Five great 2014 articles on number systems

The scholarship on numbers is, as always, disciplinarily broad and intellectually diverse, which is why it’s so much fun to read even after fifteen years of poking at it.  This past year saw loads of great new material published on number systems, ranging from anthropology, linguistics, psychology, history of science, archaeology, among others.  Here are my favourite five from 2014, with abstracts:

Barany, Michael J. 2014. “Savage numbers and the evolution of civilization in Victorian prehistory.The British Journal for the History of Science 47 (2):239-255.

This paper identifies ‘savage numbers’ – number-like or number-replacing concepts and practices attributed to peoples viewed as civilizationally inferior – as a crucial and hitherto unrecognized body of evidence in the first two decades of the Victorian science of prehistory. It traces the changing and often ambivalent status of savage numbers in the period after the 1858–1859 ‘time revolution’ in the human sciences by following successive reappropriations of an iconic 1853 story from Francis Galton’s African travels. In response to a fundamental lack of physical evidence concerning prehistoric men, savage numbers offered a readily available body of data that helped scholars envisage great extremes of civilizational lowliness in a way that was at once analysable and comparable, and anecdotes like Galton’s made those data vivid and compelling. Moreover, they provided a simple and direct means of conceiving of the progressive scale of civilizational development, uniting societies and races past and present, at the heart of Victorian scientific racism.

Bender, Andrea, and Sieghard Beller. 2014. “Mangarevan invention of binary steps for easier calculation.Proceedings of the National Academy of Sciences 111 (4):1322-1327.

When Leibniz demonstrated the advantages of the binary system for computations as early as 1703, he laid the foundation for computing machines. However, is a binary system also suitable for human cognition? One of two number systems traditionally used on Mangareva, a small island in French Polynesia, had three binary steps superposed onto a decimal structure. Here, we show how this system functions, how it facilitated arithmetic, and why it is unique. The Mangarevan invention of binary steps, centuries before their formal description by Leibniz, attests to the advancements possible in numeracy even in the absence of notation and thereby highlights the role of culture for the evolution of and diversity in numerical cognition.

Berg, Thomas, and Marion Neubauer. 2014. “From unit-and-ten to ten-before-unit order in the history of English numerals.Language Variation and Change 26 (1):21-43.

In the course of its history, English underwent a significant structural change in its numeral system. The number words from 21 to 99 switched from the unit-and-ten to the ten-before-unit pattern. This change is traced on the basis of more than 800 number words. It is argued that this change, which took seven centuries to complete and in which the Old English pattern was highly persistent, can be broken down into two parts—the reordering of the units and tens and the loss of the conjoining element. Although the two steps logically belong to the same overall change, they display a remarkably disparate behavior. Whereas the reordering process affected the least frequent number words first, the deletion process affected the most frequent words first. This disparity lends support to the hypothesis that the involvement or otherwise of low-level aspects of speech determines the role of frequency in language change (Phillips, 2006). Finally, the order change is likely to be a contact-induced phenomenon and may have been facilitated by a reduction in mental cost.

MacGinnis, John, M Willis Monroe, Dirk Wicke, and Timothy Matney. 2014. “Artefacts of Cognition: the Use of Clay Tokens in a Neo-Assyrian Provincial Administration.Cambridge Archaeological Journal 24 (2):289-306.

The study of clay tokens in the Ancient Near East has focused, for the most part, on their role as antecedents to the cuneiform script. Starting with Pierre Amiet and Maurice Lambert in the 1960s the theory was put forward that tokens, or calculi, represent an early cognitive attempt at recording. This theory was taken up by Denise Schmandt-Besserat who studied a large diachronic corpus of Near Eastern tokens. Since then little has been written except in response to Schmandt-Besserat’s writings. Most discussions of tokens have generally focused on the time period between the eighth and fourth millennium bc with the assumption that token use drops off as writing gains ground in administrative contexts. Now excavations in southeastern Turkey at the site of Ziyaret Tepe — the Neo-Assyrian provincial capital Tušhan — have uncovered a corpus of tokens dating to the first millennium bc. This is a significant new contribution to the documented material. These tokens are found in association with a range of other artefacts of administrative culture — tablets, dockets, sealings and weights — in a manner which indicates that they had cognitive value concurrent with the cuneiform writing system and suggests that tokens were an important tool in Neo-Assyrian imperial administration.

Sherouse, Perry. 2014. “Hazardous digits: Telephone keypads and Russian numbers in Tbilisi, Georgia.Language & Communication 37:1-11.

Why do many Georgian speakers in Tbilisi prefer a non-native language (Russian) for providing telephone numbers to their interlocutors? One of the most common explanations is that the addressee is at risk of miskeying a number if it is given in Georgian, a vigesimal system, rather than Russian, a decimal system. Rationales emphasizing the hazards of Georgian numbers in favor of the “ease” of Russian numbers provide an entrypoint to discuss the social construction of linguistic difference with respect to technological artifacts. This article investigates historical and sociotechnical dimensions contributing to ease of communication as the primary rationale for Russian language preference. The number keypad on the telephone has afforded a normative preference for Russian linguistic code.

XLent LInguistics

As was correctly answered in the comments to the previous post, I am now 40 years old (XL) and the next time that my age in Roman numerals will be the same length as my age in Western (Hindu-Arabic) numerals will be when I am 51 (LI).    49 is not a correct answer in this case because the Romans did not habitually use subtraction in this way; irregular formations like IL (49) and XM (1900) do sometimes occur irregularly, but normally one cannot ‘skip’ a power.  I can only be subtracted from V and X; X can only be subtracted from L and C; and C can only be subtracted from D and M.

As any schoolchild can tell you, one of the purported disadvantages of Roman numerals is that their numeral-phrases are long and cumbersome (e.g., 37 vs. XXXVII).  And of course, for many numbers that is true.  But for many other numbers (e.g., 2000 vs. MM) the Roman numeral is  equal in length or shorter than its Western numeral counterpart.     Note, in particular, that round numbers tend to be those that are shorter in Roman numerals; this is because, since the Roman numerals don’t have a 0, that numerals that in Western notation would have a 0 have nothing in their Roman counterpart.

Among numbers whose Roman numeral and Western numeral notations are exactly the same length, there doesn’t initially appear to be much of a pattern:

1, 5, 11, 15, 20, 40, 51, 55, 60, 90, 102, 104, 106, 109 …

but then we see a new sequence emerge –

111, 115, 120, 140, 151, 155, 160, 190 …

which are just the numbers in the sequence from 11-90 with an added C on the front, which makes sense, since you’re just adding a 1 in front of them, similarly.

You could be forgiven in thinking that these numerals come up frequently, since we’re in the midst of a giant cluster of years with this property –

… 2002, 2004, 2006, 2009, 2011, 2015, 2020 …

but then after that, it’s another 20 years before 2040.

Unsurprisingly, numerals containing 3 rarely have equal-length Roman equivalents and numerals containing 8 never do.     So once you get past 3000, these numbers become extremely rare.    Roman numerals don’t have a standard additive representation for 4000 and higher; you can write 4000 as MMMM, but normally one would expect a subtractive expression with M (1000) subtracted from 5000.  There are Roman numerals for 5000, 10000, 50000, 100000, etc., but they are extraordinarily rare, and the Romans during the Empire instead tended to place a bar (or vinculum) above an ordinary Roman numeral to indicate multiplication by 1000; thus,  IV=4000.  The addition of this feature creates a real conundrum: does the vinculum count as a sign or not?    If it does, then IVI=4001 has four signs; if not, then IVII=4002 does.

I’ll leave this aside and stop here to save all of our brains.   Thanks for playing!



The Case of the Missing Pi Day 4s

Yesterday was Pi Day, 3/14 (those who prefer days before months can have Pi Approximation Day, 22/7) and in celebration of this momentous annual event, I invited several of my American colleagues (who have learned to tolerate my numerical eccentricities) over to my house in Canada for an International Pi Day Pie Party, which was a great success.  And, of course, as befitting this event, we had Pie, complete with Pi (to two decimal places) on top:

It's blueberry!So far, so good.  (And for the record, it was very good).  There was only one problem: the local dollar store I went into had a very odd distribution of candle numerals: it had tons and tons of 0, 1, 2, and 9, some 3s, but no 4s, 5s, 6s, 7s, or 8s.  As a professional numbers guy, and also as a guy who needed a 4 for his pi(e), this was deeply disconcerting.

After a moment, I figured out why. Ordinarily, when stores buy products that come in different varieties from wholesalers, the default is to order the same amount of each variety.   In this case, the store had obviously ordered an equal amount of each numeral, but they were being purchased by consumers at different rates.    Now, there is nothing about the properties of the natural numbers that would lead to this observed distribution (if it were Benford’s Law in action, it would be 1 and 2 that would be in short supply). Rather, the explanation is a social one:   Many parents do not buy birthday candles for their child’s first, second, or third birthday, because, while, as my (thankfully childless) brother noted, “Babies love fire!”, parents of toddlers do not.    At the other end, by the time your kid is about 9, and certainly by the double digits, they’ve probably outgrown the ‘giant novelty numeral candle’ phase of their lives.  Ages 4-8 are the sweet spot, and thus these sell out much more quickly.

I also note that, for adults, decadal birthdays like 20 and 30 tend not to attract much numerological attention, whereas 40, 50, and 60 certainly do (not so sure about 70 and 80), and by 90 most of the clientele is deceased.    This doesn’t explain why there were so many 0s available – perhaps purchasers are aware of this phenomenon and order extra zeroes, but don’t take account of differential demand for the tens digits.

Now, if we lived in a perfect world where suppliers and store owners had full information about their stock and made perfectly rational decisions, purchasers would notice such discrepancies and perhaps order more of the missing numerals.  The local dollar store, however, does not occupy such a world.  Fortunately, this being Windsor, Ontario, there was another dollar store across the street, and while it also had a skewed distribution, lo and behold, it did have one lonesome 4 for purchase (seen above).   Thus my Friday Pi Day pi display supply foray was saved.  Yay! (Try saying that in Pig Latin.)

Actually, this is not the first time I had encountered this phenomenon.  Back in 2008, when American gas prices first regularly began to hit $4.00 a gallon, the New York Times reported on, of all things, a shortage of numeral 4s, because their number sets were purchased with an equal distribution across all ten digits (presumably with extra 2s and 3s purchased individually to deal with those dollar amounts).  Once that leading digit got to 4, there was a temporary shortage, leading to some store owners writing their own makeshift 4s until new ones could arrive.

Thus, while we think of linguistic and symbolic resources like numerals as being effectively infinite, in contexts like these, you can indeed have shortages and surpluses.   Thankfully, now that we’re on to the Ides of March and our Pi Day shortage is dealt with for another year, I can store these candles for future use, if I want.  The pie, on the other hand, has gone to a better place.  Because, while you may sometimes need to ration your fours, let’s hope we never live in a world where we have to ration pie.

What’s so improper about fractions?

Yesterday, as part of the Wayne State Humanities Center brownbag series, I gave a talk entitled, “What’s so improper about fractions? Mathematical prescriptivism at Math Corps”, based on my long-term ethnographic research in Detroit.   For those of you who might be interested, you can watch the video below (or on Youtube itself), and the powerpoint is available for download here.

Cistercian number magic of the Boy Scouts

“You know that in ancient times religion, astronomy, medicine, and magic were all mixed up so that it was difficult to tell the beginning of one and the ending of the other and to-day the Gypsies, hoboes, free masons, astronomers, scientists, almanacs, and physicians still use some of the old magical emblems.  So there is no reason why the boys of to-day should be debarred from using such of the signs as may suit their games or occupations and we will crib for them the table of numerals from old John Angleus, the astrologer.  He learned them from the learned Jew, Even Ezra, and Even Ezra learned them from the ancient Egyptian sorcerers, so the story goes; but the reader may learn them from this book.” (Beard 1918: 91)

So begins the chapter, “Numerals of the Magic: Ancient System of Secret Numbers”, by Daniel Carter Beard in his 1918 volume The American boys’ book of signs, signals, and symbols, which you can download from Google Books for free.  Beard was one of the founders of the Sons of Daniel Boone in the early 20th century, which merged with the Boy Scouts of America (of which Beard was a key founder) in 1910 when that famous group was formed.  Beard wrote a number of popular books intended for boys in the Scouting movement, including this one.   Scouting books today do not, as a rule, make reference to esoteric Egyptian sorcery or Freemasonry or ‘John Angleus’ (who is Johannes Engel (1453-1512)) or ‘the learned Jew, Even Ezra’ (Abraham ibn Ezra (1089-1164)), or, for that matter, have a chapter on number magic at all.   At least, I never heard about it, and I was in Scouts for over a decade.  But we are fortunate that this one did, because it has a couple of real treasures inside, not previously recognized as such.

Let’s take the second one first.  It appears on p. 92 immediately following the passage I just quoted:


(Beard 1918: 92)

For those of you familiar with my book Numerical Notation, these are the numerals used primarily by Cistercian monks from the 13th – 15th centuries, and thereafter described in early modern numerology and astrology for several centuries, though largely at that point as an intellectual curiosity rather than a practical notation.    David King’s wonderfully detailed Ciphers of the Monks (King 2001), which is one of the few books at that price point (somewhere around $150, if I recall) that may be worth it, lists every example the author could find of these numerals, from medieval astrolabes to Belgian wine barrels to 20th-entury German nationalist texts.    It’s extremely comprehensive.  However, it does not mention Beard’s book – and why should it? What a bizarre place to find such a numerical system!   It’s what I describe as a ciphered-additive system, which is to say that there is no zero because none is needed: there is a distinct sign for each of 1-9, 10-90, 100-900, and 1000-9000.   The Cistercian numerals are a little anomalous typologically; another interpretation of them would be that they are positional, but use rotational rather than linear position – the signs for 9, 90, 900, and 9000 (e.g.) are rotations or flips of one another, so we could consider them the same sign (9) in four different orientations.     Zero is superfluous (thus not present) because unlike linear texts, there is no ‘gap’ to be accounted for by an empty place-value.

I became curious and tried to figure out why Beard attributed these to ‘Angleus’ and to ‘Even Ezra’.    Engel’s Astrological Optics was translated into English (1655) but contains no Cistercian numerals, and King doesn’t note him as using or depicting the system.  Similarly, ibn Ezra was not a known user of the system.   And I haven’t even been able to find any other source that attributes the system to those individuals; rather, it’s almost always Agrippa of Nettelsheim or Regiomontanus who are invoked in the scholarship.  We know that Beard was a Freemason, so he may have had access to some Masonic texts that said as much, but I can’t find any such reference, and King doesn’t mention any likely sources either, although he does note that many Masons (especially in France) were familiar with the Cistercian system.    So it’s not entirely clear where Beard learned about the system (although see below), and he’s got a lot of things mixed up in the account.

The other numerical treasure in Beard’s book is even more fascinating, although it appears in the previous chapter on codes and ciphers and is less prominent, on p. 85, the ‘tit-tat-toe’ numerals:


(Beard 1918: 85)

So what we see here, again, is a ciphered-additive decimal system in which there is a ‘family resemblance’ between 9, 90, 900, and 9000 (and the other numbers so patterned), but no zero.  The signs are designed after their place in a hash / tic-tac-toe / octothorpe with the power indicated through ornamentation.  As a ciphered-additive system, it’s like the Cistercian numerals (although the signs are completely different) but instead of placing signs around a vertical staff, the signs are constructed into a box.  Note that the signs in each numeral-phrase are not strictly ordered, but are packed compactly in whatever way suits the resulting box aesthetically. This is one of the advantages of ciphered-additive systems that, if desired, for cryptographic purposes or for any other reason, the signs can be re-ordered without loss of numerical meaning.   But I know of no system quite like this, where numerals are arranged in a box-like shape, or where there is such a novel means of forming individual signs.

Beard is explicit that this system is newly designed: “The tit-tat-toe system of numerals here shown for the first time is entirely new and possesses the advantage of being susceptible of combinations up to four figures which suggests nothing to the uninitiated but a sort of Japanese form of decoration”  (Beard 1918: 84).   He claims that the alternate name ‘Cabala’ is just another name for the tit-tat-toe, which is a highly dubious claim, but he is clearly trying to invoke a connection between his newly-developed system and Jewish mysticism – in the hope that Boy Scouts will use it as a numerical code.  Ciphered-additive numerals are rare enough in the modern era – most of the systems are obsolescent at best.  So it’s fascinating to see a twentieth-century system right at the moment of its development.   It’s also fascinating to see how mystical, spiritual, and numerological knowledge from early-modern authors is incorporated into a manual for Boy Scouts and recommended for use in cryptography.

We’re not quite done, though.  Based on some of the (otherwise uncited) quotations in Beard’s book, I concluded that he was taking some of his ‘insight’ about the ‘Cabala’ from L.W. De Laurence’s Great Book of Magical Art (1915), which was a popular American book of spiritualism and Oriental mysticism at the time.  And, looking into de Laurence’s book, lo and behold, what did I find?

(De Laurence 1915: 174)

(De Laurence 1915: 174)

De Laurence, whose work is also not noted by King, gives a more standard attribution than does Beard for what we now know to be the Cistercian numerals: he attributes them to the ‘Chaldeans’, which is a very common descriptor for the system and is even found in the scholarly literature.  He doesn’t mention Angelus or Even Ezra or any other of the medieval and early modern authors who use the system, so it’s still a mystery how Beard made that attribution.  But, given that there really are not a lot of texts that discuss this system at all, I suggest that Beard encountered them through De Laurence and possibly confounded their origin with some other understandings he had picked up along the way, possibly through Masonic writings.

It’s not every day that I discover a new numerical notation system, and it’s great to do that, even when it’s one that  seems to have been developed once but never adopted more widely.   So it was neat to find the ‘tit-tat-toe’ system, even if it never appeared anywhere else.  But I also found it fascinating to track the transmission of the much more widespread (but still under-appreciated) Cistercian numerals through their roundabout path to a Scouting manual for boys.    As King’s book amply demonstrates, the system has a tendency to show up in the oddest places, so perhaps we should (ahem) ‘be prepared’ to find them anywhere.

Beard, Daniel Carter. 1918. The American boys’ book of signs, signals and symbols. Philadelphia: Lippincott.

De Laurence, L. W. 1915. The great book of magical art, Hindu magic and East Indian occultism. Chicago, Ill., U. S. A.: De Laurence Co.

King, David A. 2001. The ciphers of the monks: a forgotten number-notation of the Middle Ages. Stuttgart: F. Steiner.

The mystical Eye of Horus / capacity system submultiples

Here is a story about number systems:

The wd3t is the eye of the falcon-god Horus, which was torn into fragments by the wicked god Seth.  Its hieroglyphic sign is made up of the fractional powers of 2 from 1/2 to 1/64, which sum to 63/64.  Later, the ibis-god Thoth miraculously ‘filled’ or ‘completed’ the eye, joining together the parts, whereby the eye regained its title to be called the wd3t, ‘the sound eye’.   Presumably the missing 1/64 was supplied magically by Thoth.




This is my retelling, using many of the same phrases, of Sir Alan Gardiner’s account of the ‘eye of Horus’ symbol used for notating measures of corn and land in his classic Egyptian Grammar (§ 266.1; 1927: 197).   It’s a nice story, and it is repeated again and again, not only in wacky Egypto-mystical websites but in a lot of serious scholarly work up to the present day.   I talk about it in Numerical Notation.   But is it true? Well, that depends what you mean by ‘true’, but mostly the answer is: not really.  As I mentioned in a post back in 2010, this is certainly not the origin of the symbols.  Jim Ritter (2002) has conclusively shown that these are ‘capacity system submultiples’, which originated in hieratic texts, not hieroglyphic ones, and appear to have had non-religious meanings originally.     Even while insisting on the mythico-religious origin of the Horus-eye fractions, Gardiner himself (1927: 198) is crystal clear that all the earliest ‘corn measures’ are hieratic.  The hieratic script is very different in appearance and character than the hieroglyphs, being the everyday cursive script of Egyptian scribes, rather than the monumental and more formal hieroglyphs.   Ritter shows conclusively that in their origin, and their written form, and their everyday use, the capacity system submultiples have nothing to do with the Eye of Horus.

Ritter distinguishes this “strong” thesis from a “weak” version, in which, many centuries after their invention, the hieratic capacity system submultiples were imported into the hieroglyphic script and that some scribe or scribes wrote about them as if they could be combined into the wedjat hieroglyph.  This weak version has more evidence for it, but as Ritter points out (2002: 311), this “does not automatically mean that ‘the Egyptians’ thought like that; for example, those Egyptians whose task it was to engrave hieroglyphic inscriptions on temple walls.  Theological or any other constructs of one community do not necessarily propagate to every other; the Egyptians were no more liable than any other people to speak with a single voice.”  This is a sociolinguistically-complex, reflective view that I think is essentially correct, and which I adopt in my work (although I would rewrite it today to be even clearer, as I hope I have above).   Ritter is not fully convinced by the weak thesis either, but acknowledges that it is tenable.

Ultimately, as Ritter concludes (correctly), our willingness to buy into the ‘Horus-eye fractions’ model tells us a lot about how we view the hieroglyphs, and Egyptian writing in general, as mythically-imbued and pictorial in nature, and ultimately reflects a mythologized view of Egyptians as a ‘mystical’ people, an ideology that goes back to the Renaissance and earlier in Western thought (Iversen 1961).  But I would go further, because it is about more than just Egypt.   We like stories that give numerological explanations for numerical phenomena, regardless of their veracity, and especially where the numerical system under consideration is from societies we conceptualize as having a more mystical or mysterious relationship with the world than we purportedly do.   Very often we are projecting our image of what is going on.  This isn’t to say that Gardiner’s description is wrong – he knew the texts better than almost anyone, and correctly identifies how the system worked and the texts in which it was found.  But it’s important that when (some) Egyptians transliterated the capacity system submultiples from hieratic to hieroglyphic writing and formed them into the wedjat, they were repurposing and transforming a pre-existing set of signs that had no mystical origin whatsoever.   It deserves our attention, both for what it tells us about Egyptian life  and also for its importance for the historiography of science, mathematics, and religion in non-Western societies.

(Thanks to Dan Milton, who as the winner of the contest last week asked the question that motivates this post.)

Gardiner, Alan H. 1927. Egyptian grammar: being an introduction to the study of hieroglyphs. Oxford: Clarendon Press.
Iversen, Erik. 1961. The myth of Egypt and its hieroglyphs in European tradition. Copenhagen: Gad.
Ritter, Jim. 2002. “Closing the Eye of Horus: The Rise and Fall of ‘Horus-eye fractions’.” In Under One Sky: Astronomy and Mathematics in the Ancient Near East, edited by John M. Steele and Annette Imhausen, 297-323. Münster: Ugarit-Verlag.