Don’t trust your intuitions, they will betray you

A video at Aeon well worth checking out on what wrapping a rope around the Earth reveals about the limits of human intuition:

If you tied a rope tight around the Earth’s equator and then added a single yard of slack, would the extra material make any noticeable difference to someone standing on the ground? Yes, actually. The answer comes as a surprise to most people, but the additional bit of rope raises it high enough off the ground for our eyes to easily discern it, and our feet to easily trip over. That fact might seem trivial, but the early 20th-century philosopher Ludwig Wittgenstein believed that this chasm between human intuition and physical reality revealed something important about the fallibility of our thinking. After all, if something that seems obvious to almost everyone can be totally false, what else might we be wrong about?

See the video at Aeon (It’s about 6 minutes long.)

I have to admit that I didn’t buy this at first.  Even knowing full well that our intuitions are not to be trusted, I could not accept it until I did the calculation myself in terms of the whole Earth.

The Earth is 1,577,756,568.96 inches in circumference.  Its radius is about 251,235,122.45 inches.

Add 36 inches to the circumference, making it 1,577,756,604.96 inches, and you make the new radius about 251,235,128.18 inches, or 5.73 inches larger.  

Adding a yard of rope raises it almost 6 inches off the ground, all around the world.  Despite the mathematical proof, that still seems utterly wrong to me.  It’s like one of those visual illusions that, even after you know it’s an illusion, you can’t stop seeing it.

In both science and philosophy, we have to be very careful about our intuitions.  Often they will lead us astray.

via Eric Schwitzgebel

38 thoughts on “Don’t trust your intuitions, they will betray you

  1. It can also be seen as the power of informed and trained intuition.

    Circumference, as we know, is: C=2πR. Which is also: C/2π=R. We also know that 2π is about 6.3, so now we have: C/6.3=R.

    Which means radius is about 1/6 of the circumference. A yard is 36 inches, and 36/6 is six inches. QED

    (This mainly surprises people because we’re so not like computers. 🙂 )

    Liked by 1 person

      1. I do not understand that reply.

        My point was that uninformed and untrained intuitions about geometry are as likely to miss the mark as any ignorant opinion about something is.

        But if one knows the subject matter, one’s intuitions about it are more likely to be on point.

        As with this. I know circumference is roughly six times the radius. That’s a basic fact for me. Another way my intuition sees it is:

        C+a=2πR1 (and we’re interested in that R1 compared to C=2πR0).

        So divide both sides by 2πR to give: (C/2π)+(a/2π)=R1.

        Replace the left-most term: R0+(a/2π)=R1.

        The new radius is the old radius, plus the new amount, a, divided by . (That’s how I intuited it when I first read the post. What I wrote in my comment above was the double-check-my-thinking version.)

        So the more familiar one is with a subject — which includes knowing where the minefields are — the more useful one’s intuitions can be.

        To your point we need to be wary: Oh, most definitely, yes!

        It’s a bit like why bomb techs have a chance of living whereas the untrained usually blow themselves up immediately.


        1. Wyrd,
          My reply was to the verbiage you had in parenthesis. (Which is why my reply itself was in parenthesis.)

          On all the rest, I didn’t reply to it because I think most of us already understood the formulaic relation between radius and pi. If you’re saying that your mathematical prowess let you see the issue with clarity, good for you. But if you think that will always be true, I fear you missed the main point.


          1. “My reply was to the verbiage you had in parenthesis.”

            Ah. I guess I still don’t really understand the connection between game software and people not being computers. Why would anyone think you could use those? I just didn’t make the connection, sorry.

            “I think most of us already understood the formulaic relation between radius and pi.”

            I thought the point of your post was that most of us don’t have the necessary intuition? All I’m saying is that experience and training make one’s intuitions more acute.

            “But if you think that will always be true, I fear you missed the main point.”

            No, I agree there’s a minefield and caution is called for. I’m saying there’s a difference between a trained bomb tech and those with no experience.

            Intuitions do become more trustworthy with experience. That’s the whole point of education. Or deep learning neural nets, for that matter.


          2. “I guess I still don’t really understand the connection between game software and people not being computers. ”

            We actually discussed this a while back, so I probably took it too much for granted that you remembered. Sorry. Just as Tetris is useless for math despite being computation, the fact that human minds aren’t optimized for math says nothing about whether they are computation.


  2. Yes, that’s one of my favourite examples of non-intuitive mathematics. Also, almost anything to do with probability is counter-intuitive, which is why people are so wrong when thinking about luck and chance, and why we insist on seeing patterns that don’t exist.

    Liked by 1 person

  3. Wow, where can I buy some of this super rigid rope? Amazing stuff! It goes from tightly stretched to completely relaxed with less than 2.28 * 10^-8 linear shrinkage!


      1. [donning obnoxious contrarian hat]

        Actually, Mary’s room is quite doable. All you need is a sodium lamp. They actually have (or did have) such a room in the Exploratorium In San Francisco. The room has a full color photo on the wall, flashlights, and a jelly bean dispenser. The photo looks black and white (well, black and orange), but when you shine a flashlight on a spot you see the full color. And you can’t tell what color the jelly beans are until you take them out of the room.


        Liked by 1 person

        1. Mary only seeing orange and black doesn’t seem to have the same oomph as Mary seeing only grey scale. And learning all about the physics of color seems like it would be challenging under the low light of a monochrome sodium lamp.


  4. My hunch for why this is so hard to believe is that it is an example of a non-linear problem. We picture this problem as a rope tracing a circle, and we ask ourselves, if the radius increases by 1 yard, how much bigger is the circle? Intuitively, we don’t characterise circles by their radius, but by their area, and this is the crux of the problem. We cannot convince ourselves that a circle of radius r+1 can have a significantly larger area than a circle of radius r, since r is already so huge. Adding 1 seems to make no difference.

    But the area of a circle is pi*r*r. So the difference in area is pi*(r+1)*(r+1) – pi*r*r, which is 2r+1. So if r is large, then the change in area will also be large. It’s the r*r term that is causing us the problem.

    Liked by 1 person

      1. “I meant to say that we cannot convince ourselves that a small change in the length of the rope can create a significantly larger area!”

        That’s definitely my struggle. Along the lines of the example the video gave, it just seems like pouring a glass of water into the ocean. It seems like those additional 36 inches should get utterly lost in the immensity.

        But maybe the issue is in thinking only about the difference between the circumference of the Earth and the new circumference of the rope. Viewed in absolute terms, the additional radius is extremely small, an additional 0.000002%. It’s only because the difference is significant to us viscerally in relation to the size of our foot that it seems more significant than it is.

        Still, it remains astounding!


    1. Interesting hunch. I don’t have that area linkage — for me the linkage between circumference and radius is much stronger (just due to working with 2D circles and their associated math a lot). Maybe that’s why I don’t experience that sense of frisson that throws most more casual observers?


      1. My intuition is that most people characterise solid objects by their volume and 2D shapes by their area. If I chuck a rock at you, do you assess its threat by its radius, or how heavy the damn thing looks? (Don’t worry, I’m not planning on throwing any rocks at anyone, Wyrd. I just liked the example.)


        1. “My intuition is that most people characterise solid objects by their volume and 2D shapes by their area.”

          I can’t speak for most people, but for me parameter is a crucial concept with 2D shapes and I don’t seem (from what I can tell introspectively) to focus much on area. (I’m quite willing to concede I’m a weirdo.)

          It’s kind of why the pizza ordering thing was a thing on Twitter. In that case, people are more focused on diameter.

          “If I chuck a rock at you, do you assess its threat by its radius, or how heavy the damn thing looks?”

          Neither. I dodge any unknown object out of principle because I know I can’t possibly assess its properties in the time allotted. It’s a corollary of a central technician’s principle: If you drop something, never reflexively try to catch it — let it fall!

          Because techs drop things like sharp knives or hot soldering irons…


  5. I posted this under the wrong title Mike, I’ll try again.

    “The discovery of truth is prevented more effectively, not by false appearance things present and which mislead into error, not directly by weakness of reasoning powers, but by preconceived opinion, by prejudice.”
    Arthur Schopenhauer

    Anyone interested in solving the riddle I posted on Schwitzgebel’s blog about the chain?


      1. Mike,
        I don’t think the rope or my chain example have anything to do with intuition as such. It has more to do with an intellectual construct which then becomes a mental block. A mental block is a derivative and/or the consequence of the delusions that we construct of how we perceive the world to be. And those intellectual constructs, right or wrong, are the very mechanisms which provide stability and meaning. Any time a closely held paradigm is shattered, it de-stabilizes the very foundations which give us a transient sense of control. Because without out at least sense of control, there is no sense of self.

        That is exactly why individuals will choose to believe a delusion even when the evidence is overwhelming. It’s all about control. There are two words that underwrite our primary experience, “power” and “control”. Every scenario that one could possibly imagine about human behavior corresponds to those two simple words.


        1. Hmmm. I don’t know about “control” and “power”, but it seems to me that intuitions are mental short cuts honed to pre-existing patterns. Many of these short cuts are also embedded in our instincts. With repetition we can learn to override those instinctual short cuts with new habitual ones.

          As long as those patterns hold, the short cuts remain useful. Where things go wrong is when we attempt to use the mental short cut somewhere outside of the domain in which it was honed, or when the patterns that it was honed on shift underneath us. Often letting go of short cuts that have seemed fruitful for a long time (or that just conform to our biases) is very hard.

          Not sure if that gets into what you mean by power and control though.


          1. “…Not sure if that gets into what you mean by power and control though.”

            Absolutely counselor. Your “justification”, just like everyone else’s justification for a given model, be it right or wrong, goes right to heart of what I mean by power and control. In this particular example; for Mike, the very notion of power and control being the underlying fundamental of human behavior challenges your current model, which in turn destabilizes the self-model. Just like the example of the rope destabilized Mike, which in turn makes Mike question himself.

            Hey, I’m no psychologist, I just play one on your blog…


          2. “…is there any data that would falsify your proposition?”

            Unfortunately, no; and I really do loathe that answer Mike. All of the data supports my proposition. For any discrete system, the empirical experience of power, and the subsequent expression of that power is intrinsic to control. Control is the singular, targeted objective of power, not some of the time, not most of the time, but every time. I think that hypothesis is self evident and is clearly demonstrated throughout all of human behavior. Even the most altruistic expressions of power, or the submission to a greater power than self are driven by the need for control, a control which is essential to and reinforces the identify of the self-model.

            Make no mistake Mike, control is not a dirty little word, control is absolutely essential because it is coextensive as one with the identity of the self-model. If one wanted to harden up the soft sciences, start with this proposition.


  6. “If you tied a rope tight around the Earth’s equator and then added a single yard of slack, would the extra material make any noticeable difference to someone standing on the ground? Yes, actually.”

    Actually, no. But, like you I am quite surprised that increasing the radius by 6 inches of a circle the size of earth would only lead to an increase of 1 yard in its circumference. I would assume that your mathematics are quite correct, but the conclusion that intuition should be doubted in most situations is not correct either. The simple reason is that a mathematical proof of an abstract question does not usually apply to a question of practical reality. One has to be very careful. The question originally posed was what would happen if I had tied a rope around the earth. That is a theoretical question about our practical human experience. The intuitional answer seems much better to me than the abstract mathematical answer.

    I am certain that if I had taken the trouble to tie a rope around the earth, the length of that rope would depend on the tension that I had applied to it and the minor adjustments that would have to be made for trees, rocks, buildings, oceans and mountains that might come in the way. The idea that one extra yard would raise the rope by six inches in an obvious way seems to be quite nonsensical.

    When I saw the post on Aeon of Wittgenstein’s question, I did not bother to look into it. I generally do not not find Wittgenstein’s approach to questions very helpful in my life. The bottom line for me is that reality is a lot more complicated than the aphoristic abstractions that Wittgenstein is famous for.

    Perhaps a better way to look at it is to view intuitions as our first drafts of reality. How we try to solve the problems inherent in it is up to us. Science, logic, philosophy and mathematical abstractions are all relevant.

    Liked by 1 person

    1. I think intuitions work fairly well in day to day matters, although in this modern age where we all exist in an environment very different from the one we evolved in, we still have to be on guard. I’ve been in many business discussions where someone insisted that their intuitive understanding of the situation was correct, only to be contradicted by the actual data.

      But the further we move away from those day to day matters, the more we have to be on guard against just taking our intuitions as correct. “First drafts of reality” does strike me as a good way of looking at it since everything about the first draft is subject to revision.


    2. ‘Zactly. That was the point of my snide remark about super rigid rope.

      Moral: when your intuition conflicts with your calculation, don’t just assume that the intuition was wrong. It might reveal the falsity of some assumptions that went into the calculation.


  7. Yes, human intuition is a problematic thing, which is why we have more formal forms of reasoning and actual math. Sometimes doing the math or going through the steps isn’t about being more precise, but simply about bypassing our intuition.

    This is also why I’ve been leery of most thought experiments, especially if the conclusions “accord with intuition”.

    Liked by 1 person

  8. Maybe this will blow your minds even more.

    For grins I pulled out my arbitrary precision calculator…

    Earth’s Orbital Radius: 93,500,000 miles

    Earth’s Orbital Circumference: 587,477,826.221291335… miles

    Three feet: 0.000568181818 miles

    Increased Circumference: 587,477,826.221859517… miles

    Increased Radius: 93,500,000.000090428944 miles

    Difference: 0.000090428944 miles

    Or 5.729577 inches.

    The actual radius doesn’t matter at all.

    Because: 36 inches ÷ 2π = 5.729577 inches.


    1. You can go the other way as well. Take a circle with the radius of a hydrogen atom. Let’s call that … almost zero. Splice in the one yard and you get a radius of 5.73 inches (rounding), which is 5.73 inches more than the original radius (rounding).


      Liked by 1 person

  9. Hi Mike,
    There are a number of interesting threads that come to mind from this post.

    First, this question in my opinion has little to say about the validity of intuition—though this term too requires some definition. But what we’re dealing with is the supposition of a change on a human scale, of one yard, and it’s impacts on a scale we almost never deal with in daily life—the radius of a planet. Few of us have any intuition about planetary scale measurements.

    For starters I’d say we need an intuition about our intuition before we can assess whether or not our intuition is reliable. For instance, if you took 1,000 engineers who design bridges, and present them with a connection detail between two structural members, or gave them a design span and loading and the size of the primary steel beams, I’d take their intuition every day and twice on Sunday compared to the general public. For obvious reasons. They’ve reviewed hundreds of design calculations, if not thousands, and so have a “calibrated intuition” for such things. None of us have a “calibrated intuition” about the impact of human scale dimensions on planetary scale applications, so it’s sort of a ridiculous question. We should generally trust our intuitions about things we work with regularly, and should ideally have enough self-doubt to question our intuition about things we don’t work with regularly. Which is why I suggest we need an intuition about our intuition, which ideally all of us ought to possess, but in practice only few of us actually do.

    The question though is absurd. A rope stretched around the equator would not stretch around the equator, but would saturate with water, sink, and create a series of chords rather than a true circumferential measurement of the planet. It would also have to traverse hills, trees, buildings, lakes, roadways, bus stations, mountains, valleys, skyscrapers, skate board ramps, and all kinds of topography that our intuitions rightfully tell us make the question a poor one. And of course, the increase or decrease in tension on any fiber known to man over such distances would make the addition of 36 inches of material moot. We could probably gain miles of slack rope just by increasing tension. And lastly, no rope would magically levitate and hover over the surface of the planet given an additional yard of material without a very extensive system of supports. We could add a hundred miles of rope, and it would just lay on the ground, or the bottom of the ocean. So in a certain sense, our intuition is correct: a small increase in the length of the rope is meaningless in practical terms.

    I would argue that we should trust our intuitions in areas with which we have both empirical and theoretical familiarity. But I would add that there is no magic to mathematics. We can just as easily state absurdities in the language of mathematics as we can in the language of English, or Spanish, or whatever you prefer. Mathematics holds no special place in the universe of languages. I would also argue that mathematics provide limited value to intuition without a real world to gauge our mathematical expressions against. All sorts of mathematical expressions are grammatically correct but physically absurd. To suggest that ideas mathematically expressed are more accurate than ideas expressed in the English language is incorrect. It all depends upon context and familiarity. Many of the mathematical sentences we take for granted have been tested in the real world thousands and thousands of times.

    So, I would argue intuition is quite valuable, so long as it is applied in a field with which one is familiar on both an empirical and theoretical basis. The collapse of the Hartford Coliseum is a great example of failed mathematical reasoning, for instance. But engineers who size piping or duct systems on a regular basis inherently known that a small increase in diameter can provide a disproportionate increase in flow capacity. People who don’t work regularly with this type of phenomena will guess incorrectly.

    Intuition is a function of what you interact with regularly, I think. And the grand problem is that people develop insight in one particular area and begin to think they’re intuition is applicable generally, which is never the case.


    Liked by 2 people

    1. Hi Michael,
      I pretty much agree with everything you say. I think the main thing, as you note, is to watch our intuitions about our intuitions. I might have an intuition about what makes a robust bridge, but no one should trust it because I’m not a bridge engineer. Although my intuitions about securing an enterprise IT system would be much more credible.

      That said, I’ve been in numerous situations where even someone experienced in a certain area had intuitions about what was happening that were just plain wrong. Sometimes long held patterns shift, and if we’re not cognizant of it, our intuitions shaped under the previous pattern are going to start failing us.

      Which tells me that while our intuitions are useful, for matters where an accurate assessment of the truth is crucial (such as the foundation of bridge supports) or that are well outside our day to day experience, intuition should be a first draft understanding, but one we have to be willing to modify. This is particularly true for philosophical or scientific matters, where intuition has historically often led us astray.


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