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.

Phaistos phakery redux

(Originally published at The Growlery, 2008/08/21)

Prior to writing my previous post about Jerome Eisenberg’s conclusion that the Phaistos Disk is a recent forgery perpetrated by its excavator, Luigi Pernier, I unfortunately did not have access to the original article in Minerva magazine in which Dr. Eisenberg announced his findings (Eisenberg, Jerome M., ‘The Phaistos Disk: one hundred year old hoax?’, Minerva, July/August 2008, 9-24). Happily, once he found my post, Dr. Eisenberg commented on it and later sent me an electronic copy for my consideration. I can now report that while I previously thought I knew a lot about the disk, I now have a much better knowledge of the disk and the nature of the hoax claim. Unfortunately, I remain unconvinced, even though I admit that I do want to believe the hoax claim, but I think that the evidence from the sign-forms just isn’t strong enough, that it relies on unproven visual similarities to too great a degree. Let me explain what I mean.

In the previous post, I focused on Eisenberg’s evidence from a) the uniqueness of the artifact’s manufacture, which is unlike the Linear A tablets; b) it uses ‘movable type’ of which no other example has ever been found; c) the idea that Luigi Pernier’s rivalry with Arthur Evans would lead him to do this. In dealing with the first two, I pointed out that comparisons with the Linear A tablets aren’t necessarily that useful if in fact the PD was part of a highly specialized text genre – i.e. it would be like using a monumental inscription to proclaim a handwritten note to be a forgery. Of course we don’t know that it’s part of such a genre, or indeed what genre it could have represented at all – hence the mystery.

The one thing I didn’t focus on is the sign-forms or graphemes on the Disk. In fact, Eisenberg spends a good deal of his paper looking at resemblances between Phaistos signs and signs on other inscriptions from the ancient world in order to assert that the latter formed the models on which Pernier based his forgery. In particular, he aims to show that there are similarities between the Phaistos graphemes and authentic artifacts made much later, but that were known to 19th century archaeologists / epigraphers and thus could have been known to Pernier.

This is an unusual line of argument; it is in fact a sort of cousin to the standard techniques by which experts on scripts postulate cultural borrowings from one society to another. If we have a 10th century BC Phoenician inscription and a very early, 8th century BC Greek inscription that use many similar letter-forms, we make the reasonable inference (all right, it is more complex than this, but you get the idea) that the Phoenician script is ancestral to Greek. In particular this is the case because there is known cultural contact (e.g. trade) between the two societies, and more importantly, because there is not just a graphemic similarity but also a phonetic similarity – the signs don’t just look the same but they have the same / similar sound-values. What Eisenberg is doing, effectively, is turning these resemblances on their heads. If there are similarities between the PD signs and known inscriptions from elsewhere, then those inscriptions may have acted as a model for the forger. If the inscriptions are later in date than the PD, Eisenberg argues, it is far likelier that these artifacts served as a model for the disk’s forger than that the Disk script served as a model for the later artifacts. Similarly, if the PD shows influences from several different regional styles, this suggests that a forger just cobbled together signs from different inscriptions to make something really unique.

Now, the reason I’m unconvinced is that I just don’t think the similarities bear up, and that even where they do they don’t point unequivocally to a hoax. For instance, let’s have a look at Phaistos sign 03:

Now, this is seen by Eisenberg as being modelled after an 18th Dynasty Egyptian wall painting (16th century BC) in which the figure, a Cretan captive is facing the other direction, has extensive facial features, has hair (long, flowing hair), and has a torso with arms. The only major similarity is the two circles on the face. But this seems to go directly against the notion of the Disk as a hoax; the time is right, the captive is Cretan, so the most parsimonious explanation is that they are both genuine representations of some sort of facial decoration (indeed, as Eisenberg suggests, it may be a Cretan ‘double earring’). But, writing, “It was certainly derived from the wall painting”, Eisenberg proceeds to write as if it is now a given that Pernier did, in fact, use this as a model for sign 03 (Eisenberg 2008: 17).

When we get to one of the more unusual characteristics of the Disk – the presence of five hand-incised dots on each side of the disk, and ‘word-separating’ vertical lines – I’m in my element, because these, Eisenberg sees as being modelled after the Cretan five dots = the numeral 50 and vertical bar = the numeral 100. This is dangerous territory though – dots and lines are ubiquitous in scripts and numerical systems. And are we really to believe that Pernier needed a model to think of the idea of adding bars and dots to a forgery? These are stylistic elements found in virtually any script worldwide, and are not indicative of anything. One of the real problems with the study of writing systems is the assertion of cultural relationships based on passing visual similarities, and one of the things that we do not yet know how to do well is to know how similar two graphemes must be before a claim of diffusion can be sustained. This is the same sort of reasoning used to argue for a hoax in this case, and ultimately its inclusion greatly weakens Eisenberg’s argument, and made me look much more critically at the remainder of his claim.

But the heart of the issue is that Eisenberg is working at cross-purposes here. On the one hand, he wants us to believe that the Disk is so unique, so different from other inscriptions that it cannot possibly be genuine. On the other, he wants us to use evidence of similarities with known scripts as proof of ‘forger’s models’. While a hoax can, of course, be both unique and based on models, we’re left with the impression that virtually any similarity or difference can be evidence of forgery, and that just isn’t sound argumentation. So I’m not convinced. I do still think the idea is worthy of consideration, and I do think that it is worth trying a thermoluminescence test, not only because it can settle the hoax issue but also because it can resolve the question of the artifact’s age even if it turns out to be genuine. In this respect, I believe that Eisenberg and I are in full agreement.

In conclusion I want to thank Dr. Eisenberg for sending me this paper, and also for inviting me to the upcoming International Conference on the Phaistos Disk, which unfortunately I am unable to attend due to my new work commitments. It does highlight however the real value of blogging as a means of social interaction and information exchange.

But of course the real question remains unanswered: should it be disk or disc?

Is the Phaistos disk a phony?

(Originally posted at The Growlery, 2008/08/04)

The Phaistos Disk is one of the more enigmatic and bizarre artifacts in the field of ancient writing systems. Found in Crete in 1908 by the archaeologist Luigi Pernier and associated archaeologically with the Minoan civilization (dating to roughly 1850 – 1600 BCE), it remains completely undeciphered and has no obvious connection either to the Minoan (Linear A) script or to any other known script, deciphered or otherwise. Now, a very notable claim has been made by the American art historian / art dealer Jerome Eisenberg, an expert on forgeries, that the Disk was in fact an elaborate hoax constructed by Pernier himself, which Eisenberg has published in his own magazine, Minerva (Eisenberg 2008).

I’m not an expert on Minoan writing by any means, but my scholarly focus lies heavily in the study of ancient scripts and the anthropology and archaeology of literacy. I use Yves Duhoux’ hilariously entitled ‘How not to decipher the Phaistos Disc’ in my course on the anthropology of literacy (Duhoux 2000). Moreover, the century of scholarship on the Phaistos Disk is legendarily riddled with cranks, frauds, and loons, and as I have more than a passing interest in pseudoarchaeology, Phaistos-related material is of ongoing interest to me. Honestly, it would make a lot of things a whole lot simpler if we could just deny the disk’s authenticity – but this is no ordinary hoaxbusting exercise, and the importance of the artifact demands that we give the claim close scrutiny.

Phaistos Disk, Side A
Phaistos Disk, Side A. Source: Wikimedia Commons.

Before we get to the Eisenberg claims, we need some context. So, firstly, what do we know about the PD?
– It is a fired clay disk, roughly 16 cm in diameter and 1-2 cm thick.
– It was found at a Minoan palace site at Phaistos in southern Crete.
– It appears to have been fired intentionally (with care) to produce a permanent record, whereas other Minoan documents were fired accidentally (e.g., when buildings were burned).
– Glyphs are stamped on both sides using distinct punches or stamps, not carved/incised into the clay.
– It has 241 signs in total, consisting of 45 distinct characters / glyphs. However, the total ‘signary’ (all the signs in the system) was probably greater, since some rare signs almost certainly do not appear in this particular text.
– The ‘text’ is divided into 61 sections of up to 5 characters apiece.
– The prior two facts suggest that it may have been a syllabic writing system, recording syllables rather than single phonemes; it has too many unique signs to be an alphabet but too few to be a logographic (word-signs) or some other sort of system. However, this does not rule out the possibility that it was not phonetic writing at all (e.g., if it was a calendar or a game).
– Because the signs in the centre are slightly compressed, it seems to have been written from the outside spiralling inward.
– Judging by the overlap in some signs, it was stamped/written from right to left, suggesting that that is how it was meant to be read.
– There is no useful resemblance of the glyphs to those of any other writing system in the Mediterranean or elsewhere, although it was found in close association with a Linear A (Minoan) tablet.
– Its date is established solely through its archaeological context, and while the early second millennium is the most likely period, it may date as late as 1400 BCE.

Now, on to Eisenberg’s paper. The first thing worth noting is this is not a peer-reviewed academic venue, and the author is the founder, editor, and publisher of the magazine. A better analogy would be to think of it as editorial opinion. It also is not the result of any particular new research undertaken by Eisenberg or anyone else. In fact, as seen in the comments here, Dr. Eisenberg has been making this claim for nearly a decade, and there is no new evidence that demonstrates the likelihood that it was a forgery. Pernier, the artifact’s excavator excavator, is labelled as a forger, not on the basis of any particular evidence, but has simply been ascribed motives (rightly or wrongly) that might lead him to falsify the document. So we don’t have anything like the revelations in the early 1950s that debunked the Piltdown hoax on the basis of physical or chemical analysis; neither do we have the spectactular video evidence that revealed Fujimura Shinichi planting fake discoveries at his sites in the Japanese Paleolithic hoax in 2000 (Hudson 2005). It is a highly circumstantial case. It is nonetheless one that ought to be vetted seriously, both because it is plausible on its face and because Eisenberg has been responsible for several other (much more solid) hoax-busting episodes over the past few decades.

The starting point for Eisenberg’s claim of a Phaistos ‘hoax’ is the uniqueness of the artifact, both the object itself and the writing on it. Given that no other examples of this form of writing have been found, it is striking (pun intended) that its creator would have made 45 distinct seals to stamp into the clay rather than simply incising the signs as necessary. No actual stamps/seals resembling the signs have been found, either, suggesting that this early instance of ‘movable type’ was used to create only one artifact, and then the process was abandoned entirely. In his popular Guns, Germs and Steel, the evolutionary biogeographer Jared Diamond (1997: 239-259) asserts that the PD was indeed a very early and remarkable example of movable type, but one that could not be exploited by the Minoans because in other respects their society lacked the technology and organizational expertise to develop it further. Eisenberg’s perspective is different – he argues that the uniqueness of the artifact’s medium suggests that it is a hoax, designed by Pernier to intrigue and mystify other scholars and to boost his own prominence, and that of Phaistos, in relation to his rivals (particularly Arthur Evans).

The PD is a singular artifact and a very short text, making it literally impossible to decipher unless more examples of the writing system are found. Yet John Chadwick, whose career was built upon his work with Michael Ventris in deciphering the Mycenaean Linear B script (Chadwick 1990), was plagued by purported Phaistos decipherers and purportedly received one new solution per month; there is a fairly thorough list of purported decipherments in this Wikipedia article. Basically, every remotely plausible script tradition has been claimed as an influence, and the disk itself has been asserted to be in languages ranging from Greek to Egyptian to Basque to Atlantean (!!!). Alternately, it has been suggested to be a game board, a calendrical document, or some sort of mystical text. Unless more documents in the same script are found, no one is going to be able to resolve the matter definitively. If it were confirmed to be a hoax, however, everyone could just stop looking. Eisenberg is suggesting, in effect, that the futility of the search rests in part on Pernier’s ingenuity in creating such a mystery.

The crux of Eisenberg’s argument, however, lies in the physical properties of the artifact: the fact that it was very carefully, intentionally fired, and that it has a very cleanly cut edge in comparison to other Minoan clay tablets, and here, he finds fault with Pernier. Because it is so different from other Minoan clay artifacts in this regard, this sends up a red flag for Eisenberg suggesting that its uniqueness may be due to Pernier’s ignorance of these facts. The counterargument to this, however, would be that while Minoan clay tablets with Linear A writing are all economic documents not intended for long-term archiving, the PD, if ancient, is almost certainly of a very different textual genre and script tradition than these texts. This doesn’t disprove the notion that it may be a hoax, but neither does it act as substantial confirmation. For instance, if the disk is a gaming board, a calendar, or a devotional inscription, its makers would have a good reason to fire the clay at the time of manufacture, and a potentially good reason to cut its edges so cleanly. It simply was not the same sort of text as the copious clay economic documents. We need to answer the question, “Could the Minoans have chosen to preserve some forms of information permanently and not others?”

One potential resolution to the mystery lies in its dating. The artifact has never undergone any sort of radiometric dating, and indeed for most of the past century could not have been dated except through archaeological context, as discussed above. However, thermoluminescence dating allows archaeologists to non-destructively determine the date when clay was fired, and if TL dating were used on the disk, one could find out if it was truly of ancient manufacture. Yet this test has not been permitted by the museum that holds it (in Heraklion, Crete), because, Eisenberg claims, “no Greek scholar or politician would dare to help ‘destroy’ such a national treasure”. This is unfortunately true; museums are rarely open to this sort of inquiry, even from major scholars. Archaeology is frequently tied up in nationalistic fervor and institutional pride, and the failure to undertake a standard, well-accepted test will haunt the study of the Disk from now on, now that the claim has been made so publicly. Thus, I regard Eisenberg’s public claim as a valuable stimulus, hopefully forcing the issue of the thermoluminescence dating. It would also be highly informative even if the PD proves to be ancient, because the TL could establish whether it was an early second millennium artifact (1800-1600 BCE) or more in the range of 1400 BCE.

Ultimately, this is suggestive, and I would not exactly be astonished if Eisenberg’s claim were to be verified, and if the PD turned out to be a fake, but I cannot agree that the matter is now settled. Because literacy is not simply an ‘on/off’ phenomenon – we must deal with the possibility of different text genres, different media, and different purposes for writing – we can’t use the Linear A clay economic documents to prove the disk’s anomalous nature. A date from an independent lab would go a long way toward resolving my doubts. This would still leave the question of how it was done and by whom – remember that there is no direct evidence against Pernier. However, I for one look forward to this claim receiving greater attention over the next couple of years.

Chadwick, John. 1990. The decipherment of Linear B, 2nd ed., Cambridge University Press.
Diamond, Jared. 1997. Guns, germs and steel: the fates of human societies, W.W. Norton.
Duhoux, Yves. 2000. ‘How not to decipher the Phaistos Disc: a review article’,American Journal of Archaeology, 104, 3, 697-700.
Eisenberg, Jerome M. 2008. ‘The Phaistos Disk: one hundred year old hoax?’, Minerva, July/August, 9-24.
Hudson, M.J. 2005. ‘For the people, by the people: postwar Japanese archaeology and the Early Paleolithic hoax’, Anthropological Science, 113, 2, 131-139.

Accreditation and online scholarship

(Originally posted at The Growlery, 2008/06/08)

There’s an interesting essay in the Chronicle of Higher Education online by Gary Olson, dean of the college of arts and sciences at Illinois State University, addressing the question of certifying online research. Olson argues persuasively that online research is not taken seriously enough, and that while peer-reviewed online journals have found acceptance as ‘real’ academic work for the purposes of hiring, tenure, and promotion, other forms of work such as databases, online bibliographies, and other Internet sites remain essentially unaccredited, and thus easily ignorable within the academic mainstream. His solution is for each discipline to create its own canonization process to accredit and review this material in a manner best suited to its disciplinary conventions.

I’m about as big an advocate for online research as you will find anywhere. In particular, I find it extremely valuable to use my senior seminars (and eventually, graduate courses) as launching pads for high-quality student work that would otherwise not see the light of day, as I have done in the Pseudoarchaeology Research Archive and the Dollarware Project. An extremely important part of academic professionalization derives from taking “finished” term work, editing/revising it, and then putting it out for the world to see. If you aren’t interested in doing that (to some degree), then no matter how bright you are, you’re not really interested in being a scholar, and it’s better you should figure that out as a senior, or as a master’s student. I’m also firmly convinced that ‘even’ undergraduates (at least, the very best senior undergraduates) are capable of producing work that is of quality equal to much peer-reviewed research, and that there is an unfair prejudice against this work when it is known in advance to have been written by very junior scholars.

One potential benefit of disciplinary accreditation is that both I and my students might benefit if these projects were ‘officially’ accredited. And believe me, if such a system existed, I would be at the head of the line, submitting online projects for consideration. One concern would be, however, that if accreditation is merely seen as something that Ph.D. holders should receive for their work, then we would end up in a situation where good work ends up just as marginalized as before. Obviously, the material is still out there; Olson’s proposal is far closer to a ‘publish-then-peer-review’ model than the current ‘peer-review-then-publish’ model. But if what exists is perceived as being illegitimate, or controlled very narrowly by a small group of insiders within disciplinary societies, then what is created is a monstrosity in which these elite individuals hold power far greater than any journal editor or academic press.

In fact, the greatest challenge with the current model is that the people in charge of hiring/tenure/promotion are not part of a culture that considers these online publications to be legitimate. I’m not sure that an online peer-review/accreditation system will change that – tenure committees are free to denigrate or ignore all sorts of publications they consider to be second-rate, for instance. I understand the argument Olson is raising – that this gives deans, provosts, etc., information about the importance of a work to a discipline that they otherwise couldn’t really have. But without a general cultural change within a discipline – for instance, physics has already shifted to a model where online publications are given considerable weight – I’m not sure that this does anything but shift the error from tenure committees to the disciplinary associations themselves.

I am very strongly in favour of Olson’s proposed changes, in theory. In anthropology, this would pave the way for online site reports, field notes, photo journals, and other scholarship to receive critical attention, and to promote the publication of scholarship that otherwise might not be published because it’s not perceived as beneficial to one’s career. But the devil is in the details, as always, and what is needed is to continue to discuss these issues constructively to build a model.