CHAPTER 1

FAMILY VALUES

In a smallish London suburb where nothing much ever happened, my familygradually became the talk of the town. Throughout my teens, wherever I went, Iwould always hear the same question, "How many brothers and sisters do youhave?"

The answer, I understood, was already common knowledge. It had passed into thetown's body of folklore, exchanged between the residents like a good yarn.

Ever patient, I would dutifully reply, "Five sisters, and three brothers."

These few words never failed to elicit a visible reaction from the listener:brows would furrow, eyes would roll, lips would smile. "Nine children!" theywould exclaim, as if they had never imagined that families could come in suchsizes.

It was much the same story in school. "J'ai une grande famille" wasamong the first phrases I learned to say in Monsieur Oiseau's class. From myfellow students, many of whom were single sons or daughters, the sight of ussiblings together attracted comments that ranged all the way from faint disdainto outright awe. Our peculiar fame became such that for a time it outdid everyother in the town: the one-handed grocer, the enormously obese Indian girl, aneighbor's singing dog, all found themselves temporarily displaced in the localgossip. Effaced as individuals, my brothers, sisters, and I existed only innumber. The quality of our quantity became something we could not escape. Itpreceded us everywhere: even in French, whose adjectives almost always followthe noun (but not when it comes to une grande famille).

With so many siblings to keep an eye on, it is perhaps little wonder that Ideveloped a knack for numbers. From my family I learned that numbers belong tolife. The majority of my math acumen came not from books but from regularobservations and day-to-day interactions. Numerical patterns, I realized, werethe matter of our world. To give an example, we nine children embodied thedecimal system of numbers: zero (whenever we were all absent from a place)through to nine. Our behavior even bore some resemblance to the arithmetical:over angry words, we sometimes divided; shifting alliances between my brothersand sisters combined and recombined them into new equations.

We are, my brothers, sisters, and I, in the language of mathematicians, a "set"consisting of nine members. A mathematician would write:

S = {Daniel, Lee, Claire, Steven, Paul, Maria, Natasha, Anna, Shelley}

Put another way, we belong to the category of things that people refer to whenthey use the number nine. Other sets of this kind include the planets in oursolar system (at least, until Pluto's recent demotion to the status of a non-planet), the squares in a game of x's and o's, the players in a baseball team,the muses of Greek mythology, and the Justices of the U.S. Supreme Court. With alittle thought, it is possible to come up with others, including:

{February, March, April, May, August, September, October, November, December}where S = the months of the year not beginning with the letter J.

{5, 6, 7, 8, 9, 10, Jack, Queen, King} where S = in poker, the possible highcards in a straight flush.

{1, 4, 9, 16, 25, 36, 49, 64, 81} where S = the square numbers between 1 and 99.

{3, 5, 7, 11, 13, 17, 19, 23, 29} where S = the odd primes below 30.

There are nine of these examples of sets containing nine members, so takentogether they provide us with a further instance of just such a set.

Like colors, the commonest numbers give character, form, and dimension to ourworld. Of the most frequent—zero and one—we might say that they are like blackand white, with the other primary colors—red, blue, and yellow—akin to two,three, and four. Nine, then, might be a sort of cobalt or indigo: in a paintingit would contribute shading, rather than shape. We expect to come across samplesof nine as we might samples of a color like indigo—only occasionally, and insmall and subtle ways. Thus a family of nine children surprises as much as a manor woman with cobalt-colored hair.

I would like to suggest another reason for the surprise of my town's residents.I have alluded to the various and alternating combinations and recombinationsbetween my siblings. In how many ways can any set of nine members divide andcombine? Put another way, how large is the set of all subsets?

{Daniel} ... {Daniel, Lee} ... {Lee, Claire, Steven} ... {Paul} ... {Lee,Steven, Maria, Shelley} ... {Claire, Natasha} ... {Anna}...

Fortunately, this type of calculation is very familiar to mathematicians. As itturns out, we need only to multiply the number two by itself, as many times asthere are members in the set. So, for a set consisting of nine members theanswer to our question amounts to: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512.

This means that there existed in my hometown, at any given place and time, 512different ways to spot us in one or another combination. 512! It becomes clearerwhy we attracted so much attention. To the other residents, it really must haveseemed that we were legion.

Here is another way to think about the calculation that I set out above. Takeany site in the town at random, say a classroom or the municipal swimming pool.The first "2" in the calculation indicates the odds of my being present there ata particular moment (one in two—I am either there, or I am not). The same goesfor each of my siblings, which is why two is multiplied by itself a total ofnine times.

In precisely one of the possible combinations, every sibling is absent (just asin one of the combinations we are all present). Strange as it may sound, we caneven define those sets containing no objects. Mathematicians call them an "emptyset." Where sets of nine members embody everything we can think of, touch, orpoint to when we use the number nine, empty sets are all those that arerepresented by the value zero. So while a Christmas reunion in my hometown canbring together as many of us as there are Justices on the U.S. Supreme Court, atrip to the moon will unite only as many of us as there are pink elephants,four-sided circles, or people who have swum the breadth of the Atlantic Ocean.

Our mind uses sets when we think and when we perceive just as much as when wecount. Our possible thoughts and perceptions about these sets can range almostwithout limit. Fascinated by the different cultural subdivisions and categoriesof an infinitely complex world, the Argentine writer Jorge Luis Borges offers amischievously tongue-in-cheek illustration in his fictional Chinese encyclopediaentitled The Celestial Emporium of Benevolent Knowledge:

Animals are classified as follows: (a) those that belong to the Emperor; (b)embalmed ones; (c) those that are trained; (d) suckling pigs; (e) mermaids; (f)fabulous ones; (g) stray dogs; (h) those that are included in thisclassification; (i) those that tremble as if they were mad; (j) innumerableones; (k) those drawn with a very fine camel's-hair brush; (l) et cetera; (m)those that have just broken the flower vase; (n) those that at a distanceresemble flies.

Never one to forgo humor in his texts, Borges here also makes several thought-provoking points. First, though a set as familiar to our understanding as thatof "animals" implies containment and comprehension, the sheer number of itspossible subsets actually swells toward infinity. With their handful of genericlabels ("mammal," "reptile," "amphibious," etc.), standard taxonomies concealthis fact. To say, for example, that a flea is tiny, parasitic, and a championjumper is only to begin to scratch the surface of all its various aspects.

Second, defining a set owes more to art than it does to science. Faced with theproblem of a near endless number of potential categories, we are inclined tochoose from a few—those most tried and tested within our particular culture.Western descriptions of the set of all elephants privilege subsets like "thosethat are very large," and "those possessing tusks," and even "those possessingan excellent memory," while excluding other equally legitimate possibilitiessuch as Borges's "those that at a distance resemble flies," or the Hindu "thosethat are considered lucky."

Memory is a further example of the privileging of certain subsets (ofexperience) 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 ofchocolate cake that he devoured, his wife's enthusiastic embrace, and the pairof 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 hislap) to the peculiar (a sudden hailstorm on the mid-July afternoon that lastedseveral minutes). Most of these subsets, though, would have completely slippedhis mind.

Returning to Borges's list of subsets of animals, several of the categories poseparadoxes. Take, for example, the subset (j): "innumerable ones." How can anysubset of something—even if it is imaginary, like Borges's animals—be infinitein size? How can a part of any collection not be smaller than the whole?

Borges's taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth-century German mathematician whose important discoveries in the study ofinfinity provide us with an answer to this paradox.

Cantor showed, among other things, that parts of a collection (subsets) as greatas the whole (set) really do exist. Counting, he observed, involves matching themembers of one set to another. "Two sets A and B have the samenumber of members if and only if there is a perfect one-to-one correspondencebetween them." So, by matching each of my siblings and myself to a player on abaseball team, or to a month of the year not beginning with the letter J, I amable to conclude that each of these sets is equivalent, all containing preciselynine members.

Next came Cantor's great mental leap: in the same manner, he compared the set ofall natural numbers (1, 2, 3, 4, 5 ...) with each of its subsets such as theeven numbers (2, 4, 6, 8, 10 ...), odd numbers (1, 3, 5, 7, 9 ...), and theprimes (2, 3, 5, 7, 11 ...). Like the perfect matches between each of thebaseball team players and my siblings and me, Cantor found that for each naturalnumber he could uniquely assign an even, an odd, and a prime number. Incredibly,he concluded, there are as "many" even (or odd, or prime) numbers as all thenumbers combined.

Reading Borges invites me to consider the wealth of possible subsets into whichmy family "set" could be classified, far beyond those that simply point tomultiplicity. All grown up today, some of my siblings have children of theirown. Others have moved far away, to the warmer and more interesting places fromwhere postcards come. The opportunities for us all to get together are rare,which is a great pity. Naturally I am biased, but I love my family. There is alot of my family to love. But size ceased long ago to be our definingcharacteristic. We see ourselves in other ways: those that are studious, thosethat prefer coffee to tea, those that have never planted a flower, those thatstill laugh in their sleep ...

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.

CHAPTER 2

ETERNITY IN AN HOUR

Once upon a time I was a child who loved to read fairy tales. Among my favoriteswas "The Magic Porridge Pot" by the Brothers Grimm. A poor, good-hearted girlreceives from a sorceress a little pot capable of spontaneously concocting asmuch sweet porridge as the girl and her mother can eat. One day, after eatingher fill, the mother's mind goes blank and she forgets the magic words "Stop,little pot."

So it went on cooking and the porridge rose over the edge, and still it cookedon until the kitchen and whole house were full, and then the next house, andthen the whole street, just as if it wanted to satisfy the hunger of the wholeworld.

Only the daughter's return home, and the requisite utterance, finally brings thegooey avalanche to a belated halt.

The Brothers Grimm introduced me to the mystery of infinity. How could so muchporridge emerge from so small a pot? It got me thinking. A single flake ofporridge was awfully slight. Tip it inside a bowl and one would probably noteven spot it for the spoon. The same held for a drop of milk, or a grain ofsugar.

What if, I wondered, a magical pot distributed these tiny flakes of porridge anddrops of milk and grains of sugar in its own special way? In such a way thateach flake and each drop and each grain had its own position in the pot,released from the necessity of touching. I imagined five, ten, fifty, onehundred, one thousand flakes and drops and grains, each indifferent to the next,suspended here and there throughout the curved space like stars. More porridgeflakes, more drops of milk, more grains of sugar are added one after another tothis evolving constellation, forming microscopic Big Dippers and minuscule GreatBears. Say we reach the ten thousand four hundred and seventy-third flake ofporridge. Where do we include it? And here my child's mind imagined all the tinygaps—thousands of them—between every flake of porridge and drop of milk andgrain of sugar. For every minute addition, further tiny gaps would continue tobe made. So long as the pot magically prevented any contact between them, everynew flake (and drop and granule) would be sure to find its place.

Hans Christian Andersen's "The Princess and the Pea" similarly sent my mindspinning toward the infinite, but this time, an infinity of fractions. Onenight, a young woman claiming to be a princess knocks at the door of a castle.Outside, a storm is blowing and the pelting rain musses her clothes and turnsher golden hair black. So sorry a sight is she that the queen of the castledoubts her story of high birth. To test the young woman's claim, the queendecides to place a pea beneath the bedding on which the woman will sleep for thenight (princesses being most delicate creatures!). Her bed is piled to a heightof twenty mattresses. But in the morning the woman admits to having hardly slepta wink.

The thought of all those tottering mattresses kept me up long past my ownbedtime. By my calculation, a second mattress would double the distance betweenthe princess's back and the offending pea. The tough little legume wouldtherefore be only half as prominent as before. Another mattress reduces thepea's prominence to one-third. But if the young princess's body is sensitiveenough to detect one-half of a pea (under two mattresses) or one-third of a pea(under three mattresses), why would it not also be sensitive enough to detectone-twentieth? In fact, possessing limitless sensitivity (this is a fairy taleafter all), not even one-hundredth, or one-thousandth or one-millionth of a peacould be tolerably borne.

Which brings us back to the Brothers Grimm and their tale of porridge. For theprincess, even a pea felt infinitely big; for the poor daughter and her mother,even an avalanche of porridge reduced to the infinitesimally small.

"You have too much imagination," my father said when I shared these thoughtswith him. "You always have your nose in some book." My father kept a pile ofpaperbacks and regularly bought the weekend papers, but he was never aparticularly enthusiastic reader. "Get outdoors more—there's no good beingcooped up in here."

Hide-and-seek in the park with my brothers and sisters lasted all of tenminutes. The swings held my attention for about as long. We walked the perimeterof the lake and threw breadcrumbs out onto the grimy water. Even the duckslooked bored.

Games in the yard offered greater entertainment. We fought wars, cast spells,and traveled back in time. In a cardboard box we sailed along the Nile; with abedsheet we pitched a tent in the hills of Rome. At other times, I would simplystroll the local streets to my heart's content, dreaming up all manner of newadventures and imaginary expeditions.