Martin Kruskal, a mathematician at Rutgers university in New Jersey, has two brothers, both of whom are also mathematicians. When my older brother’s oldest child was five, Kruskal recalls, he argued with another little boy about whether there was a largest number. No doubt they were talking about the counting numbers--1, 2, 3, and so on. My nephew said there wasn’t, and his friend said there was. The next day my nephew went back to his little friend and said, ‘I asked my father about it, and he’s a mathematician, and he says there isn’t any largest number.’ And the other little boy said, ‘Well, I asked my father about it, and he says there is, and he’s a lawyer.’
Kruskal père and fils are right, of course. (So sue them.) Their assertion is easily demonstrated--whenever you think you’ve found the biggest number, just add one to it, and the new number will be bigger still.
Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it? What do you get if you divide the number 1 into infinitely many parts of equal size? And if you multiply that tiny answer by itself, will the result be bigger or smaller?
Two decades ago, using only a few simple rules and concepts, British mathematician John Conway figured out how to wring meaning from apparently nonsensical questions like these. He did it by developing a technique that produced every number you’ve ever imagined, plus untold zillions more--the surreal numbers. Conway didn’t come up with the catchy name. That was the creation of Stanford computer scientist Donald Knuth, now retired, who wrote a kooky little novella about the numbers after hearing Conway describe them in an informal talk. Knuth’s book was the first thing that appeared about these numbers, says Conway. I’m always very lazy about publishing things.
These strange denizens of the number line lurk among its crevices, so enormous or tiny that previous mathematicians never knew they were there. Yet Conway found them, and he found ways to use them in arithmetic just like their ordinary numerical neighbors. Ever since that momentous discovery, Kruskal has been busy extending and refining Conway’s work. Eventually, if all goes well, Kruskal’s efforts will leave telltale marks all over the mathematical landscape.
In some ways, Conway and Kruskal are an odd couple to be working on the same set of problems. Kruskal, now 70, founded and chaired Princeton’s applied mathematics program and worked in the Princeton plasma physics lab for years before moving to Rutgers. He describes his mathematical style as one of dogged persistence, asking questions and not being satisfied with superficial answers. He has a breathtakingly orderly mind; he keeps tabs on all his tangents, closes all his brackets, and illustrates every point. His favorite phrase is Let me give you a simple example....
Conway, in contrast, calls himself a mathematical magpie. I like things that shine, he says, and that involves quite often that they’re a bit trashy. The magpie just picks up a piece of plastic that’s covered in gold. I have taste, but I don’t exercise it very frequently. So I’m just as likely to be doing something that isn’t really worth doing as something that is. That statement gives off a whiff of disingenuousness--after all, Conway is, at 57, one of the most respected mathematicians alive. Kruskal says of his younger colleague, He’s just an ordinary human man, you understand, and he has his foibles, no doubt, but he awes me. I almost revere him for what he’s done, and for his tremendous insight. It’s a pleasure to talk to him. He understands quicker than anybody.
The two do share some traits: for example, they both started their mathematical careers early. Conway grew up in Liverpool (where his father taught high school chemistry to two of the Beatles). When I was four, he says, my mother found me reciting the powers of two. Kruskal grew up in New Rochelle, New York, where his older brother taught him algebra when he was six. The bulldog and the magpie also share an interest in origami and a passion for the simplicity of the surreal numbers, the sense they give that there’s order in the world.
To understand what these new numbers are all about, it helps to be acquainted with the old ones. At the end of the nineteenth century, mathematicians and philosophers worked urgently to give mathematics a rigorous underpinning by codifying many notions that previous generations had understood only intuitively. One of the most important of these notions was that of numbers: What exactly are they, and how can they be created and used? The mathematicians set themselves the task of building numbers with as few tools as possible.
Historically speaking, first came the natural, or counting, numbers--1, 2, 3, and so on--which caused Kruskal’s nephew such strife. Whenever you add two natural numbers, the result will be another natural. Mathematicians call this quality closure, and it gives them a warm, safe feeling. Closure is necessary for their arithmetic operations to make sense. The natural numbers are closed under addition, but the same can’t be said for subtraction: although 7 minus 5 is the natural number 2, 5 minus 7 yields no positive whole number.
To get around this difficulty, mathematicians threw in a bunch of extra numbers: the negatives (useful for describing debts) and zero. Together, the naturals, the negative whole numbers, and zero make up a set of numbers called the integers. Whenever you add, subtract, or multiply any two integers, you get another integer. So far, so good. But what happens when you try to divide them? Sometimes there’s no trouble: -34 divided by 2 is -17. But -17 divided by 2 isn’t an integer.
What now? Throw in more numbers, of course. Mathematicians dreamed up the rationals, useful for describing slices of pie or kingdom. These are fractions created by dividing any integer by any other integer except zero. (Every integer is a rational number--just divide it by 1.) With the rationals, you can perform arithmetic freely, but algebra is another matter. The rationals aren’t closed under algebraic functions such as taking square roots. (According to myth, when the Greek philosopher Pythagoras proved that the square root of 2 could not be written as the quotient of two integers, he was so disappointed he threw himself off a cliff.) Clearly what was needed was more numbers.
Next came the reals, which can be thought of as points on the number line. These consist of the rationals and the irrationals--numbers like the square root of 2, and pi, whose decimal expansion never ends and never trails off in a fixed, repeating pattern. When you write a rational number in decimal form, it will either terminate after a finite string of digits or it will repeat some pattern of digits forever.
Of course, the reals aren’t quite closed under algebraic functions, since negative numbers have no real square roots (a negative times a negative is positive). Mathematicians got around this by imagining a number that when multiplied by itself yields -1. This number, called i, gives rise to two more systems, the imaginary and complex numbers. But we won’t really be concerned with those systems, because unlike the others they’re not totally ordered.
In a totally ordered set of numbers, if you look at any pair of numbers, one will be greater than the other, and if one number is greater than another and that other is greater than a third, then the first number will also be greater than the third number. Like many mathematical properties, total ordering may seem more clear-cut than it really is. Suppose, for example, that whenever you and I play Ping-Pong, I win hands down. Whenever you and Ernestine play, you slaughter her. But Ernestine has a wicked serve I just can’t return, so whenever I’m up against her, I throw down my paddle in despair after five minutes. Who’s the best Ping-Pong player? Clearly none of us. So mathematicians would say that you, Ernestine, and I are not totally ordered under winning at Ping-Pong.
The integers, the rationals, and the reals are all totally ordered. Some of these numbers may seem kind of kinky, impossible to write without an infinitely long string of decimals. They may seem very big or very small. (Four is an extremely large number, remarks Conway. I have four daughters--not to mention two sons. Kruskal, a grandfather of triplets, comments that three can sometimes seem surreal.) But at least they’re all finite. In the 1870s a German mathematician named Georg Cantor discovered a new system of numbers, called cardinals and ordinals, which for the first time, as Kruskal puts it, include actually infinite individual numbers. Before Cantor, people thought of infinity as an undifferentiated vastness. With his ordinals and cardinals, however, Cantor showed that the vastness has structure. Some infinite numbers are bigger than others. For example, there’s an infinite number of integers and also an infinite number of points on the real number line. But the number of points is bigger than the number of integers.
Though most people have come across integers, rationals, and real numbers, perhaps in high school trig, perhaps at the grocery checkout counter, ordinals are not so familiar. But they’re vital to surreal numbers. Says Conway proudly, The real numbers are to the ordinary integers as the surreal numbers are to the ordinal numbers. It’s really quite something to have discovered the correct extensions, the correct analogue, of these things, these infinite integers. Cantor discovered infinite integers, and a century later I discovered infinite fractional numbers.
Cantor devised his ordinals to describe the sizes and orders of certain sets, which are the basic ingredient in set theory, the field he pioneered. A set, in Cantor’s time, meant any collection of distinct objects of any sort--people, pencils, numbers. To get an ordinal number, Kruskal says, you take an ordered set and you throw away everything else, all the special relationships among its elements, and you’re still left with the order and how many elements there are in the set. So the ordinal number is the prototypical ordered set--it’s just the set with all properties washed out except the total size and order.
The ordinal numbers of finite sets are simply the natural numbers; the ordinal numbers of infinite sets are called transfinite ordinals. The ordinal that describes the size and order of the set of natural numbers is called omega. So far, no problem. But if you’re planning to do much arithmetic with these babies, you’ll find they’re a washout. Adding finite ordinals works fine--you’re just adding ordinary integers. But what if you want to add a transfinite ordinal (call it n) to a finite one (call it m)? The transfinite ordinal is so much bigger than the finite one that it swallows it whole, leaving no trace; m+n=n. Cantor’s ordinals are arithmetically limited, like the natural numbers, says Kruskal ruefully. We’re back to square one!
When in doubt, throw in more numbers--the surreals, this time. In the whole intellectual history of humankind, says Kruskal, there have been only a handful of genuine totally ordered number systems: the naturals, the integers, the rationals, the cardinals, and the ordinals. In one swoop, the ordinals are extended to arithmetic and algebraic freedom in a way that was won painfully step by step through the earlier systems!
How do the surreals work? Imagine, as posture specialists like to suggest, that there’s a string attached to the top of your head, and you’re dangling from it. Feel those vertebrae line up in a nice, straight row? Now imagine that your spine, instead of stopping at your skull and your coccyx, goes on forever along that string, extending above your head past the farthest star and below your feet through the Earth and beyond, never stopping. Got that? Good. Now imagine that there’s one small problem: somewhere along your spine, you have an itch you can’t reach.
I’ll scratch your back if you scratch mine, offers a friendly alien, positioning its fingernail on the vertebra between your shoulder blades. Just tell me where.
Up, you say. The alien moves up a vertebra, but it hasn’t yet reached the itch. Up, up, up, up, up, you repeat, and each up inches the alien fingernail one vertebra higher. After 40 or so ups, you sense the alien is getting close, but at the forty-fourth vertebra, you know it’s gone too far. The itch is between vertebra 43 and vertebra 44. Down, you say. The alien moves its fingernail down half a notch, to the point midway between vertebra 43 and vertebra 44. Unfortunately, it’s gone past the itch again. Up, you direct it. It goes up a quarter of a notch this time, to the midpoint between its former spot and vertebra 44, reaching the place three-quarters of the way between vertebra 43 and vertebra 44. Right there, you say, and the alien scratches. Ah!
The spot the alien just scratched is designated by the directions you gave the alien--44 ups, followed by 1 down and 1 up. The sequence that replaces each up with a down and vice versa--that is, 44 downs followed by 1 up and 1 down--would take the alien’s fingernail to the spot 43¾ vertebrae below the place between your shoulder blades where it started. The sequence of no ups or downs would tell the alien to scratch that original spot. Up, down, down, up, down, up would take the alien’s nail to the spot 11/32 of the way between the original spot between the shoulder blades and the first vertebra.
Which itches will you be able to direct the alien to in a finite amount of time (that is, by saying up and down a finite number of times)? Whenever you say up or down, the alien moves its fingernail one vertebra, or half the distance between two vertebrae, or a quarter of the distance, or an eighth of the distance, or a sixteenth of the distance, and so on. So the finitely scratchable itches will be the ones that occur at the integers or at the dyadic fractions--fractions with a power of 2 in the denominator.
Now suppose you have an itch two-thirds of the way between vertebrae 2 and 3. Up, up, up, you say. That takes you to vertebra 3. Down, you add. That takes you to 2½, which is not far enough. Another up takes you to 2¾--too far. Another down takes you to 2⅝, too low again. Up down, up down, up down, you say, each time getting closer but never quite achieving that two-thirds mark. To reach it, you’ll have to repeat up down, up down, up down forever. At the end of eternity, after infinitely many up downs, the alien’s fingernail scratches the spot on your back two-thirds of a vertebra past vertebra 2. Ah!
If your spine is a number line, the itchy spots on your back are numbers, and since each set of directions you give the alien--those strings of ups and downs--designates one and only one itchy spot, you can think of the directions as numbers, too. These are the surreal numbers: numbers that can be designated by a string of ups and downs some ordinal number in length. Every integer is a rational number and every rational number is a real number, and in the same way, every real number is also a surreal number. However, there are many, many remarkable surreal numbers that aren’t real.
Just as you can write rational numbers as infinitely repeating decimals, you can designate any nondyadic, rationally placed itch by directing the alien with a unique, infinitely repeating string of ups and downs. (And just as you can write irrational real numbers as infinitely long, nonrepeating decimals, you can designate real, irrational itches by using infinitely long, nonrepeating strings of ups and downs.) But not all infinitely repeating strings of ups and downs designate rational numbers.
Suppose, for example, you have an itch way, way up at the end of your infinitely long spine. Up, you tell the alien. Up, up, up, up, up. You go on repeating up forever, and when forever is over, the alien scratches. That spot is called omega (and designated by the Greek letter w), just like Cantor’s ordinal that describes the collection of all positive integers. It’s not a rational number or a real number; it’s too big. Omega is the simplest surreal number larger than all real numbers.
Now suppose you have an itch the barest smidgen above zero. (Think of zero as the shoulder-blade spot where the alien’s fingernail starts out.) Up, you say. The nail moves to vertebra 1, which is way too far. Down, you say, down, down, down. You go on saying down, and each time, the alien moves its nail closer to zero, but it doesn’t quite reach the itch. You go on repeating down forever. When forever is over, the alien scratches a spot that’s above zero but beneath every positive real number. Kruskal calls that spot iota (ι); it’s the simplest infinitesimal surreal, the simplest number larger than zero but less than all the positive reals.
Two ups followed by infinitely many (that is, omega) downs will tell the alien to scratch you on the spot iota above your first vertebra. One down followed by omega ups will get you a scratch iota below the zero spot. Up, up, down, up followed by omega downs will get you a scratch iota above 1½. And so on.
Kruskal writes these strings of ups and downs using arrows pointing up and down; for example, he’d write the number ¾, up down up, like so: ↑↓↑. When a particular segment of ups and downs is repeated omega times, he puts a round cap over it, so omega would be ↑(cap), ⅓ would be ↑↓↑(cap)↓(cap), and minus iota would be ↓↑(cap).
Now it’s the alien’s turn to have an itchy spine. It reminds you of your half of the bargain. The alien’s anatomy is rather more complex than yours, though. For one thing, it has infinitely many infinitely long spines. Up, up, up, it tells you forever, and when forever is over, you arrive at the alien’s first omega spot, ↑(cap). You’re all ready to scratch, but the alien stops you. No, not quite there. Up another vertebra.
To get to the omega spot, you had to go past all the vertebrae on the alien’s first spine, so those instructions take you to its next one. (In fact, it’s useful to think of the first spine’s omega spot as the zero spot on the second spine.) The itchy place the alien specified, ↑(cap)↑, is the first positive vertebra on that new spine. Now suppose that once you arrived at omega, the alien had said down instead; that spot would be the first negative vertebra on the second spine. If the alien told you omega ups, followed by another omega ups, followed by an extra up, ↑(cap)↑(cap)↑, you’d reach the first positive vertebra on the third spine. Just as one up followed by one down gives you ½, omega ups followed by omega downs, ↑(cap↓(cap), gives you half omega. And omega ups, followed by omega ups, followed by omega ups, and so on omega times--written ↑(cap)↑(cap)↑(cap)...=↑(2cap)--will send your fingernail past all the vertebrae on all omega spines of the alien’s back.
Now what? What happens when you’ve gone beyond all omega spines? Well, did I mention that the alien has infinitely many backs? Omega of them, to be precise. To send you to a spot, say, one-third of the way between the second and third vertebrae on the negative fifth spine on the eighth back, the alien has but to tell you
Oh, and by the way, the alien has omega bodies with omega backs on each. Actually, it consists of omega populations of omega bodies, each with omega backs. Also, there are omega collections of omega populations of--well, you get the idea. There’s no end to the alien’s itchy back.
As you can tell by picturing that back, the surreals have a somewhat involved structure. For one thing, unlike the real number line, which has no holes in it, the surreal number line is riddled with gaps. There’s a gaping hole between the finite numbers and the positive infinites, for example. (That particular hole is what people often mean when they talk about infinity.) There’s another gap between the infinitesimals and the positive real numbers. There’s another that separates zero from every positive number. (That one’s not quite a gap; I call it a cut, says Kruskal. It acts a lot like zero, but it isn’t really a number.) These gaps and cuts can be represented by arrow sequences of length greater than any ordinal number, and they’re everywhere. You can find one between any pair of numbers.
For the surreal numbers to be more use than, say, the ordinals, we have to be able to perform arithmetic with them. That means defining familiar operations--such as addition, subtraction, multiplication, division, exponentiation, logarithms, and so on--so that they work as well for surreal numbers as they do for the usual reals. A concept that Kruskal calls earliness is what makes such definitions possible.
Take a surreal number and chop it off after a certain number of ups and downs. Throw out the ups and downs that come after the spot where you chopped. The number you’re left with is called a truncate of the first number. It may have an infinite number of arrows, of course: for example, ↑(cap) is a truncate of ↑(cap)↑↑↑. One surreal number is said to be earlier than another if it’s a truncate of the other.
Earlier doesn’t necessarily mean smaller. One number is defined as smaller than another if they’re not equal, and at the first place where they differ, it either stops or has a down where the other either stops or has an up. As you can see if you think about the alien’s back, ↑↑ is earlier than ↑↑↓↓↓, but bigger. And given a pair of numbers, it needn’t be the case that one is earlier than the other--think of ↑↓↓↑(cap) and ↑↓↑↓. So the surreal numbers aren’t totally ordered under earliness, they’re only partially ordered. However, it’s easy to see that if one number is earlier than another, its arrow sequence will be shorter.
The earliest number is the empty sequence, zero. You could get zero by chopping off any number before it started. The two next earliest are up and down--namely 1 and -1--since every arrow sequence starts with either up or down. Then come up up, up down, down up, and down down--that is, 2, ½, -½, and -2. And so on.
Earliness provides the surreal numbers with an order that’s independent of size. The earliness ordering has another important quality: every nonempty class of numbers has at least one earliest number (remember, not all numbers in a class can be compared for earliness--two or more numbers could be the earliest in the class). This property makes it possible for mathematicians to use the earliness ordering to find definitions and proofs and carry out other vital mathematical activities, relying on a method called transfinite induction. The principle of induction says: If whenever a statement is true for every number earlier than x it’s also true for x, then it’s true for all the surreal numbers.
Conway, Kruskal, and their colleagues used induction to define all the important operations from addition on up, and even one from calculus--differentiation. For ordinary, real numbers, these definitions give exactly the same results as conventional definitions, but, as Kruskal points out, they’re far, far simpler. And as a bonus, they also make sense for the new, wacky surreals, like omega and iota.
The definitions allow surrealists to ask questions like: Which surreal numbers can be considered analogous to the integers? Which, if any, are prime? Which are rational--that is, which can be written as the quotient of two surreal integers? That last question has a surprising answer. Conway and Kruskal each came up with a reasonable definition for surreal integers, definitions that turn out to be equivalent, and by those definitions, any real number times omega is a surreal integer. So every real number is rational in the surreal system, even the square root of 2--just write it as (√2ω)/ω. If only someone had told Pythagoras.
It wasn’t until Kruskal and his colleagues tried to define integration, an operation from calculus that tells you how much area the graph of a function encloses, that they ran into trouble. Our definition works for real-valued functions but not for surreal ones, because of the gaps in the surreal number line, says Kruskal. I’m working to fix it, but I don’t see any easy way to do so.
Martin’s project, if it succeeds, would be absolutely fabulous, says Conway. This will allow him to find exact solutions to equations that so far have had only approximate solutions.
Even without a working definition of integration, there’s a lot Kruskal can do with surreals. (You might need a bit of math beyond counting and scratching to understand his applications, so don’t worry too much about the details.) Since one of the main purposes of numbers is to measure quantities, for example, you’d expect a new bunch of numbers would allow you to measure things that never had a good yardstick before. Indeed, the surreal numbers turn out to be just the thing for evaluating how quickly functions grow. Recall from high school algebra that the graphs of certain functions head off to infinity when they reach a particular point on the x axis. But how soon do they get there? The surreal numbers, with their subtle varieties of infinite numbers, are a perfect tool for studying these asymptotic functions.
Another trick mathematicians find immensely profitable is breaking down numbers or functions into simpler components. By writing a quantity that’s difficult to get a handle on as a sum of smaller and smaller terms, you can get closer and closer approximations to the difficult quantity. When he began working on surreal numbers, Conway made a stab at describing their anatomy; later Kruskal came up with a powerful, intricate, and precise way to dissect them, based on certain surreal numbers that can’t be broken down any further. These numbers, which he calls travagances, come in two kinds: extravagances and intravagances.
An extravagance is a positive number that can’t be arrived at by performing finitely many algebraic, logarithmic, or exponential operations on earlier numbers. The earliest extravagance is omega: no amount of multiplying a finite number of finite numbers together, however big they are, or raising finite numbers to finite powers over and over again a finite number of times, or anything like that will get you to omega. And of course, there are lots--and lots and lots--of others beyond omega. If an extravagance is an enormously large (or small) number, an intravagance can be thought of as an enormously medium number, one very, very, very deeply embedded between two other numbers.
Kruskal uses travagances to study what he calls nice functions, functions of the sort that you can describe using a familiar-looking formula consisting of plain old operations. What happens to these functions when they hit surreal numbers? What about when they run into gaps in the surreal number line? Since one such gap is infinity, this is a fairly significant question. People are always trying to figure out exactly what various functions do when they cross infinity. To find answers, Kruskal plugs in a carefully chosen extravagance, which he calls a ghost. A ghost stands in for a gap in the surreal number line, but since it’s actually a number, you can manipulate it as you would any other number.
When Conway discovered the surreal numbers, he hadn’t gone looking for them. The surreal numbers actually came from games, he explains. I was trying to understand how to play go--the Japanese game, played with stones on a grid. We had the British go champion in our math department at Cambridge, and I used to watch him play games in the vain hope that someday I would understand them. I never did. But I did see that in the end, a go game decomposed into a sum of little games, and I thought it was a good idea to study this kind of sum. You could do the same thing with other games--checkers, dominoes. And then I discovered that certain games behaved very much like numbers.
My really big discovery with these games was that this was a new way of defining numbers--not only of defining new numbers, but of defining all the old ones too. And it’s much simpler than the traditional way.
Conway is tickled at the prospect that his numbers could have practical applications, particularly if he doesn’t have to come up with them himself--which might interfere with the image he projects of a mad, impractical genius. It’s half pose, half philosophy. Every surface of his office is covered with geometric puzzles and toys. He has a theory about daffodils, another about tennis balls. He collects etymologies--The word number itself is connected with a large number of other words, like economy, nimble, nemesis, numb. Tell him the date you were born, and he’ll tell you the phase of the moon and the day of the week in under two seconds. Every year I try to double my speed.
Why does he do it? Showing off, yes? But not entirely, because a lot of the things I do I don’t get much occasion to show off. For instance, my latest fad is factorizing four-digit numbers, and even people with the best will in the world are not willing to sit there for hours with you producing new four-digit numbers and listen to you go, ‘This is 69 times 38,’ and so on. I show it off to myself, really. It gives me a lovely feeling, which is akin to power. It’s also a feeling that all’s well in the world. One of my crazy ideas was to learn the names of all the stars in the sky. I don’t mean the constellations, I mean the individual stars. And so I spent some time doing it. It took about a year. I remember one great time when there was a little cloud, and I thought, I should know what’s behind that cloud. Three stars in a triangle. Well, it cleared away, and there they were. Knowing what the situation is going to be is very much like having ordered it.
Certainly practicing mathematics has its moments of elation and strong emotional moments, and understanding is one of the crucial ones, agrees Kruskal. When you have an insight--that’s one of the great things about it, it’s one of the few concrete rewards on an emotional level.
The surreal numbers yield him those rewards. For the first 20 years this has been a marvelous number system, better than the usual numbers. It is, I can’t help it. I can say so without boasting because I didn’t discover it--Conway did. It really is a major improvement in our view of numbers, and it’s just a matter of time before the world of mathematics recognizes it.