Thinking in Numbers

Preface

Like literary fiction, mathematical imagination entertains pure possibilities. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.

Family Values

When we think and when we perceive, just as much as when we count, our mind uses sets. Our possible thoughts and perceptions about these sets can range almost without limit.
Though a set as familiar to our understanding as that of "animals" implies containment and comprehension, the sheer number of its possible subsets actually swells towards infinity.
Defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few - those most tried and tested within our particular culture.
Memory is a further example of the privileging of certain subsets (of experience) over others, in how we talk and think about a category of things. Asked about his birthday, a man might at once recall the messy slice of chocolate cake that he devoured, his wife's enthusiastic embrace and the pair of fluorescent green socks that his mother presented to him. At the same time, many hundreds, or thousands, of other details likewise composed his special day, from the mundane (the crumbs from his morning toast that he brushed out of his lap) to the peculiar (a sudden hailstorm on the mid-July afternoon that lasted several minutes). Most of these subsets, though, would have completely slipped his mind.
Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.

Eternity in an Hour

With Lucy (in The Lion, the Witch, and the Wardrobe) I stepped into the large wardrobe in one of the otherwise bare spare rooms, tussled with its rows of dense and dust-fringed clothes as we fumbled our way with outstretched fingers toward the back. I, too, suddenly heard the crunch of snow beneath my shoes, and saw the fur coats give way all at once to the fir trees of this magical land, a wardrobe's depth away.
One day, on the short walk home from school, it so happened that these images came to the front of my mind. The lap posts that lined the street reminded me of the lap post I had read about in the story, the point in the landscape from which the children return to the warmth and mothballs of the professor's wardrobe.

Counting to Four in Icelandic

According to psychologists, humans can count in flashes only up to quantities of four. We see three buttons on a shirt and say "three"; we glance at four books on a table and say "four". No conscious thought attends this process - it seems to us as effortless as the speech with which we pronounce the words. The same psychologists tell us the the smallest numbers loom largest in our minds. Asked to pick a number between 1 and 50, we tend toward the shallow end of the scale (far fewer say 40 than 14). It is one possible explanation for why only the commonest quantities feel real to us, why most numbers we accept only on the word of a teacher or textbook. 40, to us, is but a vague notion; 14, on the other hand, is a sensation within our reach. 4, we recognize as something solid and definite.
The profusion of Icelandic and Chinese words for the purpose of counting appears to be an exception to the rule. Many of the world's tribal languages, in contrast, make do with only a handful of names for numbers. The Veddas, an indigenous people of Sri Lanka, are reported to have only words for the numbers 1 and 2. For larger quantities, they continue: "and one more, and one more, and one more..."
In Brazil, the Munduruku imitate quantity by according an extra syllable to each new number. They count, understandably, no higher than 5.
Few circumstances within these communities require any greater numerical precision. Any number beyond their fingertips is superfluous to the traditional way of life.
The Piraha have no specific words for measuring time or quantity. When indicating some amount, they simply hold their hand palm down, using the space between their hand and the ground to suggest the height of the pile that such a quantity could reach.
In his metaphysics, Aristotle shows that counting requires some prior understanding of what "one" is. To count 5, or 10 or 23 birds, we must first identify one bird, an idea of "bird" that can apply to every possible kind.
The Piraha tell no stories, possess no creation myths. Stories, at least as we understand them, have intervals: a beginning, a middle, and an end. When we tell a story, we recount: naming each interval is equivalent to numbering it. Yet the Piraha talk only of the immediate present. No past impinges on their actions; no future motivates their thoughts. History, they told their American companion, is "where nothing happens, and everything is the same."
Counting people directly is believed by the Kpelle to bring bad luck. In Africa, this taboo is both ancient and widespread.

Classroom Intuitions

To say that a student has no idea of a solution, I realized, is untrue. Truth is, the learner does have ideas, too many, in fact - almost all of them bad. Without the knowledge necessary to eliminate this mental haze, the learner finds himself confronted with an embarrassment of wrong answers to helplessly pick from.

Shakespeare's Zero

Shakespeare became one of the first generation of English schoolboys to learn about the figure zero.

Shapes of Speech

The pythagoreans became the first to understand the world not via tradition (religion), or observation (empirical data), but through imagination - the prizing of pattern over matter.

The Admirable Number Pi

Archimedes knew pi to only three correct places; Newton, almost 2,000 years later, managed sixteen. Only in 1949 did computer scientists discover pi's 1000th digit.

Einstein's Equations

Leibniz wrote that music's pleasure consisted of "unconscious counting" or an "arithmetical exercise of which we are unaware." ie, the numerical rations that underlie all music are grasped intuitively be our minds. Every instant the listener mentally resolves the relation between the various notes as though they were objects all laid out before him side by side in some gigantic illustration. This "grasping" of the music - however fleeting and transient - is something we can all experience as beautiful.
Pythagoras observed that all of music depended on the first four numbers and their interrelations. Ten he worshipped as the most perfect number, reflecting the unity of all things, it being the sum of 1 and 2 and 3 and 4.
Where chaos is subdued and the arbitrary averted, there lies beauty, and it is all around us.

A Novelist's Calculus

Tolstoy: "Kings are the slaves of history, the unconscious swarmlike life of mankind uses every moment of a king's life as an instrument for its purposes." Kinds and commanders and presidents did not interest Tolstoy. History, his history, looks elsewhere: it is the study of infinitely incremental, imperceptible change from one state of being (peace) to another (war).
Change appears to us mysterious because it is invisible. It is impossible to see a tree grow tall or a man grow old, except with the precarious imagination of hindsight. A tree is small, and later it is tall. A man is young, and later he is old. A people are at peace and later they are at war. In each case, the intermediate states are at once infinitely many and infinitely complex, which is why they exceed our finite perception.
Everything has its moment, its context. Earlier, in one state, you began this essay, and now later on you finish it in another. What do you think? I cannot tell you. In everyone and everything, the process of change always asserts its own meaning.

Book of Books

Nabokov compared composing a story with fitting together the pieces of a jigsaw puzzle: "Reality is a very subjective affair... You can get nearer and nearer, so to speak, to reality; but you never get near enough because reality is an infinite succession of steps, levels of perception, false bottoms, and hence unquenchable, unattainable. You can know more and more about one thing but you can never know everything about on thing: it's hopeless."
"A good reader, a major reader, an active and creative reader," says Nabokov, "is a re-reader." Initial readings, he explains, are always laborious, a "process of learning in terms of space and time what the book is about, this stands between us and artistic appreciation."
Flaubert: "How wise one would be if one knew well only five or six books." It seems to me that even this figure is an exaggeration. To learn infinitely many things, we would only ever need perfect knowledge of one book.

Poetry of the Primes

Poetry and prime numbers have this in common: both are as unpredictable, difficult to define, and multiple-meaning as a life.

All Things are Created Unequal

Mathematics and money both originated in abstraction and were invented by the ancient Greeks.
Substituting numbers for objects changed the world, for better or worse. At once everything became quantifiable.

A Model Mother

The best poems, the mathematicians determined, combined in equal parts the predictability of metre with the novelty of unusual words. To much metre made a poem banal; too much freewheeling, on the other hand, rendered it hard to follow. Convention and invention, their delicate balance, give meaning to what we say.
In a sense, we are always evaluating and predicting the other, though we may not heed the act. Often the people we scrutinize hardest are those we cherish the most. In love there is constant contemplation and the most intense desire to understand the object of our affections. There exists melancholy too, as we grow to appreciate just how little we can ever truly know for sure. Our ignorance is painful. And yet we persevere. Humbly, patiently, we assiduously observe till at last we identify ourselves in some way with the other. Anticipation becomes an act of love.

Talking Chess

The grandmaster - like the great artist - is the one who truly explores possibility's outer limits.
The Shannon number, calculated by Claude Shannon, calculates the possible number of 40-move chess games.

Selves and Statistics

The idea of death as a statistical phenomenon amenable to calculation occurs only toward the end of the 17th century, with the publication of the world's first mortality table, "An Estimate of the Degrees of the Mortality of Mankind" by Edmond Halley, in 1693.
The essence of human nature is its endless variety. Stephen Jay Gould: "All evolutionary biologists know that variation itself is nature's only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions."

The Art of Maths

The mathematician takes ideas that are valid in one area and "transplants" them into another hoping that they will take, and not be rejected by the recipient domain. Alain Connes: Metaphors are the essence of mathematical thought.