How Do Languages Count? And Are Languages with Restricted Number Systems “Primitive”?
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.
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.
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.
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.
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.
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:
|‘The man is going with the woman.’|
|‘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.
|‘We are coming.’|
|‘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’.
|‘I followed the person and the person fell down.’|
The same applies to subordination structures such as:
|‘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.
|‘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.
|‘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”:
|‘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:
|‘The person saw the water (he was looking for).’|
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.
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.