Posted by: N.S. Palmer | June 6, 2009

Why We’re All So Messed Up

By N.S. Palmer

“Metaphysics is the finding of bad reasons for what we believe on instinct.”
–F.H. Bradley

It’s not poetic, nor even grammatically correct, but it’s a question that everyone eventually asks: “Why are we all so messed up?”

I can’t presume to offer the complete answer, but I do have a little piece of it to share. It has to do with epistemology.


Epistemology. That’s the study of how we acquire and test our knowledge, as well as how our knowledge is structured and what it means “to know something.”

For example, suppose I say that “I know there is an elephant in the closet.” What exactly am I saying? Common sense gives a threefold answer:

  • I have adequate evidence to believe that there is an elephant in the closet. In other words, my belief is justified.
  • There is, in fact, an elephant in the closet. In other words, my belief is true.
  • I believe that there is an elephant in the closet. In other words, I really do believe what I claim.

Those three criteria are embodied in the usual definition of knowledge as “justified true belief.”

However, most situations are far more complicated than just having an elephant in your closet — however troublesome that might be. If you suspect that you have an elephant in your closet, it’s easy to find out. Just open the door and look. Then you know.

Harder than having an elephant in your closet

But consider a different kind of situation. This is a puzzle given by New York Times columnist John Tierney:

A puzzle given by New York Times columnist John Tierney.

The printed version of Tierney’s puzzle, scanned from a newspaper clipping.

My mother presented me with that puzzle after our weekly family dinner last Sunday night. Such are dinners at Mom and Dad’s house: It’s like being on a quiz show, except that you get a home-cooked meal and there are no cash prizes. Sitting at the dinner table are three medical doctors, an art historian, an entomologist-biochemist who looks like movie actress Anne Hathaway, a computer expert, a golf pro, a four-year-old prodigy, and me. Every one of us secretly believes that he or she is the smartest, and is determined to prove it. Over chicken and pasta, the conversation leaps from obscure diseases to medieval manuscripts, the life cycles of bugs, computer software, and the latest doings of Tiger Woods. Over dessert, an impromptu lesson in Spanish, German, or Hebrew for the boy. But I digress.

(Note: If you want to have a go at Tierney’s puzzle on your own, stop reading here. I’m going to start talking about the solution.)

My mother wanted to know the solution to the puzzle, which The New York Times columnist avowed had stumped him. She gave the puzzle to me because I’m the mathematics geek of the clan, am single, and therefore have the most free time. The figures looked at first glance like ancient numerals, perhaps from the Babylonian, Chinese, or Indian number systems. I wasn’t sure which. When I got home and looked them up, I found that they weren’t ancient numerals.

Then I latched onto the last word of the hint in the puzzle description: “knots.” An area of mathematics called “knot theory” investigates, among other things, what happens when you manipulate knot-like geometrical figures in certain ways. I didn’t know much about knot theory, and it wasn’t a perfect match anyway, but it was close enough for me to theorize that:

  • The second and third figures on row 1 were rotations and twists of the first figure.
  • The fifth figure on row 1 was a rotation of the fourth figure.

I concluded that there were two series of three figures each. For the answer, I needed a figure on row 2 that was a rotation or twist of the fourth and fifth figures on row 1:

  • The fourth figure was a triangle pointing downward.
  • The fifth figure was a horizontal line, as if the triangle had been rotated toward us in 3-D space and we were seeing it from the side.

Rotating the figure one more time, toward us in 3-D space, gives us a triangle pointing upward. That’s answer (c), which I triumphantly emailed to my mother and the other family members who had been present at dinner.

The Web to the rescue

I decided to check the Web to see if Tierney had posted an answer. However, the Web version of his puzzle included some vital information that wasn’t in the printed clipping:

The Web version of Tierney's puzzle.

The Web version of Tierney’s puzzle.

The most important new information was that row 1 consisted not of five figures, but of four figures and a blank. And the horizontal line in fifth place, which didn’t look too different on the newspaper clipping, was clearly a different color on the Web page.

I took another look at the hint. It says something about mirroring, also part of knot theory. The next solution was easy. The first figure on row 1 is the numeral 1 paired with its mirror image. The second is the numeral 3 paired with its mirror image, the third is 5, and the fourth is 7. So the series is 1-3-5-7, consecutive odd numbers. The next figure should be the numeral 9 paired with its mirror image. And that makes the answer either (a) or (d), assuming that Tierney drew the nines in an eccentric way.

Well, duh. That wasn’t hard.

One of my brothers was unimpressed both with the puzzle and with my two solutions. He emailed three more puzzles that he thought might be more challenging:

Steve's puzzle #1. Is it too hard for the SATs?

Steve’s puzzle #1. One possible answer is (D) because the line of figures moves from left to right, and (D) is the only one to the right of the last figure on row 1.

And a second puzzle:

Steve's puzzle #2

Steve’s puzzle #2. One possible answer is (A) because all the other objects are partly or entirely black.

And a third puzzle:

Steve's puzzle #3

Steve’s puzzle #3. One possible answer is (A) because all the others are waving their hands.

Yes, Virginia, there really is a point to all this.

The point of the discussion is this. Tierney’s puzzle deals with a very limited set of simple geometrical facts. It asks the reader to explain those facts and to predict another very simple geometrical fact. But even in that simple situation, different answers are possible. The answer you get depends not merely on the evidence, but on your background and interests. It also depends crucially on which pieces of evidence catch your attention.

The same applies to my brother Steve’s puzzles. For most people, the answer to puzzle 1 is obviously “B” because it would complete a series of progressively larger squares. Likewise in puzzle 2, the spatula is the only item that isn’t an animal, while in puzzle 3, the television character of “Barney” is the only item that isn’t a dictator.

However, other interpretations are possible. In puzzle 3, for example, the Soviet dictator Josef Stalin “does not belong” because he’s the only person not waving at least one hand. The fact that different solutions are possible makes it hard to design such puzzles for standardized tests, such as college aptitude tests. What such puzzles really measure is test-takers’ ability to find the solution that the test-designers expect. Truly creative people sometimes score badly on such tests because their solutions, though logical, are unexpected* and therefore “wrong.”

And all those puzzles represent very simple situations with very limited data sets that you evaluate to get a solution. Real-life situations involve hundreds or thousands of pieces of information that are connected, or not, in ways that are often unclear.

Real-life situations are even more prone to have multiple, equally-logical explanations. Which explanation you choose depends on your prior assumptions about the situation, about people, and about how the world works. It depends crucially on which pieces of evidence you spotlight, which pieces you ignore, and which pieces you outright reject. For example:

  • Conservative economists who look in good faith at official economic data think it shows clearly that we must give more tax cuts to the rich and get rid of regulations that inconvenience giant corporations.
  • Liberal economists who look in good faith at official economic data think it shows clearly that we must increase taxes on the rich and more aggressively regulate corporate misbehaviour.
  • In the Middle Ages, doctors liked to treat illnesses with leeches. They noted that people treated with leeches often got better. But so did people not treated with leeches, a fact which the doctors’ theory caused them to discount in evaluating the merits of leech therapy.
  • Scientists who contemplate the symmetry and elegance of the physical universe often get a feeling of transcendence. They conclude that it’s because the physical universe is all that exists and is just wonderful.
  • Christians who contemplate the works of God and read the New Testament get a feeling of transcendence. They conclude that it’s because God is infinitely good, Jesus is right there with him, and that only Christians can go to Heaven.
  • Muslims who contemplate the works of God and read the Qu’ran get a feeling of transcendence. They conclude that it’s because Allah is running things, Mohammed was right, and that only Muslims can go to Heaven.

A little intellectual humility

What it really comes down to is this: Most of what we consider our “knowledge” either consists of, or is based on, WAGs (wild-a**ed guesses) that we simply follow until something better comes along.

It’s relatively harmless until our WAGs combine with our arrogance to make us demonize and try to destroy people who disagree with us. What we need is a touch of humility: The awareness that however sure we are that we’re right, we still might be wrong. We should accordingly proceed with caution, giving due respect to the viewpoints, rights, lives — and the feelings — of others.

Of course, I might be wrong about that. But I’m sticking with it until something better comes along.

*There is a relevant story about Nobel laureate physicist Niels Bohr, a pioneer of 20th-century atomic theory. When Bohr was a student, a professor supposedly gave the following problem on a test: “Use a barometer to determine the height of a tall building.” Obviously, the professor was expecting students to solve the problem by measuring the atmospheric pressure at the bottom and top of the building, but Bohr had a different solution. “Go to the top of the building. Tie a long piece of string to the barometer. Lower it to the ground. Measure the amount of string that you lowered over the side of the building, then add its length to the height of the barometer. The sum is the height of the building.” It’s not the expected answer, but it’s a correct answer.

Copyright 2009 by N.S. Palmer. May be reproduced as long as byline, copyright notice, and URL ( are included.


  1. One can also triangulate with the barometer using shadow length and ratios.
    One can also use the barometer as the measuring device (a known) and count in increments up the side of the building (unknown). Thus, for example, the building could be 17,156 barometers tall.

    And I’d like to point out an alternate to your conclusion. You state “The awareness that however sure we are that we’re right, we still might be wrong.”
    The awaremess that however sure we are that we’re right, does not mean that others are not also right.
    (Not so much that we’re wrong, but that, like in your tests, more than one answer is possible, thus making each answer right, even if the answer does not match ours or equate to the same result.)

    • Right you are, though it’d be a rare hardware store that stocked enough barometers to measure a building by your second method. πŸ™‚

      Your final point is both perceptive and important. Statements can be inconsistent without being contradictory. If I say that X is black and you say that it’s blue, we can both be wrong: it might be pink. But if I say that X is black and you say that “it’s not true that X is black,” then one of us must be right even if X doesn’t exist. We often make the mistake of assuming that if one explanation is wrong, then the next one we find must be right. It’s a psychological quirk of how our minds work.

      Something similar often happens to novice chessplayers. They think for a minute or two about their next move, ruling out various possibilities until they realize that they’re running short of time. Then, they make the next move that occurs to them, on the assumption that if it’s not one of the “wrong” moves they’ve just been contemplating, it must be a “right” move.

  2. I knew if I waited a while you’d produce something fascinating again. πŸ™‚ The work you did trying to figure our Tierney’s puzzle was wonderful to read. I felt such envy as I read how you tracked down the right answer, something I could only have dreamed of. And then, to turn around and show us it could be something else… what a revelation.

    Something like that happened to me in a university literature course. There was a line of poetry that went, “I told my love/I told my love”… the professor explained that “to tell” could also mean to quantify something. So was the poet telling his lover of his love per se, or that he had measured its relative value? And suddenly it came to me, and I asked, “Couldn’t it be both?” And I realized that’s where he was steering us. That’s when I first became aware of the real subjectivity of “meaning”.

    It’s really something to see it so visually demonstrated here. It took me a while but I got where you were going by the time Stalin was sitting on his hands. But then, it took me to my third essay to stop arguing against Deconstructionism in that lit course. πŸ™‚

    Thanks for this. πŸ™‚

  3. Thanks so much for your kind words!

    However, knowing you even as casually as I do, I think that you’re being far too modest about your own ability. You could easily have solved Tierney’s puzzle, though maybe without the detour into ancient number systems. That was fun even though it didn’t lead to a solution.

    That sounds like an interesting lit course you took. Do you happen to remember what poem your quote came from? I don’t know that one.

    • I had to look it up… it’s Never Seek to Tell Thy Love, by William Blake: It was an interesting course; one of only a handful of courses I took in university that really made an impression on me. The professor was great. He had an Oxford accent and it was clear he had a ball teaching and unwrapping big ideas for us. The last I heard he was teaching somewhere in Continental Europe. It was a privilege to be in his orbit for a year.

  4. It also depends on what your definition of elephant is. πŸ™‚

  5. I fiond it interesting that you steer towards the ‘wrong’ in inconsistent. I guess more to the point of what I was trying to make, using your eaxmples is that Barney and Stalin are both right answers, therefore there is oftentimes more than one ‘right’ answer.
    (In your puzzles above, so long as a valid logical explanation can be made — so Hussein doesn’t belong because he’s the only one with a crowd or distinct bacground; Barney doesn’t belong because he’s the only one that’s more than a bust portrait; Barney doesn’t belong because he’s the only one without a mustache; Barney’s the only one that’s naked (for a dinosaur); Barney’s the only one with contemporary relevancy; Stalin’s the only one without hands; Stalin’s the only one I’ve never actually heard a recording of his voice; Barney is the only one distinctly lit from the left; Hitler is the only one with a microphone)

    So if you said Barney, and your brother said Stalin, both of you are right even though your answers do not match. Neither of you are wrong. You know with your awareness of being sure that you are right… and so is he.

  6. Reuven makes a point I wouldn’t have gotten at one time. Given a set a data, the “right” answer really comes down to what question it is you’re asking about it — and that includes all the implied ones that the questioner himself may not even be aware of. I think that’s one of the reasons why people in different civilizations are so prone to qualifying one another as “evil”. It’s easy to assume one’s own standards and presumptions are universal, and so anyone straying from them must be doing so willfully and contrarily. Sadly, in our species, that seems to make it so much easier to walk into a crowd of them wearing a bomb, or guide missiles into their homes from hardened bunkers on the other side of the planet.

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