Plants (humans?) are incredibly cool, but don’t do math

There’s a fascinating article on BBC News today, about a really interesting study that proposes that an internal mechanism in the Arabidopsis thaliana plant (which is used widely in scientific experiments as a model organism) regulates starch consumption in the absence of sunlight in a way that requires the plants to be able to mathematically “divide” the numbers of two different types of cells.  Now I’m not a botanist and I can’t say whether the result is correct, but I do take issue with the claim that “They’re actually doing maths in a simple, chemical way”.  The last quote from the article is more accurate: “This is not evidence for plant intelligence. It simply suggests that plants have a mechanism designed to automatically regulate how fast they burn carbohydrates at night. Plants don’t do maths voluntarily and with a purpose in mind like we do.”

All sorts of natural processes can be modelled using mathematics – so, for instance, Fibonacci patterns appear in a variety of plants in the operation of phyllotaxis (the arrangement of leaves on stems).    We don’t say that these plants ‘do math’.    And the same principle applies above to the new finding above.  It’s incredibly cool that these mathematical patterns emerge, and it’s a very interesting question why they emerge biochemically.  But that raises an even more interesting question: what do we mean when we say that humans ‘do math’?

Humans are organisms and thus part of the physical world, and so lots of the things they do unconsciously or without explicit reflection can thus be modelled mathematically.    But this is not the same as saying that all humans do mathematics.  This seems to be what is being suggested in the last quotation: that ‘doing math’ involves conscious, explicit, purposeful reflection on the mathematical aspects of reality.    Being able to throw a curveball is not ‘doing mathematics'; being able to model the trajectory of a curveball is.  And the overlap between the sets of humans able to do each task is minimal.

Let me give another example related to the plant study above.  A child has a pile of 23 candies and wants to divide it among some gathered group of five kids including herself.  She starts to her right giving one candy to each friend, continuing to pass them out until they’re all gone.    When the process is complete, each child will have 4 candies and the three to the right of the distributor will have 5 each.  We could, if we wished to, define ‘division’ as ‘the process of dividing up a group of objects among another group’ and then say ‘thus, the kids are dividing 23 by 5 and getting 4 with a remainder of 3′.  But I think most of us would be reluctant to argue that the first child understands division, or knows how to divide.    Even though distributing the candy is a conscious decision, and even though it requires some general process (one candy to one child), it does not require that the child be able to do mathematics.

For the same reason, I sometimes have some skepticism when my colleagues in ethnomathematics describe the mathematics of some human activity in terms of fractal geometry or the Fibonacci series.   It is, of course, possible that people have some awareness of the processes behind their activities, and ethnographically, when they can talk about that, it is very interesting.   For instance, if the child above says “Well, I know I have 23 candies and so they won’t go evenly, so there are going to be some left over at the end,” then we do indeed know that the child has some explicit knowledge of division.    I worry, in fact, that because so many natural processes result in such sequences, that we confuse the result with the conscious awareness of the process.  In doing so, we fail to investigate the explicit mathematical knowledge that humans do actually encode in all sorts of things they do, and we falsely attribute a sort of explicit consciousness to activities that have no explicitness underlying them (in humans, animals, plants, and even in nonliving things).

Screws, hammers, and Roman numerals: An allegorical complaint

Let’s imagine that you have a toolbox in your garage, full of all sorts of different useful things, and I’m your annoying neighbor.  One day I drop by while you’re working.  I rummage around, pick up a screwdriver, and say to you, “Gosh, that’s not a very good hammer, is it?”  Naturally, you protest that it isn’t a hammer at all.  Next, I hold the screwdriver by the head instead of the handle and say, “Well, of course, you could use it like this to bang in nails, but it would be very cumbersome.”  You look at me, wondering whether I didn’t hear you properly, and say, “No, really.  It’s not a hammer. I have a hammer, but it’s in the trunk of my car, and that’s not it.”  I turn to you and say, “Well, I’ve never seen your hammer, and it would really be a lot easier if you just used the handle of a screwdriver to bang in nails.  Except that it’s no good for that.”

Now let’s turn from this surreal Pythonesque world to another scenario.

You’re an epigrapher and you find some inscriptions with some Roman numerals.  You look at them and say, “Gosh, those things aren’t very good for math, are they?”  Of course, the writer is dead, so he/she doesn’t say anything.  Next, you fiddle around with the numerals and think to yourself, “Well, look at that!  You could use those for arithmetic if you wanted to, but it would be very cumbersome.”  Again, the writer is not around to protest, although as it turns out, someone else dug up an abacus a few kilometers away.   You think of that, though, and say, “Well, it would really be a lot easier if they had just used numerals to do arithmetic, except that their numerals are no good for that.”

So this is the world I live in, and this is the battle I fight.

The problem is a cognitive and ideological one. We are so attached to the idea that numerals are for arithmetic that it’s very hard to stop and ask whether number symbols were actually used for doing calculations in a given society.  There’s essentially no evidence that Romans or anyone else ever lined up or computed with Roman numerals on papyrus or slate or sand or anything else, while there’s abundant evidence that they used an abacus along with finger-computation.  This should give us pause, but our cognitive bias in favour of the numeral/math functional association overpowers it.    For almost all numerical notation systems used over the past 5000 years, there’s precious little evidence that numerals were manipulated arithmetically.  You might have a multiplication table, or you might write results, but you wouldn’t line up numbers, break long numerals into powers to work with them, or anything of the sort.   And since we don’t know that much about abaci and other arithmetic technologies, even though they were obviously used for arithmetic, we assume (wrongly) that they certainly could never be equally good as written numbers.  And thus we conclude (finally, wrongly, again) that Romans were hopeless at arithmetic.   We might even blame their (purported) lack of mathematical proficiency on their lack of a ‘good’, ‘efficient’ numeral system.

It’s a casual, all-too-easy ethnocentrism, and hard to detect.  It’s not the nativistic, “our ways are good, your ways are bad” ethnocentrism that we mostly know to avoid.     Because arithmetic as it is presently taught almost everywhere relies on the structure of the positional decimal numerals, lined up and manipulated as needed, it takes on a naturalness that is deceptively difficult to untangle.   Yes, the Roman numerals are quite difficult to use if you presume that the way to use them is to break them apart, line them up, and do arithmetic in something like the way we were taught.   This isn’t to say that the functions of technologies aren’t relevant, but if we decide in advance what their functions must be, we are likely to miss out on what they actually were, and our judgements will be compromised.

To hammer the point home: if we do that, we’re screwed.

Not the earliest zero, rediscovered

A rather unfortunate effort in Discover by Amir Aczel, ‘How I Rediscovered the Oldest Zero in History’ more or less effaces his solid legwork with shoddy theorizing and ahistorical claims.  Supported by the Sloan Foundation, Aczel (a popular science writer) went to Cambodia and tracked down the location of the Old Khmer inscription from Sambor, which is dated 605 in the Saka era (equivalent to 683 CE), which obviously contains a zero.    While the Hindu-Arabic-Western numerical tradition is seen to emanate from India, all of our earliest unquestioned examples (the late 7th century ones) of the zero are from Southeast Asia, and Sambor is the earliest one.  Because things have been rough in Cambodia for a long time, his work tracking it down and ensuring that it would be protected deserves a lot of credit.

If he had stopped there it would have been fine. Unfortunately, in an effort to bolster the importance of his claim, Aczel spends quite a lot of time justifying this as the first zero anywhere, ever, neglecting Babylonian and Maya zeroes from many centuries earlier.  To do that he needs to whip out all sorts of after-the-fact justifications of why those zeroes don’t really count, because Babylonians didn’t use their zero as a pure placeholder, or because Maya zeroes, well actually he just ignores those until the comments (but don’t read the comments – really, folks, that is the first rule of the internet).   Just for kicks, and regardless of the fact that it has nothing to do with zero, he starts off with a lengthy diatribe about how the Roman numerals are ‘clunky’ and ‘cumbersome’ and ‘inefficient’, which as long-time readers of this blog, or anyone who has read Numerical Notation, will know, is an utterly ridiculous, ahistorical claim that is divorced from how such numerals were actually used over two millennia.

I have come to terms with the fact that I will probably be spending the rest of my career pointing out that absolute judgements of the efficiency of numeral systems run the gamut from ‘missing the point’ to ‘completely ahistorical’ to ‘rabidly ethnocentric’.  While Aczel’s piece is not the worst of the sort, it certainly doesn’t deserve much praise.  Which is a shame, since that Sambor inscription really is the first known zero in the Indian tradition (to which our own Western numerals owe their origin) and it’s great that he’s been able to reconfirm its location in a politically perilous part of the world.

It’s just ones and zeroes: the representational power of binary notation

This recent Saturday Morning Breakfast Cereal strip illustrates a ridiculous, but ultimately profound, issue around how we think about numbers and computers:

Most of us who use computers, regardless of age, do not actually think that there are little physical tokens that look like ‘1’ and ‘o’ physically bouncing around inside the CPU or residing on the hard drive.    We know that that can’t be true.   In some sense, we (hopefully) understand that ‘1’ and ‘0’ are symbols of ‘on-ness’ and ‘off-ness’, conventional representations using binary (a two-state numerical system) of the foundation of  modern electronic circuitry.  And yet, when we talk about how computers ‘think’, we inevitably end up talking about 1s and 0s. Which is why we chuckle when the same idea is used in the Onion article ‘Microsoft Patents Ones, Zeroes‘ or in the Futurama movie Bender’s Big Score, which relies on the conceit of a  series of ones and zeroes tattooed upon someone’s butt that, when read aloud, opens up a time-travelling portal.

http://slurmed.com/movies/img/futurama_movie_bender_tattoo_binary_code.jpg

We laugh because, at some level, we know that computers are not really reading ones and zeroes off a page.  But if not, what do we think they’re doing? I think it would be fascinating to figure out what the cultural model is that underlies this – that it would be a nice ethnographic question to ask, “What does it mean when people say that computers use 1s and 0s?”   You would surely get a lot of responses from computer scientists that talk about switches and logic gates, and some blank stares, but it would be very interesting to see how ordinary, average, computer-literate users talk about binary as a language that computers understand.

Like any good geek dad, I spend a lot of time trying to stop my son from spending all day watching Youtube videos of video games, and the solution, fortunately, seems to be that he also likes watching a lot of Youtube videos about science and technology, and so he introduces me to some cool ones, and we watch them together.    So take a few minutes to check out this recent video from the fantastic Computerphile channel, where James Clewett talks about the importance of abstraction as a means of allowing us to talk about  what’s going on in everyday computing in an understandable way:

Let’s focus on the segment starting at around 0:59: “Look, a transistor is just a switch, and that switch can be open or closed, and the electrons travelling down the wire, they’re either there or they’re not there, which is a 1 or a 0,  and in Numberphile we talk about 1s and 0s a lot, so we won’t go back into that, but it’s just numbers travelling down a wire.”

Clewett, who obviously does understand exactly what is going on, starts with a discussion of switches (real objects) which can be in one of two states, on or off, and then moves to electrons (real objects) either being present or absent, then makes an abstracting discursive move to talking about 1s and 0s, which are not real physical objects, but an abstract representation of the states of switches or the presence/absence of electrons.  And then, within twenty seconds, he’s moved to ‘just numbers travelling down a wire’, which is a highly concrete representation indeed, but clearly not a literal one.  And even though we and he know that numbers are abstractions of the properties of the world – that the numbers are not actually little objects moving down a wire – this seems to be a very central way of thinking about how computers think.  We can’t seem to do without it for very long.

I wonder whether this is tied in to the metalinguistic idea that entities need language to communicate or to think – that we need a metaphorical, language-like understanding of how computers process information, and so we build up this understanding that is close to how we imagine a thinking entity must process information, even though we understand at some other level that it cannot actually work this way.    It may be the most apt metaphor for understanding off/on switches (or digital information generally) but it is still a metaphorical understanding constrained by how we think entities that process information analogously to humans must work.

Numerals in webcomics

Over the past few years, I have been informally collecting and curating a set of comics (mostly online webcomics) relating to my main research interest in numerals, number systems, and numeracy.     While I am led to understand that not everyone in the world appreciates my particularly nerdesque sense of humour, it seems reasonable to suppose that if you’re reading this blog, then you might be like me and find these to be hilarious and/or thought-provoking.    Here are some of my favourites; reader contributions are very welcome (along with suggestions of other comics where I might find good material in the future).

 

Married to the Sea

Card-counting: http://www.marriedtothesea.com/122811/card-counting.gif

Mortgage industry: http://www.marriedtothesea.com/022309/mortgage-industry.gif

Number Two Number Four: http://www.marriedtothesea.com/121506/number-two-number-four.gif

 

Saturday Morning Breakfast Cereal

Balls constants: http://www.smbc-comics.com/index.php?db=comics&id=2982

Conversation Trick #57721: Self-referential phrases: http://www.smbc-comics.com/index.php?db=comics&id=2448

Fouriest: http://www.smbc-comics.com/index.php?db=comics&id=2874

Polish hand magic: http://www.smbc-comics.com/index.php?db=comics&id=1914

Too many zeroes: http://www.smbc-comics.com/index.php?db=comics&id=1768

 

Toothpaste for Dinner

10 types of people: http://www.toothpastefordinner.com/052503/10-types-of-people.gif

Happy New Year 2008: http://www.toothpastefordinner.com/010208/happy-new-year-2008.gif

Swedish binary: http://www.toothpastefordinner.com/022210/swedish-binary.gif

Synaesthesia emergency: http://www.toothpastefordinner.com/111808/synaesthesia-emergency.gif

Unaphobia: http://www.toothpastefordinner.com/110310/unaphobia.gif

 

XKCD

1 to 10: http://xkcd.com/953/

Binary sudoku: http://xkcd.com/74/

Code Talkers: http://xkcd.com/257/

ISO 8601: http://xkcd.com/1179/

License Plate: http://xkcd.com/1105/

Money: http://xkcd.com/980/

Number line: http://xkcd.com/899/

Numerical sex positions: http://xkcd.com/487/

One two: http://xkcd.com/764/

Words for small sets: http://xkcd.com/1070/

Numerals inside the Great Pyramid

A couple of weeks ago all the news was about some new red ochre markings found in a shaft on the interior of the Great Pyramid at Giza (a.k.a. the Pyramid of Khufu), identified using an exploratory robot. That was pretty cool. But if you’re a professional numbers guy (as I am) you’ll be doubly excited to learn that it is probable that those marks are hieratic numerals. If this interpretation is correct, these are almost certainly mason’s marks used to indicate some quantity involved in the construction. Other than the fact that I would like all news outlets to stop calling them hieroglyphs (they aren’t – the hieratic script is a cursive Egyptian script that differs significantly from the hieroglyphs, and the numerals look nothing alike), this is really cool. I do want to urge caution, however: this does not imply that the Great Pyramid was designed along some sort of mystical pattern or using some numerological precepts. It actually doesn’t tell us even that the marks indicate the length of the shaft (as Luca Miatello suggests in the new article) – it could just as easily be 121 bricks in a pile used to make a portion of the pyramid. I am also not 100% convinced of the ‘121’ interpretation – the 100 could be a 200, very easily, or even some other sign altogether, for instance. But the idea that numerical marks using hieratic script would be made by the pyramid-makers is entirely plausible and helps show the role of hieratic script in the Old Kingdom. Although it’s hardly going to revolutionize our understanding of Egyptian mathematics, it may well help outline the functional contexts of the use of numerals in Old Kingdom Egypt.

An anti-hiatus?

Apparently the English language has a lexical gap – it has no good term for ‘the end of a hiatus’ (‘resumption’ and ‘recommencement’ hardly suffice semantically), but, in any case, my apologies for having been largely absent here the past couple of months. Our department of nine full-time faculty has just finished three simultaneous job searches, which for those of you in academia, will give you a very good sense of what I’ve been up to.

And a very happy Pi Day (3/14) to all of you who celebrate! I didn’t do much special at 1:59 pm – perhaps I should have toasted Archimedes or something like that. I’m fonder of Pi Approximation Day anyway (July 22), since 3 1/7 is much closer to pi than 3.14.

This week’s World Wide Words (the e-magazine authored for 15 years by the inestimably talented lexicographer, Michael Quinion) featured one of my favourite numerical words, chronogram, meaning number-riddles in which a date is encoded in text using Roman numerals. Quinion mentions in passing the “three big books” of James Hilton from the 19th century, but this does little justice to the 1500+ pages of chronograms Hilton compiled over two decades. The first two volumes available for free download from Google Books (vol 1 – 1882; vol 2 – 1885; vol 3, 1895, is inexplicably only in ‘snippet view’).

Lastly, here is some good advice for those who are (rightly) considering charitable donations in support of victims of the Sendai earthquake.

Numeration and Numeracy in Cognition, Language, and Culture

Last month, at the Society for Anthropological Sciences annual meeting in Charleston, SC, I organized a panel of some really interesting material on the broad topic of numeration. I want to take this opportunity (again) to thank all the presenters for their attendance and hard work. The abstracts (as also published in the conference program) were as follows:

Toward a cognitive, historical, linguistic anthropology of numerals
Stephen Chrisomalis (Wayne State University)

For over a century, the study of numeration, number systems and allied topics has been a key part of the comparative study of thought, language, and culture. The anthropology of numbers and mathematics has traditionally been a locus for unilinear evolutionary thought linked to notions of primitivity. The papers in this panel constitute a call for a culturally-grounded cognitive science of numeration within four of the disciplines of cognitive science (anthropology, linguistics, philosophy, and psychology).

Recent research in language evolution, linguistic relativity, and cultural aspects of mathematical cognition draw attention to the need for anthropologists to re-engage with this new agenda. First, the cross-cultural study of numerals allows the investigation and evaluation of universal and particular aspects of numeration and their relationship with social organization. Because numerals have multiple modalities (e.g., verbal, graphic, gestural), examining patterns in number systems beyond linguistics allows us to evaluate to what extent number concepts can be separated from language, including universal grammar. Finally, just as the cognitive anthropology of plant and animal taxonomy contributes to ecological and environmental anthropology, the cognitive anthropology of numerals and mathematics underpins economic anthropology and the anthropology of science.

Spatial-numeric associations in literates and illiterates
Samar Zebian (Lebanese American University, Beirut Lebanon)

Several independent studies have reported a cognitive association between small numbers and the left side of space and larger numbers andthe right side of space among individuals who read and write from left-to-right (SNARC effect). These associations are reversed for individuals who read and write from right-to-left. The SNARC effect has widely been taken as evidence that numbers are conceptualized as points along a mental number line, however there is growing evidence that this systematic spatial performance bias related to writing directionality is an instance of strategic processing rather than a reflection of inherent spatial attributes of numbers. In an attempt to explain the “deeper” origins of these associations researchers are examining the linkages between number and finger counting. The current study examines whether finger counting practices reveal consistent spatial-numeric associations and whether there are any spillover effects to other tasks that involve object sorting and counting and other non-counting but quantitative tasks such as line bisection and speeded parity judgment. If, in fact, finger counting practices and not the directionality of writing set up spatial-numeric associations than we should be able to observe the same type of spatial biases in literates and illiterates. Preliminary evidence suggests that the finger counting practices of literates and illiterates are not same and furthermore that the spatial biases found in finger counting are not observed across tasks.

Zero’s beginnings: the Mayan case
John Justeson (SUNY, Albany)

This paper addresses linguistic and (Mayan) historical evidence concerning the origins of a numerical concept of zero. Comparative linguistic evidence suggests that zero is not part of basic numerical cognition; rather, it develops out of computing practices of mathematical specialists. Specifically, while zero is often assumed to be prerequisite to the invention of positional notation, it seems on the contrary to emerge as a notational device within such systems. This is clearly the case in Mesoamerica. A system of place-value notation arose in Guatemala and Mexico among Mayans and epi-Olmecs by 36 BCE, with no symbol corresponding to a zero coefficient. Although data is limited, circumstantial evidence is consistent with the following scenario for the emergence of a numerical zero: Mayan calendar specialists developed discourse practices, associated with calendrically-timed ritual events, that used the word “lacking”; the associated dates were represented in a new, non-positional system of notation, which replaced positional notation except in calculating tables; the sign for “lacking” was transferred from the new notation into these tabular positional notations; as a side effect of the algorithms that specialists used to add and subtract positional numerals, the “lacking” symbol was reinterpreted numerically.

Methodological reflections on typologies for numeral systems
Theodore R. Widom and Dirk Schlimm (McGill University)

Past and present societies worldwide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. In the present paper we provide a general framework
within which the efficacy of these typologies can be assessed relative to certain desiderata. Using this framework, we discuss the two influential typologies of Zhang & Norman and Chrisomalis, and present a new typology which takes as its starting point the principles by which numeral systems represent multipliers (the principles of cumulation and cipherization), and
bases (those of integration, parsing, and positionality). We argue with many different examples that this provides a more refined classification of numeral systems than the ones put forward previously. We also note that the framework can be used to assess typologies not only of numeral systems, but of many domains.

Social relationships as a lexical source for numeral terms in Amazonia
Cynthia Hansen and Patience Epps (The University of Texas at Austin)

Due to the relatively high degree of etymological transparency found in the numeral systems of Amazonia, it is possible to see the range of lexical sources from which the numeral terminology emerges. In this paper, we present the range of strategies used to create numeral terms below 5, based on an extensive survey of the numeral systems of close to 200 Amazonian languages conducted by the authors. More specifically, we discuss a strategy that is well-attested in Amazonia but that is not attested elsewhere in the world: a ‘relational’ strategy where terms for 4 (and sometimes 3-10) are built using a social relationship term, such as ‘sibling’ or ‘companion’. We propose that this strategy mirrors a gestural counting strategy found throughout the region where fingers are grouped in pairs.

Cultural variation in numeration systems and their mapping onto the mental number line
Andrea Bender and Sieghard Beller (University of Freiburg)

The ability to exactly assess large numbers hinges on cultural tools such as counting sequences and thus offers a great opportunity to study how culture interacts with cognition. To obtain a more comprehensive picture of the cultural variance in number representation, we argue for the inclusion of cross-linguistic analyses. In this talk, we will briefly depict the specific counting systems of Polynesian and Micronesian languages that were once derived from an abstract and regular system by extension in three dimensions. The linguistic origins, cognitive properties, and cultural context of these specific counting systems are analyzed, and their implications for the nature of a (putative) mental number line are discussed.

Review: von Mengden, Cardinal Numerals

This review appeared originally in the LINGUIST List at http://linguistlist.org/issues/21/21-5213.html

AUTHOR: von Mengden, Ferdinand
TITLE: Cardinal Numerals
SUBTITLE: Old English from a Cross-Linguistic Perspective
SERIES: Topics in English Linguistics [TiEL] 67
PUBLISHER: De Gruyter Mouton
YEAR: 2010

Stephen Chrisomalis, Department of Anthropology, Wayne State University

SUMMARY

This monograph is a systematic analysis of Old English numerals that goes far
beyond descriptive or historical aims to present a theory of the morphosyntax of
numerals, including both synchronic and diachronic perspectives, and to
contribute to the growing linguistic literature on number concepts and numerical
cognition.

The volume is organized into five chapters and numbered subsections throughout
and for the most part is organized in an exemplary fashion. Chapters II and
III, where the evidence for the structure of the Old English numerals is
presented, will be of greatest interest to specialists in numerals. Chapter IV
will be of greatest interest to specialists in Old English syntax. Chapter V is
a broader contribution to the theory of word classes and should be of interest
to all linguists.

The author begins with an extensive theoretical discussion of number concepts
and numerals, working along the lines suggested by Wiese (2003). Chapter I
distinguishes numerals (i.e., numerically specific quantifiers) from other
quantifiers, and distinguishes systemic cardinal numerals from non-systemic
expressions like ‘four score and seven’. As the book’s title suggests, cardinal
numerals are given theoretical priority over ordinal numerals, and nominal forms
like ‘Track 29′ or ‘867-5309′ are largely ignored. Cardinal numerals exist in
an ordered sequence of well-distinguished elements of expandable but
non-infinite scope. Here the author builds upon the important work of Greenberg
(1978) and Hurford (1975, 1987), without presenting much information about Old
English numerals themselves.

Chapter II introduces the reader to the Old English numerals as a system of
simple forms joined through a set of morphosyntactic principles. It is
abundantly data-rich and relies on the full corpus of Old English to show how
apparent allomorphs (like HUND and HUNDTEONTIG for ‘100’) in fact are almost
completely in complementary distribution, with the former almost always being
used for multiplicands, the latter almost never. This analysis allows the
author to maintain the principle that each numeral has only one systemic
representation, but at the cost of making a sometimes arbitrary distinction
between systemic and non-systemic expressions. This links to a fascinating but
all-too-brief comparative section on the higher numerals in the ancient Germanic
languages, which demonstrates the typological variability demonstrated even
within a closely related subfamily of numeral systems.

Chapter III deals with complex numerals, a sort of hybrid category encompassing
various kinds of complexities. The first sort of complexity, common in Old
English, involves the use of multiple noun phrases to quantify expressions that
use multiple bases (e.g. ‘nine hundred years and ten years’ for ‘910 years’).
The second complexity is the typological complexity of Old English itself; the
author cuts through more than a century of confusion from Grimm onward in
demonstrating conclusively that there is no ‘duodecimal’ (base 12) element to
Old English (or present-day English) — that oddities like ‘twelve’ and
‘hundendleftig’ (= 11×10) can only be understood in relation to the decimal
base. The third is the set of idiosyncratic expressions ranging from the
not-uncommon use of subtractive numerals, to the overrunning of hundreds (as in
modern English ‘nineteen hundred’), to the multiplicative phrases used
sporadically to express numbers higher than one million. Where a traditional
grammar might simply list the common forms of the various numeral words, here we
are presented with numerals in context and in all their variety.

Chapter IV presents a typology of syntactic constructions in which Old English
numerals are found: Attributive, Predicative, Partitive, Measure, and Mass
Quantification. In setting out the range of morphosyntactic features
demonstrated within the Old English corpus, the aim is not simply descriptive,
but rather, assuming that numerals are a word class, to analyze that class in
terms of the variability that any word class exhibits, without making
unwarranted comparisons with other classes.

In Chapter V the author argues against the prevalent view that numerals are
hybrid combinations of nouns and adjectives. While there are similarities,
these ought not to be considered as definitional of the category, but as results
of the particular ways that cardinal numerals are used. Because it is
cross-linguistically true that higher numerals behave more like nouns than lower
ones, this patterned variability justifies our understanding the cardinal
numerals as a single, independent word class. It is regarded as the result of
higher numerals being later additions to the number sequence — rather than
being ‘more nounish’, they are still in the process of becoming full numerals.
They are transformed from other sorts of quantificational nouns (like
‘multitude’) into systemic numerals with specific values, but retain vestiges of
their non-numeral past.

EVALUATION

This is an extremely important volume, one that deserves a readership far beyond
historical linguists interested in Germanic languages. It is not the last word
on the category status of cardinal numerals, cross-linguistic generalizations
about number words, or the linguistic aspects of numerical cognition, but it
represents an exceedingly detailed and well-conceived contribution to all these
areas. While virtually any grammar can be relied upon to present a list of
numerals, virtually none deals with the morphosyntactic complexities and
historical dimensions of this particular domain that exist for almost any
language. Minimal knowledge of Old English is required to understand and
benefit from the volume.

The specialist in numerals will be struck by the richness and depth of the
author’s specific insights regarding numerical systems in general, using the Old
English evidence to great effect. Because it is one of very few monographs to
be devoted specifically to a single numeral system, and by far the lengthiest
and theoretically the most sophisticated (cf. Zide 1978, Olsson 1997, Leko
2009), there is time and space to deal with small complexities whose broader
relevance is enormous. The volume thus strikes that fine balance between
empiricism and theoretical breadth required of this sort of cross-linguistic
study rooted in a single language.

With regard to the prehistory of numerals, we are very much working from a
speculative framework, and where the author treads into this territory, of
necessity the argument is more tenuous. It may be true that for most languages,
the hands and fingers are the physical basis for the counting words, but
Hurford’s ritual hypothesis (1987), of which von Mengden does not think highly,
is at the very least plausible for some languages if not for all. These issues
are not key to the argument, which is all the more striking given that they are
presented conclusively in Chapter I.

A potential limitation of the volume is that, by restricting his definition of
numerals to cardinals (by far the most common form in the Old English corpus),
the author is forced into an exceedingly narrow position, so that, ultimately,
ordinals, nominals, frequentatives, and other forms are derived from numerals
but are not numerals as a word class, but something else. But the morphosyntax
of each of these forms has its own complexities — think of the nominal ‘007’ or
the decimal ‘6.042’ – that deserve attention from specialists on numerals.
Numerals may well be neither adjectives nor nouns, but omitting the clearly
numerical is not a useful way to show it. Similarly, the insistence that each
language possesses one and only one systemic set of cardinal numerals is
problematic in light of evidence such as that presented by Bender and Beller
(2006).

When comparing with other sorts of numerical expressions, e.g. numerical
notations, the author is on shakier grounds. It is certainly not the case, as
the author claims that the Inka khipus had a zero symbol, and it is equally the
case that the Babylonian sexagesimal notation and the Chinese rod-numerals did
(Chrisomalis 2010). Similarly, the author seems to suggest that in present-day
English, any number from ‘ten’ to ‘ninety-nine’ can be combined multiplicatively
with ‘hundred’, whereas in fact *ten hundred, *twenty hundred, … *ninety hundred
are well-formed in Old English but not in later varieties.

It is curious that von Mengden does not link the concept of numerical ‘base’ to
that of ‘power’, but rather to the patterned recurrence of sequences of
numerals. Rather than seeing ’10’, ‘100’ and ‘1000’ as powers of the same base
(10), they are conceptualized as representing a series of bases that combine
with the recurring sequence 1-9. But a system that is purely decimal, except
that numbers ending with 5 through 9 are constructed as ‘five’, ‘five plus one’
… ‘five plus four’, would by this definition have a base of 5 even though powers
of 5 have no special structural role and even though 5 never serves as a
multiplicand. This definition is theoretically useful in demonstrating that Old
English does not have a duodecimal (base-12) component, but as a
cross-linguistic definition will likely prove unsatisfactory.

Because the Old English numerals are all Germanic in origin, with no obvious
loanwords, it is perhaps unsurprising that language contact and numerical
borrowing play no major role in this account. Yet on theoretical grounds the
borrowing of numerals, including the wholesale replacement of structures and
atoms for higher powers, is of considerable importance cross-linguistically.
Comparative analysis will need to demonstrate whether morphosyntactically,
numerical loanwords are similar to or different from non-loanwords.

The author has incorporated the work of virtually every major recent theorist on
numerals, and the volume is meticulously referenced. There are a few irrelevant
typos, and a few somewhat more serious errors in tables and text that create
ambiguity or confusion, but no more than might be expected in any volume of this
size.

This monograph is a major contribution to the literature on numerals and
numerical cognition. Its value will be in its rekindling of debates long left
dormant, and its integration of Germanic historical linguistics, syntax,
semantics, and cognitive linguistics within a fascinating study of this
neglected lexical domain.

REFERENCES:

Bender, A., and S. Beller. 2006. Numeral classifiers and counting systems in
Polynesian and Micronesian languages: Common roots and cultural adaptations.
Oceanic Linguistics 45, no. 2: 380-403.

Chrisomalis, Stephen. 2010. Numerical Notation: A Comparative History. New York:
Cambridge University Press.

Greenberg, Joseph H. 1978. Generalizations about numeral systems. In Universals
of Human Language, edited by J. H. Greenberg. Stanford: Stanford University Press.

Hurford, James R. 1975. The Linguistic Theory of Numerals. Cambridge: Cambridge
University Press.

Hurford, James R. 1987. Language and Number. Oxford: Basil Blackwell.

Leko, Nedžad. 2009. The syntax of numerals in Bosnian. Lincom Europa.

Olsson, Magnus. 1997. Swedish numerals: in an international perspective. Lund
University Press.

Wiese, Heike. 2003. Numbers, Language, and the Human Mind. Cambridge: Cambridge
University Press.

Zide, Norman H. 1978. Studies in the Munda numerals. Central Institute of Indian
Languages.

As I was going through the Times…

Recently, there has been a “Puzzle Moment” in the science section of the New York Times, with an eclectic mix of articles combining scientific pursuits with cognitive and linguistic play of various sorts. One that caught my eye is ‘Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus’ by Pam Belluck [1], which is an account of the well-known Rhind Mathematical Papyrus. The RMP is an Egyptian mathematical text dating to around 1650 BCE, and is one of the most complete and systematic known accounts of ancient Egyptian mathematics. It’s a fascinating text, written in the Egyptian hieratic script rather than the more famous hieroglyphs, and it gives us considerable insight into the economy, social organization, and technical practices of the Second Intermediate Period.

The central conceit of the Times article is that the well-known “As I was going to St. Ives” poem-puzzle has its earliest ancestor in the RMP. This is vaguely true in that the RMP has a section involving repeated multiplication by seven, resulting in an addition problem. But Ahmes the scribe, despite his insistence that his text would reveal “obscurities and all secrets”, was not writing a mystery, but an exercise that formed part of scribal training, in an era where the literacy rate was at most 1-2%. While one can argue fairly that this is not a ‘real’ problem, and that the structure of it is meant to hold the learner’s attention through its repetitions, to call it a puzzle is only true in the broadest possible sense.

I’m a professional numbers guy, not an Egyptologist, but the article we are presented with not only tells us nothing new about the Rhind. I was very pleased, on the one hand, to see Marcel Danesi, whose work may be familiar to many readers of this blog, commenting on the widespread cross-cultural and cross-historical interest in puzzles (not only numerical puzzles, but including them). It’s not often enough that linguistic anthropologists get quoted in the Times. And like Danesi, I have broadly universalist sympathies. But I disagree with Danesi, who has made this claim about the RMP elsewhere, in his The Puzzle Instinct (Indiana, 2004, pp. 6-7) that it was “shrouded in mystery” or that “mystery, wisdom, and puzzle-solving were intrinsically entwined in the ancient world.”

The better example of numerical play in Egyptian scribal traditions mentioned in the Times article is the Horus eye, or wedjat, a combination of six symbols whose constituent parts signify the fractional series {1/2, 1/4, 1/8, 1/16, 1/32, 1/64} which when summed totals 63/64, or nearly one (see below). As the Egyptologist Sir Alan Gardiner reckoned it, the remaining 1/64 would be provided by Thoth who would heal the Eye and thus produce unity. It’s a nice story, and at least at some periods or for some writers, this narrative may have been relevant.


Source: wikimedia.org

But the Horus-eye illustrates one of the central problems in the transliteration of Egyptian texts, namely that while the vast majority of Egyptian mathematically-relevant texts are written in the cursive hieratic script, they are transcribed, and all-too-frequently theorized, as if they were hieroglyphs. This transcriptional practice leads us to think of the Rhind as a hieroglyphic text that just happens to be in hieratic in the original, but in the case of the Horus eye it couldn’t be more misleading. The Horus symbols in the Rhind don’t look like the above image, and more generally, the hieratic numerals look nothing like, and behave nothing like, the hieroglyphic numerals. We now call of these six Horus eye components by the less evocative name of ‘capacity system submultiples’ in recognition of the fact that these components were originally nonpictographic, part of a metrological system of grain measurement, and only at a much later date were they composed into the wedjat-eye. This isn’t to say that the Egyptians weren’t numerically playful, but they weren’t especially playful in the Rhind.

In short, the RMP is not an especially good example of numerical play in Egypt, and certainly not an especially relevant example from a cross-cultural perspective. It illustrates, to be sure, that mathematical texts are not purely functional or economic documents, but include semiotic and linguistic elements far beyond their pragmatic use. But this is not new knowledge about the Rhind or about mathematics. And it runs a grave risk of othering a document whose function was largely pedagogical, and is thus not so different than, for instance, the ‘ready-reckoners’ of early-capitalist sixteenth-century England.

I am thrilled to see numerical texts treated as objects of inquiry beyond the facile ‘Did they get the answer right?’ I am sympathetic to Danesi’s claim that puzzles and riddles have universal salience. Yet I worry that, at least in the case of the Rhind, the link to puzzle-like behavior is so far-fetched that it turns our best glimpse into Egyptian sociomathematical practice into an inappropriately arcane and obscurantist account. This ‘mysteries of lost Egypt’ nonsense should have been set aside decades ago.

If you wanted to pull out some cross-cultural examples of numerical play, you could easily find lots of better examples, from well-covered territory such as Hebrew gematria practices, to the richly evocative varnasankhya systems of number-word associations in premodern South Asian texts, to the complex cluster of quasi-cryptographic numerical systems used by Ottoman administrators and military officers. Or if you were really stuck on Egypt, you could investigate the cryptic numerals used on late Egyptian votive rods and Ptolemaic inscriptions, richly infused with homophony. (For a more extensive discussion of these and others, see my Numerical Notation: A Comparative History (Cambridge, 2010). There is a rich, although disciplinarily diverse, comparative body of material on numerical practices including puzzles, but the Rhind just isn’t part of it.

(Crossposted to the Society for Linguistic Anthropology blog)