Top 10 Time/Space Manipulation Puzzle Games

You, like I, might wish to bend time and space and wake up after the election results are in. Barring an unsafe quantity of alcohol, we can’t. But second best, here are 10 Time/Space manipulation puzzle games* which can take your mind off things.

*It turns out I accidentally put 11 on the list. [Insert pun about bending the laws of arithmetic, along with time and space.]

I tried to figure out what I like about puzzle games, and I think I’m rating these for a combination of:

  • Creative time/space manipulation mechanic
  • Clever puzzles
  • Sense of Discovery
  • Aesthetics

10a. HyperRogue

Of course the lines are parallel; you can tell by the way they curve away from each other.

HyperRogue is a turn-based game on a hyperbolic plane: a non-Euclidean surface in which the angles of a triangle add up to less than 180 degrees. (The opposite of being on a globe, which is non-Euclidean in that the angles of a triangle add up to MORE than 180 degrees.)

I found myself frequently getting lost but having a great time doing it.

The game itself is simple but not just window dressing for cool geometry. It’s a rogue-like game, so you’re expected to die and start over frequently. That might be best, since it keeps the focus on the experience and not on the progress or final goals.

Pro: Cool introduction to hyperbolic geometry, great way to pass some time

Cons: I found it difficult to “get good at” so I mostly wandered around and occasionally did well before being trapped by monsters.

10b. 5D Chess with Multiverse Time Travel

Not normal Chess, not my 1D Chess variant, not 3D Chess. Nope. Five dimensions. I thought this was a silly gimmick at first, but the rules are surprisingly coherent, if… mindbending. The result is more logic puzzle than chess.

Treating time as a dimension that can be traversed works. A rook — which can move as far as you want in one direction — can go horizontal, vertical, or stay in the same position but go back in time. Knights (shown below) can:
A) Move regularly — 2 steps and then one step horizontally/vertically
B) Move two spaces horizontally/vertically and then one space back in time or
C) Move two spaces back in time and then one space horizontally/vertically.

“I was playing a game against a human opponent online, and at one point they sent a queen back in time from one of the ten timelines currently in play to put five of my past kings into check at once. I sent one of my own pieces even further back to stall, and they proceeded to send one of their queens back to the start of the game to try and beat me before I even got to that point”

“If you have no idea what I just wrote, I hardly do either, this game makes my head hurt and yet eventually your mind’s eye is opened to the cosmic structure of the universe –

or you just go insane


Review by TheSpookiestUser

Pros: Actually pulls off a time-travel board game!

Cons: Steep learning curve means it’s easy to give up. Even the very positive reviews on steam have a lot of people who only played half an hour.

9. Induction

I’m a sucker for these self-interaction type puzzle games. Each level involves moving around, rewinding and creating one or more clones, and cooperating with… your previous self.

The game is minimalist and doesn’t have elaborate graphics or context but the puzzles themselves are intricate and clever.

Full disclosure: I haven’t finished, but what I’ve played has made me appreciate the thought that went into the design.

Pros: Great puzzles which actually use time manipulation

Cons: It’s pure puzzling without any skin or context, if you care

8. Monument Valley & Monument Valley 2

Monument Valley is MC Escher in a delightful little puzzle game. The puzzles themselves aren’t very difficult, but they’re not so straightforward that I found them boring. There isn’t much plot or exploration, it’s true. That’s not why you play – you play Monument Valley for the beautiful experience.

Pros: Absolutely beautiful aesthetics, delightful to play through

Cons: I wish it were longer, and while the whimsical feeling is great there isn’t much plot or exploration

7. Return of the Obra Dinn

The first games on the list were somewhat one-dimensional (pun intended) but we’re starting to get into games with more breadth.

In Return of the Obra Dinn, you play as an insurance agent for the East India Trading Company (stick with me.) The titular ship left with 60 people on board and returned with… none. You’re tasked with investigating what the heck happened to everyone. Someone sent you a mysterious pocketwatch which can take you back to someone’s moment of death, allowing you to hear the last few seconds and then explore the ship at that frozen time.

Imagine an interactive logic puzzle murder mystery, but there are 60 people whose fates you need to ascertain, deducing their identity based on the snippets of dialogue, their uniforms, who they hang out with, their accents and personal belongings…

I love the sense of discovery you get as you learn more about each crewmember and passenger, allowing you to go revisit earlier memories and make new deductions. In the best tradition of murder mysteries there will be new revelations which shift your understanding of things you thought you knew.

The graphics aim to evoke oldschool monitors, and somehow it works to make the game surprisingly immersive. Since the action during time-travel flashbacks are frozen, the sound design is key — and it’s fantastic. The music is amazing, and the sound effects and voice actors bring things to life… sometimes a bit too well.

Pros: Fantastic story, Clever conceit, Beautiful music, Repeated sense of dawning realization.

Cons: Given that everyone dies or disappears, things can get gruesome, especially the sounds. The graphics might not appeal to everyone.

6. Antichamber

I saw someone refer to Antichamber as “Portal on Acid” and they’re not wrong. It’s a first-person trip through an Escher-esque world where going in a circle might not lead you back to where you started.

You traverse the surreal world manipulating cubes of matter to open doors, build bridges or staircases, mark your path, or anything else you think would be helpful.

What sets Antichamber apart for me is the nonlinear element of discovery. Solving a particularly tricky puzzle can open a new area on the map, which absolutely floods my brain with dopamine.

There are a few different matter-manipulating guns you can find, each of which gives you a new capabilities and a new perspective on how to approach puzzles. I ended up revisiting placed I’d been stuck multiple times and occasionally being able to solve them — the ultimate variable interval reinforcement.

Pros: Great (very) nonlinear game, easter eggs to discover, epiphanies to have, creative puzzles which have multiple different solutions if you’re clever (or stubborn and patient).

Cons: This is an independent game and they didn’t put their focus on the graphics. While there’s a sense of progression, it isn’t plot-based.

5. Contrast

In Contrast, you explore 1920s Noire France as a child’s imaginary friend who can transition between the regular world and the shadow realm. Unlike Lord of the Rings, this is literally your shadow — you become your 2D shadow on the wall and can jump on top of other shadows to climb somewhere new before popping back into the world of depth.

Moving the light source changes where shadows are cast on walls, creating some neat interplay between manipulating physical objects and their 2D representation.

Pros: Fantastic 1920s Noire aesthetic, creative mechanics, rewards for exploring the French city

Cons: Could be longer, and there were a couple puzzles that didn’t fit perfectly with the rest of the mechanics.

4. The Talos Principle

The only reason this game isn’t ranked higher in this list is that the time manipulation component is limited.

The Talos Principle has stunning graphics and music, a slowly-revealed plot about the nature and fate of humanity, conversations with an ambiguously-moral AI named Milton, a God-figure named Elohim trying to preside over a virtual world on the fritz, and of course, clever puzzles of varying difficulty.

For the most part the mechanics are straightforward: connect lasers, move boxes, unlock doors, etc (but eventually you can rewind time and solve puzzles in tandem with your previous self, my justification for putting it on this list.) The puzzles are so well designed that simple elements fit together in countless new ways, giving a fantastic depth-to-complexity ratio.

Where the game goes above and beyond is how much it rewards exploration and experimentation. Not only are the worlds filled with easter eggs, they also contain a more challenging set of “star” puzzles which often require you to think outside the box — finding clever ways to bring equipment from one puzzle into another, creating a staircase to get on top of a wall you thought was irrelevant, discovering an extra block tucked away in the bushes… I love it.

Pros: Beautiful graphics and music, a wide variety of non-repetitive puzzles, rewards for exploring and being creative.

Cons: While most of the puzzles are purely about figuring out the solution, a few require fast reactions/controls. It can be frustrating to know exactly what to do but not quite be able to do it.

3. Fez

In Fez, you play as a character in a 2D world who is granted the ability to see the third dimension. Once you can look at the world from different directions, you realize that some platforms are closer together than you thought — just in a different dimension. What had looked like a simple line is actually a detailed wall with a door. A stick is really a sign with cryptic symbols from ancient civilizations.

The graphics aren’t impressive, but the worldbuilding sure is. Because things aren’t always obvious until you’ve looked at them from different perspectives, there are countless secrets tucked away and a surprising amount of thought put into the backstory.

As I explored, I found myself scribbling little notes and codes down on scraps of paper that slowly took over my desk. Over and over, I pieced together hints that gave me a new understanding of what had been in front of me all along. I’ve heard the phrase “Epiphany addiction” and it seems wholly appropriate for Fez.

Pros: Clever mechanics, huge nonlinear world, impressive amount of lore and worldbuilding, numerous perspective-shifts, tons of hidden secrets

Cons: The huge nonlinear world actually deterred me at first — I wasn’t sure where to go and put the game down for a while before being convinced to give it a second try.

Honorable Mentions:

Before getting to the Top 2 Time/Space Manipulation Puzzle Games, here are quick-hit puzzlers that didn’t quite make this list:

  • The Swapper: Create clones and swap your consciousness between them to solve puzzles. Great game by the makers of Talos Principle, but didn’t have enough time/space manipulation to make the cut.
  • The Witness: Another beautiful game which I consider the spiritual successor to Myst. But I couldn’t quite justify putting it on the list without more of a time/space connection.
  • The Bridge: I should love this game. It’s MC Escher and time-manipulation and neat graphics and everything I like! …but the controls were a bit too “floaty” for me and I found it frustrating. Your mileage may vary.
  • OneShot: Doesn’t so much break the fourth wall as stroll through where you expected the fourth wall to be, and starts complementing your furniture. There’s some elements of time travel but not really time manipulation.
  • TimeSpinner: A great Metroidvania game where you jump between the past and present, and can stop time mid-battle to get better positioning. It has the time-manipulation, but not the “puzzle game” part.
  • Baba Is You: The most meta of all games. You alter the logical rules of your world by changing the words that spell the rules out. You might win by setting “Wall is floating” and go under it, or “Wall is You” and turn into the wall, moving around to get to the goal.

The Top 2 Time/Space Manipulation Puzzle Games

2. Braid

The King/Queen of time manipulation games.

On its face it’s a simple platformer where you jump on monsters and collect puzzle pieces. Oh, and you can rewind time to undo if you miss a jump.

Oh, and then later you can keep some objects going forward in time while you go backward. Oh, and in some levels time will go forward when you move to the right and backward when you move left. And later you can make time move at different speeds in different parts of the map. Oh, and then… Somehow the different mechanics aren’t gimmicks – the puzzles actually work.

It also features the best “Backward Time” levels I’ve seen: Depending on where you’re standing, a dead monster either died because it fell into a pit of spikes or because you jumped on it. Both timelines are coherent, and allow you to change where the monster came from, as long as he ends up dead at the end.

Pair that with beautiful graphics, a wonderful classical soundtrack that was chosen to be interesting at different speeds both forward and backward, and some hidden super-difficult puzzles, and you have one of my favorite games ever.

Pros: Great puzzles with the best time-manipulation mechanics, great graphics, great soundtrack.

Cons: There’s more of an aesthetic than a plot, and some people find it pretentious. I played with a controller, but wonder if playing on a keyboard would be difficult.

1. Portal 1 & 2

The King/Queen(s) of space manipulation games.

Portal 1 was a fantastic game, and Portal 2 was somehow even better. I don’t think it can be called a cult classic anymore now that it’s so popular, but it’s the source of lines like “The cake is a lie.”

There are so many good things to say I’ll just rattle them off:

  • Innovative physics-warping mechanics
  • Fantastic puzzles which increase in difficulty at a good pace
  • Hilarious writing
  • Exciting plot
  • Great graphics and soundtrack
  • A long single-player campaign
  • An entirely separate two-player Cooperative campaign!
  • An online community which has generated its own puzzles, adding even more value

Pros: Everything above.

Cons: It’s a first-person puzzle game, requiring use of a controller or a keyboard/mouse. This is only a problem because I keep recommending the game to people who aren’t usually into video games, but think they’d love Portal anyway.

Other thoughts:

I’ve been meaning to play The Stanley Parable, and if I could get Perspective or The Outer Wilds to work on a mac I would probably give them a chance to be on the list.

If you have other recommendations or disagreements, please let me know! …politely.

What Would a Rational Gryffindor Read?

In the Harry Potter world, Ravenclaws are known for being the smart ones. That’s their thing. In fact, that was really all they were known for. In the books, each house could be boiled down to one or two words: Gryffindors are brave, Ravenclaws are smart, Slytherins are evil and/or racist, and Hufflepuffs are pathetic loyal. (Giving rise to this hilarious Second City mockery.)

But while reading Harry Potter and the Methods of Rationality, I realized that there’s actually quite a lot of potential for interesting reading in each house. Ravenclaws would be interested in philosophy of mind, cognitive science, and mathematics; Gryffindors in combat, ethics, and democracy; Slytherins in persuasion, rhetoric, and political machination; and Hufflepuffs in productivity, happiness, and the game theory of cooperation.

And so, after much thought, I found myself knee-deep in my books recreating what a rationalist from each house would have on his or her shelf. I tried to match the mood as well as the content. Here they are in the appropriate proportions for a Facebook cover image so that you can display your pride both in rationality and in your chosen house (click to see each image larger, with a book list on the left):

Rationality Ravenclaw Library

Rationality Gryffindor Library

Rationality Slytherin Library

Rationality Hufflepuff Library

What do you think? I’m always open to book recommendations and suggestions for good fits. Which bookshelf fits you best? What would you add?

Colbert Deconstructs Pop Music, Finds Mathematical Nerdiness Within

Stephen Colbert channeling Kurt Godel

And here I thought I didn’t like pop music. Turns out I just hadn’t found the songs that invoke questions about the foundations of logic and mathematics. Fortunately, Stephen Colbert brings our attention to the fascinating – and paradoxical! – pop song “That’s What Makes You Beautiful” by One Direction. Watch Stephen do his thing deconstructing the lyrics with glorious nerdy precision before we take it even further (the good part starts at 1:54 or so):

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For those of you who can’t watch the video, here’s the nerdy part, hastily transcribed:

Their song “That’s What Makes You Beautiful” isn’t just catchy, it has a great message. “You don’t know you’re beautiful. That’s what makes you beautiful.”

First of all: great dating advice. Remember girls, low self esteem – very attractive to men. Guys always go for the low hanging fruit, easy pickings.

Second: the lyrics are incredibly complex! You see, the boys are singing “You don’t know you’re beautiful, that’s what makes you beautiful.” But they’ve just told the girl she’s beautiful. So since she now knows it, she’s no longer beautiful!

But – stick with me, stick with me, oh it goes deeper! – but she’s listening to the song, too. So she knows she’s not beautiful. Therefore, following the syllogism of the song, she’s instantly beautiful again!

It’s like an infinite fractal recursion, a flickering quantum state of both hot and not. I mean, this lyric as iterated algorithm could lead to a whole new musical genre. I call it Mobius pop, which would include One Direction and of course the rapper MC Escher.

They say the way to a man’s heart is through his stomach but honestly, talking about recursion, fractals, and flickering quantum states does far more to win my love.  We can find intellectual stimulation in anything!

And there’s more – we can go nerdier!

Stick With Me, Stick With Me, Oh It Goes Deeper

Let’s analyze the dilemma a bit further:

  1. She can’t KNOW she’s beautiful because, as Stephen points out, that leads to a logical contradiction – she would no longer be beautiful.
  2. She can’t KNOW that she isn’t beautiful, because that also leads to a logical contradiction – she would be beautiful again.
  3. It’s impossible for the girl to know that she is or isn’t beautiful, so she has to be uncertain – not knowing either way.
  4. This uncertainty satisfies the requirements: she doesn’t know that she’s beautiful, therefore, she’s definitely beautiful and can’t know it.

It turns out she’s not in a flickering state of hot and not, she’s perpetually hot – but she cannot possibly know it without logical contradiction! From an external perspective, we can recognize it as true. From within her own mind, she can’t – even following the same steps. How weird is that?

Then it hit me: the song lyrics are a great example of a Gödel sentence!

Gödel sentences, from Kurt Gödel’s famous Incompleteness Theorems, are the statements which are true but unprovable within the system.  Gödel demonstrated that every set of mathematical axioms complex enough to stand as a foundation for arithmetic will contain at least one of these statements: something that is obviously true from an outside perspective, but isn’t true by virtue of the axioms.  (He found a way to coherently encode “The axioms do not prove this sentence to be true.”)  This raises the question: what makes a mathematical statement true if not the fact that it can be derived from the axioms?

Gödel’s findings rocked the world of mathematics and have had implications on the philosophy of mind, raising questions like:

  • What does it mean to hold a belief as true?
  • What are our minds doing when we make the leap of insight (if insight it is) that identifies a Gödel sentences as true?
  • How does this set us apart from the algorithmic computers, which are plagued by their own version of Incompleteness, the Halting Problem?

I had no idea pop music was so intelligent!

Was the boy band comparing her, not to a summer’s day, but a turing-complete computer?  Were they glorifying their listeners by reminding us that, according to some interpretations of Incompleteness Theory, we’re more than algorithmic machines?  Were they making a profound statement about mind/matter dualism?

I don’t know, but apparently I should turn on the radio more often.

[For related reading, see various analyses of Mims’ “This is Why I’m Hot”]

As they say in the Sirius Cybernetics Corporation: Share and Enjoy!

Time Travel Art: Fantastic Video

It takes a lot for art to excite me.  But when I stumble upon a time-travel-themed YouTube video inspired by Dr. Who and Edward Gorey… I feel the urge to share it with everyone.  Yes, it’s every bit as cool as it sounds. The video is by ‘MaryDoodles’, who produces time-lapse videos of herself painting – but this time she found a way to work time-travel into the delivery as well as the content. You’ll just have to watch it:

I found myself hitting ‘replay’ over and over, figuring out how the hell she did it, spotting new things I’d missed (did you see all the things tying the scenes together?  How about the TARDIS?), and trying to piece the plot together. Alas, she’s not giving hints about the story she intended:

The discussions on Youtube are quite fun to watch as people try and piece together the order of events and what happened. I’ve decided to hold my tongue on this matter. There is an intentional order to the story but the fact that other people are seeing different orders of events, character relations and catalysts I figured I’d just leave my opinions out of it. It’s almost like a personality test when you hear someone’s take on the video. There are those that take the “glass is half empty” approach while others say the “glass is half full”. Happy ending? Tragic ending?”

I’m still making up my mind. Curiosity is a wonderful emotion, and I think the ambiguity is actually a positive factor.

Stereotypical koans take the ambiguity too far – “What is the sound of one hand clapping?” is just a broadside assault on semantics – but this is pleasantly confusing to me, like a puzzle.

My Little Pony: Reality is Magic!

(Cross-posted at 3 Quarks Daily)

You probably won’t be very surprised to hear that someone decided to reboot the classic 80’s My Little Pony cartoon based on a line of popular pony toys. After all, sequels and shout-outs to familiar brands have become the foundation of the entertainment industry. The new ‘n improved cartoon, called My Little Pony: Friendship is Magic, follows a nerdy intellectual pony named Twilight Sparkle, who learns about the magic of friendship through her adventures with the other ponies in Ponyville.

But you might be surprised to learn that My Little Pony: Friendship is Magic’s biggest accolades have come not from its target audience of little girls and their families, but from a fervent adult fanbase. I first heard of My Little Pony: Friendship is Magic from one of my favorite sources of intelligent pop culture criticism, The Onion’s AV Club, which gave the show an enthusiastic review last year. (I had my suspicions at first that the AV Club’s enthusiasm was meant to be ironic, but they insisted that the show wore down their defenses, and that it was “legitimately entertaining and lots of fun.” So either their appreciation of My Little Pony: Friendship is Magic is genuine, or irony has gotten way more poker-faced than I realized.)

And you might be even more taken aback to learn that many, if not most, of those adult My Little Pony: Friendship is Magic fans are men and that they’ve even coined a name for themselves: “Bronies.” At least, I was taken aback. In fact, my curiosity was sufficiently piqued that I contacted Purple Tinker, the person in charge of organizing the bronies’ seasonal convention in New York City. Purple Tinker was friendly and helpful, saying that he had read about my work in the skeptic/rationalist communities, and commended me as only a brony could: “Bravo – that’s very Twilight Sparkle of you!”

But when I finally sat down and watched the show, I realized that while Purple Tinker may be skeptic-friendly, the show he loves is not. The episode I watched, “Feeling Pinkie Keen,” centers on a pony named Pinkie Pie, who interprets the twitches in her tail and the itches on her flank as omens of some impending catastrophe, big or small. “Something’s going to fall!” Pinkie Pie shrieks, a few beats before Twilight Sparkle accidentally stumbles into a ditch. The other ponies accept her premonitions unquestioningly, but empirically-minded Twilight Sparkle is certain that Pinkie Pie’s successes are either a hoax or a coincidence. She’s detemined to get to the bottom of the matter, shadowing Pinkie Pie in secret to observe whether the premonitions disappear when there’s no appreciative audience around, and hooking Pinkie Pie up to what appears to be a makeshift MRI machine which Twilight Sparkle apparently has lying around her house, to see whether the premonitions are accompanied by any unusual brain activity.

Meanwhile, Twilight Sparkle is being more than a little snotty about how sure she is that she’s right, and how she just can’t wait to see the look on Pinkie Pie’s face when Pinkie Pie gets proven wrong. Which, of course, is intended to make it all the more enjoyable to the audience when — spoiler alert! — Twilight Sparkle’s investigations yield no answers, and Pinkie Pie’s premonitions just keep coming true. Finally, Twilight Sparkle admits defeat: “I’ve learned that there are some things you just can’t explain. But that doesn’t mean they’re not true. You just have to choose to believe.”

Nooo, Twilight Sparkle, no! You are a disgrace to empirical ponies everywhere. And I’m not saying that because Twilight Sparkle “gave in” and concluded that Pinkie Pie’s premonitions were real. After all, sometimes it is reasonable to conclude that an amazing new phenomenon is more likely to be real than a hoax, or a coincidence, or an exaggeration, etc. It depends on the strength of the evidence. Rather, I’m objecting to the fact that Twilight Sparkle seems to think that because she was unable to figure out how premonitions worked, that therefore science has failed.

Twilight Sparkle is an example of a Straw Vulcan, a character who supposedly represents the height of rationality and logic, but who ends up looking like a fool compared to other, less rational characters. That’s because the Straw Vulcan brand of rationality isn’t real rationality. It’s a gimpy caricature, crafted that way either because the writers want to make rationality look bad, or because they genuinely think that’s what rationality looks like. In a talk I gave at this year’s Skepticon IV conference, I described some characteristic traits of a Straw Vulcan, such as an inability to enjoy life or feel emotions, and an unwillingness to make any decisions without all the information. Now I can add another trait to my list, thanks to Twilight Sparkle: the attitude that if we can’t figure out the explanation, then there isn’t one.

Do you think it’s possible that anyone missed the anti-inquiry message?  Hard to imagine, given the fact that the skeptical pony seems mainly motivated by a desire to prove other people wrong and gloat in their faces, and given her newly-humbled admission that “sometimes you have to just choose to believe.” But just in case there was anyone in the audience who didn’t get it yet, the writers also included a scene in which Twilight Sparkle is only able to escape from a monster by jumping across a chasm – and she’s scared, but the other ponies urge her on by crying out, “Twilight Sparkle, take a leap of faith!”

And yes, of course, My Little Pony: Friendship is Magic is “just” a kids’ cartoon, and I can understand why people might be tempted to roll their eyes at me for taking its message seriously. I don’t know to what extent children internalize the messages of the movies, TV, books, and other media they consume. But I do know that there are plenty of messages that we, as a society, would rightfully object to if we found them in a kids’ cartoon – imagine if one of the ponies played dumb to win the favors of a boy-pony and then they both lived happily ever after. Or if an episode ended with Twilight Sparkle chirping, “I’ve learned you should always do whatever it takes to impress the cool ponies!” So why aren’t we just as intolerant of a show that tells kids: “You can either be an obnoxious skeptic, or you can stop asking questions and just have faith”?

Review: The Book of Mormon

(Re-posted with permission from my article in Issue 56 of The Philosopher’s Magazine)

Even if you’ve never watched a single episode of South Park, you’re probably aware that the show’s creators, Trey Parker and Matt Stone, love nothing more than a good bout of sacred cow tipping. Show me an ideology, political, religious or otherwise, and I’ll show you an episode of South Park that lampoons it with the show’s trademark blend of incisive satire and potty humour. So it was surprising that South Park’s terrible twosome wound up creating a smash- hit Broadway musical, which they have been describing, in interviews, as being pro-faith.

Well, the “smash-hit” part isn’t surprising. The Book of Mormon pulls off the impressive trick of winking at the clichés of musical theatre and embracing them at the same time. (After all, the clichés are clichés for a reason – they work.) The story follows two young Mormon men paired together for their mission to Uganda: Kevin, who’s used to being the golden boy and needs to learn that everything’s not always about him, and Arnold, a hapless schmuck who needs to learn some self-confidence. They’re an odd couple and, like all odd couples thrown together under unusual circumstances, they’re going to have to learn to get along. There’s also a sweet Ugandan ingénue, a villainous warlord threatening her village, and a whole lot of really catchy songs. It’s no wonder the musical garnered nine Tony awards this year, including Best Musical, and that it’s been selling out its shows since it opened in previews in February.

But to hear Parker and Stone refer to The Book of Mormon as “pro-faith” was surprising, especially given how often they poke fun at Mormonism. Mormon beliefs can seem so ridiculous to outsiders, in fact, that Parker and Stone wisely realise they don’t need to do much active mocking – instead, they simply step back and let the scripture speak for itself. “I believe,” one missionary warbles in a climactic number reaffirming his commitment to his faith, “that God lives on a planet called ‘Kolob’! And I believe that in 1978, God changed his mind about black people!” With raw material like this, parody is both unnecessary and impossible.

And it’s not just Mormonism that gets skewered. It’s also the self-images of all believers who like to see themselves, and their motivations, as more saintly than they really are. Against a back- drop of war, poverty and disease, one missionary wonders, “God, why do you let bad things happen?” and then adds what is, for many people, the true concern: “More to the point, why do you let bad things happen to me?” There’s only one thing Parker and Stone enjoy sinking their talons into more than absurdity, and that’s hypocrisy.

So in what sense is The Book of Mormon “pro- faith?” Well, it’s affectionate in its portrayal of Mormons as people, most of whom come off as well-meaning, if goofy and often naive. Parker and Stone have made no secret of the fact that they find Mormons just too gosh-darned nice to dislike. But what they’re mainly referring to when they call their musical “pro-faith” is the message it sends the audience home with: that religion can be a powerful and inspiring force for good, as long as you don’t interpret scripture too strictly.

By the end of The Book of Mormon, Africans and missionaries alike are united together in a big happy posse that preaches love, joy, hope and making the world a better place. Having learned by now that it’s more important to help people than to rigidly adhere to dogma, Kevin sings, “We are still Latter Day Saints, all of us. Even if we change some things, or we break the rules, or we have complete doubt that God exists. We can still all work together and make this our paradise planet.”

That’s an appealing sentiment, especially to the sort of theatregoer who prides himself on being progressive and tolerant. It means we can promote all the values we cherish – happiness, freedom, human rights and so on – without ever having to take an unpopular anti-religion stand. But is it plausible? How, exactly, can religion make the world a better place?

I don’t know and, apparently, neither does The Book of Mormon. The central confusion you’ll notice in the musical is that it keeps conflating two very different kinds of “faith”. One could be called “figurative faith”, the warm and fuzzy kind that emerges at the end of the show, which is explicitly about bettering the world but seems to be faith in name only, as it doesn’t involve any actual belief in anything. “What happens when we’re dead? Who cares! We shouldn’t think that far ahead. The only latter day that matters is tomorrow,” the villagers sing. Once you strip away God, and an afterlife, and the requirement of belief in particular dogma, it’s not clear that what’s left bears any resemblance to religion anymore. With its progressive values and its emphasis on the here-and-now rather than the hereafter, it’s basically just humanism.

The other kind of faith in The Book of Mormon is literal faith, but for the most part, it doesn’t actually help anyone. Ugandan sweet- heart Nabalungi believes in salvation in earnest – she’s under the impression that becoming Mormon means she’s going to be transported out of her miserable life to a paradise called “Salt Lake City”, which she imagines must have huts with gold-thatched roofs and “a Red Cross on every corner with all the flour you can eat!” she sings rapturously. But she ends up crushed when she eventually learns that, no, she doesn’t get to leave Uganda after all. (“Of course, Salt Lake City’s only a metaphor,” her fellow tribe members inform her, apparently in figurative faith mode at that point.)

To be fair, there is one example of the power of literal faith in The Book of Mormon. When a villager announces his plans to circumcise his own daughter, and another is about to rape an infant in an attempt to cure himself of AIDS, Arnold manages to stop them by inventing some new scripture for the occasion. “And the Lord said, ‘If you lay with an infant, you shall burn in the fiery pits of Mordor,’” he “reads” from the Bible. (Being a science fiction and fantasy nerd, and having slept through most of Sunday school, Arnold falls back on what he knows.) So I suppose that counts as a point in favour of faith’s power to help the world, albeit conditional on the bleak premise that the only way to get people to stop raping babies and mutilating women is to threaten them with Hell … or Mordor.

Of course, the fact that The Book of Mormon’s views on faith are less than fully coherent doesn’t detract much from the pleasures of its tart- tongued satire, story, and songs. There are just a handful of moments that might raise a philosopher’s eyebrow, such as when everyone sings, in the exuberant final number, “So if you’re sad, put your hands together and pray, that tomorrow’s gonna be a Latter Day. And then it probably will be a Latter Day!” It almost feels churlish to ask “Wait, how does that work?” when everyone onstage is having such a good time singing about joy and peace and brotherhood; nevertheless, one does wonder. Maybe that will be covered in the sequel.

Lies and Debunked Legends about the Golden Ratio

In my eyes, there’s a general pecking order for named mathematical constants. Pi is at the top, e gets a good amount of attention, and Tau, like a third-party candidate, sits by itself on the fringes while its supporters tell anyone who’ll listen that it’s a credible alternative to Pi. But somewhere in the middle is Phi, also known as the Golden Ratio. It’s no superstar, but it gets its fair share of credit in geometry and culture.

I was first introduced to Phi as a kid by watching the charming video Donald in Mathmagic Land. One of the things I remembered over the years is that the Greeks used the Golden Ratio in their paintings and architecture, particularly the Parthenon. Thanks to the power of the internet, I can share this piece of my childhood with you:

How brilliant and advanced of the Greeks, right? But there’s one problem…

It’s probably not true. My faith was first shaken reading Keith Devlin’s The Unfinished Game, where he entertained a quick digression:

Two other beliefs about this particular number [Phi] are often mentioned in magazines and books: that the ancient Greeks believed it was the proportion of the rectangle the eye finds most pleasing and that they accordingly incorporated the rectangle in many of their buildings, including the famous Parthenon. These two equally persistent beliefs are likewise assuredly false and, in any case, are completely without any evidence. For one thing, tests have shown that human beings who claim to have a preference at all vary in the rectangle they find most pleasing, both from person to person and often the same person in different circumstances. Also, since the golden ratio is actually not a ratio of two whole numbers, it is impossible to construct (by measurement) a rectangle having that proportion, even in theory.

What?! Donald, I trusted you! It was tempting to tell myself that the Greeks could have found ways to approximate the ratio, and that this is just one source, and I’ve heard it so many times it must be true, and la la la I don’t want Donald to have lied to me.

But I looked into it a bit more, checking out what Mario Livio had to say about it in his book The Golden Ratio. He acknowledges that it’s a very common belief, but ultimately backed Devlin up:

The appearance of the Golden Ratio in the Parthenon was seriously questioned by University of Maine mathematician George Markowsky in his 1992 College Mathematics Journal article “Misconceptions about the Golden Ratio.” Markowsky first points out that invariably, parts of the Parthenon (e.g. the edges of the pedestal [in a provided figure]) actually fall outside the sketched Golden Rectangle, a fact totally ignored by all the Golden Ratio enthusiasts. More important, the dimensions of the Parthenon vary from source to source, probably because different reference points are used in the measurements… I am not convinced that the Parthenon has anything to do with the Golden Ratio.

So, was the Golden Ratio used in the Parthenon’s design? It is difficult to say for sure… However, this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon. [emphasis mine]

Alas, claims about the Greeks using Phi in their architecture seem overrated. Some sites bring you celebrity gossip, we bring gossip about celebrated mathematical constants. Welcome to Measure of Doubt!

Watching the video again, I can’t tell exactly how they decided where to overlay the Golden Rectangles. How much of the pedestal do we include in the rectangle? How much of the pillar? Does the waist start here, or there? It seems a bit arbitrary, as though we’re experiencing pareidolia and seeing the Golden Rectangle in everything.

Talk about disillusionment.

RS#36: Why should we care about teaching the humanities?

Episode #36 of the Rationally Speaking podcast is out, and this one’s a lively debate between me and Massimo about the value of humanities departments in universities. While I don’t deny the huge amount of enjoyment we get from arts and literature, I express skepticism about many of the typical justifications for requiring humanities courses. Those justifications strike me as either (1) overly vague and subjective (“the humanities make you a complete person”) or (2) making contrived claims about the practical benefits of studying the arts (“the humanities build critical thinking skills”) to which I usually want to reply, “If that’s your goal, there are much more direct ways to pursue it than studying literature.”

Rationally Speaking #36: Why should we care about teaching the humanities?

The Game Theory of Story Endings

Do happy endings really make you as happy if you see them coming a mile away? When we watch a trashy action flick or a fluffy romantic comedy, aren’t the conflicts less interesting because we know it’ll all end happily ever after? Someone has to bite the bullet and write a sad ending to give plausibility to the threat of unhappiness. It’s disincentivized because sad endings are more challenging and risk upsetting the audience, but someone has to do it.

Steven E. Landsburg muses about this in The Armchair Economist:

I am intrigued by the market for movie endings. Movie-goers want two things in an ending: They want it to be happy and they want it to be unpredictable. There is some optimal frequency of sad endings that maintains the right level of suspense. Yet the market might fail to provide enough sad endings.

An individual director who films a sad ending risks short-term losses, as word gets around that the movie is “unsatisfying.” It is true that there are long-term gains, as viewers are kept off their guard for future movies. Unfortunately, most of those gains may be captured by other directors, because movie-goers remember only that the murderer does sometimes catch up with the heroine in the basement, and do not remember that it happens only in movies with particular directors. Under these circumstances, no individual director may be willing to incur costs for his rivals’ benefit.

A solution is for directors to display their names prominently, so that viewers know when a movie was made by someone unpredictable. Viewers, however, may find it in their interests to retaliate by covering their eyes when the director’s name is shown.

If you can be associated more strongly with unpredictability, you reap more benefits. You’re also more strongly associated with the unhappy ending, which might turn audiences away.

One way to ease the blow of an unexpected sad ending is to make deaths triumphant, defiant, or heroic. Think of how Spock died in The Wrath of Khan (No, I’m not going to give a spoiler alert for a 30 year old movie). Sure, people die in Star Trek all the time – when Kirk, Spock, and fresh-faced, red-shirted Ensign Jimmy beam down to explore a planet for life, we all know one of them isn’t going to make it back. But to kill a main character is more significant. And it was done in a touching way. They got the unpredictability without upsetting their audience.

I genuinely respect Joss Whedon for his willingness to throw curve balls like this in his story lines. He’s developed a reputation for having sympathetic characters die, leave, or change sides – often without warning. Rather than watching Buffy, Firefly and Serenity thinking “So, how is it all going to work out this time?” we’re forced to think “Is it going to work out this time?”

TV Tropes has a name for all this – Anyone Can Die:

This is where no one is exempt from being killed, including the main characters (maybe even the hero). The Sacrificial Lamb is often used to establish the writer’s Anyone Can Die cred early on. However, if the Lamb’s death is a one-off with no follow-up, it’s just Killed Off for Real. To really be Anyone Can Die, the work must include multiple deaths, happening at different points in the story. Bonus points if the death is unnecessary and devoid of Heroic Sacrifice.

In game theory situations, reputation plays a large role. TV Tropes mentions building a ‘Anyone Can Die’ cred, which can be achieved through repeated interactions. In a TV series or multiple films by the same director, you get a feel for whether the good guys always prevail. But even within a single story, early and repeated signaling can make the remainder of the plot more intense. When a major character is killed off without it being a Heroic Sacrifice, that’s a powerful signal that anything can happen. The musical Into the Woods will always have a special place in my heart for mastering this dynamic.

But there’s another route. Historical dramas can increase society’s perception of “sadness plausibility” without anyone taking a hit for being a downer. Nobody’s going to feel unsatisfied that Titanic, The Great Escape, or Butch Cassidy and the Sundance Kid have sad endings. (Or if they do, they can take it up with reality for writing a depressing script. It’s not easy to keep those separate in our brains; we just get the overall sense that sometimes stories have sad endings. And that perception helps us enjoy all the other movies we watch.

Funding for which arts?

Jesse’s recent post, about how words like “art” don’t have universally agreed-upon, precisely-defined meanings, reminded me of a book I just read by one of my favorite bloggers, economist Tyler Cowen at Marginal Revolution. He’s gotten a lot of media and internet attention in the past couple of months for his e-book The Great Stagnation, but the one I’m referring to is an older volume called Good and Plenty: The Creative Successes of American Arts Funding.

Good and Plenty makes the case that although the U.S. government ostensibly spends little on arts funding, compared to Europe for example, we also have huge de facto, indirect subsidies of art through copyright law and tax policies that encourage private giving. Those indirect subsidies are actually much more conducive to a thriving arts sector than Europe’s centralized funding, Cowen argues.

One of the many things I like about Cowen is his cheeky but serious way of challenging the ostensible reasons people have for their beliefs, or societies have for their institutions or laws. (This is something his friend and colleague Robin Hanson does even more persistently, by the way, over at another of my favorite blogs, Overcoming Bias.) The form of the argument is basically: You claim that the reason you do Y is because of X. But if you really believed X were the case, you would also do Z, which you don’t.

So in addition to the question of how we fund art, Cowen also asks why we’ve chosen to define “art” (for funding purposes, at least) to include certain things but not others. For example:

  • What about fashion? Clothing design exhibits beauty and creativity just like sculpture, and a beautiful piece of clothing arguably has higher positive externalities than a beautiful sculpture (because the owner wears it in public, other people get to appreciate its beauty, unlike a privately owned sculpture which sits at home).
  • What about sports? Many sports showcase stunning displays of grace and power, just like dance. And sports games, and seasons, have a lot in common with drama: protagonists, antagonists, triumph and defeat. “The drama in sports, of course, is real rather than staged, but presumably this should contribute to its aesthetic merit,” Cowen says.
  • What about toys? “Most young children are deeply concerned with matters aesthetic,” Cowen says. “They are fascinated by colors, shapes, sounds, and textures… The toy presents the child with an aesthetic package, so to speak, which the child either loves or rejects. When children they love a toy, they are truly passionate about enjoying its aesthetic qualities.”

Yet our government doesn’t subsidize fashion, or toys, or sports (at least not for artistic reasons; local governments may fund stadiums and so on , but that’s for other reasons, like economic development). So that would seem to cast doubt on the reason people generally give to justify funding the arts — that they provide people with enjoyable and moving aesthetic experiences — since those reasons should also suffice to justify subsidizing things like sports and fashion and toys, which we don’t.

Of course, the big differences between the traditional arts and Cowen’s examples are that the former have (1) more cachet, historically, and (2) less commercial viability. But we still have to explain why we should spend the money to prop up the traditional arts when there are other sources of aesthetic enjoyment that don’t require any propping.

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