Draft:Yoruba mathematics
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Yoruba mathematics emerged from The Yoruba people of West Africa, with a population of over 50 million, and is incorporated into their cosmology, culture, and daily life. This tradition demonstrates the capacity for abstract reasoning and advanced mathematical concepts in oral societies, challenging the long-held Western notion that mathematical sophistication requires a written language system.
Numbers and Counting
[edit]The Yoruba numeration system, described by the Rev. Samuel Johnson as "one of the most peculiar number scales in existence"[1], is a fascinating example of a vigesimal system (base-20) with a remarkably high degree of subtractive principles. The system uses distinct terms for numbers one to ten, twenty, thirty, two hundred, and four hundred. Numbers beyond these are formed through a complex process of multiplication, addition, and subtraction.[2]
For instance, numbers eleven through fourteen are formed by adding one to ten, two to ten, and so on. Numbers fifteen through nineteen are expressed by subtracting five from twenty, four from twenty, etc. This subtractive pattern continues throughout the system, creating a unique linguistic and conceptual approach to numeration.[2]
| Number | Cardinal Form |
|---|---|
| 1 | Ọ̀kan |
| 2 | Èjì |
| 3 | Ẹta |
| 4 | Ẹrin |
| 5 | Àrún |
| 6 | Ẹfà |
| 7 | Èje |
| 8 | Ẹjọ |
| 9 | Ẹsan |
| 10 | Ẹwa |
Subtractive Logic
[edit]The subtractive principle likely originated to facilitate counting with the fingers of one hand.[3] For example, after representing multiples of ten with gestures, one finger at a time could be extended to represent 21, 22, 23, and 24. Extending the fifth finger then signifies subtraction from 30, enabling the representation of numbers 25 through 29 by retracting fingers one by one. This pattern suggests a deep understanding of numerical relationships and a practical approach to mental computation.[2]
Use in Currency
[edit]The Yoruba people were frequent traders, using cowrie shells as their primary currency for centuries. This required the development of advanced arithmetic skills to manage large sums in a low-valued currency system.[4] As reported by travelers like James Welsh and later Osifekunde, a Yoruba man sold into slavery in Brazil, the cowrie counting system involved a hierarchy of units, often based on groupings of 40 (string), 200 (bunch), 2000 (head), and 20,000 (bag).[2]
The use of cowries as currency highlights the practical application of Yoruba mathematics in daily life. It also challenged European biases, exemplified by L. L. Conant's dismissive view of African numeration systems as mere "familiarity with numbers" rather than evidence of "mental power."[5] The Yoruba people, like many other African societies, demonstrated a sophisticated understanding of currency, exchange rates, and complex computational methods.[2]
Ifa divination
[edit]Yoruba cosmology and mathematics are deeply intertwined in the Ifa divination system, which uses a binary system to generate a vast corpus of knowledge. At the foundation of Ifa is the concept of "twoness"—a fundamental duality that permeates Yoruba art, culture, and cosmology.[6] This duality is embodied in the Supreme God (Olorun) and the Earth Goddess (Onilè), who together are called "Eeji Oni-Ilè," meaning "The Two Who Possess the Universe."[6]
Binary Codes
[edit]The Ifa divination system utilizes 16 palm nuts, representing the 16 primordial deities (Orisha), each with a binary code assigned to it (two vertical lines or one vertical line). By casting these nuts and recording the configurations of single and double lines, the diviner generates a sequence of eight binary digits, forming an "Odu," or "road of Ifa." There are 256 (28) possible Odu, each associated with verses addressing specific life situations.[6]
This binary coding system, coupled with the poetic verses, reveals a profound understanding of symbolic representation, probability, and the use of a minimal set of symbols to encode vast amounts of information. It demonstrates a form of abstract reasoning strikingly similar to the binary logic underlying modern computing systems.
Algebraic Structures and Hexadecimal Codes
[edit]Recent mathematical research has highlighted the algebraic properties embedded in the Yoruba system of classifying knowledge. The pair {II, I}, representing "nothing" (zero) and "unity" (one), respectively, is argued to constitute a Boolean algebraic structure.[7] This structure exhibits closure, meaning that any combination of the two elements through a mathematical operation, like addition mod 2, results in another element within the set.[6]
Hexadecimal System
[edit]Furthermore, the 256 chapters of the Ifa corpus are organized in a hexadecimal system (base-16), similar to the system used in modern computing. Each chapter is formed by combining two of the 16 primordial deities, creating a code analogous to a two-digit hexadecimal number.[6]
Formal Group Structure
[edit]Olagunju et al. provides a formal mathematical analysis of the 16 principal Odu Ifa, demonstrating their algebraic properties. They represent these Odu as 4x2 matrices with binary entries (0 and 1) and show that they form an abelian group under matrix addition modulo 2. This "self-inversed group," as they term it, exhibits closure, associativity, an identity element, and intriguingly, each element acts as its own inverse. This algebraic characterization deepens our understanding of the Ifa divination system, revealing a structured mathematical framework underlying this ancient practice.[8]
The presence of these algebraic structures and the use of a hexadecimal system in the Ifa corpus suggest a level of mathematical sophistication previously unacknowledged in oral societies. This challenges the assumption that such concepts are exclusive to written language systems, demonstrating the potential for advanced mathematical thinking in non-literate cultures.
Geometry, Space, and Time
[edit]Within the Yoruba-Idaàcha dialect, a western branch of the Yoruba language, who practice a more ancient form of Ifa, mathematical concepts are connected with cosmological beliefs and spatial reasoning. The concept of "four" holds a sacred status, derived from the four corners of the world ("orita"), which represent the intersection of Earth and Sky forces.[9]
Circle and Center
[edit]This notion of "four" extends to the geometric understanding of a circle. The center of the circle, called "eerin" (meaning "four"), is conceived as the point where the four corners of a rectangle converge. The circle itself ("agbo") is seen as the dissolution of these corners into a unified center. This conceptualization reflects the Yoruba worldview that emphasizes the interconnectedness and interdependence of elements within a cyclical system.[9]
Time Zones and the Eightfold Division
[edit]The Yoruba-Idaàcha concept of time is also structured around the number eight. The day and night are each divided into four time zones, resulting in an eightfold division of the day. This division, termed "iwon ri iwon ogun mejo" (meaning "measurement has seen measurement eight times"), echoes the practices of other ancient civilizations like the Egyptians and the Babylonians, who recognized the significance of eight in their astronomical observations.[9]
Architecture and Spatial Organization
[edit]Yoruba architectural design reflects this understanding of geometry and cosmology. Circular layouts ("agbo") were often used in family compounds, with the chief's house at the center and the wives' houses forming a perimeter. This symbolized the chief's authority and the family's unity. Some villages also adopted circular layouts, promoting community cohesion. Rectangular structures ("ogba"), meanwhile, were favored for their practicality in house construction and urban planning. They could be easily expanded and adapted to changing needs, and rectangular grids were sometimes used for street layouts in larger towns. Often, Yoruba architecture blended circular and rectangular elements, creating both functional and symbolically rich spaces.[9]
Sacred Numbers and Symbolism
[edit]Beyond the practical applications of mathematics, the Yoruba people hold certain numbers to be sacred, imbuing them with symbolic and spiritual significance. The number four is particularly important, reflected in the four-day week, the four cardinal directions, and the four major deities.[1]
Symbolic Significance
[edit]The powers of four, namely 16 (42) and 256 (44), are central to the Ifa divination system. Other significant numbers include 40, 200, 41, and 201, often associated with specific rituals, offerings, and symbolic objects.[9]
The prevalence of these sacred numbers and their integration into religious practices, mythology, and even proverbs highlights the holistic nature of Yoruba knowledge systems. Mathematics is not simply a tool for practical calculations but is very connected with cosmology, religion, and cultural expression.
References
[edit]- ^ a b Johnson, Samuel (1921). The History of the Yorubas. London: Routledge and Kegan Paul.
- ^ a b c d e Zaslavsky, Claudia (Autumn 1970). "Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria". The Two-Year College Mathematics Journal. 1 (2): 76-99.
- ^ Armstrong, R. G. (1962). "Yoruba Numerals". Nigerian Institute of Social and Economic Research Conference Proceedings.
- ^ Crowder, Michael (1968). West Africa Under Colonial Rule. Evanston, Ill.: Northwestern University Press.
- ^ Conant, Levi Leonard (1896). The Number Concept. New York: Macmillan.
- ^ a b c d e Segla, Aimé (2016). "Viewing Formal Mathematics from Yoruba Conception of the Sky". Journal of Astronomy in Culture. 1 (1): 9-21.
- ^ Sègla, A. (2004). "The Yoruba Logic of Knowledge and Arithmetic". Preprint.
- ^ Olagunju, Adeyemi Sunday; James, Adewale Adeyemi; Adeyefa, Emmanuel Oluseye; Joseph, Folake Lois (2023). "Algebraic characterization of Ifa main divination codes". Scientific African. 20: e01729. doi:10.1016/j.sciaf.2023.e01729.
- ^ a b c d e Segla, Aimé Dafon (2008). "The Cosmological Vision of the Yoruba-Idaàcha of Benin Republic (West Africa): A Light on Yoruba History and Culture". In Holbrook, J. (ed.). Astrophysics and Space Science Proceedings. Springer. pp. 189-207.