[Author’s note: The following is an imaginary flashback to my high school days.]
Beads of blood appeared on my forehead the moment I heard the assignment. With a smile, my high school English teacher had said, “Class, I want each of you to write a four-stanza poem. It will be due on Friday.”
FRIDAY??!! Yeah, Friday of what YEAR? To me, the assignment was impossible and mind-boggling. She just as well could have asked me to move Pike’s Peak three miles to the south using only a teaspoon. I wondered, how on earth do you write a poem? Do you start with some words that rhyme—like June, moon, soon, loon, dune, maroon, spoon—and try to string them together into a meaningful arrangement? Do you start out with some kind of rhythm in your head (Ta Da, ta dah, ta DAH!) and try to figure out words that go with it? I hadn’t the faintest idea, and I sure didn’t know anything about exotic syntax like “iambic pentameter.”
Even worse, I could see the members of the literature club were already in a twitter. No doubt they were thinking, “No problem, we’ll have two or three sample poems ready in time for tonight’s club meeting.” Ordinarily, I didn’t pay any attention to the literary types; they actually read books and poems for pleasure, and I thought they belonged to the literature club because they couldn’t get dates.
Later that afternoon, in my favorite class, Algebra II, the teacher tossed out an assignment: create a new linear equation by Friday. I noticed that some of the literary types seemed to be worried about what seemed to me to be a simple assignment.
“No problem,” I thought, “I still have time this afternoon to create a couple of example equations for review at my math club meeting tonight.” Our math club consisted of students who really enjoyed math and science. We liked to read math books in our spare time and were teaching ourselves calculus. The members of our club also didn’t get many dates, but that’s only because we were too busy learning new types of mathematics.
I sailed through the math assignment and struggled mightily with the poem, finally turning in a very lame “creation.”
Over forty years have passed since that (mythical) experience in high school – I haven’t written any more poems, but I’ve written hundreds of equations. I studied mathematics in undergraduate and graduate school, taking courses such as linear algebra, calculus, differential equations, partial-differential equations, matrix methods, complex variables, topology, and statistics. I’ve used mathematics throughout my career in industry and academia and I’ve had no problem creating new mathematical equations for a wide variety of applications. I’m not a math superstar; I consider myself more of a “blue collar worker” in the mathematical world.
As I think back to my high school experience, however, it’s dawned on me that perhaps the “literary types” had as much difficulty in conceiving how to create a new equation as I did in creating a new poem. While non-mathematicians may be dubious of this, I can assure you that creating a new equation is quite simple. I can show you how in just a few steps.
First though, I have to clear up some misconceptions about the process. I suspect that many think that creating an equation starts with selecting some random symbols—like e, m, and c—and trying to put those together with the correct syntax. This would be like Einstein contemplating different variations of e = mc2. Can you imagine Einstein, as he was working through his famous theory of relativity, creating the mass-energy equivalence equation by tossing around various combinations like, let’s see, how about, e = mc3; no, perhaps, e = mc1/2; no, that’s not it, perhaps e = m1/3c5 + п….Or, perhaps, you may think that the process of creating an equation involves thinking first about the arcane syntax of equations, then trying to insert symbols to develop an equation.
While I have no way of knowing the mental processes that Einstein used, I do know that good mathematicians do not begin by playing around with various symbols to create a new equation, nor do they think about the syntax of equations. Instead, they start with a basic idea that they want to express, and the resultant equation is a means for expressing that interesting idea in a short-hand form.
For example, there is an equation that many people have encountered in basic physics called Hooke’s law. Right away, the idea of a law may seem intimidating. You may wonder, did this “law” come down mandated from the heavens? No, in this case, the law is simply an observation that seems to hold true for a wide number of physical situations involving the elasticity of objects.
Hooke’s law relates the force on an object to the amount of movement or deformation of the object. The classic example is a weight connected to a spring. Due to gravity, the weight exerts a force on the spring that we’ll label by the symbol, F. The amount that the spring stretches (which we label using the symbol, x) is related to the force exerted by the weight, by the equation, F = -k x, where k is a constant associated with the properties (stiffness) of the spring. The equation is a useful shorthand way of expressing a common observation that when we push or pull on material it responds in proportion to the exerted force.
Other questions come to mind when we see Hooke’s law. When does it apply and when doesn’t it? We know, for example, that if we put a very small weight on a spring, nothing happens (i.e., the spring doesn’t stretch). Similarly, if we place a very heavy weight on a spring, the spring will stretch completely straight and perhaps break. So, for a material such as metal (in the spring), Hooke’s law describes some behavior but not for every possible weight, and Hooke’s law doesn’t apply at all for some materials such as silly putty. While not all-encompassing, Hooke’s law does change the way we think about the world and gives rise to questions about how things behave. This is the power of equations.
So, how does one create a new equation like Hooke’s law? It really is easier than you might think; to illustrate my point, let’s create one now. Consider the common observation that large adult dogs don’t live as long as small dogs do. Large breeds, like Newfoundlands, usually weigh from 100 to 120 pounds, but only live for 6 or 7 years. By contrast, small breeds, like Miniature Poodles, weigh only 8 to 10 pounds but may live 15 to 17 years.
How can we express this observation in an equation that we can name, Dog’s law? We want an expression that allows us to input a value for the dog’s weight, Wlbs, and provide as output the dog’s potential adult life span, Lspan. We have noticed that lifespan in dogs is inversely proportional to weight (as weight increases, lifespan decreases), so the equation should have a form; Lspan = 1/ Wlbs.
But this doesn’t work as stated, since the equation would predict that a 100-pound dog would only live 1/100 of a year and a 6-pound dog would live for merely two months. Let’s try instead, Lspan = 400/ Wlbs. This would predict that a 100-lb dog would live 4 years, and that a 40 lb dog would live 10 years. We’re getting closer. However, it would also predict that a 10 lb dog would live 40 years. This doesn’t quite work, but one more iteration and we ought to have it.
How about the expression, Lspan = 6 + 200/ ( Wlbs + 15)? This equation predicts that a 5-lb dog will live 16 years and that an 85-lb dog will live 8 years. The equation “says” that a dog’s lifespan is inversely proportional to the dog’s weight, but that even the heaviest dog will likely live at least 6 years, and the smallest dog will not typically live beyond about 18 years. As with Hooke’s law, this equation provides a shorthand way to express a common observation and leads to questions like, why does this apply to dogs but not cats (who seem to live forever)? Why does this relation hold for dogs, but not across species (e.g., why do fruit flies live for a few hours but elephants live for many years)?
So how does the creative process for writing equations relate to writing poetry? Recently, as part of a writer’s group, I’ve had the opportunity to observe some poetry writers in action. I’m beginning to realize that they use a similar process when developing a poem as I do in writing a new equation: they start with an idea that they want to convey and then give shape to it in a way that will cause others to think about the concept and be forever changed after they’ve read the poem. Because they’ve read hundreds or thousands or poems, and written many poems themselves, they are inherently familiar with the rules and nuances governing rhyming schemes and language syntax, and instead are able to focus their attention on the idea at hand and determine how to present it in a powerful way.
For examples, I give you the following poems:
Do not go gentle into that good night,
Old age should burn and rave at close of day;
Rage, rage against the dying of the light.
– – -Dylan Thomas
I was born to catch dragons in their dens
And pick flowers
To tell tales and laugh away the morning
To drift and dream like a lazy stream
And walk barefoot across sunshine days.
– – – – James Kavanaugh “Sunshine Days and Foggy Nights”
These poems change our way of thinking about aging and our very idea of how to participate in life. Just as a great equation changes our view of the world, leads to new questions, and is verifiable by experimentation, a great poem can change our view of the world, lead to new questions about life, and is verifiable by living the human experience.
So after forty years, I have come to realize that poetry is at least as important as equations, and I now appreciate the creative process for poetry. In writing this post, I hope I have shed light for my more literary-minded readers on how simple it can be to create an equation and that they will have gained an understanding of the creativity that goes into developing mathematical equations. I also hope our faculty members keep this in mind in our continuing discussions about the role of mathematics as part of IST education. We need to accommodate both engineering/mathematical/computer science “types” as well as “literary” types. Both types of thinking are required in the modern information age. Moreover, while we teach our students appropriate mathematics (and associated thinking), we also need to teach our mathematically minded students how to write and think in a “literary” sense.
At last, over forty years after my high school English teacher’s assignment, I am ready to begin writing a new poem, this time with less trepidation about the process and more appreciation for the parallels between creativity in mathematics and creativity in writing. Now, to get started. Let’s see…if only I could find some words that rhyme with “equation”…