Visit theallusionist.org/numbers to listen to this episode and find out more information about topics therein
This is the Allusionist, in which I, Helen Zaltzman, boop language's nose. Boop!
On today’s show we have the return of Stephen Chrisomalis, professor of anthropology and linguistics and studier of numbers. He came on the show before to talk about words like zillion and kajillion, indefinite hyperbolic numerals - now he’s here to talk about definite non-hyperbolic numerals. Well they’re less definite than you might hope.
Just to remind you, the new Allusionist live show impends! 4 September at 4.30pm at the London Podcast Festival, get your tickets get your tickets now - you can also watch online for up to a week after, for your geographical and timezone convenience. Ticket links are at theallusionist.org/events, and if you saw or listened to previous live Allusionists like No Title or WPM, you know it’s going to be a very good time.
On with Stephen Chrisomalis and numbers.
STEPHEN CHRISOMALIS: We tend to imagine that numbers are an absolutely essential thing without which any sort of society couldn't exist, but there is actually reasonably good evidence many societies, both historically, and even some societies in the present day, get by without a great need for verbal number. But the kind of things that we imagine must happen every day, the counting of everyday objects, simply doesn't happen through language. That’s not to say that they're not able to distinguish between three stones and four stones, or they're not able to think about their children as individuals; the relevant question is not "How many children do I have?" but "Who are my children?" And so even when it comes to language, there's some pretty good evidence to suggest that verbal numbers are not as necessary as we imagine them to be.
And every number system has an endpoint. There's some number beyond which you can't count. I've heard, “What about googol? There's this number of googol,” and that's true, there is this, oh, that's one followed by a hundred zeroes, that's true. But this is not a real number in the sense that, if you ask somebody, "What's the googol minus one?" they don't have a number word for that one. And after a certain point, all the numbers just blend together.
HZ: We’re not saying, "Get rid of numbers" - they are very useful - just that they are not as foundational to our communications as the modern number-user might think.
STEPHEN CHRISOMALIS: When we think about what the purpose of numbers is, after a certain point, cease to have the purpose that we imagined they have, that prototypical purpose of counting the exact number of some thing. And they start to serve this purpose of - which is a social purpose and it's an ideological purpose - and it can be overwhelming, in fact. We know that to think about something like the national debt of a country, those numbers actually don't have any meaning to most people, only a very, very small number of specialized people, among whom I am certainly not counting myself, can really have comprehension of what does it mean have a national debt of some particular size. And so when we come to numerical notations, the same argument holds true.
HZ: Presumably, with both numerals and verbal numbering, what you're trying to aim for is avoiding ambiguity. Unless ambiguity is the point is the point.
STEPHEN CHRISOMALIS: Yes, I think that's right. I think avoiding ambiguity is central to a lot of communication. But it's exactly as you say, that sometimes ambiguity the point; that sometimes what you want to do is befuddle or confound or perplex. And of course, that's one of the reasons why you would use a very large number, is sometimes to astound the reader or the listener with the magnitude of what's being said. You don't want them to know exactly the number, that's not the relevant point; but what you want to do is you want to overwhelm them. In other words, sometimes the purpose of using numbers is to create an emotional response. And so I use this term 'conspicuous computation’ to refer to this fact of: sometimes we use a big number for the sake of using a big number, for the sake of demonstrating our power and our control over big numbers. And in a world where access or control over big numbers is limited, or where it's unevenly distributed - IE where not everyone has access to those tools - that using a big number, even if you don't know exactly what it means, creates that emotional response. And that's a socially situated, emotional and cognitive response to numbers. It has nothing to do directly with the calculation of an exact quantity and avoiding ambiguity. So, yes, absolutely, avoiding ambiguity is useful lots and lots of the time - except when it's not.
HZ: Why do we need both number words and number numbers?
STEPHEN CHRISOMALIS: It's very hard, I think, to imagine that we didn't have to do this at all. We could have just not done it. We could have either just used number words for everything, or used vague expressions. And the idea that we have these things, these communicative tools, at our disposal that are not strictly linguistic - they kind of are an add on to language that they emerge in written or literate contexts, right? The users of numerical notation are generally literate - but they're kind of a little adjunct system. And to then ask why did we do this at all? Why did we need this? And part of the reason is things like conciseness: it is often more concise to write something with a numerical notation than it is to write it all out in words. But not always.
HZ: For example, it's much quicker to write the year in numerals rather than words. 2021 is four digits, rather than fifteen letters and a hyphen. But numerals are not always the more efficient way to write a number.
STEPHEN CHRISOMALIS: So for instance, if I were to ask a copy editor how to write the number 4.6 billion, the answer is not, in most English texts, four comma six zero zero, comma zero zero zero comma zero zero zero comma... No, you don't write it out that way. You just write 4.6 in numerical notation, and then you stick the word on the end, billion. That's what almost every style guide will tell you to do. And so it's not the case that numerical notation is always more concise. Sometimes it's actually this mix that's the most concise way. But then again, conciseness isn't the only thing we think about. For instance, there's a standard rule in writing that you shouldn't start a sentence with a number in numerical notation. So if I say, “365 days make up one year,” that a lot of style guides will say you should write that out ‘three hundred and sixty-five’, rather than putting 365. I find this a bit finicky and awkward. I think normally I would want to just write ‘365’, You actually find that it's pretty rare in printed books today; mostly they recommend that you always start with a word.
HZ: Why? Well, the convention in written English is to start a new sentence with a capital letter, so that the reader knows that is a new sentence.
STEPHEN CHRISOMALIS: But the numbers are simply one set of numbers, there's not a capital. If you're accustomed to seeing capitals at the start of a sentence, well, we don't have an option with the number - you can't make the three in 365 bigger, that would definitely be weird.
HZ: You could put a top hat on it.
STEPHEN CHRISOMALIS: Oh, that's a great idea. A little hat for numbers.
HZ: Maybe some other places have writing systems that include hats.
STEPHEN CHRISOMALIS: There are certainly systems, for instance in the Greek alphabetic numerals. When we talk about classical antiquity, we often think about the Roman numerals, but the Greeks had a number system, the alphabetic numerals, where they use the letters of their alphabet in order to represent one through nine, and 10 through 90, and 100 through 900. So for instance, one is α, 10 is ι and 20 is κ, 30 is λ. And so you could combine these letters in various ways. But there’s the risk there that you would get confused between a number and a word. And so the Greeks would often mark, maybe by putting a line over top of the number, to say this one's a number.
HZ: I’ll accept that as a hat.
HZ: The numeral system we do currently have has done very well for itself: it’s in widespread use across so many different languages, despite there being plenty other possibilities.
STEPHEN CHRISOMALIS: There are still a lot of numerical notation systems, well over a hundred used worldwide historically over the past 5,000 years. I think when we look at the present day, we tend to see a very, very small number of numerical notations, in particular the familiar signs, 0 through 9, which is called the Hindu Arabic or Arabic or Indo Arabic numerals.
HZ: Though Stephen would prefer it not be called Arabic numerals.
STEPHEN CHRISOMALIS: I don't like the term Arabic numerals. I think it renders invisible kinds of other systems that are interesting and important in their own right. I think it's important to remember that there are other numerical notations used throughout south Asia and southeast Asia that are like ours structurally, but are unlike ours graphically. And I don't want to erase those all by calling those all Indo Arabic notation. No, it's not all. I don't want to deny for a second the system of numbers 0 through 9 that we use and that you're familiar with is from an Arabic ancestor, and ultimately from a south Asian Indian ancestor. It started probably in the early centuries CE - it's a little hard to know exactly when but 4th, 5th century.
But all of these notations and all of the things that have happened in the 1500 years since then are relevant. And in particular, I think understanding that it was in fact because of the social systems of Western Europe that this system became global. That it's really important to really emphasize that this didn't just happen by chance, it didn't happen because of its greater efficiency; it actually happened largely because that system happened to be the one that spread out alongside globalisation and modern industrial capitalism. And that, more than anything else, explains why many, many, many of the 100 or so systems that have ever been used declined or ceased to be used or became retained only for vestigial purposes, really over the past 300 or 400 years. It was actually at a particular moment in time, really in Western Europe, starting in the 15th and 16th centuries, that this one system - which had previously been very, very peripheral, had really not had much to do with the global history of numbers - all of a sudden took off. And its present day universality is really a product of what we know of as the spread of Western European society through colonialism and imperialism and globalization.
HZ: Our system is the Starbucks of numerals??
STEPHEN CHRISOMALIS: Why? For the same reason an obscure-ish set of relatively small, historically relatively unimportant societies in Western Europe became very, very important, starting at around the same time. These notations don't just spread randomly, they spread along with social and economic and cultural institutions. And if those institutions become the vector through which new notations spread, then those notations then become universalized. But then we tell ourselves a story. We tell ourselves a story that it was natural or normal or inevitable that this notation was going to win out. And then we look for reasons why it must have been the best all along. Fnding those reasons, we then denigrate all of the systems that came before it as clearly inadequate or inferior in some way.
HZ: Ah, the colonial mindset, but for numbers.
STEPHEN CHRISOMALIS: Exactly right. Often it is the case that what makes a notation useful is that it's usable or understandable to lots of people. And so this is what we call a frequency dependent bias. The idea is that if something is used by lots of people, especially a communication system, that it is useful for communication only insofar as it's used to communicate widely. But the other side of things is that if you can restrict the community to whom you want to speak, that you're really speaking to exactly the people who you want to hear your message, and excluding those who aren't able or who are explicitly excluded learning that structure and learning that system. And that's as true in language as it is in a writing or a notation system. We think about the way in which education systems structure what we learn and what we know, and then that knowledge is then used to include or exclude people from access to positions of power and authority.
HZ: Roman numerals did very well, considering how much longer they lasted than the Roman Empire.
STEPHEN CHRISOMALIS: Yeah. And of course, I think it's really important to note that we still use Roman numerals all over the place. We use them to denote anything or to number anything that we want to assign prestige: kings, popes, Super Bowls; these are the sorts of things that we value, and we still use Roman numerals for those purposes. But we also use them whenever we need a secondary notation, for instance the preface or the foreword of a book will be paginated in Roman numerals. It's really useful to have a second notation whenever you need it. There's no reason that it needed to survive in these ways, but it has.
HZ: They weren't used for calculations, they weren’t used for arithmetic, right?
STEPHEN CHRISOMALIS: They weren't, but that's also true of basically every numerical notation prior to the past couple of hundred years, that we imagined that the natural and normal way to do calculation is on pen and paper. And that's because that's how we all learned it; we all learned in school from a very young age that number systems get written down, lined up in columns and manipulated, then you get the result. But throughout most of history and throughout most contexts, that simply wasn't the way it was done. Calculations were actually usually done materially or through some sort of device, like an abacus or like a board on which pebbles were spread or something like that. And we actually know very well that was how the Romans did calculation, and it wasn't an inferior way at all. It's actually an excellent way.
HZ: The Roman numerals weren’t always the letters they are now: I V X L C D and M.
STEPHEN CHRISOMALIS: They certainly are not, and historically have not always been those letters. In fact, M is actually the latest one to really become firmly a letter. You can still find all kinds of texts from the 16th and 17th centuries, where instead of an M, the symbol for a thousand in Roman numerals is sort of a set of parentheses or brackets with a vertical line in between them. It's a sort of representation of a thousand that has nothing to do with an alphabetic representation. So what originally had happened was in the very, very early period of the Roman numerals, the Roman numerals were essentially an outgrowth of a kind of tallying system or an outgrowth of a kind of way of representing numbers. And the V for five is actually basically just the top half of the X for 10. And the L for 50 is basically the bottom half of what we think of as a C for hundred. And the D for 500 is really just the right half of this weird M thing that wasn't actually an M at all. So these were abstract symbols that then later became accommodated to the alphabetic system. So there was no reason that it had to do so, but it did so.
HZ: So if you had been thinking, "C means hundred because the Latin word 'centum' begins with C, and M is a thousand because M for 'mille': those are intelligent deductions, I am proud of you; but, unfortunately, they’re wrong.
STEPHEN CHRISOMALIS: There's no connection between X and 10, or L and 50. There's nothing there. So these are really graphic resemblances, rather than using the acrophonic principle, which is the idea that the first letter of something represents thing.
HZ: Who got to decide that M was going to be added to the Roman numerals?
STEPHEN CHRISOMALIS: Of course, no one person decides any of these things. There were always Ms around; even in classical antiquity, it was occasionally used. But it was a very slow process, and as I say it wasn't until, the 16th and 17th centuries that these older signs disappeared entirely. We now tend to think of the Roman numerals as extremely static, and there's a good reason for that, because if you were to go to the Coliseum, you could still go and look over the various archways and you can see the numbers, in the same way that if you go to any modern stadium, you can see which gate you're at or which section you're at. And yeah, the lower Roman numerals are generally similar to the ones they were 2000 years ago.
But in lots of other ways, they’ve changed, they've transformed and they've responded to social context; but more importantly, they've responded to the needs of writers and readers. And that's what all notations do: that's what writing systems do, and that's what numerical systems do, is that they aren't designed by a single person and then persist for all eternity, rather that they're constantly responsive to the needs and goals and interests of sets of readers. So asking the right question is a matter not just of asking the question in the abstract, which notation is best, but being able to situate ourselves a historical context and ask: for these people, for these users, what did they value? What was important to them? Rather than assuming what we now are able to study represents that entirety.
HZ: The way we express numbers, with numerals and letters, just keeps on shifting. To give an example, here's how what I'd currently call ‘1.2 million’ has been rendered over the past couple of hundred years.
STEPHEN CHRISOMALIS: Throughout the 18th and 19th century, of the most common ways that you would say what I would say as 1.2 million, and I think you would as well, would be twelve hundred thousand. Now, to me, that's actually not grammatical. When I hear ‘twelve hundred thousand’, it takes a while for me to parse that. I can say twelve hundred, for one thousand two hundred, but it's not grammatical, it doesn't feel right to me, twelve hundred thousand.
And so this historically common form, that was actually the most common form the 19th century for writing, at least, that number - we don't always know about speech, right? We don't know what was happening in spoken English - but in written English, we have a very good idea that twelve hundred thousand was actually the most common form, and that the 1.2 million form was actually really a 20th century phenomenon. But it's the height of arrogance to imagine that a thousand or two thousand or three thousand years from now, we're still going to be using the same notation. It's only had currency for about a hundred or two hundred years. It's only even been the predominant system Europe, in Western Europe, for about four hundred or five hundred years.
So on what basis would we possibly imagine that no other developments are ever going to occur in this field? And yet there are lots and lots of claims, explicit claims, out there that we're just at the end, this is the last number system that we're ever going to have, forever. I'm of course presuming in all of this that we actually survive as a species long enough to see this change. But I think if we're still around in a thousand years, the chances that we're still using this notation as the universal global notation are almost zero, because that's just never been the case historically. It's never been the case historically that a single system has just dominated the entire world and been used by everybody for that length of time. So it's a check against taking a very ethnocentric, particular view of the present and then extrapolating it to a time in the far distant future.
HZ: Stephen Chrisomalis is professor of anthropology and linguistics at Wayne State university in Detroit, Michigan. And if you want to go deep into numbering systems, you can read his book Reckonings: Numerals, Cognition and History. And in today’s Minillusionist, we pop back in on the Roman and Greek numbering we do use - except for when it makes us look like losers.
MINILLUSIONIST
HZ: Although Greek numerals don’t get as much play as Roman numerals, we do have a lot of Greek alphabet in our mathematics: such as in equations, in the celebrity irrational number π - chosen as it’s the first letter of ‘periphery’ - and in K, meaning thousand, as in Y2K.
STEPHEN CHRISOMALIS: If I say, "I make $92K a year" - which I don't, but if I were to say, "I make $92K," I think that would be perfectly understandable to most English speakers. But really that usage only emerged in the 1960s, in computing and electronics, and then sort of moved into money and other in other figures in the 1970s and 1980s. But now I think it's quite commonplace. And so you can see in various video games, I think the NBA basketball games, instead of saying ‘2018’, they'll say ‘2K18’. It's no shorter, it's no simpler. The Greeks never used K in this sort of way to attach it to some other number; and it wasn't even a K anyways, it was a χ.
HZ: χ is the Greek letter that looks like an elongated X. Our modern use of K doesn't even refer to the Greek K, kappa.
STEPHEN CHRISOMALIS: Not a kappa at all, χίλιοι starts with a χ, but it kinda got switched into a K in English and really in European languages in general. There are weird words like chiliastic, which sort of means millenarian or apocalyptic.
HZ: Oooh, useful.
STEPHEN CHRISOMALIS: But no, for kilo-, the common metric prefix, it's of course switched to a K, then the K became of reappropriated and repurposed for this purpose of, numbering as a sort of short form for thousand. It's alphabetic and numerical all at the same time. And this is quite new, and these sorts of innovations are constantly happening, both in number systems and in language in general.
HZ: Maybe Greek can make a comeback after all. Still everything to play for.
STEPHEN CHRISOMALIS: Maybe so, maybe so.
HZ: Back to the Greek numerals' nemeses, Roman numerals, which still get a lot of play: on clocks, on tattoos, on production credits to delay you figuring out how old the tv show you just watched is - and also in the Super Bowl. Except for the fiftieth Super Bowl. Why was that called Super Bowl 50 and not Super Bowl L?
STEPHEN CHRISOMALIS: So that's a great question. And they've never said exactly. But traditionally, the Super Bowls were always numbered using Roman numerals, and actually like clockwork, it's predictable that every January, the last week of January or so, you can always predict there's going to be a news article about the Roman numerals in the context of the latest Super Bowl, because it's one of the more prominent contexts where you can see big Roman numerals splayed everywhere on TV and media and stadiums, they're everywhere. And for the most part, that has occurred unproblematically. But then a few years ago, when Super Bowl 50 happened, there was the idea that maybe we shouldn't use Super Bowl L, we should use 50. And that was exactly what happened.
HZ: But the next year, the Super Bowl returned to Roman numerals - LI, LII, LIII and so on, and they continue to use them. So why not L? One explanation was given by the NFL’s creative director Shannon Melvin, who was aesthetically displeased by the L. He said at the time: “It’s very asymmetrical, and three-quarters of the letter is negative space. It’s like, what do you do with this thing to make it look attractive? I’ll take an X any day of the week. Or any other letter for that matter.” Another possible reason is because L that sign you make with your hand against your forehead and you’re not being particularly complimentary.
STEPHEN CHRISOMALIS: So there's a couple of explanations. One is that L has an evocative response, at least for English speakers, of loser. Uh, and "you take the L" is "you take the loss". And it's not a good look for a sporting event. But also I think the importance is it's a really short number, that 50 is one of the cases where the Roman numeral L is actually shorter than the corresponding Western numeral. You just have one sign. And one of the reasons why you use a Roman numeral sometimes is sometimes you want a nice long Roman numeral. You want it to be sitting on the cornerstone of a building to indicate its age and antiquity. And so this has the same sort of cognitive and emotional purpose. And I don't think we're ever going to be told exactly why. But it really is the case that there's something of the dignity of Roman numerals that doesn't accrue in modern days to these really short ones like L. I suppose we'll have to wait another 50 years and see what happens at Super Bowl C to see if they change it around.
HZ: Well, let's check back in for when that happens. If we're alive.
STEPHEN CHRISOMALIS: I promise that I'll come back in another 50 years. If we're both still around then, I think I would have to.
HZ: I appreciate.
The Allusionist is an independent podcast and it’s you listeners who keep this engine running - thanks to all of you for listening, for recommending the show to people, and for funding the operation at Patreon.com/allusionist - you patrons get a 20% discount on tickets to the London Podcast Festival show too. As well as the boundless satisfaction of funding DIY creators.
Your randomly selected word from the dictionary today is…
gastrolith, noun: 1. Zoology: a small stone swallowed by a bird, reptile or fish to aid digestion in the gizzard. 2. Medicine: a hard concretion in the stomach.
Try using gastrolith in an email today.
This episode was produced by me, Helen Zaltzman. The music is by Martin Austwick of palebirdmusic.com. Our ad partner is Multitude. To sponsor an episode of the show, contact them at multitude.productions/ads - get it done, because there are not many spaces left for this year.
Stay in touch: find @allusionistshow on Twitter, Instagram and Facebook.
And to hear or read every episode and find more information about the topics therein, to see the full dictionary entries for the randomly selected words, and browse a lexicon of every word ever covered in the show, visit theallusionist.org.
