Viewing Formal Mathematics from Yoruba Conception of the Sky

Home , Babalawo, Isha people, Oko (Orisha)

Aimé Dafon Segla Université d’Abomey-Calavi

Abstract: social science take place, they only operate Yoruba Cosmology resembles a generative system at the in terms of furtive descriptions of ideas and foundation of concepts. The traditional thought, which achievements that have been superseded. derives from the reality of the identical pair incorporated In such studies, described achievements from cosmology into real life, exemplifies all kind of ex- and thoughts are best understood as histor- isting knowledge, culture and practices. Previous studies ical markers and anthropological guides; by the author show in some detail the scientific interests in Yoruba cosmology. The present paper aims to view rarely are they logical, well portrayed, dy- formal mathematics through the interpretation of Yoruba namic transformations. For the substance sky knowledge. It attempts to demonstrate that linguistic of the state of the debate, we first recall the codes elaborated from the Yoruba conception of the sky status of Western Science, which is identi- are binary and hexadecimal codes that imply Algebraic fied by reference to its theories and system- Boolean structure or Group Structure. The system is in atic and dynamic character, as opposed to accord with classical mathematical properties of Group non-Western Science, which is associated Structure. Finally, the review and interpretation of Yo- with traditional practices, has no theory, ruba cosmology may suggest the possibility of formal rea- and has a static character. soning without any writing system in human history. The problem that has always been raised is that of To challenge and differentiate the dis- knowing if symbols and configurations in the Yoruba in- tinction between static and dynamic sys- terpretation of the sky are the result of implicit premedi- tems of thought, we need to give and show tated mathematical development. It is clear that behind sufficient parameters. What orthodox or the implicit oral discourse described hereafter lies a cor- positivist critiques say is not only that there responding semiology elaborated with the corresponding is no dynamic trend of knowledge in non- formal mathematical reasoning that we claim. Western Science, but that there is a lack of visibility and an impossibility of demon- Keywords: history: concept and logic; alge- strating the presence of tangible outward bra: Boolean-group structure; astronomy: interpretation; formal mathematics; Yoruba cos- signs and key concepts like objectivity, cau- mology; knowledge: tacit-implicit sality, deduction and induction. While on the Western side, the practices—symbols Introduction and cosmological as well as social back- As Abiola Irèlè observed in The African grounds—are cut out, making knowledge Scholar (1991) and further in The African Im- there appearing thoroughly instrumental, agination (2001), scientific and technical systematic and dynamic; on the African achievements of Classical African Cultures side the practices are still reduced to their are under-studied. The focus of scholar- cosmological and social backgrounds, fre- ship in African Studies is directed only to- quently narrowed down to the symbolic di- wards the humanities and the social sci- mension. In these conditions, for modern ences. Hountondji (1990) also argues in anthropology, it is, for example, impossible Scientific Dependence in Africa Today (1990) to sort out from African belief accurate sci- that, even when studies on humanities and entific knowledge and techniques like Journal of Astronomy in Culture 1(1), 2016, pp. 9-21 Copyright © International Society for Archaeoastronomy and Astronomy in Culture 10 mathematics. This makes Western Science art, Nok iron civilization and technology. look like it arrives at speculative and mod- It is well known that Yoruba art, together ern theorizing mysteriously from nowhere. with that of Benin, originated in the Ifè Yet, science in Europe also originated from classical period (1000 to 1400 CE), had a myth and rituals and moved from magic tradition of excellence, and is comparable and religion only progressively; theoretical to the works of art of classical Greece and constructions in Western cultures logically Rome (Willet 1967). also started from basic human traditional knowledge. Beliefs and Knowledge In the present essay, using founda- As Lawal (2008) points out, in “Ejiwapo: tional archaeology of concepts taken from the dialectics of twoness in Yoruba art and Yoruba history and culture, we try again to culture,” the concept of “twoness—ejiwapo make clear the dynamism of African tradi- meaning ‘our two(s) are many’; the idea of tional thought by bringing a case study of the reality of an identical pair derived from formal knowledge extracted from folk be- cosmology”—exemplifies the idea of the lief. By the way, included here is a demon- reality of nature, constantly characterized, stration again that African science can in faith, by two aspects: “spirit/matter, vis- prove the accuracy of the stock of tradi- ible/invisible, male/female, good/evil, es- tional knowledge and technologies (Sègla, sence/existence.” About Yoruba’s cosmo- 2015a, 2015b). We first present the Yo- logical vision, studies by Morton-William ruba cosmological model and its elements. attest to a “twoness mental model.’’ Mor- We then show their scientific interests and ton-William (1964) reported from the Yo- further, we present our mathematical in- ruba Oyo country: “…The House of the terpretation of the Yoruba conception of Sky is the domain of the supreme God, the universe and the sky. Olorun or Olodumaré (Olorun means ‘Sky – Owner’)… The Earth is the domain Yoruba Ethnoastronomy of the Goddess Onilè, Earth-Owner, who Who are the Yoruba? is sometimes simply called Ile...” This ob- The Yoruba are about 50 million people servation from the Oyo Yoruba, apart living in West Africa, Nigeria, Benin Re- from some variability, is available for all public and Togo. Millions of others in the other Yoruba. Cuba, Brazil (Bahia) and the United States From these observations, the Earth are descendants of Yoruba slaves. The or- Goddess is recognized to be the conceptual igins of the Yoruba are lost in history. counterpart of the Sky God, since Earth Their ancient history, sometime between and Sky are in relation of a Female/Male 3000 BCE and 500 CE (Horton 1979; duality. They are coeval, each of them be- Obayemi 1979), is well known by the study ing characterized by four correspondent of the pre-dynastic (pre-Oduduwa) Yoruba spirits: matter, water, air and fire. Between linguistic groups now mainly in Benin Re- them is the World of the humans (Ilé Aiyé), public, Togo or in the eastern parts of Yo- governed by the Orisha in Yoruba history ruba land in Nigeria (Sègla 2003, 2006). and culture. Thus, Orisha are the Su- Their recent history began with the foun- preme God functionaries and ministering dation of Ifè between 500 and 800 CE, and spirits in the World /Ilé Aiyé/ (Awolalu later with the foundation of Oyo. Heritage 1979). This means that the Orisha are in- is widespread with strong knowledge of so- termediary deities between the Supreme cial organization, agriculture, medicine, God and humans. The Orisha, who are

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Figure 1. Yoruba cosmological model reproduced from Morton-William (1964).

sent by the Supreme God to govern hu- tory lead us to the main proposal for a uni- man life, thus constitute socially con- versal scientific theory because they show structed institutions for scientific practices how a people’s conception of the sky con- (Sègla 2015a). structed science (Sègla 2015a), whereby Most prominent social institutions, un- Orisha (or Vodun, as named by the Yo- der the authorities of the intermediary Or- ruba’s neighbors, the Fon) is conceived as isha, that enable humans to penetrate nat- a variant of several existing world views. ural phenomenon and establish their inner They give evidence of a “science of the lo- laws are: Obatala, the knowledge of form- cal” within a common universal scientiﬁc ing a child in womb (fertility); Shango, thun- theory (Sègla 2015a). However, a detailed der and lightning; Ogun, use of iron; Oko, study of the Orisha is not our concern fertility of farm, forest and environmental here. This article aims to concentrate on science; Shapona or Sapata, curing of small- mathematical interpretations of the Yo- pox; Esu, philosophy, part of the divine ruba cosmological model. It is therefore in- which tests and tries out people; and Orun- teresting to ask why and how the original mila/Ifa, the whole encyclopedia that gath- coders count sixteen most important inter- ers all oral knowledge produced within mediary deities (the Orisha)—also called every existing Orisha (Sègla 2015a). There the Primordial Deities—and why and how are many Orisha; scholars count thou- the Sixteen Primordial Deities have under sands and thousands of them, but accord- their control the other Two Hundred and ing to oral tradition and very well devel- Forty Secondary Deities. oped studies (Bascom, 1969; Abimbo 1975, 1977), the most famous Orisha are Scientific Interests the Sixteen (16) so-called Fundamental or We propose that the study of the Yoruba- Primordial and the Two Hundred Forty Idààcha Divinatory Calendar (Sègla 2006, (240) Secondary Orisha. The total gives 2008) gives evidence of a pre-Ifa (Encyclo- two hundred and fifty-six Orisha. pedia of Yoruba Knowledge) divination Studies of the Orisha in Yoruba his- system that remained immune from to-

JAC, Vol. 1, No. 1, 2016 12 day’s ‘confusing’ and permuted Ifa hegem- world, one must take into consideration ony, which has already been observed by the Two Masters of the Universe, that is Maupoil (1943). In modern days, semantic Eeji Oni-Ilè or ‘the Two who own the truthfulness from the original Ifa World’ (or ‘two possessors of the universe’) knowledge system’s coding still shares as Morton reported (i.e. the Sky or Su- claims of aboriginal status with a cluster of preme God and the Earth Goddess), the associated cults. Or it shares evidence of search for solutions in society could only archaic ritual language forms that echo come through a probabilistic approach. earlier spoken dialects—Ifè/Icha Tutu Being able to invent a classifying mode that (the Ifè/Icha of earlier time, close to to- integrates all the factors that allow the day’s Idààcha dialect of Yoruba)—that re- highest rate of success for humans neces- surfaces in the voice of possessed devotees sarily implied intellectual activity and for- in Idààcha lands (Sègla 2006, 2008). Be- mal reasoning. Naturally, with a new gen- ginning in ancient times ca. 200 BCE and eralist dashboard, the Orisha Priests could, particularly from the Classical Yoruba pe- insofar as they faced new elements of dis- riod of 500 – 700 CE at the latest (Sègla eases, consult the contents of the chapters 2006, 2008)—on the basis of archaeologi- of the classifying corpus (encyclopedia). As cal findings in Ilé-Ifè (Obayèmi 1979; Hor- with Vygotsky (1974) who studied concepts ton 1970), the society had to meet a grow- and representations of autochthonous peo- ing number of complex demands in terms ple of Siberia, it is clear that original Yo- of healings, techniques, predictions, epi- ruba people also used inscriptions to repre- demics, agriculture, celebrations, augury sent concepts they had of the universe and and more. In these urgent conditions, cir- cosmos. Referring to the two deities who cles of initiates invented a mode of classifi- own the universe, the Supreme and the cation of at least 256 chapters of prescrip- Earth Goddess, the first sixteen inscriptions of all kinds of knowledge. This clas- tions are coded. Being only two at the orig- sification grouped together all the accumu- inal creation, the original coders gave one lated knowledge from all fields within 256 of the two the inscription code of ‘II’ and chapters, each of which is under one of the the second one was given the inscription intermediary Gods (Orisha), sent on earth code of ‘I’ (Sègla 2004). From there, the by the Supreme God (Olodumaré / Olo- sixteen Main Deities on which the Yoruba run) to help humans face real life. The clas- Ifa Encyclopedia is based are named. They sification is conceived in a way that each of are represented in the Table 1. the Sixteen Primordial Deities stands for a kind of Big Living Room governing again Astronomy and Interpretation: fifteen other rooms under its control. The mathematics and the concept Yo- original coders, therefore, obtained 16 ruba People have of the uni- Great Chapters of knowledge correspond- verse/sky ing to the 16 first Primordial Deities (Or- Base Two: The foundation of the Encyclopedia of isha) and a further two hundred and forty Yoruba Knowledge other chapters corresponding to the Sec- The original coders inscribed the Sixteen ondary Deities, giving a total of 256 chap- Elders (the Sixteen Primordial Deities) us- ters of the whole encyclopedia of ing two main foundational cosmological knowledge. concepts, the Supreme God and the Earth Above all, insofar as cosmogony Goddess. As they are two, the original cod- teaches that in order to succeed in the real ers understood they have to encode all ab-

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Table 1. Sixteen main signs at the basis of Yoruba Ifa Encyclopedia of Knowledge Sign Name Sign Name Sign Name Sign Name II II I I II I II I oyèku ika obara irosun II II II II II II II II

II II I I II I II I okanran ofun odi irètè II II II II I I I I

II II I I II I II I otuurupon iwori osè ogunda I I I I II II II II

II II I I II I II I owonrin osa otua ogbè I I I I I I I I stract figures and the Sixteen Primordial dial Deities as presented in the Table 1 are Orisha starting from a base two. And in really consecutively abstract mathematical base two, the symbols that help encode modern figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, every other number are ‘nothing’ and 10, 11, 12, 13, 14, and 15. And yet, oral ‘one’—the number ‘two’ itself not being in- traditions confirm that: cluded. The original coders then coded ‘nothing’ by II and ‘one’ by I, which means - oyeku means the ‘number that disap- they counted two by two, “they bless in peared, that died,’ standing here for ‘0’ in pair’’ as reported from one verse from Ifa modern mathematics; Encyclopedia of Knowledge, the Eiji Ogbè - okanran means ‘may one shine!’ standing Verse which is the 256th Chapter, i.e. the for ‘1’ in modern mathematics; last one: - ika means ‘the five fingers,’ standing for ‘5’; One ese Ifa belonging to this Odu re- - then ogbè meaning ‘completed life, ac- veals that the ancient Yoruba used the complishment’ stands for ‘15,’ making binary system to develop the Ifa Liter- sense because 15 is the last symbol in base ary Corpus. Like the people of Queens sixteen. land, the Ifa people counted “one,” “two,” “two,” “two” and “one,” “two The following sign: two”... I bless in pairs... I no longer I I bless one by one... (Opeola 1986) I I

I I In fact, being able to inscribe the symbols I I of base two (one and nothing), the original then, can be transcribed as: people had the power to inscribe all the other numbers. Not surprisingly, the in- (II) (II) (II) (II) scribed Sixteen Intermediary and Primor- that is:

JAC, Vol. 1, No. 1, 2016 14 0 0 0 0 (base 2) ‘the five fingers.’ Finally, ogbe which is 15 as inscribed in base 2 is really sixteen in base which is equivalent to: 10, say ‘the summit,’ meaning that we 0x23+ 0x22 + 0x21 + 0x20 = 0 (base 10) reach the Sixteenth Primordial Deity as we have a total of Sixteen Primordial Deities the Yoruba oyèku. as defined by Orisha theory.

While: Hexadecimal Codes in Yoruba Inscriptions: the I I other 240 Secondary Orisha Encoding. I I To code the other 240 Secondary Orisha, I I the original coders cross-composed at a I first stage each Primordial Deity with him- can be transcribed as: self and at the second stage each Primor- dial Deity with each of the 15 other Pri- (II) (II) (II) (I) mordial Deities. The first stage composi- which is tion (‘double composition’) gives 16 and the second (‘hybrid composition’) gives 0x23+0x22+0x21+0x20 = 1 in the decimal 240. They consequently have 256 com- system (the okanran, ‘1 shines’). posed Inscriptions. The Orisha Ency-

clodedia of 256 Chapters of Knowledge Similarly: suggests deliberate cognitive performances (I I) (I) (I I) (I I) as a result of their cosmological mental model, which have a broad concern with is divinatory calendars, numeration systems 0x23+1x22+0x21+0x20 = 4 and medical prescriptions. In fact, a chapter of knowledge that has been composed (ika, ‘the 5 fingers’). in this way, like Ogunda.Ogbè (composed And by Ogounda and Ogbè is like 14|15 in a hexadecimal code system. That is: (I) (I) (I) (I) is I I I I 1x23+1x22+1x21+1x20 = 15 I I (the ogbè, ‘the accomplishment, the sum- I I I mit’). Ogounda appearing first at the left side We can observe here that the semantics are and Ogbè following at the right side. in accord with the formal mathematics and Ogounda is ascendant having under its the logic. Indeed, 0 in base 2 is 1 (one) for control Ogbè. That is why the original the Yoruba original coder. And one shines, coders also inscribed Ogounda appearing say oyeku giving 2 that is okanran, which is 1 at an upper position as shown below. in base 2 and 2 in base ten in modern The left inscription or upper side being mathematics. Consequently, and logically, ascendant, we have ogunda.ogbè giving in 4 which is ika as inscribed in base 2 is cau- base ten (10) = 239, that is the 240th chap- tiously 5 in base ten (10). It makes sense ter of the General Encyclopedia of when ika is named by the original coders Knowledge.

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I faith, the Orita. He can henceforth begin I the divination. I A similar practice among the Chalde- I I ans has been reported by Oppenheim: ------I A sign that portends evil in the sky is I (also) evil on earth; one that portends I evil on earth is evil in the sky [. . .]. I These are the things you have to consider when you study the two collec- To address the origin of this system, let tions… (Swerdlow 1998). us recall briefly that The Supreme God who owns the Sky, and the Goddess Onilè who Because the eight cowry shells appear on owns the Earth, are called “Eeji Oni-Ilè,” two sides of the divining chain and are which means ‘The Two Who Possess the aligned (Sky and Earth giving 2x4 spirits; Universe,’ as noted above. Each of the two and two strokes representing concave side, forces, ‘the Earth force’ and ‘the Cosmic one stroke for convex side), viewed by the force,’ has four symmetric correspondent original coder from the left to the right (al- signs on Earth and in the Sky. These are ways facing geographic north), from the ‘fire,’ ‘earth material,’ ‘air’ and ‘water’ end of the chain to the center (the higher (Sègla 2006). Earth and Sky being in a re- power value conferred to the low position, lationship of a Female/Male duality, the while the low value is conferred to the po- system gives a total of eight signs (4 + 4 = sition that declines towards the top; as in 8), which interact and give inspiration for the Euclidean division), there is a resource the diviner to use an eight-cowry-shell rope for the hexadecimal system to occur. In- for prediction. Again, each sign also having deed, in Ifa, it is the divining chain as a me- two sides, the concave side and the convex dium and form of inscription that allows a side in virtue of the dialectics of the “twon- place value numeration system. ess’’ in Yoruba art and culture, there is ev- In fact, it is precisely because each of idence of a total of 256 configurations or the two poles of the universe has 4 corre- probabilistic possibilities each time the di- sponding spirits that the 8 spirits ‘Earth viner throws down the ‘divining chain’— and Sky’ implies the instrumentalization of or opèlè in the Yoruba language. Earlier re- the divinatory chain. It is because the cow- search reported the way the present-day ries, in concave or convex modes, are Babalawo (the old man who possesses the aligned on each branch of the rosary that knowledge), in the right tradition of the the hexadecimal codes appeared in the im- forecaster, original coder of the Ifa corpus, agination of the original coder. It is these facing the divining chain and geographic factors of inference and causality and only north, calls for four elements that charac- these that are necessary to render the re- terize the two owners of the universe, the sults valid and formal. The concept or no- earth and the sky: these elements being tion of order here is fundamental. It is the matter or earth, water, air and fire (Sègla chosen order that is one of the causes with- 2006). The Babalawo reproduces them by out which the results that are obtained marking out the four cardinal points which would be impossible. symbolizes the beginning of the creation in

JAC, Vol. 1, No. 1, 2016 16 Figure 2. A forecaster in front of a divinatory rosary with eight nuts. The striped ovals represent the ‘empty’ side with a value of II; the white ovals represent the convex side with a value of I.

One example of possible configura- converted into a decimal representation tions among 256 is given in Figure 2. Using and the right side gives the integer [14]. the above stated principles of viewing When they are associated, [1] and [14] (writing) the configuration after the diviner give, in modern hexadecimal notation, not throws the divining chain opèlè (from left to ‘one hundred and fourteen’ but 1E, where each right), the diviner obtains the following: digit 0-9 corresponds to its decimal equivalent, but 10 is represented by ‘A’, 11 by Left side / Right side ‘B’, 12 by ‘C’, 13 by ‘D’, 14 by ‘E’ and 15 by ‘F’. (II) (II) (II) (I) / (I) (I) (I) (II)

1 / 14 = 1E The ‘Left side’ of the configuration ap- 16 pears as the first stage before the ‘Right It is interesting to note that, in Yoruba Ifa side.’ A comparative reading through Ordus, when we consider (II, I) and not modern mathematics—when the couple (0,1), the left viewed side is named by the (concave, convex) or (II, I) is compared to original coder as Okanran, while the right the binary couple (0, 1)—allows us to con- viewed side is given the name Ogunda. They sider the couple (II, I) as a Boolean Alge- respectively correspond to the linguistic braic Structure. And thus, we can write the code of (II) (II) (II) (I) and (I) (I) (I) (II). This following for each viewed side (left and means that the composed word [Okanran- right): Ogunda]/Yoruba Ifa linguistic codes and

1E /Hexadecimal codes are the same. 0 0 0 1 / 1 1 1 0 16 In modern mathematics, we can trans- or late 1E16/Hexadecimal into the more common decimal system, to give the inte- 0x23+0x22+0x21+1x20 / ger thirty (30) [1x161 + 14x160 = 30 ]. 1x23+1x22+1x21+0x20 10 Okanran- Ogunda is also thirty 30, in Yo-

ruba Ifa Ordus. In modern mathematics, each side is writ- There is a second way of calculating ten with binary codes using four (4) posi- using modern mathematics: if we do not tions. The left side gives integer [1] when view two sides but we view each on one

Segla, A. 17 unique line, using eight positions and not in England—research that originated the two times four positions (2 x 4), we arrive discovery of Boolean algebra. It is there- at the same result. Reading and viewing fore interesting to explore the parallel be- from left to right, the transfer from tween the subtle and elaborate Yoruba uniquely binary codes to decimal codes way of thinking and the computer ‘ma- also gives the integer 30. chine language’ that also has a hexadecimal base at the foundation of the sharpness (II) (II) (II) (I) (I) (I) (I) (II) and precision of the prescriptions of the 256 chapters that help solve problems in 0x27 + 0x26 + 0x25 + 1x24 + 1x23 + 1x22 Classical Yoruba society. In machine lan- + 1x21 + 0x20 = 30 guage, a structurally identical system represents all numbers and all alphabetical Ifa Ordus would systematically count by characters that is also the base of high dividing the divining chain into two, thus, modern computing, in the manner of privileging hexadecimal codes, Okanran- structured and constituted Yoruba “archi- Ogunda. tecture of thought” (Verger 1997). Verger But Okanran-Ogunda being 3010, in- (1972, 1997) shows quite well how the Ba- dicates the 31st chapter of the Ifa corpus. In balawo/Baba oni awo/Ifa priest (the man fact, because we deal here with a positional ‘who owns the knowledge,’ ‘who owns the numerical system, the 256 configurations secret’), after throwing the divinatory possibilities are not listed from One to Two chain facing the sick man, refers to the cor- Hundred and Fifty-Six but from Zero to responding chapter in the Encyclopedia of Two Hundred and Fifty-Five (Sègla 2004). knowledge, which contains the appropri- It means that 255, as the abstract figure of ate medical prescription. He then can pre- Ifa, indicates the 256th chapter of Ifa Cor- scribe. For example, Verger (1972) who pus. In fact, 255 in Ifa is linguistically was accepted and initiated by a Yoruba coded Ogbe-Ogbe = FF in modern hexa- Convent in southern Benin Republic, city decimal notation or Eiji-Ogbe, Ogbe- of Ketu, reported: Meji, which means Two-Ogbe / Two-Fif- teen, the exact equivalent of the integer 1. Prescription against cough: this pre- 255, or (I) (I) (I) (I) /(I) (I) (I) (I), Ogbe- scription in classified or arranged in Ogbe, in Ifa. the Chapter named Obara.Ofun, that The 256 Yoruba linguistic codes in Ifa is (I) (II) (II) (II) / (II) (I) (II) (I) or are thus all hexadecimal and are therefore 1000/0101 or 8516 which equals 13310 convertible to binary codes. The entire Ifa and so represents the 134th Chapter of Encyclopedia is linguistically coded this Verse of the Encyclopedia. The 134th way (Sègla 2004). verse says: mix the leaf of ojà (given bo- The US army developed and used the tanical name which is hannoa klaieneana) same principle to improve its computer with orombo (lemon); drink the obtained data system and organization in the 1960s. mixture when pronouncing the follow- It is the same principle which has been ing incantation: “ojà, take out from my generalized and is currently at the basis of mouth this bad cough!” (Verger 1972). the great computer data and audio revolu- And, ojà really takes it out and away! tion. It is this very principle that was the basis of the research of English mathema- 2. Prescription number 56 is (II) (II) (I) tician Boole at the end of the 19th century (I) / (II) (I) (I) (I), or 3716, and 5510, thus

JAC, Vol. 1, No. 1, 2016 18 represents the 56th Chapter of the En- theory (Thiry 1998; Piaget 1968; cyclopedia, and exposes clearly how to Permingeat & Glaude1991). The dyna- strengthen the body with additional mism of the dualistic principle, which is blood. The Verse which is called also not haphazard, generated the 16 Primor- imu èjè po l’ara, ‘making blood many in dial Deities and the 256 Chapters of the body’ is arranged under the Verse Knowledge. We see here how a parallel Owonrin.Osa. It says: ewé apèjè pè ejè can be drawn to Boolean algebra with per- pupo wa si ara mi, or ‘leaf of apèjè, call fect structure of a class that we name {II, blood a lot into my body’ or ewé amùjè I}. Having gotten a total of 256 Chapters mù èjè wa si ara mi, ‘leaf of amùjè, bring of Knowledge all inscribed by the means of blood to my body’ (Verger 1972). The II (nothing) and (I) (one), again, the math- two leaves, apèjè and amujè, when ematical structure of class represented by used separately, bring blood to the {II, I} is consistent with the major property body, really! (apèjè is not identified of a class structure that is theoretically taxonomically; but amujè is ‘harungana deﬁned in modern mathematics as: madagascariensis’) When elements of a class are com- It is interesting to see how the name given posed by means of a mathematical op- by the ancestral original coders marries eration, the combination of elements very well with their function in real life. In from the class always returns an ele- the case of cough, ojà (the leaf used) means ment of the class (group) (Thiry 1998; in Yoruba ‘to break, rupture, cut,’ so ‘cut- Permingeat & Glaude 1991). ting’ or ‘breaking’ the cough. In the case of blood bringing, the leaves amujè or apèjè This finding is also characteristic of mean respectively ‘bringer of blood’ and formal reasoning in Yoruba linguistics and ‘caller of blood.’ It may be that the leaves architecture of thought. Indeed, within the strengthen the body by reinforcing the rate 16 first signs, all formed couples of non- of hemoglobin in blood. All the other Yo- consecutive signs are complementary two ruba ancestral medical prescriptions found by two. These are (Sègla n.d.): 0 and 15, in the Encyclopedia are equally clear and oyeku/ogbè (0000/1111 = 0F); 1 and 14, referenced to Ifa Ordus (Verger 1972, okanran/ogunda (0001/1110 = 1E); 2 and 1997; Sègla n.d.). 13, otuurupo/iretè (0010/1101) = 2D); 3 and 12, owonrin/irosun (0011/1100) = 3C); 4 Yoruba Ifa System of Inscription, a Boolean Group and 11, ika/otua (0100/1011 = 4B); 5 and Structure 10, ofun/osè (0101/1010 = 5A); 6 and 9, The materiality of the zero (0) in Yoruba iwori/odi (0110/1001 = 69); and 7 and 8, history and culture implies the evidence of osa/obara (0111/1000 = 78). Class {II, I} conceived as an algebraic And again, when they are associated group structure. If stroke (II) is ‘nothing’ by a group operator, say ‘or’ which is ‘+’ (zero) and stroke (I) represents ‘unity’ as in- in Boolean algebra, they all give the sign corporated from Yoruba cosmology, then Ogbè 1111, which is 15 or F in modern the pair {II, I} is able to represent all other computer data theory (Sègla n.d.). existing figures. This attests to the concept There is a second way the original cod- of the whole that cannot exist without its ers coined configurations. This way is not constituent parts, as is theoretically ex- often practiced but when viewed, gives pressed in modern mathematics’ group more evidence of the zero in Yoruba num-

Segla, A. 19 ber concept development (Sègla n.d.). And Consequently, we can conclude that, because the zero in Yoruba number theory in contrast to classical theses, this work was missing in their works, McClelland calls for the possibility of formal and ab- (1966), Longe (1983) and Opeola (1986) stract mathematical constructions in oral who first attempted to see a number theory societies without any written language sys- in Ifa Ordus misinterpreted the entire fig- tem. We then question the common West- ure’s semantics and their corresponding ern view that states that theoretical mathe- hexadecimal and binary codes in the Ifa matical knowledge was exclusively trans- Encyclopedia (Sègla n.d.). mitted by written systems which show a We can thereby conclude that the Yo- tendency for definition and formalism. ruba system of classifying knowledge is The problem that had always been raised originally constructed in binary. Evidently, was that of knowing if inscriptions and config- these are the mental constructs that attest urations in classifying Yoruba knowledge are the to a proof of cognitive performance. In result of implicit premeditated mathemati- conclusion, the form of abstraction here cal development. Behind the development observes the same assumption that is at the and the oral discourse studied here, lies a basis of modern computing. semiology elaborated with a formal corresponding reasoning that we claim. Again, Conclusion it is possible to demonstrate the accuracy Astronomy and interpretation through the of the stock of African implicit/tacit history of a Yoruba ancestral technique knowledge by concentrating on logical and clarify how cosmology, in shaping human structural transformations in traditional well-being, dictates human cognitive capa- African thought and systems of knowledge. bilities. A revisiting of Yoruba cosmology in light of a new indexical approach References demonstrates the archetypal genius of the Abimbola, W. Ifa system in terms of techniques of episte- 1975 Sixteen Great Poems of Ifa. Unesco. mological compression and expansion in 1977 Ifa Divination Poetry. Nok Publishers Ltd, which a minimal structure of symbolic New-York-London, Lagos. forms—a relationship between one and Awolalu, O. J. 1979 Yoruba Beliefs and Sacrificial Rites. Long- two, Sky and Earth—encapsulates an en- man, London. tire cosmogony. This cosmogony is itself Bascom, W. represented in terms of an unfolding series 1969 Ifa divination: Communication between Gods of numbers, from two to sixteen and from and Men in West Africa. Indiana University sixteen to two hundred and fifty-six, with Press. an infinite range of literary texts of all sorts Hountondji, P. of prescriptions including medical ones, as- 1990 Scientific Dependence in Africa Today, Re- sociated with each number that emerges search in African Literatures 21, 3. through this numerical unfolding. These Horton, R. associations demonstrate the process of the 1979 “Ancient Ifè: a reassessment,” in Journal of the Historical Society of Nigeria 9(4), pp. 69- creation of infinite possibilities of meanings 149. from the most limited initial means which Irele, A. F. are Sky God (Olorun), its correspondent 1991 The African Scholar, Transition n°51-69, on Earth (Earth Goddess, Onilè), and its Indiana University Press. ministering intermediary spirits or deities on Earth (Orisha).

JAC, Vol. 1, No. 1, 2016 20 2001 The African Imagination, Literature in Africa 2004 “De la Cosmologie à une Théorie du and the Black Diaspora, Oxford University Nombre: Le corpus Ifa revisité comme un Press. questionnement de la mathématique Lawal, B. orale,” in Max-Planck Institute for History of 2008 “Ejiwapo: The dialectics of twoness in Science Report Series 256, Berlin. Yoruba art and culture,” in African Arts 2008 “The Cosmological vision of the Yo- 41(1), pp. 24-39. ruba-Idààisha of Benin Republic (West Af- Longe, O. rica): A Light on Yoruba History and Cul- 1983 “Ifa Divination and Computer Sci- ture,” in Astrophysics and space science proceed- ence,” Inaugural Lecture, University of ings - African Cultural Astronomy: Current Ar- Ibadan. chaeoastronomy and Ethno astronomy Research in Maupoil, B. Africa. Holbrook J et al (ed). Springer, pp 1943 La géomancie à l’ancienne côte des esclaves, In- 189-207. stitut d'Ethnologie, Musée de l'Homme, 2015a “Yorùbá Ethno astronomy: Or- Paris. isha/Vodun or how people’s conception McClelland, E. M. of the sky constructed science,” in Clive 1966 “The significance of number in Ifa Ruggles (ed). International Handbook of cul- odu,” in Journal of the International African In- tural astronomy, Springer Science New- stitute 36, pp. 412-430. York. Morton-William, P. 2015b “Chasing the Shadow of the Sun: The 1964 “An Outline of the Cosmology and Batammaliba’s Binary System of Time,” Cult Organization of the Oyo Yoruba,” in in Inspiration of Astronomical Phenomena VIII: Africa 34, p. 243-261. City of Stars. B. P. Abbott (editor). Astro- Obayemi, A. nomical Society of the Pacific (ASP) Con- 1979 “Ancient Ifè: Another Cultural-Histor- ference Series, Vol. 501 New-York. ical Interpretation,” in Journal of the Histor- n.d. IFA, Le Calculateur Yoruba: Ency- ical Society of Nigeria 9(4), pp. 151-185. clopédie de la connaissance ancestrale. Oppenheim, A. L. Presses de l’Université d’Abomey-Calavi, 1974 “Divination and Celestial Observation République du Bénin, forthcoming. in the Last Assyrian Empire,” in Centaurs, Sègla, D. A. and A. Boko International Magazine of the History of Science 2006 “De la Cosmologie à la Rationalisation and Medicine 14, pp. 97-135. de la vie sociale: ces mots Idààisha qui par- Opeola, M. S. lent ou la mémoire d’un type de calendrier 1986 “From Ifa Divination to Computer Sci- Yoruba ancient,” in Cahier d’Etudes Afri- ence,” a paper presented at the Third In- caines 181, pp. 11-50. ternational congress of Orisa Tradition Swerdlow, N. M. and Culture, Department of Education 1998 The Babylonian Theory of the Planets. and Curriculum Studies, Univ. of Ilé-Ifè. Princeton University Press, Princeton. Permingeat, N. and D. Glaude Thiry, P. 1991 Algèbre de Boole: Théories, Méthodes de Cal- 2007 Notions de Logique 3e Edition: méthodes en culs, Applications avec Exercices. Deuxième Sciences Humaines. De Boeck, Bruxelles. Tirage Corrigé. Masson, Paris. Verger, P. Piaget, J. 1972 “Automatisme verbal et communica- 1968 Le Structuralisme: Le Point des Connaissances tion du savoir chez les Yoruba,” in Actuelles. Presse Universitaire de France. L’Homme, revue Française d’Anthropologie XII: Paris. 2. Sègla, D. A. 1997 Ewé, le verbe et le pouvoir des plantes chez les 2003 “The Scientific Mind and Cultural Ar- Yoruba. Maisonneuve et Larose, Paris. ticulation in an Oral Society: language as Vygotsky, L. S. a mirror,” Social Science Information, SAGE Publications, 42(3), pp. 339-374.

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