A rebuttal on the beauty in applying math
On August 18th, 2017, the PBS NewsHour had a short piece Math is amazing and we have to start treating it that way by Dr. Eugenia Cheng. Dr. Cheng's actionable proposal was the introduction of specialized math teachers into K-5 education. This is a worth-while idea, allong the lines of having specialized music teachers. I know I profitted from having good math teachers in 3rd and 5th grade (Mrs. Nichols, Mr. Gregory). Alas, the idea does not free us up from the financial constraints that have forced us to cut music teachers from our schools, and if we have to cut math instruction in the same way some day, we're worse off than we're starting.
For me, the essay is wrong. There are a number of factual issues that may mislead the general reader (see below). But primarily, I strongly disagree with Dr. Cheng's perspective that "The usefulness of math is a burden". I believe the usefulness of mathematics is what makes it fun and worth-while.
While we each may find beauty in abstractions like the paintings of Jackson Pollack and the jazz scatting of Ella Fitzgerald, abstraction is not our only source beauty. Our world itself is beautiful, complex, and full of mysteries from the flight of butterflies, down to the functioning of the atom and out to the shape of galaxies of stars millions of lightyears across. But much of this beauty is difficult for us to comprehend and express. Rachel Carson needed 3 books to complete her expression of our ocean's beauty in the 1950's and 60's, and since then we've learned so much that more volumes could be easily be added. But some of this beauty cannot even be expressed at length -- common language is just not expressive enough for us to explain how waves roll accross the ocean without moving the water itself, or how the rotation of the earth creates the spinning of a hurricane.
Mathematics enriches our language in ways that let us comprehend and express more of this beauty. We've created mathematical equations that describe the flow of fluids and the pull of gravity, and let us express the interconnection of the spinning of a hurricane to the rotation of the earth, and how the motion of a little-league baseball player's home run mirrors the motion of the moons of Jupiter. Mathematics can open our eyes to the pieces of the world we could not see otherwise, like how the glimmer's of an emerald arrise from the arrangement of the electron's in its atoms. It is an essential element of the discoveries of science because it can say so many things that we'd never needed words for before. And in saying these things, math broadens our understanding and empowers us to do and build and engineer what we'd not yet even imagined.
In 1764, Charles Mason and Jeremiah Dixon were trudging through the wilds of colonial America to map out what we now call the "Mason-Dixon line" now forming the state boundaries between Pennsylviania, Maryland, and Delaware. The land was so wild and rugged, and the distances were so great, that despite several attempts, nobody had yet been able to reliably establish these boundaries. Part of the difficulty was that over these long distances, the familiar rules of Euclidean geometry no longer worked. The earth, after all, is a ball and not a flat plane. But thanks to their careful procedures and smart application of mathematics, Mason and Dixon succeeded where others had failed. Part of their success was due to the use of beautiful little formula called the "Spherical Law of Sines", which allowed them to calculate distances from angle surveying in much the same way that it would have been done in Euclidean geometry. This formula traces back to the works of men in the Medieval middle east and ancient Rome, who developed their formulas to help themselves make sense of the stars in the night sky.
Another classic example was the work of Joseph Fourier describing how heat moved through an iron bar when one end was stuck in a furnace and the other end was stuck in a bucket of water. This was an important problem for things ranging from the construction of wine cellars to the firing of cannons. Fourier found a way to solve this problem but also many other practical problems for the motion of heat and waves. But his contemporaries railed against his work, complaining that it was not sufficiently abstract and rigorous. Yet today, Fourier's analysis is a fundamental part of things like the MP3 music file format that made distribution of music across the internet possible.
History is full of beautiful stories like these where applying mathematics smartly changed the world, a little or a lot. I find beauty and joy in the math of these applications just as much as I do stories like those of Fermat's last theorem. I think some of our students will as well, and may be inspired to go on to become the engineers, scientists, and mathematicians of the next generation when we embrace the beauty of practical things.
Factual issues
Briefly, some "factual" issues with Dr. Cheng's essay are the following...
"Calculus depends on irrational numbers that the Egyptians started wondering about thousands of years ago" -- this wrong in spirit and fact. Spiritually, calculus was and is the study of smooth lines and shapes like ellipses and projectile motion, and very far from the aspects of geometric number theory that give rise to "irrational numbers". In fact, calculus itself can be and was developed without reference to irrational numbers.
It's claimed that math stopped the plague. While we do use mathematical descriptions to help us understand and manage and prevent infectious disease epidemics, the importance of this math in saving lives is small compared to radical progress non-mathematicians have made in public health, vaccines, drugs, and treatment.
And the pony express was not put out of business by math, but by telegraph thanks in large part to Henry and Morse and others who sent a telegram from DC to Baltimore in 1844, 16 years before James Clark Maxwell untangled the mysteries of electromagnetism with a descriptive set of math equations.