08G:001:039

Due: Nov. 1, 1999

__Left-Brained Poetry__

For eons, artsy people have created poetry. They spewed their emotions out onto the medium before them and created a rustle in the hearts of all who heard or read what the poet had manifested. However, for just as long, mathematicians have fashioned great works of logic, the antithesis of emotion. Yet the truly good mathematicians were able to shape their words and arguments in just such a way, that some of the theorems they produced were just as beautiful and awe-inspiring as the works of their right-brained brothers. Mathematical theorems serve as a type of poetry cast in logic, yet brimming with style.

The foremost property common to both poetry and mathematical theorems is elegant beauty. Like a poem, a theorem contains a certain amount of gracefulness and simplicity. It is concise and does not contain superfluous information. The mathematician does not write with the intent to confuse, rather to explain to the best of his ability. Poetic proofs lead the reader on a roller coaster of logical propositions and validations much the same as the emotional experience gained from a superb set of poetry. The twists and turns of poetry and mathematics often lose the first-time reader, yet each divergence is not without purpose. The experienced reader welcomes each bend as he allows the writer to guide him through the steps of reasoning. The English mathematician G. H. Hardy put it best when he said:

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind – we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.”It is chiefly in this way that mathematics is truly analogous to poetry.^{1}

One can also consider mathematics as a type of poetry that has no hues;
it is only black and white. Therefore, instead of painting a picture full
of beautiful color that may or may not contain a definite brightness, the
theorem forcefully engulfs our eyes with a monochromatic light that represents
the truth. As an example, *reductio ad absurdum*, the proof by contradiction,
is one of the most common styles of argument in a mathematical proof. In
this method, the mathematician assumes the proposition is false and then
reaches a logical contradiction (such as 1 = 2) based on this assumption.
With it then being impossible that the proposition is false and there being
no in between, it must be true. This facet of mathematics mirrors the indispensable
veracity of poetry.

In addition, a poetic theorem enables us to see the world around us in another way. Some of the greatest works of poetry contain simple ideas that when stated bluntly seem trivial. Yet when written out in full application and explanation, the simple idea will stick in our hearts for some time. Similarly, some mathematical statements may seem trivial. However, upon producing a proof of the statement, one gains a much deeper insight into the fact contained in what we took for “common sense.” When Euclid proved that the shortest path distance between any two points was a straight line, most of his Greek contemporaries did not even care. To them, the idea was so diminutive that it was self-evident to the ass that was located at point A and traveled directly to point B, where its food was located, instead of meandering here-and-there first. However, Euclid’s poetic proof showed that even this simple idea rested on yet simpler ideas. Euclid went even further and demonstrated that everything in geometry rested on only five basic facts. In so doing, Euclid accomplished what poets have tried to do for ages: explain a large, abstract idea strictly in terms of a small number of indisputable truths.

In addition, while poetry expands the vision of our personal realities, the poetry of mathematics defines the frontier of our physical being. Through rules and rigorous thought, constraints can be found in our own tools. One example is the construction of a regular polygon with an odd number of sides using only a compass and an unmarked straightedge. The Greeks could construct an equilateral triangle and pentagon. Yet, it was not until 1796, that Carl Friedrich Gauss showed that it was possible to construct a 17-gon. Further, in a beautiful line of reasoning and deduction, the young Gauss showed that there were only 120 different odd-sided regular polygons that could be constructed with the given tools. The style of proof was excitingly new and unique from anything else at the time. By using some of the simplest properties of algebra, Gauss was able to state an innovative fact about our physical world; similar to the way a revolutionary poet uses the attributes of his words to relay a fresh idea about our society.

At the same time, mathematics can take us into an exotic fiction land unlike any other ever imagined. Just as poetry can whisk the reader off to some far land using a different set of words, the mathematician can create a new universe by changing some of the simplest rules. When Gauss decided to alter one of Euclid’s five defining truths, he found that he could construct a bizarre space in which the sum of the interior angles of a triangle did not equal 180 degrees, where the distance between two parallel lines was not necessarily constant, and the shortest path distance between two points was no longer straight. At first, his ideas seemed whimsical and full of nonsense, but as time elapsed, scientists showed that the physical universe we existed in was not Euclidean. Gauss’s non-Euclidean approach to geometry, analogous to the power of poetry, opened the minds of countless individuals, allowing them to approach the world in a different state of mind.

Another way in which mathematical theorems are comparable to poetry is through the writer. Both mathematicians and poets must possess bounding intellect and vast understanding. They look at the subject at hand and totally engross themselves within it. Another mathematician, Howard W. Eves, once remarked:

“An expert problem solver must be endowed with two incompatible qualities – a restless imagination and patient pertinacity.”This is true of both mathematicians and poets. A poet blankets himself in whatever feelings he wants to transmit so that he can find the sounds, words, and rhythms that adequately describe the essence. Likewise, a mathematician shrouds himself in myriad examples to discover the extraordinary functions, constants and procedures that simplify the whole concept and force underlying truths to surface. Only after a large amount of comprehension can either bring life to their visions.^{2}

Moreover, when the poets and mathematicians share their inspirations with others, the response is extremely similar. If the poem or theorem deals with a minor truth, the average person may only be mildly impressed. They sense the tone and pulse of the poem or get the general implications of the proof. However, if the masterpiece confronts a more difficult conception, often the response is one of initial confusion and distaste. Still, after studying the work, and extracting exactly what the mathematician or poet was trying to say, the reader can find himself in awe from the grace and insight of the author.

An additional coincidence of poetry and mathematics is their ability
to stand the test of time. The age of epic poems such as Homer’s *The
Iliad* and *The Odyssey* are comparable to that of Euclid’s *Elements*.
Each endures an unknown number of centuries. Such poems and theorems must
share some intrinsic property that allows them to last for so long. G.
H. Hardy also addressed this aspect of mathematics when he stated:

“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”If what Hardy says is true, then in some way one could consider mathematics better than poetry since it has a greater longevity.^{3}

Finally, no paper can fully deliver the poetic aspect of mathematics without some sort of example. The wonder, elegance, exactness, genius, and power of a mathematical proof can only be properly described by being displayed. Words alone cannot give mathematics and poetry justice. Therefore, Euclid’s proof that there are an infinite number of primes (numbers only divisible by 1 and themselves) follows:

“Assume, said Euclid, that there is a finite number of primes. Then one of them, call itEuclid’s clever use of theP, will be the largest. Now consider the numberQ, larger thanP, that is equal to the product of the consecutive whole numbers from 2 toPplus the number 1. In other words,Q= (2 ´ 3 ´ 5 ´ ... ´P) + 1. From the form of the numberQ, it is obvious that no integer from 2 toPdivides evenly into it; each division would leave a remainder of 1. IfQis not prime, it must be divisible by some prime larger thanP. On the other hand, ifQis prime,Qitself is a prime larger thanP. Either possibility implies the existence of a prime larger than the assumed largest primeP. This means, of course, that the concept of ‘the largest prime’ is a fiction. But if there is no such beast, the number of primes must be infinite.”^{4}

Therefore, mathematics serves as a different breed of poetry. From its
elegance and beauty to its durability and truthfulness, mathematics parallels
poetry like nothing else. Mathematics adequately matches the imagination
and creativity it takes to create poetry. In fact, there is a legend about
David Hilbert, a renowned mathematician, who one day noticed that a certain
student had stopped attending class. When told that the student had decided
to quit mathematics and become a poet, Hilbert replied, “Good – he did
not have enough imagination to become a mathematician.”