Thinking in Numbers: How Maths Illuminates Our Lives (2 page)

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!’ they would exclaim, as if they had never imagined that families could come in such sizes.

It was much the same story in school. ‘
J’ai une grande famille
,’ was among the first phrases I learned to say in Monsieur Oiseau’s class. From my fellow students, many of whom were single sons or daughters, the sight of us together attracted comments that ranged all the way from faint disdain to outright awe. Our peculiar fame became such that for a time it outdid every other in the town: the one-handed grocer, the enormously obese Indian girl, a neighbour’s singing dog, all found themselves temporarily displaced in the local gossip. Effaced as individuals, my brothers, sisters and I existed only in number. The quality of our quantity became something we could not escape, it preceded us everywhere: even in French, whose adjectives almost always follow the 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 I developed a knack for numbers. From my family I learned that numbers belong to life. The majority of my maths came not from books but from regular observations and interactions day to day. Numerical patterns, I realised, were the matter of our world. To give an example, we nine children embodied the decimal system of numbers: zero (whenever we were all absent from a place) through to nine. Our behaviour even bore some resemblance to the arithmetical: over angry words, we sometimes divided; shifting alliances between my brothers and 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 when they use the number nine. Other sets of this kind include the planets in our solar system (at least, until Pluto’s recent demotion to the status of a non-planet), the squares in a game of noughts and crosses, the players in a baseball team, the muses of Greek mythology and the Justices of the US Supreme Court. With a little 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 high cards 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 taken together they provide us with a further instance of just such a set.

Like colours, the commonest numbers give character, form and dimension to our world. Of the most frequent – zero and one – we might say that they are like black and white, with the other primary colours – red, blue and green – akin to two, three and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a colour like indigo – only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-coloured 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 recombinations between my siblings. In how many ways can any set of nine members divide and combine? 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 it turns out, we need only to multiply the number two by itself, as many times as there are members in the set. So, for a set consisting of nine members the answer 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, 512 different ways to spot us in one or another combination. 512! It becomes clearer why we attracted so much attention. To the other residents, it really must have seemed that we were legion.

Here is another way to think about the calculation that I set out above. Take any 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 at a particular moment (one in two – I am either there, or I am not). The same goes for each of my siblings, which is why two is multiplied by itself a total of nine times.

In precisely one of the possible combinations, every sibling is absent (just as in one of the combinations we are all present). Mathematicians call such collections without members an ‘empty set’. Strange as it may sound, we can even define those sets containing no objects. Where sets of nine members embody everything we can think of, touch or point to when we use the number nine, empty sets are all those that are represented by the value zero. So while a Christmas reunion in my hometown can bring together as many of us as there are Justices on the US Supreme Court, a trip 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.

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. Fascinated by the different cultural subdivisions and categories of an infinitely complex world, the Argentinian writer Jorge Luis Borges offers a mischievously tongue-in-cheek illustration in his fictional Chinese encyclopaedia entitled
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 this classification; (i) those that tremble as if they were mad; (j) innumerable ones; (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 distance resemble flies.

Never one to forego humour in his texts, Borges here also makes several thought-provoking points. First, 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. With their handful of generic labels (‘mammal’, ‘reptile’, ‘amphibious’, etc.) standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic and a champion jumper, 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 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. Western descriptions of the set of all elephants privilege subsets like ‘those that are very large’, and ‘those possessing tusks’, and even ‘those possessing an excellent memory’, while excluding other equally legitimate possibilities such as Borges’s ‘those that at a distance resemble flies’, or the Hindu ‘those that are considered lucky’.

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.

Returning to Borges’s list of subsets of animals, several of the categories pose paradoxes. Take, for example, the subset (j): ‘innumerable ones’. How can any subset of something – even if it is imaginary, like Borges’s animals – be infinite in 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 of infinity provide us with an answer to the paradox.

Cantor showed, among other things, that parts of a collection (subsets) as great as the whole (set) really do exist. Counting, he observed, involves matching the members of one set to another. ‘Two sets
A
and
B
have the same number of members if and only if there is a perfect one-to-one correspondence between them.’ So, by matching each of my siblings and myself to a player in a baseball team, or to a month of the year not beginning with the letter J, I am able to conclude that each of these sets is equivalent, all containing precisely nine members.

Next came Cantor’s great mental leap: in the same manner, he compared the set of all natural numbers (1, 2, 3, 4, 5 . . .) with each of its subsets such as the even numbers (2, 4, 6, 8, 10 . . .), odd numbers (1, 3, 5, 7, 9 . . .), and the primes (2, 3, 5, 7, 11 . . .). Like the perfect matches between each of the baseball team players and my siblings and me, Cantor found that for each natural number 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 the numbers combined.

Reading Borges invites me to consider the wealth of possible subsets into which my family ‘set’ could be classified, far beyond those that simply point to multiplicity. All grown up today, some of my siblings have children of their own. Others have moved far away, to the warmer and more interesting places from where 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 a lot of my family to love. But size ceased long ago to be our defining characteristic. We see ourselves in other ways: those that are studious, those that prefer coffee to tea, those that have never planted a flower, those that still 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.

Eternity in an Hour

Once upon a time I was a child, and I loved to read fairy tales. Among my favourites was ‘The Magic Porridge Pot’ by the Brothers Grimm. A poor, good-hearted girl receives from a sorceress a little pot capable of spontaneously concocting as much sweet porridge as the girl and her mother can eat. One day, after eating her 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 cooked on until the kitchen and whole house were full, and then the next house, and then the whole street, just as if it wanted to satisfy the hunger of the whole world.’

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

The Brothers Grimm introduced me to the mystery of infinity. How could so much porridge emerge from so small a pot? It got me thinking. A single flake of porridge was awfully slight. Tip it inside a bowl and one would probably not even spot it for the spoon. The same held for a drop of milk, or a grain of sugar.

What if, I wondered, a magical pot distributed these tiny flakes of porridge and drops of milk and grains of sugar in its own special way? In such a way that each 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, one hundred, one thousand flakes and drops and grains, each indifferent to the next, suspended here and there throughout the curved space like stars. More porridge flakes, more drops of milk, more grains of sugar are added one after another to this evolving constellation, forming microscopic Big Dippers and minuscule Great Bears. Say we reach the ten thousand four hundred and seventy-third flake of porridge. Where do we include it? And here my child’s mind imagined all the tiny gaps – thousands of them – between every flake of porridge and drop of milk and grain of sugar. For every minute addition, further tiny gaps would continue to be made. So long as the pot magically prevented any contact between them, every new flake (and drop and granule) would be sure to find its place.

Hans Christian Andersen’s ‘The Princess and the Pea’ similarly sent my mind spinning towards the infinite, but this time, an infinity of fractions. One night, 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 turns her golden hair to black. So sorry a sight is she that the queen of the castle doubts her story of high birth. To test the young woman’s claim, the queen decides to place a pea beneath the bedding on which the woman will sleep for the night. Her bed is piled to a height of twenty mattresses. But in the morning the woman admits to having hardly slept a wink.

The thought of all those tottering mattresses kept me up long past my own bedtime. By my calculation, a second mattress would double the distance between the princess’s back and the offending pea. The tough little legume would therefore be only half as prominent as before. Another mattress reduces the pea’s prominence to one third. But if the young princess’s body is sensitive enough 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 detect one-twentieth? In fact, possessing limitless sensitivity (this is a fairy tale after all), not even one-hundredth, or one-thousandth or one-millionth of a pea could be tolerably borne.