Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
What question? Well, it's the same question all 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 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:
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
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? Would 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 of a musical piece. 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 ugly and elegant proofs. 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:
In Einstein's words:
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. He writes, ... 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, like 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.
For, say, 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 each 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
As usual, we can define f(x) = x2 - a with the derivative
then leads to
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
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 y0 and
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 yk+1 and zk+1. By direct verification, (4) also holds. Each of the means is bounded by yk from below and by zk from above. So indeed, y's grow while z's decrease. Further,
a superfast convergence. Note that (5) can be rewritten as
which ultimately reduces to
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.
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,
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. Thus it is thus bound to become 0 after a finite number of iterations.
(Note: in the symmetric variant in which 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 E. 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:
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.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at email@example.com
Copyright © 1996-2000 Alexander Bogomolny