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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/   
    
Associated Topics:
High School About Math
High School Conic Sections/Circles
High School Geometry

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