The Truth about Counting
“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”
“I don’t know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction?”
-Lewis Carroll, Through the Looking Glass
The hour was cool, and the sage Kronos and his eager young mathematical protégé Simplicius had eaten their fill and drunk their wine. But the meal had provided food for the mind as well as the body. Kronos had let fall, over sips of the strongest coffee Simplicius had ever tasted, that he had begun to think about a new and radical way of understanding numbers. For reasons he had not yet explained, he had taken to call his ideas non-Euclidean arithmetic. As the two strolled into the garden, a dialogue began, and it led in such provocative directions that, when the two parted, Simplicius rushed home to write down what he could remember of the exchange. In what follows, Simplicius and I have tried to edit those recollections for clarity, but the words remain, as much as memory can make them, the words of the mathematically minder philosopher and his curious apprentice.
SIMPLICIUS. Let’s start with the obvious questions. What is non-Euclidean arithmetic?
KRONOS. That depends on your point of view. Some people might consider it simply an unorthodox new way of thinking about ordinary numbers. Others might look at it and see the death of God.
SIMP. The death of God?!
KRON. Or at any rate, the death of the kind of quasi-religious thinking with which mathematicians approach their work. Ask them about the nature of the objects they study – numbers, points, lines, sets, spaces – and you’ll get an entire theology. The doctrine is Platonism: the idea that certain ideal objects (in this case, mathematical ones) are “out there” somewhere, existing prior to human beings and their culture, untouched by change, independent of energy and matter, beyond the confines and necessities of space and time, and yet somehow accessible to the minds of mathematicians.
SIMP. Do mathematicians really believe that? Or is it just a way of talking that lets them get on with their research?
KRON. They believe it, deeply. In 1996, in the Times Literary Supplement, our friend Brian Rotman reviewed Conversations on Mind, Matter, and Mathematics, a transcript of discussions between two illustrious French scientists: the mathematician Alain Connes and the biologist Jean-Pierre Changeux. It’s a fascinating book. At the outset Connes proclaims his belief in a “raw and immutable reality: – a mathematical realm that exists “independently of the human mind.” Mathematics, he says, is the exploration of an “archaic reality”, the mathematician “develops a special sense…irreducible to sight, hearing or touch, that enables him to perceive a reality every bit as constraining as physical reality, but one that is far more stable…for not being located in space-time.” That may be slightly more intense and more poetic that the way other mathematicians would put it, but most of them would agree with Connes.
SIMP. How does the biologist react to those ideas?
KRON. He is flabbergasted. How, he demands, can Connes be a materialist and still think like that? How could a physical brain make contact with such an “archaic reality”? But Connes doesn’t budge an inch. He happily admits that the “tools devised by mathematicians to understand mathematical reality” are a human invention, formed (or “tainted,” as he puts it) by history. But that has no effect on the “raw reality” the tools are used to investigate. In fact, he suggests, the physical world is created out of the mathematical one – an idea that the numerical mystic Pythagoras propounded more than a century before Plato.
SIMP. So mathematical Platonism has pretty deep roots!
KRON. Deep but not firm. In fact, I’ve convinced its days are numbered. Mathematicians will cling as long as they can to the belief that their art brings them into contact with an eternal world. But that idea can’t hold up forever. Not with computers around.
KRON. For two decades computers have been helping create a new kind of mathematics – experimental mathematics. Thanks to the explosion in graphics and imaging software, mathematicians can construct mathematical realities and then manipulate and visually explore them. Now they can produce previously undrawable diagrams such as fractals and chaos maps; they can visualize topological surfaces whose existence was unsuspected before they were seen on a screen; they can discover features that precomputational mathematicians never could have imagined.
Computers are even starting to change what is meant by a mathematical proof. Twenty-one years ago mathematicians were shocked when a computer proved the so-called four-color theorem – the theorem that asserts that no more than four colors are needed to color in all the regions of any map in the plane. The computer had to check millions of details on some 2,000 complicated maps that no human mathematician is ever likely to see. Traditionally, giving a proof had always meant giving a convincing logical narrative that a mathematician could follow step by step. The proof of the four-color theorem certainly wasn’t that.
Now even more advanced methods are coming into play. One of them, known as automated reasoning, made headlines just a year ago when a computer settled a difficult sixty-year-old problem known as the Robbins conjecture. Probabilistic proof procedures have also been invented, in which a computer runs spot checks on a complicated proof to determine how likely the proof is to be correct. The catch – as the name implies – is that probabilistic proof is never totally certain. The use of such methods has led some mathematicians to predict the coming of a “semi-rigorous mathematical culture.”
SIMP. All very interesting. But why should that threaten the idea of Platonic objectivism?
KRON. For one thing, because it’s clear that the computers themselves are not perceiving mathematics; they are constructing it. By giving mathematicians access to results they would never have achieved on their own, computers call into question the idea of a transcendent mathematical realm. They make it harder and harder to insist, as the Platonists do, that the heavenly content of mathematics is somehow divorced from the earthbound methods by which mathematicians investigate it. I would argue that the earthbound realm of mathematics is the only one there is. And if that is the case, mathematicians will have to change the way they think about what they do. They will have to change the way they justify it, formulate it and do it.
SIMP. It seems to me that ideas like this have popped up before.
KRONOS. Yes. More than a hundred years ago the German mathematician Leopold Kronecker declared that “God made the integers; the rest is the work of Man.” He meant that mathematicians can take it for granted that integers exist, but they must prove the existence of other kinds of numbers (fractions, irrational numbers and so on).
Early in the twentieth century the Dutch logician Jan Brouwer carried Kronecker’s concern about existence several steps further. Brouwer sharply criticized the methods by which mathematicians prove the existence of mathematical objects. One of the most powerful techniques of classical mathematics was (and is) to prove the existence of something by disproving its nonexistence – that is, to assume that it does not exist and then show that the assumption leads to a contradiction.
One famous theorem, for instance, attributed to Euclid, proves that the number of primes is infinite by assuming the contrary: namely, that there is some largest number prime p. That assumption enables Euclid to prove that there must also be a prime larger than p, a contradiction. The only way to wriggle out of the contradiction, then, is to deny the original assumption, and that means there is no largest prime, proving the theorem.
That’s not good enough, Brouwer said. If numbers and other mathematical objects do not exist in some kind of Platonic realm – if, that is, they are constructed – then the only acceptable existence proof must be a recipe for constructing them. Brouwer spent many years developing such constructive existence proofs. Then, in the 1960s, the American mathematician Errett A. Bishop, then at the University of California, San Diego, showed that most of classical mathematical analysis can be proved constructively. The bonus was the constructive proofs tend to be much more informative that traditional proofs are.
SIMPLICIUS. So mainstream mathematicians were glad to see them?
KRONOS. No. They ignored them. The main reason is that most of twentieth-century mathematics has devoted itself to exploring the properties of infinite sets – the whole numbers, the real numbers – as if they were completed (rather than constructed) entities. Seen in this way, such sets resist the methods of Brouwer and Bishop. Loosely speaking, if you accept constructivism, you have to abandon the idea of an infinite set.
SIMPLICIUS. Yes, I can see that would be going too far.
KRON. On the contrary, it doesn’t go far enough – not in the case of experimental mathematics. As mathematicians increasingly embrace the computer, the entire theology of “out thereness” will go out the window. And because computers are real objects – at least, they are objects that must be potentially realizable in this universe – mathematics must learn to confront the realities of computation: real time, real storage, real energy, real error, real instructions, and real, material implementation. In other words, it will run up against real-world limits – finite limits.
Non-Euclidean arithmetic is an attempt to show how mathematics might cope with those limits. Its key idea is a new concept of integer – one that doesn’t go on forever.
SIMP. Why do you call it non-Euclidean?
KRON. It’s analogous to non-Euclidean geometry. In the classical geometry of Euclid, points and lines are supposed to reside on an infinitely extended, already existent, everywhere identical plane. Their rock-bottom properties are described by simple, seemingly self-evident axioms. The fifth of Euclid’s axioms states that through a point external to a given line, one and only one line can be drawn that is parallel to the given line. That seems intuitively obvious if you look at a point and a line in the plane.
But early in the nineteenth century mathematicians discovered that they could formulate internally consistent systems of geometry by assuming that more than one parallel line, or alternatively, no parallel lines, could pass through the external point. Those so-called non-Euclidean geometries turned out to be ideal for describing lines and points on spheres and other nonplanar surfaces. In fact, as Einstein showed, the geometry of the universe itself, on a cosmic scale, is non-Euclidean, even if on a human scale its divergence from flat, Euclidean geometry is so small as to be unnoticeable. But even if non-Euclidean geometries had turned out to be useless, their mere existence was enough to shatter the idea that the Euclidean plane was some kind of uniquely privileged Platonic realm.
Euclidean arithmetic, then, is just familiar, classical arithmetic. I call it Euclidean because it rests on the Platonic idea of numbers as an already existent, infinitely extended series of objects, each different from its neighbor by an identical unit. It treats all numbers, even the impossibly large ones, as if they behaved exactly the way the familiar, local numbers do. But what if arithmetic, like geometry, is only locally Euclidean? Why should mathematicians assume that, say, arithmetic operations with extremely large numbers – numbers you can’t even write down without special notation – work just the way that they do with numbers of more “ordinary” size, numbers you (or a large computer) might be able to compute with in standard decimal or binary notation? Non-Euclidean arithmetic rejects Platonic ideas about numbers by rejecting one key part of the Euclidean arithmetic scheme.
SIMPLICIUS. The concept of infinity.
KRONOS. Right. Specifically, it does away with the ad infinitum principle that allows one (whoever and whatever “one” is taken to mean) to go on doing something endlessly. In this case the something is counting: the exact identical adjoining of a unit over and over. The basic idea goes back to Aristotle. In his view, infinity means being able to do what you’ve just done again, and again, and again and so on. No individual step ever “reaches” infinity, but the process transcends finitude because potentially it can go on without limit. Non-Euclidean arithmetic says you can’t do that.
SIMP. But why not? There’s always a next number. Say you name a number n. I can come back with n + 1. How can anyone rationally deny my ability to add one to n and get n +1.
KRON. It’s true, being able in principle to do something ad infinitum is deeply embedded in Western thought. It appears utterly natural, obvious, undeniable. To see how strange it really is, you have to step aside and ask: Who or what is adding or continuing ? Who or what is assigned the task of endless counting? You have to look closely at the language used by mathematicians – how they operate with and respond to signs, and especially how they interpret injunctions and open-ended instructions like “go on adding a unit.”
In his book Ad Infinitum Rotman spends several chapters doing just that. Simply put, he concludes that counting is an idealization of certain simple procedures people undertake in the physical world: laying stones in a row, stringing beads, making marks with a pencil. We mathematicians idealize the procedure, the marks, and the person or machine that does the counting.
Now, in mainstream mathematics, that idealized doer – the Agent, as Rotman puts it – can count ad infinitum, and the process at every step will always be the same. That’s because the Agent lives in a frictionless, airless Platonic world where it can count as long as it pleases without cost or effort. But suppose you decide not to idealize quite that much. Suppose you decide there is a cost to counting – that as you go along, counting gets increasingly difficult until it stops altogether. In the real world, your calculator batteries might run down. Or your pen might run out of ink. In any case, eventually you will reach a point at which what you are doing has changed so much that it no longer constitutes counting. It really doesn’t matter what kind of limit there is, or how big it is. There needn’t even be a precise numerical boundary. The important thing is that a boundary is there.
If you then introduce the arithmetic operations of addition, multiplication and so on, in the usual way, it turns out that you can study the kind of arithmetic that takes place under those conditions, and you can even draw up axioms that govern its properties. And when you do, you get an intriguing alternative to the classical view – a more realistic alternative, I would agrue.
SIMPLICIUS. What exactly do you get?
KRONOS. You get a number system – a number line, if you will – like the one for Euclidean arithmetic but with some important differences. For example, the line is bifurcated into two kinds of numbers: countable ones and uncountable ones. Within the countable numbers, arithmetic works the way you learned it in elementary school, except that each operation will inevitably specify numbers that can’t be reached by counting. For example, if you count upward, you’ll reach a countable number x such that x is not countable. Before you get there, you’ll have reached a countable number y such that y is not countable. And before that, you’ll have reached a countable number z such that z to the zth power is not countable. I call the countable numbers iterates and the uncountable ones transiterates.
SIMPLICIUS. So x + x and y + y and Zz are off-limits.
KRON. Not exactly. You can still use them to name numbers, just as you can still write the symbols for 10 100 or 10 10 100 or any of the staggeringly huge numbers that mathematicians have devised for specialized applications (see side bar on next page). You just can’t reach those numbers by counting. The transiterates are numbers that you can name but not reach. That distinction gives rise to some unusal properties. For example, suppose you come from a culture in which counting stops when you run out of fingers. In that case, the transiterates start with the number 11. It’s clear that addition is commutative when the result is an iterate: three plus seven is the same number as seven plus three. But are transiterates communtative as well? Is four plus seven the same number as seven plus four?
SIMP. It would have to be, wouldn’t it? That’s how numbers behave. Seven plus one is the same as one plus seven. Seven plus two is the same as two plus seven, and so on.
KRON. (aside) One, two, three, dot, dot, dot. (to Simplicius) But the trouble comes with your “and so on.” In both your examples you are iterating, which is exactly what can’t go on happening. “Seven plus one” means “count to seven, and count one more.” With seven plus four you are dealing with a number our fictional culture can’t count to. It’s just as inaccessible as 10 100. If you like, you can state an axiom requiring the transiterates to be commutative, but nothing forces you to do that. Similarly, you could state an axiom that says that if a is greater than b, and c is greater than d, then a + c is greater than the transiterate 7 + 4. Setting things up that way would force the transiterates into some kind of order. But it’s not necessary to do that, either.
In a sense, then, the transiterates aren’t really on the number line at all. You might imagine them floating off to the side somewhere, as if the number line fans out into a spray of disordered numbers.
SIMP. Whereas within the iterates, normality prevails.
KRON. Not entirely. You still have the rational numbers, or fractions, to worry about. For example, ordinary rational numbers are what mathematicians call dense: if you take any two of them, you can always find another one in between. That’s easy to see: just take their average.
But which fraction lies between 1/9 and 1/10? The only possible candidates have transiterate denominators greater than 10 – and transiterates, as you have just seen, have no fixed address. Similar problems crop up when you try to put fractions in order by size. Is 4/6 the same fraction as 2/3? “Obviously,” you say. But not when you remember that, by definition, a/b = c/d if and only if a x d = c x b. In this case, the relevant products, 4 x 3 and 2 x 6, are both transiterates, so there’s no way of telling whether they are equal. Hence fractions, too, float free of the number line.
If the Euclidean number sequence is an unswerving line, a laser beam shining down a long hall to infinity, I picture the non-Euclidean number sequence as a line of iterates with the transitereates cascading out of them. Or, to evoke another image, it’s a circulatory system, narrowing into capillaries as one gets closer to any given rational number, and billowing out at the far end into a transiterate heart. Only in between the extremes are things comfortably normal.
SIMPLICIUS. And all that is supposed to be more realistic than the way we ordinarily think about numbers?
KRONOS. Well, the number 10 isn’t very realistic, of course, but the basic idea is. If numbers are constructed rather than discovered, and if they are rooted in the real universe rather in some Platonic heaven, then whatever they are made from has got to be a limited resource. So numbers must run out sooner or later.
SIMP. But even granting, provisionally, that your iterates stop somewhere, that “somewhere” has to be a lot higher than 10, doesn’t it? Where is it, really?
KRON. There are several ways of approaching that question, depending on your assumptions. The English astrophysicist Sir Arthur Stanley Eddington, for instance, attached a kind of numerological significance to the number of particles in the universe, which he calculated to be around 10 80. One might think of that number as the “somewhere, “ which would have a crude intuitive appeal – you stop counting when you run out of things to count. But I think it’s possible to be more sophisticated than that.
For example, you could imagine the counting being done by an ideal computer, which occupied a bounded portion of the universe and used the theoretical minimum of energy. As it happens, thermodynamics and information theory provide formulas for how much energy such a computer would consume to count up to any given number n. In theory, the computer could keep counting until it had consumed all the energy in the universe. No one knows how much energy that is, but physicists have estimated U, the mass-energy of the visible universe, at around 10 75 joules. On the basis of that figure, an ideal computer could count to about 10 96 before running out of energy.
Actually, you could do better than that. If you imagine that the machine is as big as the universe, it wouldn’t be necessary to expend the energy needed to store all the preceding numbers. The state of the universe itself could serve as the computer’s memory, and one would only need to allocate energy to store the current number. In that case, counting would halt whenever the universe-computer needed an amount of energy equal to U to count one single step, from n to n + 1. Even under such extravagant conditions, the computer couldn’t get beyond 10 10 98 – which, you could say, is the outer horizon of all counting in this universe.
As I’ve said before, though, the crucial thing about a limit to counting is not where the limit lies but that it exists. If you tried to count that far from inside the universe, using a real computer with real energy requirements, you would use up more and more of the fabric of the universe trying to get there. Classical arithmetic doesn’t reflect that reality. Non-Euclidean arithmetic does.
SIMP. So in order to face reality, everyone should scrap the old arithmetic and switch to the new, more realistic one.
KRON. Oh no, not at all, any more than schools should stop teaching Euclidean geometry. I’m merely promoting non-Euclidean arithmetic as another possibility that can be coherently imagined, and that might prove useful.
SIMP. Useful for what?
KRON. Potentially, for keeping physics mathematically honest, or for finding an arithmetic appropriate to the materiality of computer-inflected mathematics. That assumes, of course, that the people in those disciplines are interested.
I also think that non-Euclidean numbers might be useful for studying time. Mathematicians have long suspected a link between numbers and time. For example, the nineteenth-century Irish mathematician William Rowan Hamilton confessed: “ I do not find myself able to frame a distinct conception of number, without some reference to the thought of time…I cannot fancy myself as counting any set of things without first ordering them, and treating them as successive.” The idea of temporal analogue to non-Euclidean space – in terms of spatializing time, of imagining it spread out on a surface – has intrigued me for some years now. The way transiterates fan out from the non-Euclidean number line may provide a way of working it out.
In particular, the distinction between iterates and transiterates may provide a tool for investigating the opposition between parallel and serial actions, between doing lots of things at once instead of one things after another. I’ve been fascinated by that problem, as part of a study about the effects of technology on human consciousness.
SIMPLICIUS. Those sound like deep philosophical waters.
KRONOS. Yes. We are really talking about philosophical questions. In a time of questioning and skepticism about every source of authority, mathematics alone is still widely seen as a font of timeless, indubitable truths. That privileged position was bound to come under fire sooner or later. In that sense, non-Euclidean arithmetic might well be the vanguard of a postmodern mathematics. I can’t say where that would lead. But if I’m right that modern mathematical rigor isn’t written in stone and that sociocultural reality (in this case, the sequence of numbers) is constructed, the next step is to start on the social reconstruction of reality.
SIMP. What would you call that? Post-constructivism?
KRON. Enough, enough.