Lateral thinking

One of the biggest impediments to clear thinking is the fact that it's so hard for us to keep in mind that we could be wrong.

As journalist Kathryn Schulz put it:

I asked you how it felt to be wrong, and you had answers like humiliating, frustrating, embarrassing, devastating.  And those are great answers.  But they're answers to a different question.  Those are answers to the question, "How does it feel to find out you're wrong?"  But being wrong?  Being wrong doesn't feel like anything...  You remember those characters on Saturday morning cartoons, the Coyote and the Roadrunner?  The Coyote was always doing things like running off a cliff, and when he'd do that, he'd run along for a while, not seeing that he was already over the edge.  It was only when he noticed it that he'd start to fall.  That's what being wrong is like before you've realized it.  You're already wrong, you're already in trouble...  So I should amend what I said earlier.  Being wrong does feel like something.

It feels like being right.

We cling desperately to the sense that we have it all figured out, that we're right about everything.  Oh, in theoretical terms we realize we're fallible; all of us can remember times we've been wrong.  But right here, right now?  It's like my college friend's quip, "I used to be conceited, but now I'm perfect."

The trouble with all this is that it blinds us to the errors that we do make, because if you don't keep at least trying to question your own answers, you won't see your own blunders.  It's why lateral thinking puzzles are so difficult, but so important; they force you to set aside the usual conventions of how puzzles are solved, and to question your own methods and intuitions at every step.  This was the subject of a study by Andrew Meyer (of the Chinese University of Hong Kong) and Shane Frederick (of Yale University) that appeared in the journal Cognition last week.  They looked at a standard lateral thinking puzzle, and tried to figure out how to get people to avoid falling into thinking their (usually incorrect) first intuition was right.

The puzzle was a simple computation problem:

A bat and a ball together cost $1.10.  The bat costs $1.00 more than the ball.  How much does the ball cost?

The most common error is simply to subtract the two, and to come up with ten cents as the cost of the ball.  But a quick check of the answer should show this can't be right.  If the bat costs a dollar and the ball costs ten cents, then the bat costs ninety cents more than the ball, not a dollar more (as the problem states).  The correct answer is that the ball costs $0.05 and the bat costs $1.05 -- the sum is $1.10, and the difference is an even dollar.

Meyer and Frederick tried different strategies for improving people's success.  Bolding the words "more than the ball" in the problem, to call attention to the salient point, had almost no effect at all.  Then they tried three different levels of warnings:

1. Be careful!  Many people miss this problem.
2. Be careful!  Many people miss the following problem because they do not take the time to check their answer.
3. Be careful!  Many people miss the following problem because they read it too quickly and actually answer a different question than the one that was asked.

All of these improved success, but not by as much as you might think.  The number of people who got the correct answer went up by only about ten percent, no matter which warning was used.

Then the researchers decided to be about as blatant as you can get, and put in a bolded statement, "HINT: The answer is NOT ten cents!"  This had the best improvement rate of all, but amazingly, still didn't eliminate all of the wrong answers.  Some people were so certain their intuition was right that they stuck to their guns -- apparently assuming that the researchers were deliberately trying to mislead them!

[Image licensed under the Creative Commons © Nevit Dilmen, Question mark 1, CC BY-SA 3.0]

If you find this tendency a little unsettling... well, you should.  It's one thing to stick to a demonstrably wrong answer in some silly hypothetical bat-and-ball problem; it's another thing entirely to cling to incorrect intuition or erroneous understanding when it affects how you live, how you act, how you vote.

It's why learning how to suspend judgment is so critical.  To be able to hold a question in your mind and not immediately jump to what seems like the "obvious answer" is one of the most important things there is.  I used to assign lateral thinking puzzles to my Critical Thinking students every so often -- I told them, "Think of these as mental calisthenics.  They're a way to exercise your problem-solving ability and look at problems from angles you might not think of right away.  Don't rush to find an answer; keep considering them until you're sure you're on the right track."

So I thought I'd throw a few of the more entertaining puzzles at you.  None of them involve much in the way of math (nothing past adding, subtracting, multiplying, and dividing), but all of them take an insight that requires pushing aside your first impression of how problems are solved.  Enjoy!  (I'll include the answers at the end of tomorrow's post, if any of them stump you.)

1.  The census taker problem

A census taker goes to a man's house, and asks for the ages of the man's three daughters.

"The product of their ages is 36," the man says.

The census taker replies, "That's not enough information to figure it out."

The man says, "Okay, well, the sum of their ages is equal to the house number across the street."

The census taker looks out of the window at the house across the street, and says, "I'm sorry, that's still not enough information to figure it out."

The man says, "Okay... my oldest daughter has red hair."

The census taker says, "Thank you," and writes down the ages.

How old are the three daughters?

2. The St. Ives riddle

The St. Ives riddle is a famous puzzle that goes back to (at least) the seventeenth century:

As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven kids,
Each kid had seven cats,
Each cat had seven kits.
Kits, cats, kids, and wives, how many were going to St. Ives?

3.  The bear

A man goes for a walk.  He walks a mile south, a mile east, and a mile north, and after that is back where he started.  At that point, he sees a large bear rambling around.  What color is the bear?

4.  A curious sequence

What is the next number in this sequence: 8, 5, 4, 9, 1, 7, 6...

5.  Classifying the letters

You can classify the letters in the English alphabet as follows:

Group 1: A, M, T, U, V, W, Y

Group 2: B, C, D, E, K

Group 3: H, I, O, X

Group 4: N, S, Z

Group 5: F, G, J, L, P, Q, R

What's the reason for grouping them this way?

6.  The light bulb puzzle

At the top of a ten-story building are three ordinary incandescent light bulbs screwed into electrical sockets.  On the first floor are three switches, one for each bulb, but you don't know which switch turns on which bulb, and you can't see the bulbs (or their light) from the place where the switches are located.  How can you determine which switch operates which bulb... and only take a single trip from the first floor up to the tenth?

Have fun!

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Different kinds of impossible

Many of us engage in magical thinking -- attributing causal relationships between actions and events that are simply (often accidentally) correlated.  Superstitions are magical thinking; as nice as it would be if you could influence the win/loss ratio of your favorite team by wearing a particular shirt, the universe just isn't put together that way.

Where it gets interesting is that there are different degrees of magical thinking. A clever piece of research from the online journal PLoS-One, carried out by psychologists John McCoy of the University of Pennsylvania and Tomer Ullman of Harvard, illustrates that even those of us who engage in magical thinking seem to be intuitively aware of how impossible different false causations are.

So we can, like the White Queen in Through the Looking Glass, believe in six impossible things before breakfast.  [Image is in the Public Domain]

The paper, entitled "Judgments of Effort for Magical Violations of Intuitive Physics," asks test subjects to perform a simple task.  First, imagine a world where magic is real, where conjuring a spell could make things happen that are impossible in our world.  Then, they were asked to judge how difficult those spells would be.  What the researchers found is that the bigger the violation of physics required for the spell to work, the greater the effort by the conjurer must be.  The authors write:

People spend much of their time in imaginary worlds, and have beliefs about the events that are likely in those worlds, and the laws that govern them.  Such beliefs are likely affected by people’s intuitive theories of the real world.  In three studies, people judged the effort required to cast spells that cause physical violations.  People ranked the actions of spells congruently with intuitive physics.  For example, people judge that it requires more effort to conjure up a frog than to levitate it one foot off the ground.  A second study manipulated the target and extent of the spells, and demonstrated with a continuous measure that people are sensitive to this manipulation even between participants.  A pre-registered third study replicated the results of Study 2. These results suggest that people’s intuitive theories partly account for how they think about imaginary worlds.

After all, to levitate a frog using ordinary physics has already been achieved.  Frogs, like humans, are mostly water, and water is diamagnetic -- when exposed to a strong magnetic field, the constituent atoms align, inducing a magnetic field of the opposite polarity and triggering a repulsive force.  So it doesn't take any particular violation of physics to levitate a frog, although imagining a situation where it could be done without a powerful electromagnet is more of a reach.

Conjuring a frog out of nothing, though?  This is a major violation of a great many laws of physics.  First, if you imagine that the frog is coming from the air molecules in the space that it displaces when it appears, you have to believe that somehow oxygen, nitrogen, and the trace gases in the air have been converted to the organic molecules that make up living tissue.  Just getting from lightweight gaseous elements to the iron in the frog's hemoglobin isn't possible in the lab -- iron, in fact, is formed in the cores of supergiant stars, and only dispersed into space during supernova explosions.  (Pretty cool that the molecules that make up you were once in the ultra-hot cores of giant stars, isn't it?  Carl Sagan was spot-on when he said "We are made of star stuff.")

So there are different sorts of impossible.  You'd think that once you've accepted that the regular laws of physics don't apply -- that you're in a world where magic really happens -- you'd decide that all bets are off and anything can happen.  But our intuitive understanding of the laws of physics doesn't go away.  We still are, on some level, aware of what's difficult, what's impossible, and what's ridiculously impossible.  The authors write:

[P]eople’s ranking of the spells in all our studies were not affected by exposure to fantasy and magic in the media.  We suggest that the media does not primarily affect what spells are seen as more difficult, but rather that people bring their intuitive physics to bear when they engage with fiction.  That is, in line with previous research on myths and transformation, systems of magic are perceived as coherent to the extent to which they match people’s intuitive theories.  People perceive levitating a frog as easy not because they know it’s one of the first charms that any young wizard learns at Hogwarts, rather young wizards learn that spell first because readers expect that spell should be easy.

 
In his 1893 essay The Fantastic Imagination, the novelist George Macdonald wrote, “The natural world has its laws, and no man must interfere with them …but they themselves may suggest laws of other kinds, and man may, if he pleases, invent a little world of his own.”  It seems people’s little worlds do not stray far from home.

What's especially interesting to me about this study is that being an author of speculative fiction, tweaking the laws of physics is kind of my stock in trade.  I've messed around with time travel (Lock & Key), alternate/parallel worlds (Sephirot), machines that act as psychic amplifiers (Gears), and ordinary people gaining knowledge of the future (In the Midst of Lions), to name a few.  It's fascinating to think about my own writing -- and figure out which of the crazy plot points I've invented were impossible, and which were really impossible.

At least it's reassuring that the evil superpowerful shapeshifters in Signal to Noise fall into the latter category.

****************************************

Different kinds of impossible

Many of us engage in magical thinking -- attributing causal relationships between actions and events that are simply (often accidentally) correlated.  Superstitions are magical thinking; as nice as it would be if you could influence the win/loss ratio of your favorite team by wearing a particular shirt, the universe just isn't put together that way.

Where it gets interesting is that there are different degrees of magical thinking.  A clever piece of research appeared in the online journal PLoS-One last week, carried out by psychologists John McCoy of the University of Pennsylvania and Tomer Ullman of Harvard, which illustrates that even those of us who engage in magical thinking seem to be intuitively aware of how impossible different false causations are.

So we can, like the White Queen in Through the Looking Glass, believe in six impossible things before breakfast.  [Image is in the Public Domain]

The paper, entitled "Judgments of Effort for Magical Violations of Intuitive Physics," asks test subjects to perform a simple task.  First, imagine a world where magic is real, where conjuring a spell could make things happen that are impossible in our world.  Then, they were asked to judge how difficult those spells would be.  What the researchers found is that the bigger the violation of physics required for the spell to work, the greater the effort by the conjurer must be.  The authors write:

People spend much of their time in imaginary worlds, and have beliefs about the events that are likely in those worlds, and the laws that govern them.  Such beliefs are likely affected by people’s intuitive theories of the real world.  In three studies, people judged the effort required to cast spells that cause physical violations.  People ranked the actions of spells congruently with intuitive physics.  For example, people judge that it requires more effort to conjure up a frog than to levitate it one foot off the ground.  A second study manipulated the target and extent of the spells, and demonstrated with a continuous measure that people are sensitive to this manipulation even between participants. A pre-registered third study replicated the results of Study 2.  These results suggest that people’s intuitive theories partly account for how they think about imaginary worlds.

After all, to levitate a frog using ordinary physics has already been achieved.  Frogs, like humans, are mostly water, and water is diamagnetic -- when exposed to a strong magnetic field, the constituent atoms align, inducing a magnetic field of the opposite polarity and triggering a repulsive force.  So it doesn't take any particular violation of physics to levitate a frog, although imagining a situation where it could be done without a powerful electromagnet is more of a reach.

Conjuring a frog out of nothing, though?  This is a major violation of a great many laws of physics.  First, if you imagine that the frog is coming from the air molecules in the space that it displaces when it appears, you have to believe that somehow oxygen, nitrogen, and the trace gases in the air have been converted to the organic molecules that make up living tissue.  Just getting from lightweight gaseous elements to the iron in the frog's hemoglobin isn't possible in the lab -- iron, in fact, is formed in the cores of supergiant stars, and only dispersed into space during supernova explosions.  (Pretty cool that the molecules that make up you were once in the ultra-hot cores of giant stars, isn't it?  Carl Sagan was spot-on when he said "We are made of star stuff.")

So there are different sorts of impossible.  You'd think that once you've accepted that the regular laws of physics don't apply -- that you're in a world where magic really happens -- you'd decide that all bets are off and anything can happen.  But our intuitive understanding of the laws of physics doesn't go away.  We still are, on some level, aware of what's difficult, what's impossible, and what's ridiculously impossible.  The authors write:

[P]eople’s ranking of the spells in all our studies were not affected by exposure to fantasy and magic in the media.  We suggest that the media does not primarily affect what spells are seen as more difficult, but rather that people bring their intuitive physics to bear when they engage with fiction.  That is, in line with previous research on myths and transformation, systems of magic are perceived as coherent to the extent to which they match people’s intuitive theories.  People perceive levitating a frog as easy not because they know it’s one of the first charms that any young wizard learns at Hogwarts, rather young wizards learn that spell first because readers expect that spell should be easy. 

In his 1893 essay The Fantastic Imagination, the novelist George Macdonald wrote, “The natural world has its laws, and no man must interfere with them …but they themselves may suggest laws of other kinds, and man may, if he pleases, invent a little world of his own”. It seems people’s little worlds do not stray far from home.

What's especially interesting to me about this study is that being an author of speculative fiction, tweaking the laws of physics is kind of my stock in trade.  I've messed around with time travel (Lock & Key), alternate/parallel worlds (Sephirot), telepathic, energy-stealing aliens (Lines of Sight), and mythological creatures becoming real (The Fifth Day), to name a few.  It's fascinating to think about my own writing -- and figure out which of the crazy plot points I've invented were impossible, and which were really impossible.

At least it's reassuring that the evil superpowerful shapeshifters in Signal to Noise fall into the latter category.

***********************************

In 1919, British mathematician Godfrey Hardy visited a young Indian man, Srinivasa Ramanujan, in his hospital room, and happened to remark offhand that he'd ridden in cab #1729.

"That's an interesting number," Ramanujan commented.

Hardy said, "Okay, and why is 1729 interesting?"

Ramanujan said, "Because it is the smallest number that is expressible by the sum of two integers cubed, two different ways."

After a moment of dumbfounded silence, Hardy said, "How do you know that?"

Ramanujan's response was that he just looked at the number, and it was obvious.

He was right, of course; 1729 is the sum of one cubed and twelve cubed, and also the sum of nine cubed and ten cubed.  (There are other such numbers that have been found since then, and because of this incident they were christened "taxicab numbers.")  What is most bizarre about this is that Ramanujan himself had no idea how he'd figured it out.  He wasn't simply a guy with a large repertoire of mathematical tricks; anyone can learn how to do quick mental math.  Ramanujan was something quite different.  He understood math intuitively, and on a deep level that completely defies explanation from what we know about how human brains work.

That's just one of nearly four thousand amazing discoveries he made in the field of mathematics, many of which opened hitherto-unexplored realms of knowledge.  If you want to read about one of the most amazing mathematical prodigies who's ever lived, The Man Who Knew Infinity by Thomas Kanigel is a must-read.  You'll come away with an appreciation for true genius -- and an awed awareness of how much we have yet to discover.

[Note: If you purchase this book using the image/link below, part of the proceeds will go to support Skeptophilia!]

Right in the gut

I know I've said it before, but it bears saying again: the strength of science lies in its reliance on hard evidence as the sine qua non of understanding.

I've tried to embrace this outlook myself, insofar as a fallible and biased human can do so.  Okay, so every day I poke fun at all sorts of odd beliefs, sometimes pissing people off.  But you know what?  You want to convince me, show me some reliable evidence.  For any of the claims I've scoffed at.  Bigfoot.  Ghosts.  ESP.  Astrology.  Tarot divination.  Homeopathy.

Even the existence of god.

I'm convinceable.  All you have to do is show me one piece of irrefutable, incontrovertible evidence, and I'm sold.

The problem is, to my unending frustration and complete bafflement, most people don't approach the world that way.  Instead, they rely on their gut -- which seems to me to be a really good way to get fooled.  I'm a pretty emotional guy, and I know my gut is unreliable.

Plus, science just doesn't seem to obey common sense at times.  As an example, consider the Theory of Relativity.  Among its predictions:

• The speed of light is the ultimate universal speed limit.
• Light moves at the same speed in every reference frame (i.e., your own speed relative to the beam of light doesn't matter; you'll still measure it as traveling at 300,000,000 meters per second).
• When you move, time slows down.  The faster you move, the slower time goes.  So if you took off in a rocket ship to Alpha Centauri at 95% of the speed of light, when you came back from your trip you'd find that while twelve years or so would have passed for you, hundreds of years would have passed on Earth.
• When you move, to a stationary person your mass increases and your length in the direction of motion contracts.  The faster you move, the more pronounced this effect becomes.

And so on.  But the kicker: all of these predictions of the Theory of Relativity have been experimentally verified.  As counterintuitive as this might be, that's how the world is.  (In fact, relativistic effects have to be taken into account to have accurate GPS.)

None of which we would know now if people relied solely on their gut to tell them how things work.

Despite all this, there are people who still rely on impulse and intuition to tell them what's true and what's not.  And now a study jointly conducted by researchers at Ohio State University and the University of Michigan has shown conclusively that if you do this, you are more prone to being wrong.

[image courtesy of the Wikimedia Commons]

Kelly Garrett and Brian Weeks decided to look into the connection between how people view evidence, and their likelihood of falling for incorrect information.  They looked at survey data from almost 3,000 people, in particular focusing on whether or not the respondents agreed with the following statements:

• I trust my gut to tell me what’s true and what’s not. 
• Evidence is more important than whether something feels true.
• Facts are dictated by those in power.

They then correlated the responses with the participants' likelihood of believing a variety of conspiracy theories.  Unsurprisingly, they found that the people who relied on gut feelings and emotions to determine the truth were far more likely to fall for conspiracies and outright untruths.

"Misperceptions don’t always arise because people are blinded by what their party or favorite news outlet is telling them," Weeks said.  "While trusting your gut may be beneficial in some situations, it turns out that putting faith in intuition over evidence leaves us susceptible to misinformation."

"People sometimes say that it’s too hard to know what’s true anymore," Garrett said.  "That’s just not true.  These results suggest that if you pay attention to evidence you’re less likely to hold beliefs that aren’t correct...  This isn’t a panacea – there will always be people who believe conspiracies and unsubstantiated claims – but it can make a difference."

I'd say it makes all the difference.  And in the current political environment -- where accusations of "fake news" are thrown around right and left, and what people consider to be the truth depends more on political affiliation than it does on rational fact -- it's more than ever absolutely essential.