Tens of thousands of years ago, long before humankind hit on the the nifty trick of preserving language with marks on clay or papyrus, our ancestors notched tally marks on animal bones to count … things. We don’t know what the things were. Take for instance the 40,000-year-old Lebombo bone found fifty years ago in southern Africa. We can make guesses about what the person who carved it was counting1, but we’ll never know. That’s the thing about tally marks (and the numerals that replaced them much later): their power to describe huge swaths of reality stems from an essential vagueness.
The notion of quantity shorn of context – that is, the advent of the concept of Number – was the greatest mathematical revolution of all time, the one that made all subsequent developments possible. I don’t have much to say about it because we know so little about it, but since most great advances involve trade-offs I want to mention two of the hazards made possible by the abstract number concept.
One hazard is logistical, and is exemplified by the famous failure of NASA’s Mars Climate Orbiter in 1999. A subcontractor submitted data in which force was measured in pounds rather than newtons, and NASA didn’t catch the mismatch. If the numbers stored in NASA’s computers had had units attached, the discrepancy would have been caught. But in twentieth-century data-processing2 the focus was on speedy operations using compressed representations of data, with all the fat (such as units of measurement) trimmed away. Programs were far removed from the meaning of the data being manipulated.
The other hazard is moral. Picture a harbormaster of the 1720s looking at a ledger he has just received from the captain of ship newly arrived from Africa. The ledger informs him that 213 units of cargo were lost at sea. He multiplies the number 213 by the typical price each unit of cargo would fetch at auction to compute the total loss of value of the ship’s contents. His arithmetic is flawless, unimpeachable. He does not pause to consider that the 213 units of cargo are enslaved people who died on the trip. The blankness of the numeral invites detachment from the reality it refers to. Numbers can numb us, even blind us.
Yet, offsetting the disadvantages of abstraction we have the universality of arithmetic. People have disagreed about many things (such as who made the world and why, and how those who inhabit that world are supposed to comport themselves while passing through) and have gone to war against those who disagreed with them, but propositions about arithmetic have not been the sort of thing that made people kill other people. (Well, there’s been a bit of a culture war about 2+2=4 lately, but it’s a war of words, not weapons, and 2+2=4 is really a proxy for other things.) Different cultures give different names to the counting numbers3, but all of them agree about the facts governing those numbers. And one of the most curious facts is that there is no last number. You can run out of sheep or sacks or any other thing you care to count, and you can even run out of names for numbers and be forced to invent new ones, but you’ll never run out of numbers themselves. No matter how big a number you have, you can always add 1 to it, and that new number will be bigger. With this insight, we see a wondrous and somewhat terrifying vista opening up before us, a vista of numbers without end – an infinite stairway with a bottom but no top, vanishing into the clouds.
HOW DO WE KNOW?
It’s often a good idea to ask the question “How wrong might we be?” and the related question “Why are we so sure that we’re right?” Even in situations in which doubt seems a bit silly, employing our capacity for doubt is good exercise, and sometimes fun. The stairway of counting numbers is no exception. Consider the short story “The Secret Number” by Igor Teper, which invites us to believe that there is a counting number between three and four that They don’t want you to know about. In Teper’s story, published in 2000, the secret number is called “bleem”, but I remember hearing about a similar secret number called “bleen” back in the 1970s.4
Wondering how wrong we might be was a serious matter for 19th century mathematicians. Reeling from the discovery of non-Euclidean geometries and other challenges to mathematical intuition, many mathematicians thought that, while it was all well and good to add new stories atop the edifice of mathematics, someone needed to go down below and make sure that the foundation was sound and that the basement wasn’t flooded.
One such mathematician was Giuseppe Peano (1858-1932). Although he’s remembered for his mathematics, he was also extremely interested in language;5 one of his pet projects was the development of an international language, specifically a kind of uninflected Latin. He thought that by reviving and reconfiguring the moribund lingua franca of the Renaissance, he could create a shared vehicle of expression for modern thinkers and writers. As it happens, his work on the foundations of arithmetic (in collaboration with Richard Dedekind and others) also involved a revival of something ancient: those tally marks we started with, or rather, the unary numeral system that underlies them.
In Peano’s system (the version that starts from zero rather than one) the numbers are named 0, S0, SS0, SSS0, etc., to be pronounced respectively as “zero”, “the successor of zero”, “the successor of the successor of zero”, “the successor of the successor of the successor of zero”, etc. (The names of numbers get tiresome very quickly!) Note that this system is in its essence a fancy version of using tally-marks; replace the S’s by notches and omit the trailing 0 and you’re back in the Paleolithic period.
If your goal is to work with numbers for practical, everyday purposes, such as ordering fish for a party of penguins, the unary representation notation is awkward to use.
But if like Peano you want to establish mathematics on a firm foundation, then the simplicity of unary makes it a winner. Consider for instance the task of adding numbers together. In the modern Hindu-Arabic system, to add two numerals you have to line them up and add digits systematically, using your short-term memory (or some external extension of your short term memory) to propagate carries appropriately. In contrast, to add two unary representations, you just write them side by side.
Well, Peano addition is actually more complicated than that, because the injunction “Write them side by side” is not expressed in the language of the Peano axioms. Peano’s prescription for adding numbers is a little more subtle; it’s a recursive6 procedure that expresses the answer to addition problems in terms of the answers to other addition problems. If you want to add two numbers in Peano’s system (call them M and N), there are two cases. If N is zero, then M+N is just M. Otherwise, if N isn’t zero, N has some S’s in it; in this case, M+N (the thing we want to compute) is just M+N′ with an S stuck in the front, where N′ is just N with an S removed.
The process is less confusing than it sounds; an example should clarify the definition. Let’s compute two plus two, or rather SS0 plus SS0. Putting M = SS0 and N = SS0, we find that we’re in the case where N isn’t zero, so the “Otherwise” clause tells us that SS0 + SS0 is S(SS0 + S0) (that is, S followed by whatever SS0 + S0 turns out to be). But what’s the value of the expression inside the parentheses? That is, what’s SS0 + S0 ? Putting M = SS0 and N = S0, we find that we’re again in the case where N isn’t zero, so the “Otherwise” clause tells us that SS0 + S0 is S(SS0 + 0). And now, putting M = SS0 and N = 0, we’re (at last) in the case where N is 0, so SS0 + 0 is SS0, so (heading back upstream) SS0 + S0 is S(SS0) = SSS0, and SS0 + SS0 (the thing we were originally trying to compute) is S(SSS0) = SSSS0, which is Peano’s name for four. And two plus two indeed equals four.
Likewise, one can give a procedure for multiplying Peano numerals: if N is zero, then M×N is just 0, while if N isn’t zero, M×N is M×N′ plus M, where N′ is N with an S removed as before.
In the context of Peano’s work, these two recursive algorithms shouldn’t be thought of merely as ways of performing calculations. Rather, these algorithms are to be treated as definitions. Much as Euclidean geometry starts with just a handful of basic notions like points and lines and builds up more complicated objects like trapezoids and angle bisectors in terms of the basic notions, Peano’s approach starts with only with the idea of succession, and a few axioms governing it, and builds everything up from there: addition, multiplication, subtraction, divisions, primes, you name it. (And along the way, we can easily banish bleem and bleen.)
A TRIP TO THE BASEMENT
I first encountered the (to my young self, audacious) idea of defining arithmetic purely in terms of the successor operation S in Kershner and Wilcox’s 1974 book The Anatomy of Mathematics (a reworking of Edmund Landau’s classic 1930 work Grundlagen der Analysis7) when I was a teenager. I found it thrilling. In a way, reading the book was reminiscent of going down into the depths of the house I grew up in, back when I was a child. The basement was a mysterious place, and I had the impression that if you flipped the wrong switch you could make the house explode.8 The basement contained a boiler that gave us hot water, and a valve that controlled the flow of water throughout the house, and circuit breakers that controlled the electrical fixtures in the parts of the house where we spent most of our time. Going into the basement of mathematics, and seeing where all the valves and circuit breakers were, provided all the thrill of my childhood basement with none of the imagined danger. I’d long known that addition and multiplication were commutative (M+N = N+M and M×N = N×M for all counting numbers M and N), but after reading The Anatomy of Mathematics I knew how to prove it.
The book went on to build up the real number system in stages, but that’s not what I want to focus on today. Rather, I want to discuss the subtle way in which Kershner and Wilcox shifted my mental image of what a counting number was, from decimal numerals to something more abstract like “successor-of-successor-of-. . . -successor-of-zero”. The change didn’t happen on a conscious level, and given an arithmetic problem I’d still solve it the way I’d been taught to solve it in grade school (as opposed to, say, translating the problem from decimal to unary, solving it in unary, and then translating the answer back to decimal). But on some deep level, in reading the book I imbibed the Platonist Kool-Aid that whispered (if a beverage can be said to whisper) Beneath it all, this is what counting numbers are.
I was probably also affected by what Kershner and Wilcox didn’t do in their book. Specifically, they didn’t build a bridge between Peano’s way of writing counting numbers and the Hindu-Arabic decimal system I’d grown up with. (As the authors say in their Preface, “The reader does not even need to know the sum of 7 and 5; incidentally, if he does not know this sum he will not learn it from this book.”) The authors go so far as to prove the quotient-and-remainder theorem, which says that given positive integers N and D (for “divisor”) there exist unique non-negative integers Q (for “quotient”) and R (for “remainder”), with R between 0 and D−1, satisfying the relation N = D × Q + R. For instance, if N = 213 and D = 10, we get 213 = 21 × 10 + 3. Armed with the quotient-and-remainder theorem, Kershner and Wilcox could have recursively defined the decimal expansion of N as the decimal expansion of the quotient Q followed by the digit representing the remainder R, in the case where the divisor D is ten. But from their point of view, that would’ve been a cruel anticlimax − like bringing someone to a majestic castle, leading them up to a high balcony with an panoramic view of towering cliffs and a surging sea and then telling them “Did you know you can see your house from here?”
You might say that I’d been prepared for Kershner and Wilcox’s book by my early immersion in the pedagogical experiment known as the New Math. New Math had lots of problems in terms of both its design and implementation, but it worked very well for me (as one might have expected, given that it was largely designed by mathematicians). In particular, the New Math laid emphasis on the difference between numbers and numerals. Numerals were the things you wrote down; numbers were the things numerals denoted. What exactly was a number, then? New Math invited me to ask the question. The Anatomy of Mathematics answered it.
GETTING THE PICTURE
Peano proposed different versions of his axiom system during his lifetime, and the people who came after him introduced more. Some versions use just three axioms; others use a dozen. Some versions include axioms governing equality (e.g., “if M=N then N=M”); others presuppose them. Some versions incorporate addition and multiplication from the get-go and include axioms governing them, rather than building them up from the successor operation.
I’m going to join the ranks of Peano’s successors (sorry!) by presenting my own version, phrased in the language of combinatorics. A directed graph (sometimes called a digraph for short) is a bunch of nodes connected by arrows, like this:
The specific digraph we’re trying to characterize is the one you get if you create a node for each of the counting numbers, with an arrow pointing from each counting number to its successor.
You’ll notice that I’ve stripped off the names of the numbers; all that remains is the structure of succession, depicted as an abstract stairway.
Here are the axioms that single out this particular architecture:
Axiom 1: Every node has exactly one arrow pointing out of it.
Axiom 2: Every node has exactly one arrow pointing into it, except for one special node, which has no arrows pointing into it.
Axiom 3: If you color the nodes blue and red, where the special node from Axiom 2 is colored blue and at least one other node is colored red, then there must be an arrow from a blue node to a red node.
(This last axiom is equivalent to the axiom of mathematical induction. If it looks unfamiliar, here’s a paraphrase, a sort of contrapositive, that might be more reminiscent of the versions you’ve seen: If you color the nodes blue and red, where the special node from Axiom 2 is blue, but there is no arrow from a blue node to a red node, then all the nodes are blue.)
Once we’ve characterized the digraph with these axioms, we can put the labels back in if we like. That is, we can define 0 as the special node with no incoming arrows, define 1 as the target-endpoint of the unique arrow pointing out of 0, define 2 as the target-endpoint of the unique arrow pointing out of 1, etc.
Then we can define addition and multiplication as described above, prove the important properties of those operations, and if we like, do some calculations in the guise of proofs. Here for instance is Kershner and Wilcox’s proof that 2 times 2 is 4:
Notice that this proof uses just addition and multiplication. We’ve thrown away the scaffolding embodied in Peano’s successor-operation S. We’re back where we started, but also, in a way, somewhere else. We’ve arrived at a deeper understanding of what lies beneath the counting numbers.
WHAT WE CAN KNOW
The twentieth century weekly radio show “A Prairie Home Companion” had a recurring feature called “The news from Lake Woebegone”, in which host Garrison Keillor would describe fictional happenings during the past week in his iconic, nonexistent home town of Lake Woebegone, Minnesota. Each week he’d end the news segment with the same tag-line: “And that’s the news from Lake Woebegone, where all the women are strong, all the men are good-looking, and all the children are above average.” That last line gave humorous expression to the fact that most parents think their children are objectively special, and it even gave rise to a new bit of psychological jargon. But curiously, a version of the Lake Woebegone fallacy applies to the counting numbers, not as a fallacy but as a fact – specifically, the fact that every counting number is smaller than average.
When I say smaller, I don’t just mean smaller. I mean much, much smaller. Even a number that’s biggish by human standards, such as a trillion, is smaller than nearly every other number. Replacing a trillion by an even bigger number – go ahead, take your pick – will kick this paradox down the road (or rather up the staircase) but won’t abolish it. No matter what counting number N you pick, I claim that it’s atypical because there are infinitely many numbers that are bigger than N and only finitely many that are smaller. Am I saying that the “average” counting number is infinite? Sort of. But there are no infinite counting numbers. Every counting number is finite. The infinite stairway doesn’t have a top; it’s not that kind of stairway.
In the face of an imaginary structure of such daunting and paradoxical infinitude, you might think that human reason would be powerless. We can never explore more than the tiniest bit of the stairway, so how can we possibly make brash assertions about what is or isn’t to be found in its upper reaches?
Yet we can know some things about the stairway, and know them for certain. And the firm foundations provided by Peano and others give us the tools with which to do it.
For instance, we can know that no matter how high up the stairway we go, we’ll never find two consecutive counting numbers whose sum is even. To assert this so boldly is not hubris; our conclusion is just a consequence of the meaning of the word “even”, and the basic axioms and theorems governing the stairway. If we’re standing on the tread marked N, then the next tread is marked N+1, and the sum of these two numbers is N+(N+1), which can also be written as 2×N + 1; if you divide this sum by 2, you’ll get N with 1 left over.
Perhaps you feel that the mathematical universe is just a little bit emptier because of its lack of two consecutive counting numbers whose sum is even, and more broadly because some of the things we can imagine existing turn out to be impossible, but here is some consolation. We just saw that if you add together two consecutive counting numbers, the sum is never divisible by two. On the other hand, if you add together three consecutive counting numbers, the sum is always divisible by three. And if you add together four consecutive counting numbers, the sum is never divisible by four; while if you add together five consecutive counting numbers, the sum is always divisible by five. This alternating pattern continues forever: if K is even, the sum of K consecutive counting numbers is never divisible by K, while if K is odd, the sum of K consecutive counting numbers is always divisible by K. This beautiful numerical pattern (not hard to prove, by the way)9 grows from a soil compounded equally of existence and nonexistence, of nevers and alwayses.
If math were the kind of game in which you could find counting numbers satisfying whatever combination of properties you like, then math would be poorer, not richer. Math – the kind I love, anyway – is about the beautiful patterns at the border between the possible and the impossible. If everything were possible, there’d be no contrast, no patterns, and no math. The phantoms of the stairway – the numbers that on shallow thought seem as if they might exist, but on deeper thought are seen to be self-inconsistent – live on as meaningful absences, subsumed into patterns that have a life of their own.
Some of the phantoms of the stairway do live; they just live elsewhere. The stairway is full of beauty and wonder but it isn’t the only place to be. There may not be a counting number N with the property that N plus N+1 is even, but there are rational numbers with that property. There may not be a rational number whose square is 2, but there are real numbers with this property. There may not be a real number whose square is −1, but there are imaginary numbers with this property. The study of number systems is the study of elsewheres, of places-to-be that resemble the infinite stairway while differing from it in important ways. And for the next couple of years, this blog will focus largely (though not exclusively) on some of those elsewheres.
Thanks to Ori Gurel-Gorevich.
R. B. Kershner and L. R. Wilcox, The Anatomy of Mathematics.
Edmund Landau, Grundlagen der Analysis. A translation by F. Steinhardt is available on the web at https://salamon.sdsu.edu/Math534A/LandauReading.pdf.
#1. Not all archeologists agree that the Lebombo bone was used for counting; some consider the more recent Ishango bone to be the oldest surviving mathematical artifact. I personally want to believe that the Lebombo bone is indeed the sole surviving evidence of some ancient person’s pressing need to record the number twenty-nine. Do its twenty-nine notches count days in a lunar month? Some scholars think so. Perhaps women in that society used arithmetic to keep track of their menstrual cycles, maintaining control over their reproductive capacity until patriarchs arose and imposed a tally-ban.
#2. Newer programming languages, especially object-oriented ones, permit metadata alongside data, so perhaps newer computer programs won’t be subject to the sort of problem NASA fell victim to.
#3. One charming counting system is the “Yan, tan, tethera, methera, …” count used by shepherds in some parts of England. What I find most delightful about this system is the way it was used mostly, perhaps exclusively, for the counting of sheep. My guess is that if in the heyday of the system you had used it for some other purpose, say counting out coins, people would have thought you were being metaphorical for comic effect. Nowadays “yan, tan, tethera, methera, …” survives mostly among knitters counting stitches, and they are probably not thinking of stitches as individual sheep, since the amount of wool in a single stitch is closer to one microsheep.
#4. George Carlin apparently heard about bleen too, because in one of his routines he announced “The Nobel Prize in mathematics was awarded to a California professor who has discovered a new number! The number is bleen, which he claims belongs between 6 and 7.” If anyone can trace bleen/bleem back to the 1970s or earlier, please post to the Comments!
#5. Wikipedia calls Peano a glossologist rather than a linguist; can someone explain the difference to me?
#6. In my use of the word “recursive”, I’m hinting at the real technical crux of Peano’s system, which is the axiom of induction. Check out Kershner and Wilcox’s book if you want to see how the axiom of induction lets you devise and work with recursive definitions in Peano’s system.
#7. I love the paradoxical pair of admonitions Landau offers his student-readers in his preface: “Please forget what you have learned in school; you haven’t learned it.” “Please keep in mind everywhere the corresponding portions of your school work; you haven’t actually forgotten them.”
#8. I think in hindsight that my father must have said that the pipes could explode if I turned off the heat in winter when we left for vacations, but the main thing I took from his warning was the word “explode”.
#9. The sum of K consecutive counting numbers is equal to K times the average of the numbers; this product is a multiple of K precisely when the average is itself a counting number. If K is odd, then the average is just the number in the middle; but if K is even, then there are two numbers in the middle, so the average is the number halfway between those two consecutive counting numbers, which is not itself a counting number.