African Mathematics: From Bones to Computers

African Mathematics: From Bones to Computers by Mamokgethi Setati and Abdul Karim Bangura  is a useful introduction to how various African cultures across time have incorporated mathematical concepts and logic that demolish stereotypes of the 'primitive African' one encounters nearly everywhere in Western academic and popular culture. Largely functioning as an overview and review of other scholarship on the subject of math in Africa, we learn about how numeration systems, fractals, geometry, algebra, Combinatorics, the Fourier Transform, logic, mathematical tiling, magic squares, Number Theory, and various other branches of mathematics can be seen in Africa's written and oral cultures, as well as material culture (hair braiding, pottery, murals, urban and village layout, textiles, crafts, arts, architecture, board games, riddles, and counting systems). Indeed, reading this book will challenge all of one's preconceived notions of 'Africa,' and it integrates the Maghreb and Egypt into the story of African mathematics. Indeed, the widespread use of base-2 calculations in African mathematics from prehistoric times to Ancient Egypt and the Yoruba numeral system would seem to point to some perhaps continent-wide common legacies in mathematics.

Indeed, the Lebombo Bone and Ishango Bone seem to be early prehistoric evidence of doubling system in African counting, the use of base-2 calculations, and perhaps even some sort of calendar for lunar purposes or menstruation. As the authors suggest, this is perhaps all rooted in how early hunter-gatherers had to geometricize their labor and subsistence activities, which in turn led to some forms of mathematical thought and observation. Other early evidence of important African contributions to mathematics can be found in Ancient Egypt, where base-2 calculations were prevalent, as well as strides in geometry, algebra, recognition of the Pythagorean Theorem, unit fractions, the creation of a calendar of 365 days, arithmetic, volume equations (such as the Egyptian formula for the volume of a truncated square pyramid, which was correct!), a complex numeration system that changed over time with new scripts (hieroglypic, hieratic, demotic), and accurate counting for censuses, tax collection, and maintaining an army. The authors suggest that ancient Egypt's centralization likely fueled the need for improved mathematics, since the maintenance of a strong state and the completion of monuments such as temples and pyramids obviously required some advanced mathematics. And this was no static systems of thought, it changed with the new of more efficient writing systems for describing numbers, and also incorporated fractals in temple architecture. Although not discussed by the authors in great detail, ancient Egyptian mathematics and science would have also required a centralized system of measurement for everything from measuring temples and volume to recording the Nile (nilometer). Ancient Egypt's written mathematical tradition survives from the Ahmose Papyrus, Rhind Papyrus, and Moscow Papyrus, which show that by at least the Middle Kingdom, ancient Egyptian mathematics must have been one of the most advanced systems in the world, as well as numerous example problems with answers that reveal the depth of mathematical knowledge.

Besides ancient Egypt, the Maghrebi tradition of matematics from the 9th through 19th centuries, largely drawing from the work of the Algerian scholar Djebbar, shows how various North African mathematicians contributed to the field. Indeed, they were part of the Islamic world and the 'Arab' tradition of mathematics, but also reflected an African contribution, since Berber dynasties like the Almohads cultivated an intellectual climate or became scholars themselves. It is this tradition of Arabo-Islamic North African mathematics and astronomy that contributed to mathematical manuscripts in Arabic across Muslim Spain, North Africa, Egypt, and parts of 'sub-Saharan Africa.' The Maghrebi tradition gave the world scholars such as Abu al-Qasim al-Qurash, Al-Hassa, Ibn al-Yasamin (who was black), Ahmad Ibn Muncim, and Ibn al-Banna who advanced algebra, innovated the use of symbolic writing of fractions (the origins of the horizontal bars used in fractions in Europe came from the Maghreb), studied the operations and order of algebra, abstract manipulation of polynomials, writing of equations, magic squares, Number Theory, Combinatorics, enumerations, and astronomy.

These writings represent a vast North African mathematical, scientific, and philosophical written tradition that, part of the broader Islamic world, was nonetheless very much an African creation, with reverberations in nearby Europe, Africa south of the Sahara, and Moorish Spain. In fact, translations of these writers' works into Latin and Greek by often Jewish scholars, or exposure to these innovative mathematicians by European students directly shaped the future course of mathematical knowledge. Indeed, if Ron Eglash is correct, the transmission of African divination systems through geomancy during this era in the Middle Ages, established the foundations for binary and computing centuries later in Europe. Speaking of divination, an exploration of how local Berber and West African influences impacted this Maghrebi mathematical tradition warrants further study, particularly if Eglash's convincing theory about the impact of African binary in divination proves correct. However, the text states that the Maghrebi mathematical treatises and manuals were copied and influential in West Africa, influencing writings in Timbuktu, but the decline of the Maghreb politically, economically, and socially after the 15th century may have impeded further developments. I must locate some studies by Ahmad Kani of precolonial mathematical and scientific writings of West Africans to complete the picture of how Islamic and Arab-African mathematical systems contributed to the study.

We know that Muslim Fulani scholar Muhammad ibn Muhammad al-Fulani al-Kishnawi discovered the formula for calculating the magical constant in magic squares, and that he lived in Cairo for a number of years, where he died after performing the pilgrimage to Mecca. Al-Kishnawi was from Katsina, a Hausa city-state in precolonial Nigeria, and according to Paulus Gerdes, magic squares were common among Fulbe/Fulani groups in West Africa, worn as amulets. It would seem the association of magic squares with actual 'magic' and numerology was quite common in the Islamic world and among relatively non-Islamized West Africans, as well as Muslim kingdoms in the Western and Central Sudan regions of West Africa (mathematics, astronomy, medicine, horticulture, and a variety of influences reflecting local and North African mathematical traditions) were utilized in the last centuries of the precolonial era. Perhaps other instances of mathematics in the Sahel and savanna regions of West Africa merit further discussion, especially in architecture (Eglash discusses fractal elements in Senegal), divination (which Eglash discusses in Bamana sand divination), West African sculpture and art, or the megaliths of the Senegambia. Perhaps, like Nabta Playa in Egypt or similar structures in other parts of Africa, the megaliths serve an astronomical function, similar to how indigenes of the Canary Islands devised lunar calendars and one rooted in the Canopus star, an ancient cosmological system in northwestern Africa.

Besides the veritably ancient traditions and contributions to mathematics from ancient Egypt and the medieval Maghreb, what of the rest of the African continent? Surprisingly absent from this study is the architectural and mathematical traditions of ancient Aksum, Meroe, Nubia, Berber (except for the indigenes of the Canary Islands) and other well-known African societies of Antiquity. But due to the desire for brevity and accessibility, one cannot expect everything in this otherwise excellent book. The reader is introduced to a variety of mathematical concepts that are present in African material culture and languages from across the continent, such as the complicated Yoruba numeral system that was based in 20 and required 'feats of mathematical manipulation' for arithmetic. Tellem textiles from the Middle Ages, for instance, reveal geometric patterns on a plane, or Bakuba raffia cloth, with geometric designs. Tshokwe (or Chokwe) sona (sand drawings) of Angola also have a geometry that can prove the Pythagorean Theorem, according to Gerdes. Kuba raffia cloth and other African decorative arts (baskets, homes, murals, pottery) and textile designs also arrive at the Pythagorean Theorem.

The plethora of Mancala-type count and capture games of the African continent also reveal some understanding of Combinatorics, a sub-field that studies the selection, arrangement, and operation of mathematical elements within finite sets. These games, clearly, as Abginya argues, require intellect and thought, especially if a mancala game with 36 counters has a total of 10 to the 24th power possibilities! The authors also argue for a highly developed mathematical tiling or tessellation tradition in Africa, which can be seen in North Africa and Muslim Spain, as well as how fractals and the Fourier Transform were influenced by Fourier's observations in Egyptian architecture. African recursive scaling, Fibonacci Sequence in ancient Egypt, and vector geometry in rural Mozambican peasant houses also reveal the extent to which mathematics is embedded in African cultures.

The material covered in the text continues to the 20th and 21st century, but because my interests are mainly in the precolonial period, I'll cease my summary here (although one should read the fascinating chapter on the English language's impact on Black South African schoolchildren learning math). This is certainly a fascinating book worth reading, even if at its best it's mostly an overview and excellent source for additional references. I certainly gained a lot, even if there were some concepts beyond my grasp. Regardless of my own limitations, anything that contributes to the study of science, technology, and maths in African history is in my interest, including this excellent introduction.