Cut The Knot!by Alex Bogomolny |
What Is Your Answer to That Question?
February 2000
A concept without precept is empty; a precept without concept is blind. Immanuel Kant |
What question? Well, it's the same question over again. The question that stumps students year in and year out and to which many a parent and a teacher would like to have a definitive answer. The question - Why to study mathematics? - appears in a multitude of guises and a plethora of variants like the one implicit in a discussion thread that started with a message on the k12.ed.math newsgroup:
A friend has a son who is a junior in HS. He's always done well in Math and, for the most part, enjoyed it. This year however, he's bored in class and becoming disinterested. As a result, his grades are dropping. He's very talented musically and loves the creativity that he finds in music. Unfortunately, he has concluded that none of that same creativity exists in math. Are there any resources that anyone might suggest that could demonstrate to him that math can be imaginative, creative and even fun?
The responses posted this time around did not include the most forceful "A math degree is one of the best degrees you can get as a preparation for an M.B.A. (or a law degree or a medical degree or, in fact, most professional degrees)." But not surprisingly they focused on applicability of mathematics to the real world as in
... The reason is that many math problems are not easily related to the 'real world'. The best combination of 'imaginative, creative or even fun ' would be practical (real world) 'math word problems'.
or
... Most of science and engineering involve creativity in applied math almost as much as the science or engineering itself.
In contrast, it does not seem to follow from the original message that the boy got to appreciate music because of the creativity inherent in its applications in the real world. (Undeniably there are some. Music has a unifying and uplifting effect on listeners. Can you imagine a wedding or a funeral without music? Will you be surprised to learn of the results of some scientific research to the effect that soothing background music enhances workers productivity at an assembly plant?) It's not spelled out explicitly what creativity the boy found in music. He might have discovered a gift for creating music, or felt an excitement of his uniquely personal rendering. I do not know. But it's unlikely a high school junior got hooked on practical aspects of music applications in the industrial (or is it information?) society.
Mathematics is often compared to art. In an assay on Humanistic mathematics, Philip Davis likens mathematics to literature. Like literature, mathematics has metaphor, ambiguity, paradox, and mystery. It has history. Mathematics has contributed mightily to philosophy. It has a sense of outcome, a feeling of rightness, a sense of catharsis.
Like music, mathematics has harmony and dissonance.
Thanks to a message by Antreas P. Hatzipolakis on the geom.college newsgroup I learned about the Einstein-Wertheimer correspondence. In one of the letters A. Einstein shares his view on proofs ugly and elegant. As an example, he proves the Menelaus theorem in two different ways. The first (ugly) proof is the first one from my discussion on the theorem. The second (elegant) one I discussed earlier on a separate occasion. There are two essential differences between the proofs:
The ugly one is based on an auxiliary construct - a straight line through one of the vertices parallel to the transversal, whereas the elegant proof handles the original configuration without modifications.
Einstein observes that the Menelaus theorem is symmetric with respect to the vertices of the triangle. However, the ugly proof gives an unjustified preference to one of the vertices, thus destroying the symmetry of the problem. The elegant proof does not discriminate between the vertices.
In Einstein's words:
Although the first proof is somewhat simpler, it is not satisfying. For it uses an auxiliary line which has nothing to do with the content of the proposition to be proved, and the proof favors, for no reason, the vertex A, although the proposition is symmetrical in relation to A, B, and C. The second proof, however, is symmetrical, and can be read off directly from the figure.
What is the thrust of Einstein's argument? Does he object to using auxiliary elements as such or may he oppose their introduction that does not jibe with the symmetry of the original problem? We can only speculate. Here's one possible reading.
The Menelaus theorem is a statement concerning a relationship of certain linear segments in a triangle cut by a transversal. However, the proof that Einstein judges elegant, exploits a formula for the area of a triangle. If not an auxiliary construct, it's an auxiliary concept nonetheless. When staring at a diagram it's not at all obvious that areas should be introduced into the argument. On the other hand, auxiliary constructs are so ubiquitous in mathematics that it's hard to imagine that Einstein would oppose them as a matter of principle. Writes he, ... we are completely satisfied only if we feel of each intermediate concept that it has to do with the proposition to be proved.
How does one know what has to do with a proposition and what has not? There is no unique answer. May I suggest that, to Einstein's ear, the line that destroyed harmonic symmetry of the vertex configuration introduced a screeching dissonance into the background tune of the Menelaus theorem.
Mathematics has no rivals among other sciences in the number of practical applications. It's also obvious that even the sum total of all its applications does not constitute mathematics. This is because, as every other science, mathematics has its own objects of study and its own methods. Even in its most practical aspects, mathematics differs from its applications.
Say, for Application Engineer, Newton's is a fast and reliable method for solving functional equations. Application Engineer highly recommendeds it to his Electronics Engineer colleague for hardwiring in a computer chip. Their concern is efficiency of implementation and hardwiring. Mathematician may spend a lifetime in search of peculiar examples and generalizations of the method. He may get curious about other instances of iterative processes and their general theory. As a consequence of universal applicability of mathematics, the two - Application Engineer and Mathematician - are likely to cross paths more than once but they will arrive at crossroads by different ways and continue in different directions.
Let's look at an example whose end result may be of interest to an application engineer, but that may delight a mathematician. This is an alternative derivation of Newton's method for finding square roots of positive numbers.
Let for the sake of certainty, a > 1. We seek the positive root of
(1)
x^{2} = a
As usual, we can define f(x) = x^{2} - a with the derivative being
x_{k+1} = x_{k} - f(x_{k})/f '(x_{k})
then leads to
(2)
x_{k+1} = x_{k} - (x_{k}^{2} - a)/2x_{k} = (x_{k} + a/x_{k})/2
Estimates can be derived from the general theory. But here is an elementary (no calculus) derivation that pulls together several useful but mostly disregarded in the pre-college mathematics ideas.
Instead of looking for x, introduce positive y and z such that
(3)
y < z
and
(4)
yz = a
For example, one can start with y = 1 and z = a. Think of these y and z as the starting points of the ensuing iterations, say y_{0} and z_{0}. Obviously,
(5)
y_{k+1} = 2y_{k}z_{k}/(y_{k} + z_{k}) and z_{k+1} = (y_{k} + z_{k})/2
The old values of y and z were distinct. Therefore, their arithmetic and harmonic means are also distinct and, as is well known, the former is the larger of the two. Thus (3) holds for y_{k+1} and z_{k+1}. By direct verification, (4) also holds. Each of the means is bounded by y_{k} from below and by z_{k} from above. So indeed, y's grow while z's decrease. Further,
z_{k+1} - y_{k+1} | = (y_{k} + z_{k})/2 - 2y_{k}z_{k}/(y_{k} + z_{k}) | |
= (z_{k} - y_{k})^{2}/2(y_{k} + z_{k}) | ||
< (z_{k} - y_{k})^{2}/4y_{k} | ||
< (z_{k} - y_{k})^{2}/4y_{0} | ||
= (z_{k} - y_{k})^{2}/4, |
a superfast convergence. Note that (5) can be rewritten as
z_{k+1} = (y_{k} + z_{k})/2 and y_{k+1} = 2a/(y_{k} + z_{k}) = a/z_{k+1}
which ultimately reduces to
z_{k+1} = (z_{k} + a/z_{k})/2,
Newton's method (2).
(I like this approach very much. Once the iterations (5) have been set up, every step of the derivation falls in its place very naturally. Practically speaking, (5), unlike (2), comes with a built-in termination condition.)
Mathematician can get his ideas from applications. Smoothing operators and functions have been used for interpolation and, more generally, data fitting. With the advent of computers, they became an important tool in image processing. Applied iteratively, they have a blurring effect on an array of pixels (integers in computer memory.) Mathematician may pick up and toss around the idea of integer iterations and eventually come up with a useless (in applications and for the time being) diversion that was discussed in one of Ivars Petersen MathTrek columns.
A number of students sit in a circle while their teacher gives them candy. Each student initially has an even number of pieces of candy. When the teacher blows a whistle, each student simultaneously gives half of his or her own candy to the neighbor on the right. Any student who ends up with an odd number of pieces of candy gets one more piece from the teacher. Show that no matter how many pieces of candy each student has at the beginning, after a finite number of iterations of this transformation all students have the same number of pieces of candy.
(Number of candies can be set either randomly or by modifying each of the circled entries. To modify a number, click a little to the right or left of its centerline. In the symmetric variant, each fellow shares equally between his/her two neighbors.)
Let's have an outline of the proof. The problem deals with a finite sequence of integers in a circular arrangement. An operation is defined on the set of such sequences. In every sequence there is the largest and the smallest number. Both may appear more than once. We may observe that, as the result of the operation,
- The largest number may not increase.
- The smallest number may not decrease.
- The number of occurrences of the smallest number necessarily decreases. When there is only one instance of the smallest number, it is eliminated after the operation.
After a finite number of steps (not exceeding the number of occurrences of the smallest number in the sequence) the difference between the largest and the smallest numbers in the sequences decreases. It is thus bound to become 0 after a finite number of iterations.
(Note: in the symmetric variant wherein a student gives one half of his or her candies to the neighbor on the right and the other half to the neighbor on the left, the argument breaks down. Do you see why?)
So, what is my answer to the old question? Like literature and music, mathematics is part of human culture. As a peculiar species, we can't help but aspire to pass on and further develop this accumulation of wisdom, traits, technology, behaviors, history that comprise whatever it is that makes us in whatever we are. As in the saying by I. Kant, shifting emphasis on creativity from mathematics to its applications renders the concepts of mathematics empty. This also makes the precepts of applications blind.
After I began writing this column, another message has been posted to the discussion thread:
*Any* fruit of human endeavor shows creativity, if you think about it. The interesting question to me is this: Why is it that a student who is only playing other people's music instinctively understands that those composers were creative, and that s/he might aspire to the same kind of creativity -- or, in English class, instinctively understands that those writers were creative, even when s/he is just reading their creations and answering quiz questions about them -- but doesn't have the same instinctive understanding that Euclid and Newton and Pascal and Gauss and Euler were creative mathematicians?
The most obvious answer has to do with the way these disciplines are taught.
The idea of the absence of creativity in mathematics is so much absurd that I can't help but think that the real culprit is the lack of creativity in mathematics education. The latter is not quite the same as math applications education. The surest motivation to study mathematics is in its intrinsic beauty and harmony. In the very least, students must gain a clear understanding of the difference between mathematics and its applications.
References
- A. Benjamin, M.B. Shermer, Teach Your Child Math, Lowell House, 1996
- G. Chang and T.W. Sederberg, Over And Over Again, MAA, 1997
- P. Davis, The Humanistic Aspects of Mathematics and Their Importance, in Essays on Humanistic Mathematics, A.M. White (editor), MAA, 1993
- A. Engel, Exploring Mathematics With Your Computer, MAA, 1993
- A.S. Luchins, E.H. Luchins, The Einstein-Wertheimer Correspondence on Geometric Proofs and Mathematical Puzzles, Math. Intelligencer 12, No.2, 35-43 (1990).
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In the symmetric variant the parity of the number of students is essential. If that number is even, the iterations can settle in a 2-cycle. For example, with four students and an intial candy distribution of 2-4-2-4, the first iteration leads to 4-2-4-2, while the next one returns to the initial state.
For this and other generalizations of the original puzzle, see the article by Iba and Tanton listed below.
References
- G. Iba, J. Tanton, Candy Sharing, The American Mathematical Monthly, Vol. 110, No. 1 (Jan., 2003), pp. 25-35