(Archive Question of the Week)
One of my favorite questions, from 2001, asked about how to convince a skeptical friend, when a clear mathematical result goes against their intuition. Why should they believe the math? It led me into thoughts about the relationship of intuition to math, whether (and when) math can be trusted, and different directions from which to approach an example so as to make it convincing rather than just amazing.
Here is the question, from Archer:
Mathematics and Intuition This problem may have more to do with math pedagogy. It is, however, a vexing problem, and experience from other teachers may offer some help. Certain "puzzlers" in mathematical recreations defy our sense of experience, leaving you wondering if the answer to a problem can really be true. One example is the well-known birthday probability problem, and the answer that 23 people in a room leads to a 50/50 probability that two will share the same birthday. Another is the problem of adding, e.g., "only" one meter to a rope around the Earth, and determining that the "gap" created between the lengthened rope and the Earth is about 16 cm. How can it be that adding such a short length to the rope will result in such a large gap? Of course, it's easy to show using simple algebra that the result is a pure value ("amount of rope added"/2pi) independent of any circumference, so that whether you do it around a superball or around Jupiter the result will be the same. My question is how to handle the intelligent non-believers. I showed this problem to a friend. She had no argument or confusion over the solution. Her responses were along the lines of, although I understand the algebra, how do you really know? Has anybody actually measured it? Aren't there situations in which a mathematical proof leads to a result that, investigated empirically, proves to be false? (Of course there probably are, or at least the possibility exists that there could be.) She just refuses to believe that intuition can be that wrong, and until somebody actually goes out there and does it and measures it, she will not be convinced. One could resort to examples. Do it with a pill bottle, then a coffee cup, then a round table top, ... if the measured results are all the same, doesn't that suggest that the circumference doesn't matter? Oh, but those are just a few instances. Such induction can't prove that it will hold for larger celestial bodies, can it? How do we respond to those who question that what seems certain might not be? And is it not a good question, by the way, to ask whether it (certainty) might not be? Are we really justified in asking others to toss aside their intuitions in favor of a few sensible jottings on a piece of paper?
Can math be wrong? Why should we believe it in situations where no one can actually check it? And how can we convince someone with that sort of attitude?
I’ve sometimes been troubled by presentations of math that focus on these unexpected results, but do it in such a way that the reader or viewer may well come away thinking that math is nonsense! Our goal must be to show why the result can be believed, not just to show what looks like magic. Archer wants to be able to do this.
Intuition can be wrong
I started by agreeing with him, and sharing our experience with such doubts:
We certainly have some experience dealing with this sort of "unbeliever," as you can see from our FAQ! People have trouble believing that 0.999... = 1, that -1 * -1 = 1, and so on; our usual approach is to give them a variety of explanations and hope that one of them might get through. And it's not just untrained people who have this problem; mathematicians have stumbled over their intuition in the past, as to whether numbers can be irrational, whether negative or imaginary numbers make sense, whether there are as many integers as rational numbers. We've learned through such experiences not to trust our intuition. The first thing we have to recognize is that our intuition can be wrong. Mathematicians (and the rest of us) need a healthy dose of humility, because this happens all the time. I suppose one of the benefits of studying math beyond mere arithmetic is that it can teach us not to trust our assumptions, or even what seems like sound reasoning, but to look closely at the logic behind what we believe. What seems true, not only in math but in all of life, may not be! That's just part of growing up.
My reference to mathematicians stumbling is an important part of math history. The Pythagoreans (followers of Pythagoras, who were something like a religious cult) assumed all numbers were rational; the proof that this was not true caused Euclid’s famous Elements to do a lot of extra work to avoid needing that assumption. And negative numbers were being questioned by mathematicians well into the 1800’s – something I remind students of when they struggle with them. But this is true of every area of life – many ideas in science were initially considered nonsense, and conspiracy theories today are often built around a refusal to believe something surprising (round earth, anyone?) even after it has been demonstrated.
Math can be wrong (sort of)
Second, we have to be convinced that math really tells the truth. It's often been pointed out that airplane pilots flying by instruments can be deceived by their own senses into thinking the plane is upside-down in a cloud, when it is really right-side up. If they try to "right" it, they will be in trouble. So they have to learn that their instruments really are reliable. Similarly, if we don't have reason to believe logic, it will not be able to convince us that our intuitive answer is wrong. Of course, one problem here is that math often _doesn't_ tell the truth - about the real world, that is. Math is based on reasoning from stated premises (axioms); as long as those are true, and we don't make mistakes in our reasoning, the results have to be correct. But those axioms deal with an ideal world, not the real one in which lines are made of atoms with a finite size, "space" may be curved by gravity, and so on. So it's easy to find cases where math gives a wrong result - not because the math itself was wrong, but because it was applied to an incompletely understood reality, or one that differs in small but important ways from our assumptions.
As I have stated elsewhere, the truth of math is not quite the same as the truth of the physical world. Math is perfectly correct when its assumptions are correct; but we don’t always know the right assumptions to make when we apply it, or we deliberately simplify our assumptions to make it possible to do the work. So sometimes, math does yield inaccurate results. Skeptics need to learn to distinguish between different kinds of errors.
The application of math to the real world is based on induction: we try something repeatedly and see that, yes, our calculations about circumference do work in the real world, so the assumptions on which they are based must be accurate. If we measured big enough circles, we would find that relativity makes it not quite work right; that means that the world doesn't quite match the Euclidean geometry on which our calculations are based. But induction does show us that it is a close enough approximation in normal cases. Math itself is not inductive, but deductive. Having accepted that the world is reasonably Euclidean, we have to accept the results of the deduction that the radius always increases by the same amount. And that's why math is useful: it makes it possible to find answers without having to check every possible case.
Here we start to deal with Archer’s specific example. On one hand, we don’t have evidence to conclude that the entire universe follows Euclidean geometry; in fact, it doesn’t. (I could also have mentioned that circles measured on the surface of the earth don’t follow C = 2πr, because of curvature.) But on the other hand, measurements can be made on the scale of the earth, and they do work. So some skepticism about the applicability of math is valid; but the “rope around the earth” would in fact come very close to the calculations.
Why the right thinking is right
So I think the best thing to do is not to focus on testing the actual solution to this problem, but to build confidence in the mathematical methods by explaining the reasoning in ways that make intuitive sense. An added benefit is that, in analyzing WHY the math does what it does, we can gain a better understanding of the whole problem. Let's try that for your example problem.
I won’t copy all the details of what I said here; but I gave a series of approaches to the “Rope around the earth” problem:
- Do the algebra with symbols rather than specific numbers, to show that it is valid, before using the large numbers that make it look surprising. (I might have added that it is often best to first use specific small numbers, because many students struggle with variables; that work can then support the version with only variables, which in turn supports the large-number surprise.)
- Check the algebraic answer, with specific numbers, to show that it makes mathematical sense.
- Look at the formula for circumference, showing that it is proportional to radius, so adding radii adds circumferences. Do this with pictures as well.
- Illustrate with a simpler problem, in this case using squares, where we can see visually why it is true. This can be extended to other regular polyhedra, and enough of those should convince you that it will be true also of a circle.
Why the wrong thinking is wrong
Now there's one final way I can see to make the answer seem reasonable: analyze why the wrong answer seems right, and correct the underlying misunderstanding. In this case, I think we are used to proportionality, and figure that adding a relatively small amount to the circumference should make only a small change in the radius. But that's exactly what happens! Your 16 cm height is a very small amount _relative to the radius of the earth_; in fact, as I've shown, it is proportional to the small change in circumference. It only seems large because we're focusing on the height above ground, rather than the distance from the center of the earth.
This, I think, is the most interesting part. Our intuition recognizes proportionality, but focuses its attention on the wrong things. Let’s state this with specific numbers. The radius of the earth at the equator is about 6378 km, and its circumference is therefore 40,074 km. (You’ll find other numbers in various places, usually referring to something other than the equator.) We’re going to need extreme precision here, so suppose the radius is exactly 6,378,000 meters, and the circumference is therefore 40074155.9 meters. Adding 1 meter to the circumference makes it 40074156.9 m; dividing by 2π gives a new radius of 6378000.159, an increase of about 16 cm as we said. Now, 16 cm (6.3 inches) seems unexpectedly large to us as a distance from the ground, on our own scale; but it is just as tiny in proportion to the radius of the earth as the 1 meter was to the length of the rope. Our proportional intuition is exactly right, except for what we are comparing to.
I don't know that all this effort is really worthwhile just to convince a friend, but for students it can be important to see math make sense. What we're doing here is building a foundation for the unexpected result. By first making the basics believable, and gradually shoring up our abstract reasoning with connections to intuitive understanding, we can make the leap of faith shorter. It may still seem surprising, but it will start to seem like a natural consequence of things we've come to know well.
Training your intuition
At the end, I quote an unarchived answer by Doctor Rick to a question about another counterintuitive result: If 100 kg of potatoes, of which 99% is water, lose moisture to a point where they are only 98% water, how much do they weigh?
Here, again, our intuition is proportional; we think that 98% isn’t much less than 99%, so the change can’t be very large. What we are missing is that we really have to focus on the 1% that is not water. This must remain the same actual mass, but become 2% of the whole – it has to double! So the total mass must be halved, to 50 kg.
Doctor Rick ties this to his own personal experience:
To me, the derivation of the equation makes the solution understandable. What does it mean to make it intuitive? Perhaps it means to train our intuitions so that next time we encounter a problem like this, we won't be fooled again. Our intuition isn't going to solve the problem; we still need to think it through carefully in order to get a quantitative solution. But at least we can learn not to jump to conclusions. One way to train my intuition is to compare the problem with something I've seen plenty of times in real life. I've painted murals, mixing the colors I want from poster paint. Often I want a light color, and a little color goes a long way. I have learned to be very careful not to add too much color to the white at first. Why? If I need 1 part blue to 100 parts white (which is not unreasonable in my experience), and I put in 2 dabs of blue instead of 1, I need to add another 100 parts of white to get the color to be what I wanted! I can't tell you how many times I've ended up with far more paint than I needed, by the time I got the color right. I've learned that it's better to throw away half the too-dark mixture, rather than try to save the whole batch by adding white. Do you see how this is the same idea as the potatoes? It's just reversed. The concentration of potato "substance" is analogous to the blue paint. In order to *increase* the concentration of the "potato substance" from 1% to 2%, evaporation must *remove* half the water (analogous to the white paint).
He had to train his intuition in order to avoid wasting paint; we can train our intuition, at least to step aside when we know a problem will challenge it! Again, the key idea here is that you can’t make the math seem intuitive until you admit that your intuition may have to change.
As I pointed out in introducing this last example,
After all, children develop an intuition gradually, starting with wrong assumptions (that things we don't see don't exist, for example), and having trouble with basic ideas like conservation of volume (that two glasses of milk must be more than one larger glass, even if they see it being poured from the one to the others). Intuition is learned. And for that purpose, perhaps an inductive approach can help - if only to shake up our assumptions and allow us to accept the mathematical result by seeing our predictions turn out wrong. But if we refuse to accept what we see, and keep asking for more evidence, then it's useless to continue with more examples. Such a person's skepticism goes too far, and until he or she learns to accept reality, there's not much we can do.
This is true of other areas of life as well! We need the humility to accept that reality may not be what we think it is.