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How Do Languages Count? And Are Languages with Restricted Number Systems “Primitive”?

Submitted by on December 1, 2013 – 4:30 pm 31 Comments |  
According to a recent article by Mike Vuolo in Slate.com, Pirahã is among “only a few documented cases” of languages that almost completely lack of numbers. Dan Everett, a renowned expert in the Pirahã language, further claims that the lack of numeracy is just one of many linguistic deficiencies of this language, which he relates to gaps in the Pirahã culture. In an earlier GeoCurrents post, I argued against the idea that Pirahã is a primitive language lacking grammatical complexity, for example, recursion. But just how rare is the absence or severe limitation of numerals cross-linguistically?

Numeral Bases map

The various types of number systems are considered in the WALS.info article on Numeral Bases, written by Bernard Comrie. Of the 196 languages in the sample, 88% can handle an infinite set of numerals. To do so, languages use some arithmetic base to construct numeral expressions. According to Comrie, “we live in a decimal world”: two thirds of the world’s languages use base 10 and such languages are spoken “in nearly every part of the world”. English, Russian, and Mandarin are three examples of such languages. The “base” is the value n such that numeral expressions are constructed according to the formula xn + y; that is, some numeral x multiplied by the base plus some other numeral. For instance, 26 in Mandarin is expressed as èr-shí-lìu ‘two-ten-six’. Around 20% of the world’s languages use either purely vigesimal (or base 20) or a hybrid vigesimal-decimal system. In a purely vigesimal system, the base is consistently 20, yielding the general formula for constructing numerals as x20 + y. For example, in Diola-Fogny, a Niger-Congo language spoken in Senegal, 51 is expressed as bukan ku-gaba di uɲɛn di b-əkɔn ‘two twenties and eleven’. Other languages with a purely vigesimal system include Arawak spoken in Suriname, Chukchi spoken in the Russian Far East, Yimas in Papua New Guinea, and Tamang in Nepal. In a hybrid vigesimal-decimal system, numbers up to 99 use base 20, but the system then shifts to being decimal for the expression of the hundreds, so that one ends up with expressions of the type x100 + y20 + z. A good example of such a system is Basque, where 256 is expressed as berr-eun eta berr-ogei-ta-hama-sei ‘two hundred and two-twenty-and-ten-six’. Other hybrid vigesimal-decimal systems are found in Abkhaz in the Caucasus, Burushaski in northern Pakistan, Fulfulde in West Africa, Jakaltek in Guatemala, and Greenlandic. In a few mostly decimal languages, moreover, a small proportion of the overall numerical system is vigesimal. In French, for example, numerals in the range 80-99 have a vigesimal structure: 97 is thus expressed as quatre-vingt-dix-sept ‘four-twenty-ten-seven’. Only five languages in the WALS sample use a base that is neither 10 nor 20. For instance, Ekari, a Trans-New Guinean language spoken in Indonesian Papua uses base of 60, as did the ancient Near Eastern language Sumerian, which has bequeathed to us our system of counting seconds and minutes. Besides Ekari, non-10-non-20-base languages include Embera Chami in Colombia, Ngiti in Democratic Republic of Congo, Supyire in Mali, and Tommo So in Mali.

Besides languages that use base 10, base 20, or some other base to construct their numerals, 12% of the world’s languages use restricted number systems. Some such languages use an extended body-part counting system. In the WALS sample, languages in this category include Eipo, Haruai, Kobon and Una, all spoken in highland New Guinea. Not unlike familiar finger-counting systems used in Europe and elsewhere, these languages use additional body parts to extend the system. For instance, in Kobon one starts counting from one’s little finger of the left hand, which corresponds to ‘one’. Numbers two through twelve are counted by the ring finger, middle finger, index finger, thumb, wrist, middle of forearm, inside of elbow, middle of upper arm, shoulder, collarbone, hole above breastbone. The count can then continue down the right-hand side of the body, from the collarbone to the (right) shoulder as 13 to the little finger as 23. It is then possible to reverse the count, starting from the little finger of the right hand as 24 back up to the hole above the breastbone as 35 and down again to the little finger of the left hand as 46. One problem with such a counting system is that the names of particular body parts can be ambiguous when used as numerals. For instance, in Kobon siduŋ  ‘shoulder’ can denote either 10 (on the left-hand side of the first pass across the body), or 14 (on the right-hand side of that pass), or 33 (on the right-hand side of the return pass across the body), or 37 (on the left-hand side of that pass), or 56 on the left-hand side of the next pass across the body, etc. But methods have been developed, optional in some languages and obligatory in others, to distinguish the second side of the body used in a count from the first, as well as to indicate which pass across the body is being used, but there is no productive means to identify anything beyond a small number of passes across the body. Extended body-part systems are thus typically rather limited in the range of numbers that they can express, but they can be used productively for quantities extending at least into the scores.

finger_counting

In regard to finger counting, it is worth pointing out the cultural differences as to how digits are indicated. In the English-speaking world and in Western Europe, fingers are extended (opened) as the count proceeds, whereas in Eastern Europe and Russia fingers are folded inwards (closed) as one counts. In the United States, Canada, and the United Kingdom, the count starts with the index finger (‘one’) and continues to the little finger (‘four’); the extension of the thumb indicates ‘five’. In contrast, for Germans, Italians, Belgians, Austrians, the Swiss, the Dutch, the Spanish, and the French the thumb represents ‘one’, the index finger is ‘two’ and so on to the little finger as ‘five’. For most Central/Eastern Europeans, the counting also starts with the thumb (‘one’) and proceeds to the little finger (‘five’), folding the digits inwards rather than extending out. In Russia, the count starts at the little finger for ‘one’ and proceeds to the thumb for ‘five’, also folding the digits inwards. The Japanese finger counting for oneself is similar to that of Central/Eastern Europeans, as the fingers are folded inwards starting at the thumb for ‘one’ and proceeding to the little finger for ‘five’, but then the action is reversed and the little finger is extended for ‘six’, and so on. However to indicate numerals to others, the Japanese use the hand in the same manner as do English speakers: the index finger becomes ‘one’ and the thumb now represents ‘five’. For numbers above five, the appropriate number of fingers from the other hand are placed against the palm. For example, ‘seven’ is represented by the index and middle finger pressed against the palm of the open hand. Thus, the gesture shown in the image on the left means ‘one’ for an American or a Brit (although the thumb would more typically be folded over the bended fingers), ‘two’ for a German, an Italian, or a Frenchman, and ‘three’ for a Russian. Such differences in finger-numbering can reveal national origins; in the Quentin Tarantino film Inglourious Basterds (sic) a British spy in wartime Germany gives himself away by ordering three beers with his index finger, middle finger, and ring finger.

Restricted Counting Systems

Going back to the various types of counting, some languages use a restricted system that does not effectively go above around 20, and some languages are even more limited, as is the case in Pirahã. The WALS sample contains 20 such languages, all but one of which are spoken in either Australia, highland New Guinea, or Amazonia. The one such language found outside these areas is !Xóõ, a Khoisan language spoken in Botswana.

Restricted_Counting Systems_Americas

Besides Pirahã, other languages with restricted numeral systems in Amazonia include Wari’, also studied by Everett, and Hixkaryana, known also for its rare Object-Verb-Subject basic word order.

 

 

 

 

Munduruku

Another language with a restricted counting system, not mentioned in WALS.info, is Mundurukú, described in an article by Pierre Pica and colleagues. A Tupi language spoken by about 7,000 people in the Amazonia (not far from the Pirahã), Mundurukú has a very small lexicon of number words: pữg/ pữg ma ‘one’, xep xep ‘two’, ebapữg ‘three’, ebadipdip ‘four’, and pữg pỡgbi ‘five’ (literally, ‘one hand’). As can be seen from the image on the left, these words are used almost exclusively for the corresponding numerals, although pữg pỡgbi is also used on occasion for ‘six’. Other ways to express certain quantities in Mundurukú include xep xep pỡgbi literally ‘two hands’, used mostly for ‘ten’, as well as adesữ/ade gu and ade/ade ma which are used for smaller and larger numbers, respectively, not unlike the English some and many.

Restricted_Counting Systems_Australia_PNG

In Australia, languages with restricted numeral systems belong to both the Pama-Nyungan family (e.g. Pitjantjatjara) and other language families (e.g. Kayardild). One of the best-described languages in this set is Yidiny, a highly endangered Pama-Nyungan language, spoken by some 150 members of the Yidindji tribe of northern Queensland. One might be tempted to think of Yidiny as a primitive language, especially because of its restricted numeral system and its limited sound inventory, with only 16 phonemes. Yet, as R.M.W. Dixon’s nearly 600-page grammar of the language shows, Yidiny is as complex, intricate, and ingrown as any language. What follows is a brief, “tip of the iceberg” sketch of some of the complexities of Yidiny.

Like other Pama-Nyungan languages, Yidiny is an ergative language, which means that the subjects of intransitive sentences have the same (zero) case ending (called “absolutive”) as the objects—rather than the subjects—of transitive sentences. The subjects of transitive sentences, in contrast, are marked with the “ergative” case. In the following example, wagu:ĵa ‘man’ has the zero (i.e. unpronounced) absolutive ending, whether it is the subject of an intransitive sentence or the object of a transitive one:

Intransitive:
wagu:ĵa buɲa:-y galiŋ
man.ABS woman-with go.PRES
‘The man is going with the woman.’
 Transitive:
buɲa:-ŋ wagu:ĵa giba:l
woman-ERG man.ABS scratch-PAST
‘The woman scratched the man.’

 

However, not all words in the language follow this ergative-absolutive pattern; particularly, first and second person pronouns follow the more familiar nominative-accusative pattern: We [not us!] are coming and We will hit you.

Intransitive:
ŋaøĵi gadaŋ  
we.NOM come.PRES
‘We are coming.’
 Transitive:
ŋaøĵi øuniø bunĵaŋ
we.NOM you(SG).ACC hit.FUT
‘We will hit you.’

 

Regardless of how the pronouns are used, Yidiny takes its ergativity seriously; for example, a subject of an intransitive and an object of a transitive can serve as the pivot of coordination, in contrast to English, where we match subjects, regardless of whether the clause is transitive or intransitive. Thus, in English I followed the person and ___ fell down means that I fell down, not the person; the missing subject of the intransitive fell down is interpreted to be the same individual as the subject of the transitive I followed the person. In Yidiny the literal counterpart of this English sentence means exactly what it cannot mean in English: ‘I followed the person and the person fell down’.

ŋayu bama banĵa:r wanda:ø
I.NOM person.ABS followed fell.down
‘I followed the person and the person fell down.’

The same applies to subordination structures such as:

buøa ĵa:wurga:ø ŋayu bunĵa-øunda
woman.ABS yawned I.NOM hit-which
‘The woman whom I was hitting yawned.’

 

One further twist in Yidiny grammar concerns the distribution of the morpheme -ĵin, which is impossible to translate into English. In normal transitive clauses such as ‘The woman scratched the man’ illustrated above, the ergative morpheme is attached to the noun that denotes the “agent” of the action (i.e. the person who did something intentionally, on purpose). Whenever this correlation is violated, the morpheme -ĵin pops up on the verb. It can happen, as in the following structure (called “anti-passive”), when the agent (here, wagu:ĵa ‘man’) is marked by absolutive rather than ergative case. In such a sentence, the object of the action (‘woman’) is marked with dative case and must appear at the end of the sentence; the semantic effect of this is similar to the insertion of the preposition at in English. Since the semantic object (‘woman’) is no longer the grammatical object of this sentence, the “agent” (‘man’) becomes the subject of a (technically) intransitive sentence and is therefore marked absolutive. Yet, he is still doing it intentionally; hence the -ĵin on the verb.

Antipassive:
wagu:ĵa giba:- ĵiø-u buɲa:-nda
man.ABS scratch-INTRAN-PAST woman-DAT
‘The man scratched (at) the woman.’

 

A similar thing happens in reflexive sentences: as there is no grammatical object, the sentence is technically intransitive, and its subject ‘man’ is marked absolutive. But since he is still the intentional agent, -ĵin pops up on the verb.

Reflexive:
wagu:ĵa giba:- ĵiø-u  
man.ABS scratch-INTRAN-PAST
‘The man scratched himself (on purpose).’

 

The same morpheme -ĵin also shows up when the transitive subject marked with ergative case is inanimate and therefore is not (and cannot be) intentional “agent”:

Inanimate subject:
ŋaøaø ginga:ŋ giba:- ĵiø-u
I.ACC prickle.ERG scratch-INTRAN-PAST
‘A prickle scratched me.’

 

Finally, -ĵin also appears on the verb to encode a chance event in which the ergative subject is not doing something intentionally but achieves some result by pure chance, as in the following minimal pair where the first sentence, without -ĵin, means that the person was looking for water, while the second sentence, with -ĵin, means that the person just happened to encounter a stream entirely by chance, when he was engaged on some other errand:

Intentional:
bama:l bana wawa:l
person water saw
‘The person saw the water (he was looking for).’
 

Chance event:
bama:l bana wawa:- ĵiø-u
person water saw-INTRAN
‘The person saw the water (unexpectedly).’

 

 

All this goes to show that Yidiny as far from being a primitive language with minimal grammar. The same is true of other languages with restricted counting systems: although most of them have not received as much attention in linguistic literature as did Yidiny, this lack of proper description reflects gaps in our knowledge rather than gaps in the languages themselves. Works such as Marjorie Crofts’ monograph on the grammar of Mundurukú and Uli Sauerland’s work on Pirahã show that there is much more that we linguists can learn from the elaborate grammars of these languages.

 

 

 

Sources:

Crofts, Marjorie (1973) Gramática mundurukú. Série Lingüística, 2. 2. Brasilia: Summer Institute of Linguistics.

Dixon, R. M. W. (1977) A Grammar of Yidin. Cambridge University Press.

Pica, Pierre; Cathy Lemer; Véronique Izard; and Stanislas Dehaene (2004) Exact and Approximate Arithmetic in an Amazonian Indigene Group. Science 306(5695): 499-503.

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  • steve

    Andamanese languages are also reported to have a counting system with just “one/few”, “two/several” and “many”, occasionally supplemented by ad-hoc finger counting. However, knowledge of them is so scant (not least since most were extinct before being properly described – beyond a few vocabularies by explorers and amateurs) that it is hard to be sure whether it is a feature of all language groups there or even if the translation is accurate. Given their purely hunter-gatherer lifestyle, however, it does make sense.

    Anyway,it is fascinating how number systems, probably more than anything else easily identifiable in language, reflect the switch to more complex agricultural, pastoral, etc. lifestyles, as well as commercial relations (which is largely a function of how isolated/integrated a certain group is). It would be interesting to investigate how each of those lifestyles influenced or determined different counting systems. For example, do sheep/goat-herding societies tend to have a certain “pattern” and rice-growing societies another one? Do hunter-gatherer groups with large trade networks or in a client relationship with agriculturalists tend to acquire a more complex number system than more isolated fellow HGs?

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thank you for sharing this fascinating information about Andamanese languages and for raising some interesting questions. More research is needed!

      • HoundsTooth

        andamanese languages are indeed worthy of future geocurrents articles! asya, you should send a team to the north sentinel island(s) to document their unique and largely unknown language!

        • http://www.pereltsvaig.com Asya Pereltsvaig

          Thanks, HoundsTooth! I am afraid I am not in a position to dispatch teams anywhere, but indeed this is languages that need more documenting and careful study.

          • HoundsTooth

            well, the last few groups that have tried to communicate with the north sentinelese have been fired upon with arrows, so it may be a language that ends up dying out before it can be catalogued. it would be an interested death wish, i must say…

  • HoundsTooth

    also, if i may add, i’d like to thank you for the info about indigenous australian languages. i myself as an english teacher and australian am quite disappointed in how LITTLE we know about these remarkable languages and cultures. i was lucky enough to study with a woman who did her TESOL teaching rounds in the ‘top end’, where english is usually the 3rd or 4th language spoken. she told our class about a language quite literally in the middle of nowhere, which has NO modality. that’s right, no conditionals, no probability and no ifs or may/mights. i have no idea how you go about arranging things in that language and or culture…i don’t think that makes them primitive (or anything as judgmental as that), just different, and that’s what makes this subject matter so interesting…

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Do you happen to know what language that was? I am really curious to look into it. Often when people say that a language doesn’t X, it is not that they cannot express those meanings, but simply that they are expressed quite differently. Do let me know if you remember the name of the language (or more precisely where it is spoken).

      • HoundsTooth

        no idea about the name, sorry…it was a remote community a few hundred kilometres ouf of fitzroy crossing. i also remember her explaining how there was a complex interbreeding structure, where from the 10 – 15 surrounding tribes, tribe A could only breed with tribe D, and this offspring could only breed with tribe M, or something similar. it warded off certain diseases (the ones inherited from genes, not the ones acquired) and they also had a complex system of ‘skin names’. i remember most of our presentations on ‘issues encountered whilst student teaching’ lasted a few minutes but her presentation was a lot longer than that…
        i’m sure they had modality, implied or otherwise, but there just wasn’t any grammatical notion of it…

        • Claire Bowern

          OK… so first of all, it’s “marriage’, not “breeding”. Breeding’s what you do with horses.
          The language is probably Walmajarri, but could be Gurindji, Gija, or a Wati language. All of those languages have ways of marking conditionals and clausal dependencies.

          • http://www.pereltsvaig.com Asya Pereltsvaig

            Thanks, Claire!

          • Gilbert Mane

            There are many benefits to taking up ballroom dancing – exercise, co-ordination, fun – and meeting new people and getting to talk to them.
            While at a recent ball in Canberra I was chatting to a young lady whom I had only just met and, as one does, I asked what she did to fill in the time between dancing.
            She was doing a PhD in threatened languages and had just written her dissertation on an Aboriginal language with only one living speaker.
            She (the dancer) had spent time with her (the indigenous final speaker) but, and I feel now I’ve let the linguisitc team down, I do not know the name of the language or its grammatical characterisitcs.

          • http://www.pereltsvaig.com Asya Pereltsvaig

            No argument from me about ballroom dancing—I used to be a professional ballroom dancer myself, back in the day! And thanks for sharing the story of this lady linguist-dancer… Fascinating!

  • nachasz

    I have never seen anyone folding fingers inwards in Poland.

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Interesting. So how is it done in Poland?

      • nachasz

        As in Western Europe by extending fingers, starting with thumb.

        • http://www.pereltsvaig.com Asya Pereltsvaig

          So like the Germans?

          • nachasz

            Indeed.

  • Thomas

    Dear Asya, thanks a lot for a fascinating article! I wanted to add a comment on the hybrid vigesimal-decimal structure of the French numbering system. Interestingly enough, this is valid for France and Québec, but to my knowledge Belgium and Switzerland use a strictly decimal system. The Gallic “soixante-dix” and “quatre-vingt-dix” become “septante” and “nonante” in Switzerland, Belgium (and Acadia?), whereas “quatre-vingts” becomes “huitante” (formerly “octante”) in Switzerland only. I would be interested in knowing the exact reasons for these differences!

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thanks for sharing this, Thomas. I am not sure about the precise developments of these French dialects, but it seems to be simply a simplification/regularization of the system.

  • Pingback: Numeracy and language | The k2p blog

  • Ygor Coelho Soares

    Interestingly, now that I’ve read about finger counting, I found out Brazilians – I don’t know about Portuguese people for sure, but I suspect they count the same way – follow the English and Western European pattern, not the Spanish or French one. Here we count with fingers extending as the count proceeds and the index finger definitely represents “one”, and the counting in general follows after the index finger until the fourth finger, so that the thumb ends up representing 5. At least that is the general counting pattern here. It is intriguing that Spanish people, who developed their culture so near to the Lusophone one, make finger counting in a different way…

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thank you for sharing this, Ygor. I’d be curious to know how the Portuguese in Portugal count, and I wouldn’t be surprised if it’s different from the Brazilian way, actually…

  • Claire Bowern

    For recent publications on restricted numeral systems in Australia, Amazonia, and North America, see https://www.academia.edu/1917177/On_numeral_complexity_in_hunter-gatherer_languages

    I also have a paper specifically on Australian numerals, in Anthropological Linguistics (http://muse.jhu.edu/journals/anthropological_linguistics/v054/54.2.bowern.html)

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thanks for sharing these links, Claire!

  • Gilbert Mane

    Dear Asya,
    Many thanks for the fascinating article. I conducted a completely unscientific survey among the staff at my school, a primary school in Sydney Australia (no children – they are already on their Summer break). The result out of about 20 adults (including some UK, NZ and South African expats): without exception everyone started with a closed fist and extended the thumb for one, index finger for two etc.
    As an aside we teach Sanskrit as a compulsory classical language starting at age five (Latin is optional). I was going to comment on your previous Proto-Indo European article, and the discussion about the use of certain sounds, by mentioning the work of the ancient Sanskrit grammarian Panini in his eight volume Ashthadhyayi.
    He sets out a complete system (called “Sandhi”) governing the euphonious modifications of sounds at the meeting point of words – eg the “y” sound that emerges when we say “three eggs”; and the change of the final “n” of London to “m”, when we say “London Bridge”. He sets out all the rules for the correct way this happens and the changes in Sanskrit spelling – eg “sat” + “chit” + “ananda” (truth consciousness bliss) becomes, as a single word: sadchidananda.

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thanks, Gilbert! Indeed, Australian finger-counting seems to be different from elsewhere in the anglophone world…

      As for Panini, he indeed was one of the first people to give a careful consideration of what we now call “grammar”. Linguists now use the term “sandhi” to describe certain phonological rules, not just in Sanskrit, but in general…

      • dw

        I grew up in England, and I start counting with the thumb. It may be a personal idiosyncrasy.

        • http://www.pereltsvaig.com Asya Pereltsvaig

          I wonder about that… Perhaps other readers would care to comment on that?

  • Kate Bellamy

    What are the other documented cases of languages without numbers?

    • http://www.pereltsvaig.com Asya Pereltsvaig

      As discussed in the post, there aren’t any languages without numbers at all, although a few (including the Piraha, as far as we can tell) have very restricted set of numbers. There’s a map in the post, and you can check WALS.info (feature #131A) for details and references to specific language descriptions:

      http://wals.info/feature/131A#2/25.5/146.1

  • Justin Barker

    Since first coming upon Pirahã and the extraordinary claims of Dan Everett I have felt that infinite counting systems in particular are in no way inherent to human behavior. I can see no reason why, across cultures and time, humans should be able to count numbers larger than those they can subitize. It’s a neat trick if you can but hardly necessary for survival. (unless you live in a pastoral society, even then, I come from a pastoral background and, in spite of years of practice, I still have a hard time accurately counting cows.)
    During one conversation one of my professors put forward the question, “Well what happens to a mother who has more than three children? Can she not keep track of them?”( Pirahã counting is basically one, a little, and a lot.) I’ve ruminated on this question over the years, and I believe that a Pirahã mother would keep track of her children as well as any mother from any other culture. Do parents keep track of their children by knowing how many children they have? You might take a head count in America or England where military precision invades the culture of the common people, but I’m sure this is not common practice around the world. So, you keep track of them by name, or not. The setup of the movie Home Alone was, I think, not implausible.

    Try it in fact. If you are a teacher try to recall how many students you had in your last class without recalling their faces or names. I may not be as smart as most but I can’t do it without naming my students and keeping track with my fingers and this only works if there were fewer than ten.

    • http://www.pereltsvaig.com Asya Pereltsvaig

      Thanks for sharing these thoughts, Justin. I agree completely that we don’t keep track of children (or friends, or students, or whover) by counting them, as much as by naming them in some way or another. And thanks for mentioning Home Alone—indeed!