1 Introduction
Metrics are systems of measurement that facilitate quantification in various forms of valuing and ordering in everyday life, speeding up communication and enhancing mutual trust in transactions. The term metric goes back to the word metron, which was meaningful in ancient Greece, where it concerned the craft of both poets and geometers. In poetry it was patterns in timing of performance, while in geometry it was patterns in spatiality that mattered. Metric as I use it here is a general word implying measures in action, an imagined mesh embedded within expressed patterns of sociomaterialising relations. Metrics are patterned connections and separations in society where social relations are mediated through involvement with the stuff of the physical world. In contemporary Yorùbá life the traditional Yorùbá metric is in widespread common use. Arithmetically, linguistically, and historically, this metric is very different than the international base ten, or decimal standard metric which permeates equally widely.
The paper opens with an ethnographic story of a lively arithmetic class involving both metrics. This setting has me homing in on metrics as they come to life in classrooms where children are schooled in the rules for using so-called natural numbers, rules for valuing in counting and measuring, and rules for manipulating number relations. We are not concerned here with statistical numbers, or economistic numbers (Verran 2015). Then, having delved into the different arithmetics and linguistic practices embedded in the workings of the decimal and Yorùbá metrics, I sum up the first part of the paper by going back to the opening ethnographic story, trying to put myself into the shoes of the teacher who designed and implemented the classroom lesson I describe. This exercise of imagination has me proposing both metrics as established linguistic-arithmetic lacings that thread as supportive dynamic mesh through the sensible sociomaterial everyday we negotiate in our ordinary lives. In this picture the numbers mediated in and expressing the patterns of metrics, dynamic meshed sets of relations, happen in ways that can be felt or experienced in a here and now. I go on to develop this working imaginary using comments from children bilingual in Yorùbá and English. The commentary of these remarks
I propose this different feel the children talk of as analogous to the different feels of iconic and indexical numbers (Verran 2010). I suggest that in use Yorùbá numbers as taught in classrooms lean towards iconicityâa rationality relating wholes and parts, whereas decimal numbers seem to be naturally indexical, expressing a rationality involving the one and the many. I have previously suggested that while indexical numbers are important in scientific knowledge making, but when it comes to trade, it is iconic numbers that matter. Thus, I am proposing that the metrics differ not only in the rationalities of their making in linguistic and arithmetic practices, but also in the rationalities of the practices of their primary uses beyond the classroom. This is not to deny that with slight modifications the numbers of each of the metrics as taught in classrooms can be repurposed to work the alternative rationality.
Alternative patterns of relations are embedded in the dual metrics of contemporary Yorùbá life, and these different patternings in making are felt in use. One pattern of relations begins with precisely defined ones or units, and indeed has purloined the word metric as its basic unit of spatiality: âthe metreâ. The pattern of relations embedded in this set of numbers comes to life most strongly using English language and English numbers. It begins in precise specification of unit. The other set of numbers, embedding a different pattern of relations comes to life in Yorùbá language. This form of quantification commences with a vague whole and achieves precise definition of units of measure in processes of material partition; here definition of unit of measure is an outcome.
Arithmetically speaking, decimal numbers are based around ten, the number at which the basic set of numbers starts to repeat with some small modification in naming to indicate the number of repeats of the set, so for example, âtwenty-oneâ starts the third repetition of the basic set. This is the internationally accepted system. In contrast, arithmetically the Yorùbá metric works through using three base points: twenty, ten, and five. While it is possible to use a base ten system when using Yorùbá (invented in the 1960s), this is fairly rare. When speaking or writing Yorùbá, it is usual to use the triple base metric, and indeed algorithms have been devised to enable computer generated text-to-speech and machine translation of Yorùbá text that includes use of the Yorùbá number system (Akinadé and á»déjá»bà 2014).
This chapter then offers a very short introduction to the duality of the contemporary Yorùbá metrics, drawing on insights developed in Science and an
Having seen and heard how bilingual Yorùbá children easily connect the two metrics in practice in lessons, I was intrigued. I went on to talk to around seventy children bilingual in Yorùbá and English asking them to tell me how they actually used the two systems when they were doing the practices involved in measuring and counting. This very complex study which can be understood as a form of conversational ethnography, involved providing actual substances to be measured and counted during discussion lasting thirty to forty-five minutes. In all, I conversed one-to-one with nearly two hundred and fifty children, some in English and some in Yorùbá. Of the seventy children bilingual in Yorùbá and English, I spoke with thirty or so of these children in English, and the other thirty plus in Yorùbá (Verran 2001).
To sum up quickly the take-away message embedded in the answers these bilingual children gave me, they said that when they were actually using the Yorùbá number system in counting and measuring, they really needed to pay attention to the purpose of the quantification, that would tell them what unit of measure to use; it is people doing particular things who make-up the units of the material being counted or measured. The sort of unit adopted in using the Yorùbá metric is based on the social reason that has people needing to measure or count. To say that more formally, the children emphasised purpose in number use as important when using the Yorùbá system. In contrast when they used English numbers they said, you need to really look at the substance,
I come back to this very general difference the children discerned after we have had a look at arithmetic and linguistic mechanisms embedded in the metrics. Inquiring into the different mechanisms involved in articulating the metrics we will see that the difference the children identified is associated with another other significant difference. The pattern of relations embedded in and expressed by Yorùbá numbers foregrounds a whole-parts trajectory in use, so particular usage begins with some sort of vague whole and ends up with precise but distinct relational parts, which acting backwards together define the beginning whole precisely. Whereas the standard international decimal numbers foreground a one-to-many relation, which in use begins in a precise one and ends up with a many which projecting beyond and outside the situation of quantification, becomes a precisely defined whole ready to be reconstrued as a precise one in a further context. Thus, there is a sense of abstraction, a carrying off to another domain associated with decimal numbers and the modern metric.
2 Contemporary Yorùbá Metrics in a Classroom Setting
The school is located on one of the narrow streets radiating off from the Oòniâs palace Ilé ifáº¹Ì in Nigeria, the roadway usually clean and relatively clear of cars. The school grounds too, much smaller than most schools, always seemed neat, rake marks showing in the dust when we arrived in the morning. Mr. Ojeniyi is dressed as usual in a clean, crisply ironed white shirt. The older children in their final years of elementary education are quiet and responsive to their teacher sitting in orderly rows, their exercise books open before them. I always enjoy Mr. Ojeniyiâs lessons, he is one of those people turned on by the aesthetic of math. Numbers and their relations clearly provoke joy and pleasure for him, which he communicates easily to the children. His lesson concerns the translation between the base ten numbers of English language metrics, and the complicated multi-base numbers of Yorùbá, which pivot around twenty, ten and five.
Mr. Ojeniyi begins his lesson in English with the statement, âYou will not understand a number unless you understand the many ways it can be dividedâ. This is certainly not the usual way that a whole number is understood in primary school mathematics. After a few sentences Mr. Ojeniyi shifts to Yorùbá as he gets warmed up in his explanation. I close my eyes to try to pick-up the Yorùbá better and to follow his reasoning. I keep losing his line of argument in my increasing agitation over the unorthodox account of what a number is
Relaxed and catching again the gist of Mr. Ojeniyiâs reasoning: he names a Yorùbá number, too complicated for me to hear. Then he explains it as a factor of twenty, plus or minus various factors of twenty, I remember these Yorùbá number names well enough to follow his account. Then helpfully, he translates it into a base ten English language number and plots out the arithmetic process traced in making the Yorùbá number using English number names and the base ten system. Then he starts again with an English language number name and converts into a Yorùbá number, using division into sets of twenty as the first and defining process. A complicated explanation switching between Yorùbá and English language in a precise, comparative performance of two very different numbering systems.
The children had clearly all followed his explanation. They then copy down in their exercise books the series of arithmetic operations Mr. Ojeniyi has elaborated for each translation on the blackboard. After two more such demonstrations are duly spoken and inscribed on the blackboardâone a Yorùbá number translated to English base ten number, the other an English number translated to YorùbáâMr. Ojeniyi names a number in English, âNineteen thousand, six hundred and sixty nineâ and writes â19,669â on the blackboard. How can this be named in Yorùbá? Mr. Ojeniyi asks for volunteers, and children loudly call out suggested alternatives. When called upon children jump out of their seats to rush to the blackboard.
One child announces, âá»Ìkáº¹Ì kan ó dÃn erinwó ó lé okaà n dÃnláà ádá»Ìrinâ; another announces âẹÌẹÌdẹÌgbà áwà á ó lé ẹgbẹÌa ó lé á»Ìkà ndÃnláà ádá»Ìrin.â A third youngster offers âá»Ìkáº¹Ì kan ó dÃn á»ÌtadÃnÃrinwó ó lé mẹÌsán.â All these answers are correct, they all name the number ânineteen thousand six hundred and sixty-nineâ but each indicates that the child got to that number by a different arithmetical path. Mr. Ojeniyi asks the class which of these is to be preferred. Few have any doubt that the first version is best, but Mr. Ojeniyi insists maybe not. Writing on the black board he demonstrates a notion of elegance in Yorùbá numbers using decimal number symbols to show the sequence of arithmetical operations mapped out in each named version, a form of modelling. Arithmetically these are three distinct versions of the same number.
| â First: 19,669 translates as | (((20,000Ã1)â400)+(â1â10+(20Ã4))) |
| â Second: 19,669 translates as | ((20,000â1,000)+((200Ã3)+(20Ã3))+9) |
| â Third: 19,669 translates as | ((20,000Ã1)â(â(20Ã3)+400)+9) |
This ethnographic story which arose out of a slightly disconcerting surprise experienced by the author, offers a form of evidence. But in attesting that âthis happened,â the story is not evidence for a scientific truth claim: it is not proclaiming something about Yorùbá children, nor schools, nor contemporary Yorùbá life in general. The work the story is doing is more of a literary nature. It is setting out the particular situation in and from which this chapter is written. In that sense it is a framing device. But as part of an chapter in social study of numbers, the story is also doing some epistemic or knowledge work. It offers a beginning set of data, it is âa sightingâ so to say, of Yorùbá numbers in situ, numbers in action along with the numbers of the modern standard decimal system, in an arithmetic lesson. It gives usâas reader and authorâa point of departure.
3 Arithmetical Practices in Number Formation
In the Yorùbá number series there are fifteen basic numbers from which an infinite series is derived. I list these in Table 7.1 in a standard counting form.
The basic set of Yorùbá number names (Abraham 1962)
| Set 1 | ókan | èjì | ẹÌta | ẹÌrin | à rùún | ẹÌfa | èje | ẹÌjá» | ẹÌsá»ÌÅ | ẹÌwà á |
| one | two | three | four | five | six | seven | eight | nine | ten | |
| Set 2 | ogún | á»gbá»Ìn | ||||||||
| twenty | thirty | |||||||||
| Set 3 | igba | irÃnwó | ẹÌẹÌdẹÌgbà áwà á or á»Ìká»Ì kan (as whole) | |||||||
| two hundred | four hundred | twenty thousand |
The elaborated tone marking on the words I have listed above, point to the aural pleasure that can be derived from listening to Yorùbá number names being used, or for many speakers, even in their being read. The words make up a sort of music score. Spoken Yorùbá numbers are melodious like all Yorùbá words, and the tones-sounds offer clues in discerning the histories of the words. I explain the linguistic origins of these words naming the basic set of numbers in my next section, here I merely note that verbsâwords denoting actionâlie at the core of these names and in that they differ from the English set of basic numbers which are nouns and name things (for example âfiveâ is a form of a very old noun meaning âhandâ). And further, the sorts of action Yorùbá verbs denote is rather different than action as remarked by English verbs. Of rope
Twenties matter in Yorùbá numbers. In counting, after twenty the core arithmetic process in the working of the Yorùbá metric is progression in jumps from whole vigesimal (twenty) point to the next whole twenty while noting the multiplication factor. Then, when you get near to the number you are targeting, you utilise the secondary base of ten, and then in a third or tertiary stage, the further subsidiary base of five comes in. Instead of the addition of ones, additions of tens, and addition of hundreds and so on, the process that is familiar in the decimal metric, in the Yorùbá metric it is multiplication which generates multiples of ogún (twenty). To be more exact, the Yorùbá verb embedded in a number in action in measuring means: âmultiple placings.â Working around
Here is a general description of how to derive Yorùbá numbers. The vigesimal points occur at twenty, forty, sixty and so on, we can understand the generation of numbers up to sixty in the following way:
The first four numbers of each vigesimal are generated through a process which is fairly familiar to base ten users through the addition of ones, say:
40 + 1 = 41,
40 + 2 = 42, etc.
After 44, we progressively take away one less at each step to generate 45 to 49. We âleapâ to:
60 â 10 â 5 = 45,
60 â 10 â 4 = 46 etc.
Then to generate 50 to 54:
60 â 10 = 50,
60 â 10 + 1 = 51 etc.
From 55 we progressively take away one less at each step to 59:
60 â 5 = 55,
60 â 4 = 56 etc.
Yorùbá numbers emerge as a nested cluster. Arriving at a particular number form by working from the nearest vigesimal can be imagined as setting out and aligning of parts, and of course, âpassageâ can be made differently through the relations of the ten and the five, different setting-out and aligning of parts is possible.
Yorùbá numbers were first written down as words, as number names, in the late nineteenth century; they became a series of inscribed names in the process of being collected as âa cultural thingâ by European anthropologists, missionaries, linguists, and teachers. In the nineteenth century, Yorùbá numbers became objects of knowledge in anthropology. The numbers which had existed solely as uttered names, part and parcel of, indeed often the sociocultural pivot of, actual on-the-ground processes of counting and measuring by Yorùbá people, came to life in a different formâthey became âmodernâ so to say. An integral part of that collection process was translation between two different metrics, and that process of writing down was part and parcel of understanding their working as a modern metric.
The education authorities have tried to âsimplifyâ the Yorùbá numeral system by removing one stage in the process, i.e. the fives-up-and-down between the decades. Now they are copying the Indo-European system and adding from one to nine from the previous decade ⦠At the same time theyâve simplified the construction of the words ⦠As theyâve left the construction of decades intact I donât know whether the new system will make it easier or just confuse things further. ⦠The bridge between the two conceptual systems they were beginning to teach before I left is only a bridge ⦠at a very superficial level.
In considering to what extent reform of Yorùbá numeration is needed, and likely to foster both the survival of Yorùbá forms, and help contemporary Yorùbá children as they work in the modern world, we should remember that
In finishing this section, I stand back to get an overview of the patterns of relations traced out by the arithmetic characteristics of the two metrics. As part of that I speculate on the two metricsâ sociomaterial origins, suggesting that the human body has been inspirational in the emergence of both. I see the normal digital complement of humans (fingers and toes arranged on hands and feet) is expressed in both, albeit construed quite differently. In Yorùbá it is the whole human being who figures with their twenty digits: four sets of five; two sets as hands, and two sets as feet. This four by five set, also seems to be a significant pattern in Yorùbá religious life (Segla 2017), and indeed four by five was also an important pattern in the cowrie shell currency in Yorùbá trading up until the end of the nineteenth century (Johnson [1921] 2010). I detect a circular moment in the Yorùbá metric, a circling inwards in engagement with the modalities of a complex sociomaterial present.
In English many have speculated that it is fingers that matter, that it was in acts of representing the world through fingers, say noting the passing of a single sheep through a gate by gesturing with a single finger raised or lowered, that numbers aroseâten suggesting that it is hands and eyes working together representing a cognised world. Numbers in the medium of fingers raised and lowered, or scratched markings on wood, as distinct from the messy complex world of sheep and their shepherds (Ifrah 1985). In this decimal patterning, singularity of fingers affords a gesture and the position in the series of gestures, little events, is what matters. This singularity and positionality contrasts with what is afforded by the four sets of five which comprise the whole human digital complement, the sense seemingly summoned up by the Yorùbá number pattern.
That sense of decimal numbers effecting a shift to an alternative medium of communication fits with the usual claim that the modern decimal one-to-many metric effects abstractionâa carrying off to another domain in translation to an alternative sociomaterial medium of communication. In such a shift what the number preserves is a relation, position in a metric scale of value. Such shifting in sociomateriality to an alternative communicative domain would in all likelihood be associated with different political and social stakes and different stakeholders. In contrast to that sense of abstraction, the Yorùbá
4 Linguistic Practices
In linguistic and philosophical studies of number there is controversy over the relation between language and number (Hurford 1987). Is number emergent within word usage, expressing the differing grammars arising in linguistic mechanisms? Or is it the other way around? Does number arising directly as cognitive mechanism drive the emergence of grammar? It is an ancient and much debated question and beyond the scope of this chapter to consider. It seems likely that both claims are right to some degree.
In skirting around this vexed issue let me instead announce that in my framing numbers are treated as words, and as such as expressions of the patterns of relations traced out in the grammars of particular languages. Thus, the Yorùbá metric, just like the Yorùbá language in its various dialects is an outcome of the forms of life that have emerged in particular places and times. Both metric and grammar express patterns of relationality, as indeed does the decimal metric and the grammar of English and its cognate languages. This does not mean that the metrics are incommensurable. Translation is feasible and indeed routine as we have already seen. As I noted in beginning the previous section, key in making good translations is recognising differences in the meaning making work of verbs in Yorùbá and English. In English one might say of a section of rope: âIt is longâ, meaning something like âThat pile you see coiled (here and now) is a long rope.â One could add, âActually it is ten metres longâ. In Yorùbá in the same situation pointing at the coils one might say âà gùnâ (literally âIt longsâ) and in providing detail, âOkùn gùn mẹÌwà á mita (literally: âRopematter longs in mode divided, and in metre (mita) mode collected, ten; Or, in a better translation âThe rope is ten metres longâ).
In following numbers in a sociomaterial inquiry, what is interesting is the ways this difference in the meaning making forms of verbs shows up in patterns of relations embedded and variously expressed in particular situations, mediated by numbers. Discerning and learning to âreadâ the ways the metrics sometimes mesh and sometimes interrupt each other, helps analysts develop a âfeelâ for the dynamic of a form of life. This for me is the sort of useful work that social study, sometimes called philosophical anthropology of numbers, can offer.
In English, numbers qualify in a second order way, they qualify a qualifier which names a property of an object. Of course, there are also numbers that tell about position in an order, the ordinal numbers, and other ways of using numbers, but here I am not aiming to give an exhaustive account of numbers and their usages, but rather to focus-up some obvious contrasts and connections between English and Yorùbá language metric usage that might be felt. I want to develop some insight into sociomaterial experience of numbers and their usage, and in this section I do that by probing the different ways numbers are used with words in talk and writing, in English and Yorùbá. This will involve me in using some grammatical concepts that readers might not be familiar with.
Readers are probably comfortable with the idea that sentences have subjects named by nouns, verbs naming actions, and objects that are somehow situated in the action. In both English and Yorùbá such object nouns are named by nouns that follow the verb, which we see above in âI cooked ten potatoes.â A translation into Yorùbá would be âMo á¹£e poteto mẹÌwà áâ (literally: âI made potatoes collected to the extent of tenâ). We can parse these sentences: âI/Mo (subject, a pronoun) cooked/á¹£e (verb) ten potatoes/poteto mẹÌwà áâ (object, noun). I use this idea that sentences have distinct parts in what follows, but I will use more precise terms than subject, verb, and object, since while Yorùbá sentences usually have these similar parts, there are also profound grammatical differences between the languages that matter when it comes to discussing how numbers sit in language use and how that is sociomaterially significant. I will refer to sentences as having designants (roughly equivalent to subjects of sentences), and predicates (which include verbs and their objects).
An appropriate translation into Yorùbá of the English sentence âThe potatoes are in the potâ would be âÃwá»n poteto wà ninú ìkòkòâ, but that sentence does not remark exactly what the English sentence does. A literal back-translation would be âThat (very) potato matter (about to be cooked) exists inside the potâ. There are no articles in Yorùbá grammar (no equivalent of âtheâ or âaâ in English), so although âà wá»nâ seems to be âthe,â it is actually an emphatic pronoun that introduce; it implies âthatâ (a pointing-at, indicating that which is about to be cooked). It emphasises the noun âpotetoâ (a loan word from English) that is at issue here. This pronoun effecting emphasis is necessary because what is routinely designated in Yorùbá sentences (the subject), what the verb says will be acted upon, is a sortal entityââmatter of potato-sort.â Matter of a particular sort is routinely conjured up in Yorùbá sentences, if one wants to say something about a particular sort of matter here and now in predicating something of it, then it is the manner in which that sort of matter manifests that will be remarked on. This is a very significant issue when it comes to the differences between Yorùbá and English number words in terms of how they are made. It explains why Yorùbá number words are elisions of introducers and verbs, why the names of the basic or âcountingâ set of Yorùbá numerals have the form of nominalised verb phrases.
Saying that the numerals are verb phrases that have been elided to form single words, identifies that the numerals function as mode or modal nouns in a grammatical sense. âModeâ here is used to point to the manner in which something manifests or is presentedâhow it is bundled. One could say in English âThe product they were selling was presented in a unified mode, despite its several internal parts being obvious.â The product being promoted in that situation
The grammar of Yorùbá language and the grammar of English have their speakers designating differently. And to say something about the subject of a Yorùbá sentence by using a number word, a speaker points to its form of spatiotemporal manifestationâthe degree of its dividedness. But when a speaker of English, who has just designated a spatiotemporal entity as the subject of their sentence, wants to add some further information in the predicate, they point to the sort of stuff it is and note the extent of that sortality.
Let me now look a little more closely at how we see the modalising work that Yorùbá number words do in language use by teasing out some particular terms that are shortened and elided to form number words in use. This analytic technique uses the rules of elision to unpack, these are linguistic norms that are well established, and widely known to Yorùbá language users. In further elaborating the linguistic uniqueness of Yorùbá number I am looking inside the quantifying number words, to discern the linguistic resources used in their formulation. This exercise is made possible by a remarkable dictionary first published in 1946 by linguist Roy Clive Abraham (1890â1963) with a second edition in 1962 (Abraham 1962). In Yorùbá life there are four distinct sets of numbers each set denoting a different situation in which numbers are used. And each of these sets includes particular elided terms. Before I consider these different sets, each of which points to purposes, I return to look into the core number set I discussed briefly in the previous section. The core set is the arithmetical set, this is the form used in teaching. The number words are verb phrases and identifying the verbs involved helps us to recognise that Yorùbá number words name with precision particular sorts of relations achieved in certain actions. Through applying the rules for elision and vowel harmony in Yorùbá, it is possible to tease the elements of the number words apart, etymologically significant insights can be gained.
Apart from the primary set of names elaborated in Table 7.1 where origins cannot be discerned, the numbers of this core arithmetic set are formed from elisions involving three different verbs. When twenties are involved the verb which becomes part of the elided verb phrase, is ó ná»Ìn (it places out). So
Passing on to consider further forms of Yorùbá number names, let me briefly point to two further verb forms deeply involved with number usage in Yorùbá, when the core arithmetic set is used to count something that is indicated. If it is currency, that is indicated by adding an elided form of the noun for money, owó into the number word. When counting other things, a form of mú an obsolete verb related in meaning to the present day mún (to take or pick up several things in a group or as one) is used. The core number word form (already a mode nounâan elided phrase with verb and introducer) is further modified when used in quantification statements in Yorùbá talk. In the collection of things set, the core number name is prefixed with âmâ and a high tone. Thus, the derived ajá méjì (two dogs): the number word here implies literally âdogmatter in the mode of being grouped in the mode of being twoâ; or I might say say âà fún mi ni ìwé mẹÌrinâ which is conventionally translated as âHe gave me four booksâ. A more literal translation is âHe gave me bookmatter in the mode of a group in the mode of fourâ.
As a further set there is an ordinal set of number words focussing up positioning and ranking, which tell of the number of times the event of a particular manifestation has occurred. There is a noun involved here ìgbà (a time) as in nÃìgbà yìà (at the present time), when attached to a number word this become elided to ẹÌáº¹Ì and the quasi verb ká»Ì is included. Coming together this generates a highly elaborated form which is a modality of a modality of a modality. A literal translation of âà gbà ìwé ìkẹta ni (He took the third book)â illustrates the triple modal nature of this form of numeral: âHe took bookmatter in the mode of collected together individual items, in the mode of three, as the third positionedâ mode noun formed by nominalising a verb phrase that already contains a mode noun.
This section of the chapter has offered analysis of how the different numbers we met in the previous section, work as words in Yorùbá and English. Likewise, considering relations between numbers and words is the final section of my book Science and an African Logic (2001). Indeed this paper can be considered
Associated with the different meanings that verbs, as words in some sense re-performing action, carry, Yorùbá has its speakers designating different types of entities in talk and writing than English. Yorùbá routinely designates matter of a particular sort, the subjects of Yorùbá sentences are characterised by have particular sorts of qualities, what is presented in Yorùbá language is a world where things have particular characteristics. In contrast English routinely designates the spatiotemporal situation of matter, the worldâs material form is focused up as what exactly it is that is being talked about, that is, being designated. In this section I have shown, that perhaps unexpectedly, this difference in verb meaning making matters when it comes to how numbers sit in and work in language use.
5 Making Something of the Metricsâ Arithmetic and Linguistic Differences
In beginning to answer the âSo what?â question that a puzzled reader who has by now waded through a great deal of perhaps tedious detail might justifiably ask, I go back to Mr. Ojeniyi in his classroom faced with a group of attentive children. He is an experienced teacher and is aware that the children sitting attentively before him likely have a lot of practical experience with numbering. In beginning to show how interpretation might make use of the detailed arithmetic and linguistic description of the internal workings of the dual metrics of contemporary Yorùbá life I have just given, I return to my ethnographic story. I identified the story as a framing and as offering a beginning in interpretation of the experienced actuality of contemporary Yorùbá life involving numbers, an exegesis, or perhaps more precisely, an eisegesis, a reading into the experience of the lesson had by the ethnographer as revealed in the story.
I begin with the figure of Mr. Ojeniyi and in particular his expressed preference for one of the Yorùbá versions of the decimal number nineteen thousand, six hundred and sixty-nine. He announced that he preferred âẹÌẹÌdẹÌgbà áwà á ó lé ẹgbẹÌa ó lé á»Ìkà ndÃnláà ádá»Ìrinâ, rejecting those numbers whose names begin with
Putting myself in the shoes of Mr. Ojeniyi, as I wrote and honed the ethnographic story I came to see him as meaning something like this. If you use the rationality of the decimal metric all three translations the children offered are correct. If you use the rationality of the Yorùbá metric the alternatives which start with á»Ìkáº¹Ì kan are not acceptable. He was clearly recognising both rationalities as equivalent, while urging the children to apply the conventional standards of each of the metrics appropriately: know the alternative standards and respect them by developing habits of working Yorùbá numbers that respect them. I take away two insights from listening carefully to Mr. Ojeniyi, from paying close attention as a learner in his lesson.
First, Mr. Ojeniyi insists that there are rationally alternative ways of knowing numbers, and goes to some pains to have the children in his class learn these rationalities well. Second, he insists that one can feel, can experience numbers, and he urges the pupils to develop such feelings and take notice of them when they are using numbers. While I never did develop a technical capacity for fluently using the rationality of Yorùbá numbers, as a learner I did take to heart both the existence of the dual rationalities as expressed in metrics, and with that awareness gradually honed my capacity to feel for different sociomaterial expressions of numbers.
Mr. Ojeniyi is certainly amongst good company in asserting these as realities of conceptual life. The eminent American cognitive scientist Susan Carey (2009) is equally explicit on these two issues. She differentiates the rationality of the metric that is the set of numbers up to somewhere between five and ten as expressing a rationality embedded in biology and evolution. The distinction here can be seen in a comparison between five and say fifty-five, which she points to as having a cultural origin. While both five and fifty-five are said, in English, to be cardinal numbers indexing value in the physical stuff of the world, indexical numbersâthe rational basis of their beingâso differs. Carey argues that such differentially felt contrasts arise in conceptsâ alternative origins.
There is another distinction between numbers that can be felt: that between cardinality where numbers work as indexes in meaning making; and ordinality where numbers work as icons in meaning making. Indexical practices begin in valuing with social orderings effected through relations between values. The alternative sequence holds in numbers being iconic, they begin in ordering
Having taken Mr. Ojeniyiâs lesson seriously, learning from it and from others delivered by his colleagues, as I elaborate in Science and an African Logic, I sought clarification from children themselves. I arranged to talk rather intensely to a large number of both English and Yorùbá speaking children, around half of who were profoundly bilingual. I discovered that the bilingual children already knew what I had just discovered. I felt myself extremely fortunate that many of these children, sensing my deficit in such matters, went out of their way to explain to me how they did it. Here is an account of what some of them said.
Imagine Afolabi watching as a small tin, which had formerly contained condensed milk, is filled with peanuts and emptied into a wide shallow plastic bowl, and then filled a second time and emptied into a thin tall glass mug. I ask him if the peanuts in the bowl and the cup are the same amount. Speaking English, he says, âThe peanuts in the bowl and the mug, they look different. But that are the same: one tin here and one tin there.â Afolabi attributes the quality of thingness (a tin of nuts) to the different presentations of peanuts in the wide flat bowl and the thin tall mug. His judgement is correct, whereas the younger child I had just spoken to was convinced that the tall mug had more nuts. Toyin, who is eleven knows too that you can solve the puzzle if you think of any âcollection of stuffâ as a thing. Nevertheless, she tells me in English you can think about it differently too: âWhen I look at it one way they look the same, then when I look at it another way they donât look the same. Thatâs when I think âItâs just a tin of nutsâ, but when I think about the bowl I can see that if I push way [indicating with her hands the diameter of the bowl], it will make them higher; itâs the same as in the cup.â Toyin is attributing two different qualities to the two collections of peanuts: thingness (âa tin of nutsâ) and volume, noting that the amounts in the bowl and mug are equal. She knows too that to think about volume you need to visualise its dimensions. Toyin could even comment on the usefulness of the different qualities. I was asking Toyin if she would think that the peanuts in the mug were equal in amount to the peanuts in the bowl if she had not watched when I emptied the tin. She replied that âIf youâre thinking about the space that the peanuts fill up, you can try to imagine if they will look the same when the peanuts in the bowl are squashed up the
âBola is a village child, while she learns English at school, she is functionally monolingual in Yorùbá. Here is what âBola (eleven years old) speaking in Yorùbá had to say. She watches as full tins of peanuts are emptied into the plastic bowl and the mug. I ask in Yorùbá if there are the same amounts of peanuts in the bowl and the cup: âǸ jáº¹Ì iye hóró ẹÌpà kan náà ló wà nÃnú ká»Ìá»Ìbù yìà à ti abá»Ì yìÃ?â (âIs there the same amount of peanuts in this cup and bowl?â) âBola laughs and replies, âá»Ìkan wà nÃbà á»Ìkan wà lóhùn-únâ (âThere is one here and one thereâ). I ask âBola if she is quite sure that there is the same amount of peanuts in the two containers, she almost scoffs, âẸyá» kan ni ẹyá» kan, à fi tà a bá pin in si méjì bẹÌáº¹Ì ni mo ni wò á»Ìokó pin in.â (âOne is one unless you divide it into two, and I watched and you didnât divide itâ). Then I ask her whether she would know they were the same if she had turned her back while I poured the peanuts out: âTó bá se pé o wo eèhìn ni gbà ti mo ni da ẹÌpà náà ni, ò bá mò pé iye kan náà ló wà ninú ká»Ìá»Ìbù áti koto náà ?â She replies, âRárá ó seése ki o ti pin in ki o si ti mú dÃè lo tà fún elòmirà nâ, (âNo you might have divided it and taken some away to sell to another personâ.) When I ask âBola if the peanuts in the bowl and the cup look the same amount, she replies âWá»n kò dá»Ìgbaâ (âThey are not equalâ). When I repeat the question putting emphasis on iye (amount), âÅ jáº¹Ì Ã³ dà bi eni pé iye kan náà ni wá»Ìn,â âBola asserts that you cannot know whether it is the same amount by just looking, âà nira láti mò bóyá iye kan náà ni wá»Ìn nipa wÃwòâ (âIt is difficult to know if they are the same size by lookingâ).
Folake is bilingual and speaks and quantifies fluently in both Yorùbá and English. Here is Folake, aged nine, explaining in Yorùbá why the coke in a bottle is the same as that contained in a wide-mouthed plastic mug filled with the contents of a second bottle of coke. âAra kan náà nà wá»Ìn tórà pé inú ìgò kékéré náà ni won fi si, o si jáº¹Ì kà o já»Ì èyìà sùgbá»Ìn à papá»Ì èyìà à ti èyìà jáº¹Ì á»Ìkan náà .â (âThey are the same because they put this there in this little bottle and that made them look like this. But the aggregate of this one [indicating the difference in width of the two containers] and this one [indicating the difference in the two heights of the liquid] is the same one.â) Although she does not name the quality involved in English language measurement of liquids (volume), in connecting up the unitary one involved Yorùbá measurement with a sense of the volume of the liquid, Folake is prepared to comment on the nature of her unitary feature, indicating that it is a unit of spacefillingness, but she still talks of it as a one. Folake is one of the bilingual children who is confidently connecting the cognitive processes of measuring in Yorùbá with measuring in English.
6
In working my way to ending my chapter, and in keeping with the autoethnographic thread by which I have woven this chapter together, I shift the scene in which I imagine numbers as coming to life; no longer a Nigerian classroom but rather an Australian river whose health as a river is compromised, and a cause of worry amongst some citizen scientists in Melbourne. Having returned to my home country, in following what I saw as a significant shift in Australian environmental governance, I began to take a close interest in the numbers involved in that work.
The two insights into the lives of numbers that I had developed as a consequence of paying close attention in Mr. Ojeniyiâs lesson, proved crucial in this new study. I felt grateful too to the kindly bilingual children who had helped me hone new understandings of numbersâ lives. Thanks to those experiences, I had developed a capacity to experience different rationalities expressed in numbers, and these skills developed in making sense of living through dual metrics in Nigeria proved invaluable. They enabled me to inquire into the differences in number use when environmental water is cared for with scienceâindexical numbers, or traded in a newly instituted water market (Verran 2010). Having developed a capacity to feel the alternative rationalities of indexical metrics and iconic metrics in Nigeria was crucial this new analytic work.
Now in concluding this chapter, I use that insight developed in studying numbers as they are used in managing Australiaâs environmental water through water trading. I use this to throw new light on the insights I developed under the tutelage of Mr. Ojeniyi and the Yorùbá children. I go one step further in interpretation than in previous commentaries on the dual metrics of Yorùbá life. I propose Yorùbá numbers can be understood as primarily iconic numbers. Of course however, by adding an extra step in the measuring process, they can be rendered in ways that have them working as indexical numbers. This contrasts with decimal numbers as taught in primary school classrooms. There iconicity is treated as a minor rationality, more or less irrelevant to the sociomaterial work of numbers. But as we have seen decimal numbers working as icons is crucial in financialisation.
I am suggesting that Yorùbá numbers as icons, intrinsically carry the feel of numbers agential in trading in a way that those decimal numbers taught in primary school classrooms as indexes do not. In having ready access to the differing rationalities of both the metrics that are significant in contemporary life, and in being trained in classrooms in using both metrics, I propose Yorùbá children become equipped equally as entrepreneurs, where iconic numbers matter, and as experimental scientists where indexical numbers are crucial.
Abraham, R.C. 1962. Dictionary of Modern Yorùbá. London: Hodder and Staughton.
Akinadé, Olúgbénga O and á»dẹÌtúnjà A. á»dẹÌjá»bÃ. 2014. âComputational Modelling of Yorùbá Numerals in a Number-to-Text Conversion System.â Journal of Language Modelling 2, no. 1: 167â211.
Armstrong, Robert G. 1962. âYorùbá Numerals.â Nigerian Social and Economic Studies, no. 1. Ibadan: Oxford University Press.
Barber, Karin. 1984. Yoruba Dun Un So: A Beginnerâs Course in Yoruba. New Haven: Yale University Press.
Carey, Susan. 2009. Origin of Concepts. Oxford: Oxford University Press.
Hurford, James R. 1987. Language and Number: The Emergence of a Cognitive System. Oxford: Basil Blackwell.
Ifrah, Georges. 1985. From One to Zero: A Universal History of Numbers. New York: Viking Penguin.
Johnson, Samuel. (1921) 2010. The History of the Yorùbás: From the Earliest Times to the Beginning of the British Protectorate. Cambridge: Cambridge University Press.
Segla, Dafon. Aimé. 2017. âCulture and Cosmos: Structural Changes In A System Of Knowledge-History, Concepts And Logic In Yorùbá Number Concept Development From Peopleâs Conception Of The Sky (West Africa).â Revue des Sciences du Langage et de la Communication 4, 360â75.
Verran, Helen. 2001. Science and an African Logic. Chicago: Chicago University Press.
Verran, Helen. 2010. âNumber as an Inventive Frontier in Knowing and Working Australiaâs Water Resources.â Anthropological Theory 10, no. 1: 171â78.
Verran, Helen. 2015. âEnumerated Entities in Public Policy and Governance.â In Mathematics, Substance and Surmise, edited by Ernest Davis and Philip Davis, 365â79. Switzerland: Springer International Publishing.