It All Adds Up

“Indigenous science.” To many the phrase seems a contradiction in terms. To a large degree the Western academy has accepted that other cultures have sophisticated systems of philosophy and religion, but science is still widely viewed as something unique to the West. A growing body of scholars has begun to question this view and is claiming that science is a multifaceted phenomenon manifesting in many cultures.

On one level we are becoming used to such claims. Acupuncture, Chinese herbalism and ayurvedic medicine have all gained huge followings in the West. Major pharmaceutical companies are also interested in “indigenous medicines,” investing millions in bioprospecting programs.

Other societies, most notably the Maya, developed their own very different systems of astronomy. Where Western astronomy is based on looking up, Maya astronomy was focused on the horizon. Where Western astronomy is centered first on the Earth, then the sun, to Maya astronomers, the most important celestial body was Venus.

Anthropologists and others have begun to examine a range of non-Western knowledge systems with a view to investigating their empirical and predictive power. In her book “Naked Science,” UC Berkeley anthropologist Laura Nader argues that in rigidly “demarcating science from other systems of knowledge,” we not only misrepresent science, we also do ourselves a disservice by failing to benefit from centuries of observation and insight by other cultures.

In “Science and an African Logic,” philosopher of science Helen Verran takes the debate surrounding “indigenous science” much further. A leader in the emerging field of what is also known as “ethnoscience”--though she herself eschews both terms--Verran focuses on the way the Yoruba people of Nigeria relate to numbers, a relationship that she suggests is quite different from Western patterns.

Between 1979 and 1986, Verran was a lecturer at Obafemi Awolowo University in Ife-Ife, Nigeria, where she taught mathematics and science teachers. She found that her African pupils would often approach mathematical problems in what, to her Western-trained eyes, seemed highly unorthodox ways. At first alarmed, then fascinated, Verran began to investigate how Yoruba people interacted with numbers in their daily lives.

It turns out that the Yoruba language has a particularly flexible way of representing numbers. Although in English and other Western languages, any given number has a unique verbal representation, in Yoruba there will be many ways of speaking a large number-- for the number 19,669, for example, Verran lists no fewer than seven distinct ways. Each variation is a verbal encoding of a different arithmetic pattern:

oke kan o din erinwo o le okaan dinlaaadorin

((20,000 x 1) - 400) + (-1 -10 + (20 x 4)) = 19,669

eedegbaawaa o le eedegberin o din okanlelogbon

(20,000 - 1,000) + (-100 + (200 x 4)) - (1 + 30) = 19,669

This flexibility comes about because, unlike our numerical system, which is based on the number 10, the Yoruba language recognizes three bases--10, 5 and 20. By playing around with different combinations of these bases, one can produce different spoken versions of the same numerical quantity. Being a good enumerator in Yoruba takes long years of training and requires an ability to quickly break down numbers into their components.

Yoruba enumerators must not only be adept at mental factoring, they must also in a sense be poets, for Yoruba people consider some verbalizations of numbers more elegant and aesthetically pleasing than others. I have no idea how the words are meant to sound, but from written examples, one cannot but be struck by the patterns and clearly musical rhythms embedded in Yoruba number names. Even on paper they look like songs.

More radically, Verran says, number words in Yoruba do not act as nouns or adjectives but as adverbs. Numbers work not as attributes of objects (as in: there are four oranges) but as modifiers of verbs. Yoruba language conceives of number not as a static abstract quality but as an active characteristic which changes the mode of what it enumerates.

In English, for example, we would say “there are four stones.” The Yoruba translation would be something like “the matter with characteristics of stoneness is manifesting here/now as a collection divided to the extent of four.”

Excited by such findings, Verran originally wrote some academic papers that presented the formalities of this numbering system. But by abstracting it, by treating this system as if it were an object which could be put under a glass and studied in isolation away from its everyday context within Yoruba life, Verran came to understand that she was distorting the phenomena.

She realized that in a deeper sense she was guilty of the same kind of colonialism and philosophical imperialism she had been so trying to avoid. It was a similar notion to that of zoologists who now understand that to comprehend animal behavior, creatures must be studied in the wild; zoos are a very poor substitute.

After this realization--for her it was a philosophical crisis--Verran embarked on a decade-long quest for a deeper understanding that would do justice to Yoruba engagement with number in the context of their lives. Crucial here was the insight that traditionally the Yoruba had no written form for their numbers; their engagement with number was never abstract but always verbal and always within the context of practical activities like trading.

Verran argues that Yoruba understanding of the existence of number is grounded in the body, those 10s, 5s and 20s reflect the human arrangement of fingers and toes.

The importance of embodiment in our own relationship to number has also been proposed by English mathematician and philosopher Brian Rotman in his “Ad Infinitum: Taking God out of Mathematics and Putting the Body Back In.” Both these proposals are radical departures from mainstream Western thinking about mathematics, which sees numbers as transcendent entities existing independently from the physical world.

The origin of number and the mysterious ability of humans to manipulate numbers (that is, to do mathematics) has been a hot research topic during the last few years. “Science and an African Logic” makes a major contribution to this debate and will no doubt keep philosophers arguing for years to come. Verran’s chapters on Yoruba life and classrooms are a delight, and anyone interested in the subject of mathematics will have much to gain from this unusual and, in the end, deeply personal book.

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Margaret Wertheim is the author of “The Pearly Gates of Cyberspace: A History of Space from Dante to the Internet.”