Last week, my interview with Grace East (my former MA advisee) went up on the CaMP Anthropology blog. Check it out if you want to read some of my thoughts about my book, Reckonings: Numerals, Cognition, and History and how it relates to broader issues. For reasons of space, the final question and answer of the interview was omitted, so with Grace’s permission, here it is:
GE: As you mention throughout the book, there’s a widespread misconception that numbers are simply tools used for computation, but you craft a larger argument that numbers actually carry immense representational potential. From your discussion of conspicuous computation (“intentional use of large numbers for their visual or psychological effect on the reader”) (42) to your later assertion that inventing number systems among colonized peoples often serves to assert “the right to be considered fully civilized” (139), numbers function as emblems of persuasion, status, identity, and more. What do you see as the possible future(s) of this disciplinary intervention and what kinds of work do you hope to see emerge from this view of numbers?
SC: Frankly, the intervention here is to insist that we see numbers as representations, as signs. A lot of cognitive scientists analyze numbers as part of the narrower subfield of numerical cognition, which is legitimate but incomplete. I’d rather encourage them to see numbers more generally, in terms of attention, perception, and memory. Those broader topics are a place where anthropologists and cognitive scientists can have much better conversations than have been possible to date. I think the topics you’ve raised, about numbers as emblems of civilization, as identity markers, and as tools of persuasion, are already well-understood among many humanists and social scientists. I’m certainly not the first to mention it. So of course there are conversations to have there as well, but they’re ones that lots of folks are already having.
What’s different in Reckonings, I think, is that at the same time, and using the same kinds of evidence, I insist that this is also a question about human cognition. To be able to go to my colleagues who are non-anthropological cognitive scientists and have a different kind of productive conversation, one that raises these issues, is an important intervention. And I don’t think it’s at all impossible, or even that difficult! Once we stop viewing cognitive science as some sort of bogeyman in some sort of turf war, we will actually find a lot of committed, humane scientists “over there” who want to have exactly the sort of conversation I’m hoping that Reckonings will encourage.
Once again, the early-career scholars in the 2022 edition of my course, Language and Societies, have written some amazing papers, for which the abstracts are linked below. The authors are undergraduate and graduate students in anthropology and linguistics at Wayne State University. Comments and questions are extremely welcome, especially at this critical juncture, when the authors are making final revisions to their papers.
Fifteen years ago today, I was the temporary occupant of an office that had been, for seven years from 1996 to 2003, a central part of my intellectual life. Since that August, I had been the very junior resident of the office on the seventh floor of the Leacock building at McGill University. It had been for thirty years, and was still, officially, occupied by my doctoral advisor, colleague, and friend, Bruce Trigger. I had come to be sitting on the other side of the desk by grave misfortune. Bruce had been diagnosed with pancreatic cancer the year before, and had retired to emeritus status. While he was still poking around the university in the fall of 2006, he wasn’t well, and despite heroic measures being taken, the prognosis wasn’t great. We all knew that. And so I had been hired to occupy his office, to teach ‘his’ history of archaeology course and a couple others as a visiting lecturer, which was fortuitous, since I had a new baby and no other income, since my postdoc at Toronto had ended and we certainly couldn’t afford to stay there. So we came back to Montreal, and I came back to McGill. And there was his office, sitting unoccupied, except now and again when he’d pop by. So that was how in 2006 I came to be on the flip side of the desk where, in 1996 as a new doctoral student, I had first met the great man himself.
Now here’s the thing you need to understand about the Leacock building at McGill. Most of the offices in Leacock are pretty small and not the best. The building is one of those 60s brutalist things, nine stories, and along the long sides (west and east) all the offices were pretty standard. But on the south-facing side, looking out across campus, down the mountain towards downtown, four offices on each floor are extra-long, maybe twice the length of an average office. You can see these great beasts jutting out here in this Google image:
So one of those, 724, that was Bruce’s. There wasn’t a lot of natural light or space, because practically every available square foot was occupied with bookshelves heavy-laden with books and journals, most of which, Bruce assured me, he had actually read. Stacks upon stacks of offprints and reprints, Bruce’s own articles and things sent to him. It was all there, a monument to an intellectual life. And all kinds of other crap. Like the inflatable sarcophagus Tutankhamun that just kind of was always, inexplicably, there in the back of the office. Or random 5¼” floppy disks, where Bruce saved most of his manuscripts (yikes! Thankfully, he also printed out everything). Or an axe, just … an axe, which we found after his death in the back of the office, as if somehow, if a fire broke out, Bruce was going to smash the seventh-floor window and then … who knows? I can’t find a photo of 724, though there must surely be one somewhere. The office pictured in his Wikipedia photo is far too empty. It doesn’t matter, because whatever its actual physical stature, it was nothing compared to the magnitude, that September in 2006, of me walking around to the other side of the office and taking a seat to use, however borrowed, as my own. To wait for Bruce to die.
Lots of people have complex, mixed feelings about their PhD advisors. Not me. It’s not that I think that Bruce was perfect — you don’t get to know someone that well, for a decade, and suppose somehow they are immune to human fallibility. But he was, for me certainly, but also, I know, for many others, a truly remarkable and supportive mentor. You don’t really have any expectation that when your advisor is also a Big Personage in the field, that they are also the kind of person who would, as Bruce often did, spend two hours with you in tutorial in the afternoon, then call you up on the phone that evening to talk some more about our excellent tutorial. Or turn around papers with detailed comments within a couple days. Or tap into his research funds to give you some summer work that I honestly think he would have preferred to do himself, because he preferred to do lots of things himself. Or — and this one is still a source of great shame — read and comment on a manuscript on the early modern abacus I’d written, from the hospital bed where he would eventually die. That paper, fifteen years later, is sitting in my metaphorical drawer, unpublished after a too-nasty bout of peer review. Or make sure to leave a detailed reference letter with the department staff, when he knew — though he wouldn’t say it out loud — that he wasn’t going to be there to see me land a tenure-track job. (Look at me, I’m still here, I did it!)
That fall started out great. Classes were good, and Bruce was, though thin, though unwell, still out and about. He’d been very sick through much of 2005-06, and hadn’t been well enough to travel to receive his Order of Canada in December 2005 (there was a private ceremony instead, in Montreal). But by the summer of 2006 he was, well, better. Well enough to attend the book launch that September of The Archaeology of Bruce Trigger, the volume that had emerged from a SAA session we had done a couple years earlier, before he was sick. I still remember the speech he gave at that event, ending with Bruce’s still-uninterpreted “cryptic pronouncement“. The book wasn’t originally intended to be a memorial, which is partly why a few years later, we put together Human Expeditions: Inspired by Bruce Trigger, to make sure no one who wanted to have a voice was left out. The introduction to that volume summarizes how I still think about his intellectual legacy. But back in September 2006, we were really not thinking about legacies, but of futures. Bruce was even musing about teaching his renowned arch-theory graduate seminar that winter (which I had taken twice as a student, once for credit, once for … fun?). If I’m not misremembering, it was even on the books for students to register.
But then it started to get cold, and Bruce was in the hospital by late October, still working, furiously, on new work, a book about conservatism in Canadian politics that I don’t think survives even in his papers (although anyone who would like to take a peek through 5.4 linear metres of the Bruce Trigger Fonds could check for me). In the spring he’d done his page proofs for the 2nd edition of History of Archaeological Thought from the hospital while undergoing chemo (it came out in September) and this was no different. We — his friends and students who were around — would make the trip west along Avenue des Pins, the road you can see behind Leacock in the image above (which apparently Google Maps still calls ‘Pine Ave’, in curious anglo tradition), to the Montreal General to chat with Bruce in his hospital bed, to talk about things. He was cheerful, every time. But we knew where things were heading.
I taught my class that morning, before I heard the news. It was his class, History of Archaeological Theory, the undergrad class he had taught so many times, that was mine only by administrative fiat, not by any right. It met down in the basement of Leacock, a windowless, joyless room. I still have the notes from that day; I still have all my notes from all the classes I’ve ever taught. It was the last class before the exam, and the topic of the day was “What I Think”, the final lecture (in those days it was pretty much all straight lectures from notes for me, which is how Bruce also did it), where I brought the class to a close by trying to situate what we’d been learning in terms of my own personal experiences and thoughts. Looking at it now, it’s a pretty pretentious class for someone who didn’t even have secure employment. I guess it was a good one though. I’m still in touch with some of the folks from that class, and they tell me I was cool back then.
I heard the news from my friend and mentor Mike Bisson, who for thirty years was one of Bruce’s best friends. You can read the obituary Mike wrote here. I had come back up to the 7th floor for lunch, and he caught me in the hall and let me know. No one was surprised; no one could believe it either. Bruce was 69.
I remember calling my wife, sitting on the other side of that desk, though I have no recollection of what she said other than “I’m sorry”. I don’t remember eating; I can’t have eaten. But I remember walking, out of the building, and down the narrow roadway that led, it can’t be more than a hundred meters, to the Redpath Museum (the building right in the foreground of the Google photo). To this day I’m not really sure why I went there. I’ve never told anyone that I went there before. I never, ever went there, except for some talk or event, even though it was literally right there next to Leacock, even though it has an amazing anthropological collection. It was, however, a place where, on a Friday afternoon in December, I could reliably be sure to be alone, to just wander around for a bit, to collect my thoughts. Which was, I suppose, what I needed. I spent some time upstairs on the second floor, in the Egyptology collection. Bruce was one of the central linkages between anthropology and Egyptology, and for a semester I even tried to teach myself some Middle Egyptian. But mostly I went there because it was a place on campus that didn’t remind me of anything.
But my day wasn’t done.
You mostly don’t remember individual classes. I mean, you remember some of the students, and you remember the courses, maybe even the room you were in, that kind of thing. Perhaps a fleeting moment, a well-timed joke. But the individual class meetings, no, not as individual instances. But I remember that class, or at least the start of it, when I came back to Leacock that day. It was my seminar on writing systems, held in the windowless interior seminar room right across from his office — my office. I don’t know why I didn’t cancel. The topic for the day was ridiculous. It was supposed to be our fun, let off steam, talk about something silly, end-of-term class. Here’s the agenda from that day (totally bizarrely, all the URLs still work fifteen years later!):
Constructed Scripts: We will be looking at three scripts: the phonetic alphabet known as Shavian; the Klingon alphabet, and Tolkien’s Elvish (Tengwar) script. The readings are mostly online; for some of them, it’s really just best to look at the Internet, which is where nerds live (not that I would know anything about that!) Readings: Hibbitt, George W. 1964. Pshaw for Shaw’s British alphabet. American Speech 39(3): 213-216. (Also take a brief look at http://www.omniglot.com/writing/shavian.htm, so you will know what it looks like); Attached Adobe Acrobat file: Klingon.pdf (2 pages); Web sites on Tengwar (Tolkien’s invented Elvish alphabet): http://ring-lord.tripod.com/tengwar/index.htm, http://at.mansbjorkman.net/tengwar.htm
And so I had to pretend to be, well, not ok, but to at least finish off the class, for my eight or so seminar students that day. I told them what had happened — the word hadn’t officially gone out, but of course it would, soon enough — and it’s not like I was in any real shape to do serious seminar work. And so here I was, explaining to a bunch of kids, 19 and 20 years old, that this man, the greatest scholar and one of the finest humans I ever knew, was gone, and to try to convey the monumentality of it all, when seriously, they probably just wanted to go home and work on their term papers which were due the next week. And then, I guess maybe we talked about Klingon? That, I don’t recall.
* * *
I stayed in that office, his office, through the spring and summer. The job fell to me — although I don’t think I would have let anyone else have it — of cataloging his library, of undoing thirty years of the habit of putting correspondence from authors in their own books on his shelves, so that it could all go to the McGill Archives. Helping redistribute books and journals to Bruce’s friends, family, and colleagues, the high school at Kanesatake First Nation, the Musée des Beaux-Arts, McGill’s own library, the McGill book sale. I still have dozens of his old reprints from the huge piles that were stacked in the office, including some underappreciated classic Bruce pieces that I still assign to students (Trigger 1975, 1976, 1981, 2003). I also have some real rarities and bizarro stuff sent from others (image above), not that Bruce believed the weirder of these things at all. Taking care of weird ephemera such as good old Tutankhamun and the Bruce Trigger Memorial Axe. In the end I don’t think much was just discarded.
I stayed at McGill another year after that, but in another office. That was ok. 724 wasn’t his anymore anyway, and I certainly had no claim on such a massive empty space. My office the next year was a bright, spacious corner office up on the eighth floor, shared with a full professor who wasn’t around much. And then from there, in 2008, I was off to my current office at Wayne State, which has its own weird ephemera that someone might go through someday. But this isn’t about me and my places.
I still bounce ideas off Bruce from time to time, fifteen years later. I think the Bruce in my head isn’t nearly as clever as I sometimes need him to be, but that’s all right; it does the job.
Chrisomalis, Stephen and Andre Costopoulos. “Bruce Trigger: citizen scholar.” In Human Expeditions: Inspired by Bruce Trigger, Stephen Chrisomalis and Andre Costopoulos, eds., pp. xiii-xx. Toronto: University of Toronto Press, 2006.
How many number systems are out there? When I finished my dissertation in 2003, I described my work as analyzing “over 100” structurally distinct numerical notations. Counting them is really impossible, because no one knows what ‘structurally distinct’ means. Does it ‘count’ as a distinct system when, in Western Europe, folks started to use numeral delimiter commas (26,000 vs. 26000) or decimal points? I was hopelessly trying to give a number, without necessarily counting the dozens of decimal, positional systems of the broader Indo-Arabic family. All those systems descended from the positional variants of the Brahmi numerals that originated in early medieval India, in which all sorts of script traditions use ten signs for 0-9 but substitute local signs. We can call those all different systems, or we can not, depending on our perspective.
But then by the time my dissertation became a full-fledged book, Numerical Notation: A Comparative History, in 2010, having been poked and prodded by no fewer than 14 peer reviewers (yes, really!!!), more systems were added. I stuck with “over 100” because, well, that’s technically true, but by that point it was many more than that. And I keep finding more. There’s so much out there that hasn’t been accounted for. I was going over some notes earlier this week and there are at least 25 notations on my ‘to add’ list not described anywhere in the synthetic / comparative literature. Probably closer to 50, and counting. Part of the challenge is that these are notations that are peripheral to the concerns of the major traditions of philology, epigraphy, and the history of science. I don’t think I missed any well-known ones! Some of them may have been used by only a handful of individuals, or for a short time. But there are a lot of them – far more than I would have guessed when I started on this wild path.
In a single article (cited only four times since publication), M.A. Jaspan (1967) described not one but two numerical notation systems used by speakers and writers of Rejang, a language of southwestern Sumatra. Other than technical reports by Miller 2011 and Pandey 2018 for Unicode encoding, basically no one has ever acknowledged or discussed them:
This first system may look unusual, but it is part of a broad tradition of aksharapallî systems, which use the alphasyllabaries (abugidas) of South and Southeast Asia, in their customary order, to assign numerical values to specific syllables (Chrisomalis 2010: 212-213). Here, the 23 signs (with the implied vowel ‘a’) correspond to 1-9, 10-90, and 100-500, and then for the higher hundreds, two signs combine additively. This system doesn’t have a zero – each multiple of each power of the base (10) gets its own sign, so it’s what I’ve classified as ciphered-additive – like Greek, Hebrew, and Arabic alphabetic numerals, or Cherokee, Jurchin, or Sinhalese, among others. Jaspan is dead wrong in writing (1967: 512) that “It has, as far as I know, no parallel or similarity to, other known systems either in South-East Asia or elsewhere.” Aksharapallî systems were once widespread throughout South and Southeast Asia, and are used for various purposes, including pagination, which is exactly what Jaspan reports that at least some Rejang writers used them for during his fieldwork in the early 1960s.
The second system is in some ways, even more striking. The system is structurally almost identical to the Roman numerals – there are signs for each power of 10, as well as the quinary halves 5 and 50. The hundreds are still additive but have some more complexities, and then the thousands don’t have a quinary component at all. These sorts of systems that rely on repeated signs within each power, and don’t use place-value, are called cumulative-additive and are very common throughout the Near East and the Mediterranean but relatively rare in East and Southeast Asia (though there are systems like the Ryukyuan suchuma that have this structure). I have absolutely no idea where it came from – unlike the first system, it doesn’t have any obvious relatives. At least for Jaspan’s consultants, it was used for keeping business accounts in the 1960s, though not widely.
The standard history of numerical notation is one where all systems gave way to a single, universalizing notation, the digits 0123456789, which spread globally without competition. And there’s certainly a point to be made there. But there is a countervailing factor, the inventive impetus under which we can expect all sorts of notations to be invented, perhaps not with global reach, but of critical importance for understanding the comparative scope of the world’s numerical systems. In my new book, Reckonings: Numerals, Cognition, and History (Chrisomalis 2020), I make the case that we are not at the ‘end of history’ of numeration – that innovation continues apace in this domain, and that focusing only on the well-known systems produces a very barren history. Cases like the Rejang numerals help produce a richer narrative – one of constant and ongoing numerical innovation.
Chrisomalis, Stephen. Numerical notation: A comparative history. Cambridge University Press, 2010.
Chrisomalis, Stephen. Reckonings: Numerals, cognition, and history. MIT Press, 2020.
Jaspan, Mervyn Aubrey. “Symbols at work: Aspects of kinetic and mnemonic representation in Redjang ritual.” Bijdragen tot de Taal-, Land-en Volkenkunde 4de Afl (1967): 476-516.
Miller, Christopher. “Indonesian and Philippine Scripts and extensions not yet encoded or proposed for encoding in Unicode as of version 6.0.” (2011).
Pandey, Anshuman. “Preliminary proposal to encode Rejang Numbers in Unicode.” (2018).
Once again, the early-career scholars in the 2021 edition of my course, Language and Societies, have written some amazing papers, for which the abstracts are linked below. The authors are undergraduate and graduate students in anthropology and linguistics at Wayne State University. Comments and questions are extremely welcome, especially at this critical juncture, when the authors are making final revisions to their papers.