirit.

*On my own campus, there are regulations against chalking, so it would not be free speech here on either side.  But apparently there are no such regulations at Wisconsin-Madison.

Similar complaints were heard by AHA!  Just because you can draw Muhammad doesn't mean you should.  I mean, you're allowed to speak up in favor of free speech, but surely free speech can fend for itself as long as we all keep super quiet about it.  (Right now I'm visualizing Free Speech Personified boxing with Muhammad.)

But on a more serious note, I must admit there is one thing I don't like about the chalking.  The vandalism or erasure of the chalk does not hurt AHA! in any tangible way, and is not even breaking any rules.  Rather than serving as an example of why we must defend free speech, it arguably serves as an example of where an attack on free speech doesn't hurt anything except free speech itself.  The chalking itself is empty and pointless.  Its worth only derives from a public reaction to reaction to reaction.

Story time!  It's November.  I'm at a rally on the anniversary of Prop 8.  People are going up to the microphone to talk about the tragedy that is Prop 8.  Everyone is wearing black. Who do I see walking by in a bright orange Hawaiian shirt, but my most hated enemy, Mr. O?  Let's just say that he is the only person I've ever defriended on Facebook out of dislike.  Mr. O shows interest in the rally, asks me what's going on.  I'm friendly.  He hasn't done anything horrible in some time, and I can't hold a grudge for very long.  He totally supports the rally.

But he has a funny way of showing it.  He asks to speak at the mic.  "Aren't there more important things to worry about than guys having buttsex?"  He got kicked off the stage rather quickly.  Later, people at the mic would passionately tell him that, no, this really is the most important issue to them.  Mr. O tells me that this is the response he was going for.  What a jerk!

I'll divide the consequences into two parts.  There is Mr. O's little speech.  And then there is the reaction to the speech at the rally.  The speech itself was terrible, a total dick move.  The passionate responses at the rally were good, though perhaps they would have been just as passionate otherwise.  And yet, Mr. O is still a total dick, regardless of the ultimate responses to his actions.

That's sort of the problem with Chalk Muhammad.  The chalk itself does nothing but offend people.  That's a neutral to negative consequence.  And then there's the reaction to the chalk, the Muslim complaints.  That's also neutral to negative.  And then there's the reaction to the Muslim reaction, the realization that free speech is under attack.  That's positive.  But it took us a bit of a chain reaction to get there.

At least AHA! is being totally transparent about their intentions.

My point is it's not completely outlandish to judge AHA! for the action by itself, without regard to the chain of responses.  Most of the credit for the responses should go to the responders themselves.  The Muslims deserve responsibility for their attacks on free speech.  People who are inspired to fight for free speech deserve credit for fighting.  I deserve credit for this essay.  Mr. O doesn't get any credit, oooooh no.

You can't make every publicity stunt perfect.  But if we recognize the flaws, we can determine what direction would be an improvement.  It would be an improvement if the action itself had some intrinsic benefit.  For example, if Mohammad were drawn on fliers or signs, they would clearly serve the purpose of advertising the group (or better yet, an event celebrating free speech).  Also, if anyone defaces a sign, that's some serious rule-breaking right there.  Signs cost money.

I suppose the chalked Muhammads implicitly serve the purpose of advertising the group, and to that extent, they're totally cool.

Wednesday, May 5, 2010

Sam Harris goes for objective morality

Sam Harris recently had a talk on TED Talks called "Science can Answer Moral Questions".  So what's that old horseman of the apocalypse doing these days, anyway?  He's writing a book called "The Moral Landscape: How Science can Determine Human Values".  He's devoting a whole book to ending "religion’s monopoly on morality and human values."

If I had just read the title, I might have agreed with Sam Harris.  It's not that hard.  First we start with some basic principle, like "fed people are better than unfed people", and then we can use science to figure out which actions will ultimately feed more people.  Easier said than done, but then no one said ethics was easy.

But Sam Harris got a lot of criticism for his talk, because he argued something much stronger.  He thinks that you can use science to decide the basic principle itself.  Sam Harris argues for the basic principle that "we should value the wellbeing of humanity".  If I ask why, Sam Harris responds that this is a "profound and profoundly stupid question."  I think he means that there's no real justification for it, but it's obvious.

As far as I'm concerned, that's basically conceding that it's not science.  Science is not in the business of justifying unjustifiable statements, whether they're obvious or not.  In any case, wellbeing is only an obvious value as long as you consider extreme cases and are unclear about the details.

But rather than continuing my apocryphal rant on metaethics, let me discuss Sam Harris' motivations.

Popular wisdom says that if God exists, then there is objective morality, and if God doesn't exist, then there isn't.  Sam Harris wants to argue that objective morality is possible even without God.

But I would go in the completely opposite direction.  You cannot derive "ought" from "is", even if the "is" includes God.  If you believe in a benevolent deity, you might claim that values come from God.  But why should we just accept these values?  Because it's "obvious" (particularly if there is a heaven and hell, since one is obviously more desirable).  But not because it's justifiable.  Same problem Sam Harris ran into.

Sam Harris' other motivation is that he would like to have objective grounds from which he can condemn burqas and nazis.  But does having an objective morality really help?

Imagine that I got in a hypothetical argument with Hitler.  There are two possibilities.  The first possibility is that Hitler and I agree on enough moral precepts that I could in principle persuade him to stop the Holocaust.  The fact that the moral precepts are not objective truths does not matter, because we already agree on them.  The second possibility is that we do not agree on enough moral precepts.  I daresay that claiming that the moral precepts are objectively true because they're "obvious" would not move Hitler.

In conclusion, Sam Harris is wrong, and for the wrong reasons.  You know, I'm not sure when I last agreed with Sam Harris on any major point.

Monday, May 3, 2010

The purpose of doubt

I watched a film from 2008 called DoubtThe trailer explains the premise: At a Catholic school, Sister Meryl Streep (I missed her character's name) somehow becomes convinced that Father Flynn was engaged in sexual misconduct with a student.  She has little or no proof, but she has her certainty.

I can't say I was all that impressed.  The movie was too slowly paced.  I don't mean that in the sense of not having enough action, but in the sense that the start and end are hardly any distance from each other.  The story simply doesn't go much beyond its premise.  Pretty much all the best moments are in the trailer.

On the plus side, that means I can discuss the theme without any spoilers.  The main theme of the story is set by a sermon by Father Flynn at the very beginning.  He says (as shown in the trailer), "Doubt can be a bond as powerful as certainty."  In other words, certainty may help to unite a community, but doubt is just as effective, if not more so.

It's hard for me to say how certainty can unite a community, mainly because I am not part of any such community.  I suppose that if everyone has unwavering faith in a particular belief, then that belief provides a point of commonality for everyone, no matter how unlike we may be in other respects.  The unfortunate result of such a community, is that anyone who doubts becomes isolated.  People with doubts often try to put up a pretense of certainty in order to stay with the community.  But this is a deeply dissatisfying resolution, because it requires hiding one's true feelings.

And that's why doubt can be a better unifier for communities.  If everyone confessed their doubts, they would realize that everyone has doubts.  Thus, doubts provide a point of commonality for everyone, and no one needs to hide their feelings just to fit in.

But I feel there is something huge missing from this discussion.  It's missing from most religious discussion of doubt.  Belief is seen as something that can unite or divide communities.  Doubt is also seen as something that can unite or divide communities.  But that's not the primary purpose of doubt.  I think of belief and doubt as tools in our quest to know what is true and what is false.  They may also function as tools to build communities, but that should be secondary.

Going back to the movie, if Sister Meryl Streep had shown the slightest doubt in her crusade, that would have made her a more sympathetic character, and bridged some of the divide between her and Father Flynn.  But more importantly, it could have been a tool to actually decide whether Flynn had commit the crime or not.  If she had shown doubt, perhaps she would have put more effort into seriously investigating the possibility.  Instead, because she already knew herself to be right, she spent more effort trying to convince everyone else.  She was unable to convince a single person, for lack of evidence.

Of course, this would only work if Sister Streep publicly showed doubt.  And that's just not going to happen as long as doubt is only something you reveal to your closest friends in order to bond with them.  We must act on doubts, or we might as well not doubt at all.