Mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius. – Harold Marston Morse
I had not intended this to be the first essay posted here. Indeed, I had not intended to write this at all. However, while preparing what was to be the first essay, and in doing so delving into many mathematical writings, I noticed a common deficiency in them. This deficiency in mathematical exposition is, I believe, the cause of many difficulties in mathematics education, and the central motive to my creation of this site. It is simply this: Mathematics writing, even popular mathematics writing is horribly insular. There is a fiery vitality to mathematics that is caged behind a web of arcane symbols, exclusive terminology, and elusive concepts. To the adept who has tamed these, the passion is evident; it seeps through the symbols, glows behind the formulas, and thrums in the theorems. However, to the outsider, or initiate it is often intractable. I do not believe the nature of this great joy is indescribable. I do not believe that language is so poor a tool that it cannot capture it.
What makes mathematics so vital? What is the nature of its joy? While I have seen the answer to these questions circumscribed and hinted at, I have never seen it clearly articulated. Yet, it is there in mathematical work. Calmly coiled in the rigor, and secretly shared by its enthusiasts, the heart of math’s fierce cadence is constant and insistent, but seldom heralded. Here I will attempt to define it. I do not pretend that the specifics of mathematical fury are constant from person to person, or that they are not intensely personal. My aim here is to identify and illustrate the common thread, to get to the root of mathematical wonder.
We are the bees of the invisible. –Rainer Maria Rilkes
Mathematics isn’t the exercises and word problems in textbooks. It isn’t the +, -,×,÷ of arithmetic, the lines and curves of geometry, the angles of trigonometry, or the sums of analysis. These things, though they are wondrous, are merely humanity’s best, and woefully inadequate hymns to mathematics. To get at the nature of mathematics, let us consider a poem (poetry being, to me, math’s closest analogue):
One day I wrote her name upon the strand,
But came the waves and washed it away:
Again I wrote it with a second hand,
But came the tide, and made my pains his prey.
Vain man, said she, that doest in vain assay
A mortal thing so to immortalize,
For I myself shall like to this decay,
And eek my name be wiped out likewise.
Not so (quoth I), let baser things devise
To die in dust, but you shall live by fame:
My verse your virtues rare shall eternize,
And in the heavens write your glorious name.
Where whenas Death shall all the world subdue,
Out love shall live, and later life renew.
Here Spenser confronts the central paradox of humanity, perhaps of all life: The present touches eternity, but it is nothing to eternity. Worse still, perhaps there is no present at all. If all efforts are washed away by the tides of time, were they ever really there? Does the present exist, or is it just an illusion, the fleeting impression of the future becoming the past? How much more truthful were the hour glasses and water clocks, where the unceasing flow of time was permanent and irreversible, than our circular clocks, where the hours will always come again?
It is a major theme of the human experience that when people are confronted with the infinite, the bold respond with dread, horror, and awe, and that the meek turn away and try to forget. There is madness in eternity. It is far too large, we are far too small, and the implication of this is too easily grasped. To frame it mathematically, in calculus, whenever the individual (1) encounters the eternal (∞), we write this:1/∞=0 ; you are nothing when compared to everything. The present is so fleeting, and so small that it may as well have never existed.
Spenser recognizes this. He sees the great threat, and responds heroically as an artist. His weapon against eternity’s indifference is his poem, and he knows it is an effective one, because he recognizes that a poem is something quite special: It is the crystallization of an idea. Idea is a surprisingly slippery concept, and importantly so. Ideas have no fetters, they are atemporal, and they represent our transcendence over time’s tyranny.
Everything changes, but ideas. They are our one constant, and they let us know who we are. You have never seen the same face in your mirror. What you have seen is a succession of similar masks. Each time you’ve combed you hair, brushed your teeth, and washed your face someone different was watching you. Sisters perhaps, maybe even twins, but surely not the same person. Her hair was a little longer, his skin a little paler, the eyes slightly duller, the wrinkles deeper, the age older, the lighting greener. However, this constant assault by strangers has not driven you to shatter all your mirrors. Somehow you have known that from day to day, hour to hour, minute to minute, the reflections you see are not portraits of different people, but are rather frames of the same person’s film. The totem that has saved you from insanity and your mirrors from destruction is the continuum of your thoughts.
Your thoughts can also resurrect the dead. This was Spenser’s mission. By reading his poem, and mirroring his thoughts at the time of its writing, you have invited Spenser into you. Spenser, his lover, and his love now live in you, and all those that have found something in his sonnet. In a way, he is more alive today, and his love burns brighter than ever, because it is no longer one heart that beats with the vigor of Spenser’s love, but the hearts of the millions that have read the poem, and the millions that will read it.
Spenser’s poem tells the story of much more than a poet on the beach. It is the David and Goliath story of man versus the infinite. It is a great act of defiance that has immortalized two humans, and amplified their love exponentially; a tremendous feat for fourteen lines.
Mathematics is pure poetry. –Immanuel Kant
Perhaps the most well-known mathematical statement is the Pythagorean Theorem: In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse, or a2+b2=c2. This modest collection of letters and numbers encapsulates something that is truly awesome. Those that understand what this is love mathematics. However, it is the great tragedy of mathematics that the symbols, chosen for their utility, we use to describe and investigate the subject obfuscate by their quotidian nature the enormity of what they represent. To understand the magnificence of a2+b2=c2 , we must analyze the essential feature of mathematics- proof.
The name given to this remarkable relationship, the Pythagorean Theorem, indicates that it is something very special. A theorem is a mathematical statement that has been proven. The concept of proof in mathematics is the defining attribute of the subject. It is not used casually, and it has nothing to do with evidence. In our legal system, we follow the Presumption of Innocence, the often stated legal right that the accused is innocent until proven guilty beyond a reasonable doubt. In the sciences we search for empirical evidence that supports a suspected correlation. In neither of these fields is a conclusion considered final and unfalsifiable no matter the extent of the evidence. The “proofs” in human endeavors only assert a high probability, never a certainty. Though they may loath to admit it, both the lawyer and the scientist are essentially Humean in their outlook. If they are being truthful, they will never contend that they know something, only that they strongly suspect it.
The mathematical proof is not evidence of the validity of a mathematical statement, it is the validity of the statement. When something has been proved its dominion becomes infinite and eternal. a2+b2=c2 is not a characteristic of any one right triangle, or any group of right triangles, or a very likely “law” of right triangles. It is a feature of all right triangles so fundamental that it defines right triangle. There is no need to test each case. It is applicable no matter the triangle. It is as applicable to the triangle formed by your closet floor, broom, and closet wall as it is to the triangle formed by the Earth, Sun and Moon during the third lunar phase.
The theorem’s universal truth is implicitly relied upon by every architect. It is the reason why one carpenter’s square can be used in every project, and each blueprint is not required to “re-invent the wheel” to provide a stable house. By proving the statement, humankind gained infinite knowledge. Mathematicians have been able to tame the numinous.
A thought is an idea in transit. –Pythagoras
It cannot be emphasized enough: We know something about everything. To make something of the infinite comprehensible, concise, and approachable is math’s most remarkable success. However, this joy is merely intellectual, and there is a much more human side to mathematics that is no less unique to the subject.
The Pythagorean Theorem was discovered (or invented) more than two and a half thousand years ago. Throughout those years, it has never lost its vigor. Each generation of mathematicians has learned it, meditated upon it, incorporated it into their mathematical corpus, and added to it. Regardless of time, place, culture, gender, religion, etc. the mathematicians of each age safeguard the work that came before them, and expand upon it. It is a unifier across history, culture and language that has not gone unnoticed by historians. As Edward Gibbon observes in The Decline and Fall of the Roman Empire: “The mathematics are distinguished by a particular privilege, that is, in the course of ages, they may always advance and can never recede.”
In most sciences one generation tears down what another had built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure. -Hermann Hankel
From a modest seed, counting, an ever growing, enormously complex structure has grown. Branches sprout, expand, branch themselves, but they never break. All the complexity, each new bud, draws its sustenance from the same trunk, and common roots. This map of the infinite, this island in the eternal is humankind’s most amazing creation.
This is grand from a cultural perspective, but it is amazingly humbling from a personal one. As a practitioner of math you are constantly aware of the long lineage of great minds that have come before you, each providing a wrung for you to grasp. For the student, the ascent of this mathematical tree is a literal walk through human history. It is a living history that to understand you must participate in. With each new lesson you are confronting and overcoming the challenges of a generation. When I’m studying Euclid, I am quite literally thinking the same thoughts of the ancient Greek geometers. This synchronicity through history, through millennia, binds us together. It makes personally clear the continuum of human experience, and reminds us of our common position as peers in existence.
The great glory of mathematics is its durative nature; that it is one of humankind’s longest conversations; that it never finishes by answering some questions and taking a bow. – Barry Mazur
A commonly expressed belief about math (traceable to Leibniz, I believe) is that it is a universal language. This is true, but I believe in a way more subtle than is often intended. The great message of all mathematical statements is the latent one that declares, “I’m conscience.” It is no accident that the man who declared, “I think, therefore I am” also revolutionized geometry, and it is no coincidence that the messages we would send into space via Voyager’s golden records would be coded mathematically. Mathematics is our touchstone and litmus test for consciousness, our vanguard against solipsism when seen in others, our Rosetta stone for the dead, and the great thread that ties us together. That is the burning heart, the vibrant core, of math.
Cogito ergo sum. -René Descartes