Mathematics and Intuition
Date: 07/10/2001 at 21:32:46 From: Archer Veledoit Subject: How to Handle Intelligent Unbelievers 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 or 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 result 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? Archer
Date: 07/11/2001 at 12:00:44 From: Doctor Peterson Subject: Re: How to Handle Intelligent Unbelievers Hi, Archer. 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. 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. 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. 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. We have a sphere of radius R, with a rope of length 2 pi R around its circumference; then we add L units to that circumference. What is the new radius of the rope? The new circumference is (2 pi R + L); the new radius is that divided by 2 pi, or R + L/(2 pi). This means that the radius is increased by L/(2 pi), which is independent of R. This seems a little less surprising, perhaps, than if we solved it with specific numbers, since we don't have a specific unexpectedly large number to reject outright; we're forced to look at the algebra. But we can still question the algebra once we see what it tells us. The next step, of course, is to check the answer: plug in actual numbers for R and L, solve for the change in R, and then add that to R to find what the new circumference will be. It will be L more than the original, showing that as long as we accept 2 pi R, our answer is right. Next, if this still seems too weird to be true, we can look into what's happening. Algebraically, the main point is that 2 pi R is linear; that is, adding the radii of two circles adds their circumferences: 2 pi (R1 + R2) = 2 pi R1 + 2 pi R2 So if we add something to the radius of a circle, the new circumference is the sum of the circumference of the original circle, and the circumference of a circle whose radius is the added amount: *********** *** | *** ** | ** * | * * | * ***** * | R1 * * |R2* *------------+------------* *---+---* * | * * | * * | * ***** * | * ** | ** *** | *** *********** *********** **** | **** *** | *** * | * ** | ** * | * * | * * | R1+R2 * *----------------+----------------* * | * * | * * | * ** | ** * | * *** | *** **** | **** *********** If you can accept that, the result follows automatically. If you can't, think about it more. To convince ourselves of this, we can start with something easier to understand than a circle, such as a square. Suppose we do the same thing: start with a given square, then add some amount to its "radius" (half side) and compare perimeters: 2 R1 +-------------------------+ | | | | | | 2 R2 | | +-------+ | R1 | | R2| | +------------+ | +---+ | | | | | | +-------+ | | | | | | +-------------------------+ R2 2 R1 R2 +---+-------------------------+---+ | | | | R2 +---+-------------------------+---+ | | | | | | | | | | | | | | | | | | R1 | R2| | | +------------+---+ 2 R1 | | | | | | | | | | | | | | | | | | | | +---+-------------------------+---+ | | | | R2 +---+-------------------------+---+ Here you can see that the additional perimeter in the larger square is the sum of the perimeter of square R1 and the four corners added, which is the perimeter of square R2. You can do this with other regular polygons, and see that it still works. Keep going, and you have reason to believe the same thing is true for a circle. 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. 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. Below I've quoted an answer Dr. Rick wrote recently that touched on this sort of issue, and I think it may help you. As he points out, one way to deal with a "non-intuitive" result is to retrain our intuition so that it agrees with reality. After all, children develop an intuition gradually, starting with wrong assumptions (things we don't see don't exist, for example), and having trouble with basic ideas like conservation of volume (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. ================================================== Question: I have heard a problem which I can solve, but I can't really comprehend it. The "potato problem" goes like this: You have 100kg (or pounds) of potatos. Like most food it's mainly made up of water - in this case 99% of its weight is water. The 1% left is "potatosubstance." You then have the 100 kg out on a warm day and of course the water in the potato begins to evaporate. When you take the potatoes inside, you get the information that now, "only" 98% of the mass of the whole potato (potato substance + water) is water. What does all the potato weigh now? Take a guess before calculating it. Isn't the answer hard to grab? I would be very greatful if someone could make it more intuitively understandable (the calculation itself isn't the problem)! Answer: Our naive intuition is that a change from 99% to 98% isn't much, so the change in weight can't be much, either. Then we think the problem through. The 100 kg of potatoes contain 1 kg of "substance." If this 1 kg is 2% of the total after evaporation, then that total must be 1 kg / 0.02 = 50 kg 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? I 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). What we learn from the two examples is this: To make a small change in the concentration of a minor component of a mixture requires a large change in the dominant component of the mixture. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ ================================================== Thanks for an interesting question. I'd like to hear back from you if anything actually helps in this specific situation, or if you have further ideas in this area. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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