Mathematics, Counting and Measurement

The Yoruba numeral system is the strongest claim Yoruba culture has to mathematical distinctiveness, and it needs no help from any of the claims usually made on its behalf. It is a base-20 system that constructs numbers by subtraction to a degree matched almost nowhere else, it derives from counting cowries rather than from finger counting alone, and it has a structural feature that specialists find genuinely remarkable: the same number can frequently be expressed in many different well-formed ways, which suggests a conception of number as combinatorial array rather than as position on a line.

That is a real and specific finding about a real and specific system. It is a great deal more interesting than the popular claims about Ifá and binary, which this corpus has already examined and largely rejected in 05-ifa/07. This file cross-references that treatment and does not repeat the overclaims.

The system

Base twenty. The base is ogún, twenty, or score 1. Twenty has two lexical forms with distinct grammatical roles: okòó is used in initial word position when twenty is being added to or subtracted from, and ogún is used with the multiplication formatives in numerical derivation 1.

One to ten are simple lexemes: ení or ọ̀kan (1), èjì (2), ẹ̀ta (3), ẹ̀rin (4), àrún (5), ẹ̀fà (6), èje (7), ẹ̀jọ (8), ẹ̀sán (9), ẹ̀wá (10) 1.

Eleven to fourteen are additive. Ọ̀kanlá (11), èjìlá (12), ẹ̀tàlá (13), ẹ̀rìnlá (14), formed as ten plus one through ten plus four 1.

Fifteen to nineteen are subtractive, and this is where the system departs from most others. Ẹ̀ẹ́dógún (15) is twenty minus five, ẹ̀rìndínlógún (16) is twenty minus four, ẹ̀tàdínlógún (17) is twenty minus three, èjìdínlógún (18) is twenty minus two, ọ̀kàndínlógún (19) is twenty minus one 1. The element dín, to decrease or be less by, is the subtraction operator and it is visible in the words.

Note ẹ̀rìndínlógún, sixteen, which is also the name of the sixteen-cowrie divination system and the number of principal odù. Sixteen in Yoruba is literally "four less than twenty".

The decades. Ogún (20), ọgbọ̀n (30), ogójì (40, two twenties), àádọ́ta (50), ọgọ́ta (60, three twenties), àádọ́rin (70), ọgọ́rin (80, four twenties), àádọ́rùn-ún (90), ọgọ́rùn-ún (100, five twenties) 1.

The pattern is worth seeing clearly. The even decades are multiples of twenty: 40 is two twenties, 60 is three twenties, 80 is four twenties, 100 is five twenties. The odd decades are formed by subtracting ten from the next even decade: 50 is (20 × 3) − 10, 70 is (20 × 4) − 10, 90 is (20 × 5) − 10 12. So a Yoruba speaker reaches fifty by going to sixty and coming back.

Higher numbers. Igba (200, twenty tens), irinwó (400), ẹgbẹ̀wá or ẹgbàá (2,000, two hundred tens), ẹgbàawàá or ọ̀kẹ́ kan (20,000, literally one bag) 1. That last is the tell: twenty thousand is one bag, because twenty thousand cowries was a bag of cowries.

Worked examples of the compounding. These are the ones that make the structure visible 2:

• 45 = (20 × 3) − 10 − 5
• 50 = (20 × 3) − 10
• 108 = (20 × 6) − 10 − 2
• 300 = 20 × (20 − 5)
• 318 = 400 − (20 × 4) − 2
• 525 = (200 × 3) − (20 × 4) + 5

Three hundred is twenty times fifteen, and fifteen is itself twenty minus five, so 300 is 20 × (20 − 5). That is nested subtraction inside multiplication, expressed in a single ordinary word that Yoruba speakers use without effort.

Why the cowries matter

The system is based on counting cowrie shells, which served as currency in the region for centuries 3. This matters for three reasons and it is the key to the whole structure.

It explains the base. Twenty as a base is usually attributed to finger-and-toe counting, and that may be part of it, but the Yoruba higher units are cowrie-trade units. Twenty thousand being called a bag is the decisive evidence: the number words are counting-house words.

It explains the object forms. Up to thirty, Yoruba has distinct numeral forms specifically for counting objects, which derive from counting cowries 1. Oókàn, one, is a contraction of owó ọ̀kan, one cowrie 1. The counting vocabulary carries the word for money inside it.

It explains the subtraction. If you are counting physical objects into heaps, it is quicker to make a heap of twenty and remove three than to count seventeen individually. Subtractive naming is what a physical counting practice with standard heap sizes produces. The system is a description of a procedure with the hands, not an abstract notation invented at a desk.

Cowries were also laid out in arrays and the arrangement has been described as functioning like an abacus 3. Mann's 1887 report to the Journal of the Anthropological Institute describes the activities of what he called the magician-calculator, the specialist who performed rapid computation with cowries 34. This is the earliest substantial European description of the system.

The genuinely unusual feature

Two properties make the system remarkable to specialists, and they should be stated in the terms the specialists use.

Unusually heavy subtraction. Subtractive number formation exists in many languages, Roman numerals and Latin undeviginti among them, but it is normally marginal. In Yoruba it is structural and pervasive: it governs 15 to 19, all the odd decades, and a large share of compound numbers. Yoruba numbers involve more than a typical amount of subtractive formation, an operation described as back-counting 3.

Combinatorial flexibility. This is the finding that goes furthest beyond curiosity. Yoruba numbers are curiously flexible in how they may be formed, permitting a given number to be expressed in an unusual number of alternative but well-formed ways; 19,669 can be expressed in eleven different ways 3. Overmann's interpretation is that this suggests Yoruba numbers are conceived as arrays of combinatorial possibilities rather than as points along a linear continuum in the way the Western number line conceives them, and she connects this to the visual arrangement of cowries in arrays 3.

That is a claim about mathematical cognition, not just about vocabulary, and it is the most interesting thing anyone has said about Yoruba numeracy. It should be flagged as an interpretation rather than a measurement: Overmann is inferring a conceptual structure from a linguistic one, which is a reasonable inference and not a demonstration. Confidence: medium.

Helen Verran's related work approaches Yoruba number from a different angle, examining how Yoruba numeric practices work in use rather than in the abstract, and it is the other substantial scholarly treatment 5. Verran's concern is with what happens when Yoruba and English numeric practices meet in classrooms, and her argument bears on whether the two systems are doing quite the same thing.

Is it hard? The mathematics literature notes the system is complicated for intermediate numbers, and raises the fair question whether the complexity carries any comparative merit, given that complicated arithmetic involves a heavy load of recall 2. That is an honest observation and it belongs here. A system that expresses 45 as three-twenties-less-ten-less-five is not obviously better for calculation than one that says forty-five, and the modern practical response has been decisive: educated Yoruba speakers today frequently use English numerals for arithmetic, prices and dates while retaining Yoruba numerals in ordinary speech and in traditional contexts. The system is under real pressure from that competition.

Counting and measurement practice

Counting groups. The four-day week and the recurrence of four, eight and sixteen in Yoruba organisation are treated in 09-social/09-time-and-the-calendar, which sets out the four-day ọ̀sẹ̀, the market cycle it governs and the cosmological rationale given for it. That file is the reference for Yoruba time reckoning and this one does not duplicate it. What belongs here is the observation that the counting system's base is twenty while the temporal and divinatory organisation runs on four and its doublings. These are two different modular structures in the same culture, and they do not derive from each other.

Length. Measurement of length used body-based units, the arm span, the forearm, the hand span and the pace, as in most pre-metric societies. This compiler located no standardised Yoruba length unit with a documented value, and does not supply one.

Volume and quantity. Trade measurement was by counted units and by standard containers: the calabash, the basket, the heap. Market practice measures by the heap rather than by weight for a great many commodities and still does, which is a system of standardised customary quantities rather than of metrology. 09-social/06-economy-and-the-market covers market practice.

Weight. This compiler found no evidence of a Yoruba balance-and-weight system comparable to the Akan gold-weight system, where brass weights of standardised mass were used with a balance to weigh gold dust. The absence is worth stating explicitly, because the Akan system is sometimes loosely attributed to West Africa generally. Yoruba trade ran on counted cowries, and counting is a different technology from weighing.

Value. The cowrie was the unit of account and the numeral system is built on it, as above. Large sums were reckoned in bags, ọ̀kẹ́, of twenty thousand.

The mathematics in Ifá, handled honestly

This corpus has already treated this at length in 05-ifa/07-ifa-as-information-system, which separates what is established from what is popularly claimed. The essentials, cross-referenced rather than restated:

What is true. Ifá is a binary addressing system over a very large memorised corpus. Each odù is eight binary positions, giving 2⁸ = 256 addresses, and the corpus of verses is retrieved through that address space. That is an accurate description and it is a serious information-management achievement in a society without writing.

What is false, and the corpus says so. The claim that Leibniz derived binary arithmetic from Ifá is refuted by chronology: his De Progressione Dyadica is dated in his own hand to 15 March 1679, and there is no documented contact between Leibniz and Ifá at any date. The claim that Eglash demonstrated Ifá to be fractal and recursive misattributes a finding: Eglash's recursion result concerns Bamana sand divination, in which later figures are generated from earlier ones by addition modulo two, and Ifá casting has no such step, since all eight positions are generated by independent physical events.

Why this file does not relitigate it. The reader who wants the full argument, the dates, the sources and the assessment of the transmission debate should go to 05-ifa/07. What matters for a section on Yoruba science is the methodological point: the genuine achievement is a 256-address randomised retrieval system over a corpus of a hundred thousand-plus verses held in memory with quality control, and it needs no borrowed credit from the history of European computing. The numeral system described above is the better example of Yoruba mathematical distinctiveness precisely because nothing about it has been exaggerated.

What can be claimed

The Yoruba numeral system is vigesimal, derived from cowrie counting, unusually and structurally subtractive, and unusually flexible in permitting multiple well-formed expressions of the same value 123. It was documented in the European literature from Mann's 1887 report onward 4 and is treated in the standard survey of African mathematics 6. It is genuinely unusual among the world's numeral systems on the subtraction and flexibility measures, and this is the assessment of specialists in numeral systems rather than an enthusiast's claim.

Yoruba measurement was customary rather than metrological, and no standardised weight system comparable to the Akan one is documented. Saying so costs nothing and protects everything else in the file.