Doorworks 2: Columna rostrata

Elogium of Gaius Duilius, Rome, Pal. dei Cons, CIL 12.125, 6.1300

The Columna rostrata was originally erected in Rome in 260 BC, commemorating the naval victory of the Roman consul Gaius Duilius over the Carthaginian fleet. The inscription boasted that over two million aes of loot were plundered. Rather than expressing the amount in numeral words, it was written using at least 22 (and possibly as many as 32 – the inscription is fragmentary) repeated Roman numeral signs for 100,000, seen towards the bottom of the inscription. The effect of this ‘conspicuous computation’ was to impress the reader with the vastness of the quantity, serving as an indexical sign of Rome’s military might.

Doorworks 1: Margarita philosophica

I spent an hour or so today putting up some material on my previously-barren office door, one page of which is the first in a new series, ‘Doorworks’.  These will normally be an image or plate of something related to my core research interest in numeration, followed by a brief description.  Since none of you (to my knowledge) are anywhere near my office door, I thought I’d post them here as well as I put them up (once every week or two), to give you a sense of the sorts of individual objects and texts that interest me. Enjoy!

Gregor Reisch, Margarita philosophica. Freiburg: Johann Schott, 1503.

This plate is an allegorical representation of Arithmetic as a female figure bedecked with Western (Hindu-Arabic) numerals. At her left hand sits Pythagoras, using the counting-board (abacus) with loose pebbles on lines. At her right hand is the sixth-century philosopher Boethius, once thought to be the inventor of Western numerals. By turning her head towards the latter man, Arithmetic indicates her favor to the new system. The discourse over the efficiency of different arithmetical techniques (e.g. Roman vs. Western numerals) reached its climax in the fifteenth and sixteenth centuries, as the newly literate middle class moved away from arithmetic as traditionally practiced in medieval universities, and as commercial arithmetic texts began to be produced in large quantities advocating the newer Western system.

On ‘Western numerals’

For the past nine years, really ever since I defended my dissertation proposal, I have been using the term ‘Western numerals’ to describe the set of signs 0-9 used in a decimal fashion with the place-value principle.  This is not standard practice (although it is not unique to me), and after someone asked me about this, I thought I’d explain myself, since I’ll undoubtedly be using the term repeatedly in my numerical posts.

In the English-speaking world, we all learn these signs under the name ‘Arabic numerals’, which reflects the fact that they were borrowed by Western Europeans from Arabs living in Spain, Sicily, and North Africa in the tenth century CE.  In the scholarly literature on numerals, these are most often called ‘Hindu-Arabic numerals’, which reflects a little more of the history of the system, because the Arabic script got its numerals from an antecedent system used in northern India as early as the fifth or sixth century CE.   The historian of mathematics, Carl Boyer, whose early work on numeral systems played an important role in my development as a ‘numbers guy’, argued somewhat facetiously that we might more properly call it the ‘Babylonian-Egyptian-Greek-Hindu-Arabic’ system (1944: 168) – although in this case I think he was wrong, and that ‘Egyptian-Mauryan-Hindu-Arabic’ would get the history straight.

The most basic problem with these formulations ‘Arabic’ and ‘Hindu-Arabic’ is that they do not adequately distinguish the set of signs 0123456789 from the set of signs ٠١٢٣٤٥٦٧٨٩ used in Arabic script from the set of signs ०१२३४५६७८९ used in the modern Devanagari script, and any number of other decimal, place-value systems, all descended ultimately from that 5th-6th century CE Indian ancestor.  To make matters more confusing, in Arabic the numerals used alongside Arabic script are called arqam hindiyyah (Hindi numerals).

The problem of ambiguity is thus a serious one.  Because several such systems are in active use (particularly the Western European 0-9 and the ‘Arabic’ set) it becomes a nightmare to try to distinguish these systems meaningfully.  We need different terms for each set of numerals.  Not only is there potential ambiguity, but using the term ‘Arabic’ or ‘Hindu-Arabic’ for 0123456789 tends to obscure the continued existence and active use of actual ‘Arabic’ and ‘Hindi’ numerals in the Middle East and south Asia.

So I talk about Western, Arabic, and Indian numerals to refer to the place-value systems used in three different script traditions.  Structurally the systems are identical, but paleographically – in terms of the history of the signs themselves – they are quite distinct. Now, one could argue that just as we talk about the ‘Latin alphabet’ we could call 0123456789 the ‘Latin numerals’ instead of ‘Western’, but this would only create confusion with the ‘Roman numerals’.  ‘Western numerals’ reflects the fact that the particular graphemes (sign-forms) developed in a Western European context and were first and most prominently used in Western Europe.

Now, there is a counterargument, that by calling them ‘Western numerals’ I am denying them their history, obscuring the fact that they derived from Indian and Arabic notations, which I certainly do not wish to do!  But I think that Boyer has a point – why stop at ‘Hindu’, since the Hindu place-value numerals derive from a non-positional system used in Brahmi inscriptions in India as early as the 4th century BCE, which in turn probably derive from Egyptian hieratic writing going back as early as the 26th century BCE!  And if we decide that the history is wrong, do we change the name?

Basically I am dissatisfied in general with the notion that we should name extant phenomena after their place of origin; it causes so many problems, including ambiguous nomenclature, that I decided to give up the practice entirely.  Hence ‘Western numerals’.

Works Cited

Boyer, Carl. 1944. Fundamental steps in the development of numeration. Isis 35(2): 353-368.

Why numerals?

Over the past few weeks in my new job, I have had many opportunities to introduce myself or be introduced as an expert in the ‘anthropology of mathematics’, which is probably the simplest and most accurate way to describe my work (although I also have strong research interests in other areas, such as writing systems and cross-cultural theory).  The comments I receive on this are mostly of two sorts:

(a) Oh, how interesting! I had never imagined there was such a thing!

(b) Oh, how interesting! Are there many people working in that field?

Both of these are perfectly understandable responses, because there aren’t really very many of us out there; perhaps a dozen working anthropologists who publish regularly on numerals, and a couple dozen more linguists.   There are many more anthropologists and linguists who at some point have written something on the subject, but aren’t specialists in the field.   But in comparison to, say, psychologists working on mathematical cognition, or to historians of mathematics, there aren’t too many of us.  And honestly, I’m okay with that, because it means that there are plenty of great questions that are completely untouched.

The question of why one would study the anthropology of mathematics is actually more interesting, from my perspective, and usually it doesn’t take much to get me onto that subject, particularly with people who answer (a).  For me, the fantastic thing about the subject is that it is so often taken for granted that there is one thing called ‘number’, or one thing called ‘mathematics’, and that there should be limited cross-cultural difference in the domain.   But at the same time, anthropologists generally work in domains where there is a lot of variability and assume that there are few or no constraints on human behavior, but in numeral systems there are all sorts of constraints, some evolutionary, some functional, and some social.  So teasing out the differences and similarities, without assuming in advance that the phenomenon is highly regular or highly variable, is fascinating stuff.   In other words, the fact that I am a comparativist (rather than a cultural particularist) and the fact that I study mathematics are closely linked.

The other really fascinating thing about number systems, for me, is that numbers can be represented using spoken or written language, but can also be represented using graphic numerical notation systems (like the set of signs, 0123456789, which laypeople generally call Arabic numerals but I call Western numerals).  So you have one system that has its origins in auditory media and is linked to linguistic abilities, and another that gets away from directly representing language and is trans-linguistic, but is nonetheless probably linked somehow to language.  So from an anthropological perspective, you have one linguistic and one non-linguistic (graphic) representation of number, which allows you to ask all sorts of interesting questions about the intersection of language and culture.

The last thing that is really cool about number is that within the domain of numerical notation, we have a pretty good database of all the numerical notation systems that have ever been used, and can without too much difficulty reconstruct the relationships between them (i.e. which systems are ancestral to which others, or which systems replaced which others).   This allows us not only to look at each system as a structured system of signs in a synchronic fashion (omitting the time dimension) but also to engage in a diachronic analysis, examining how systems interact and change over 5000 years of written history.   This is why I describe my forthcoming book as a ‘comparative history’.  But I’m writing about numbers, not about cross-cultural theory, which will have to be an essay for another day, because right now my book manuscript isn’t going to edit itself.

Was Stonehenge mathematically structured?

(Originally posted at The Growlery, 2008/06/07)

Stonehenge is never really out of the news; in fact, it’s probably the archaeological site that enjoys the most media exposure. Even so, it has been in the news quite a lot lately, what with the recent report of work by Mike Parker Pearson’s Stonehenge Riverside Project, which has given us radiocarbon dates from burials excavated in the 1920s, suggesting that the site was used for burials from around 3000 BCE, several centuries earlier than previously thought and really quite early in the British Neolithic.

I must insist, however, that Parker Pearson’s theory that Stonehenge stood in contrast to the much larger timber circle at Durrington Walls, is plausible at best but completely unproven. Based on an ethnographic analogy resulting from some earlier fieldwork he did in Madagascar, Parker Pearson has sought to revive structuralist archaeology with his contention that the two British sites were conceptually binary opposites, the stone of Stonehenge representing permanence and ancestry, with wood representing transience and impermanence. Okay, so far so good (though still ‘not proven’). However, he goes on to assert that stone is not only ancestral but also male, while wood is (quoting PP himself) “soft and squishy, like women and babies.” (1) At which point my inclination is to get out a Walloping Cod and suggest that he keep his structuralism to himself until he has archaeological evidence for all this.

But lost amidst all this highly-funded work is a new book by Anthony Johnson, Solving Stonehenge: The Key to an Ancient Enigma (Thames and Hudson, 2008), offering clues to the mathematical abilities of the builders of the monument. I haven’t read the book (which isn’t published for another week) , but an article in the Independent suggests that the geometrical knowledge of the builders was more considerable than previously believed (by some). I hadn’t heard of Johnson before (despite having a very strong research interest in the prehistory of mathematical thought, and a secondary interest in the archaeology of megaliths). He appears to be a doctoral candidate at Oxford working on geophysical techniques in archaeological survey, but has not published before on Neolithic mathematics. One does need to be cautious when dealing with topics in ancient science, which are particularly prone to attracting the attention of pseudoarchaeologists, but I don’t think that’s what’s going on here.

How, then, do we evaluate what an ancient monument can tell us about the mathematical abilities of its creators? The most important finding that Johnson is suggesting, from my perspective, is that other than the well-known solar alignments of the monument, no significant astronomical knowledge was employed in the orientation of the site. Rather, it was in geometry, and the creation of complex polygons using ‘rope-and-peg’ technology (making arcs and lines on the landscape using physical means), that the Stonehenge builders excelled, creating, over 1000+ years of the site’s history, a palimpsest of complex polygons among the various features of the site. By ‘experimental archaeology’ I take it that Johnson used this technique himself to show that using modest technology and a modicum of geometrical knowledge about the relationship between circles and squares, the monument’s shapes could have been constructed precisely. This is fascinating stuff, and gets us away from Alexander Thom’s ‘megalithic yard’ and Gerald Hawkins’ ‘Neolithic computer’ theories, both of which start from the assumption that astronomy was the function of the site.

My only major issue (prior to reading the book, which I’ll have to do over the next few months) is Johnson’s claim that “It shows the builders of Stonehenge had a sophisticated yet empirically derived knowledge of Pythagorean geometry 2000 years before Pythagoras”, mostly because Pythagoras was essentially a fiction about whose work we know almost nothing, and because it suggests inappropriately to the untutored reader that in fact the Stonehenge builders had proven the Pythagorean theorem, which is not what is being claimed. It’s not quite the same kind of error as asserting that sunflowers ‘know’ the Fibonacci sequence because their florets are arranged in such a pattern (okay … no one actually claims that, as far as I know). The point is, though, that there is always a danger in inferring specific mathematical knowledge from the outcomes of processes such as the rope-and-peg technique. Similarly, while it is plausible that “this knowledge was regarded as a form of arcane wisdom or magic that conferred a privileged status on the elite who possessed it”, we don’t actually know who exactly controlled this knowledge (and how), whether in fact the engineers/surveyors/artisans involved were part of the (as-yet incipient) social elite at the site, whether that status changed at all over a millennium or more (almost certainly!), and whether in fact geometrical knowledge was perceived as ‘magic’ in any sense.

In my ‘Prehistory of Language and Mind’ seminar, I emphasize the real dangers in attempting to hermeneutically insert oneself into the minds of prehistoric individuals based on their material culture, a caution that is worth repeating here. This is particularly true in the case of megaliths, which archaeologists approach too often on the basis of intuition, faulty ethnographic analogies (I’m looking at you, Parker Pearson…) and wishful but unsupported thinking, as Jess Beck and I show in a forthcoming publication (2). All of which is fine when one is speculating idly, or creating one’s own personalized or intuitive understanding of the past, but is pretty shoddy evidence-based scholarship. Accordingly, I’d insist that even Johnson’s work (to which I am initially positively disposed, and whose use of experimental archaeology is a definite advantage here) needs to be treated with the utmost caution, due to the exceedingly high risk of erroneous interpretations of ancient scientific abilities.

(1) Caroline Alexander. 2008. If the stones could speak. National Geographic, June 2008, p. 50.
(2) Jess Beck and Stephen Chrisomalis. Landscape archaeology, paganism, and the interpretation of megaliths. Forthcoming in The Pomegranate.

ETA: Anthony Johnson himself has now commented on the post, noting that the quotation from the news article about Pythagoras does not actually reflect his words. The blog for his book can be found at Sarsen56.

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