American Mathematical and Information Sciences
by Dr. Ron Eglash http://www.rpi.edu/~eglash/eglash.htm
Note: This paper was presented at Howard University's AFRICON electronic conference, Jan 1995. This article has been turned into a book, see http://www.cohums.ohio-state.edu/comp/eglash.dir/afbook.htm
The study of African cultural influences in mathematical sciences is challenged not only by the pragmatic difficulties of racist historical elisions, but also on several theoretical fronts. First, any social study of science must overcome the stereotype of science as a culture-free process, and this is exacerbated in the quantitative or exact sciences. Second, static notions of cultural knowledge transmission need to be supplemented, since the constructive, inventive aspects of culture are at least as important as continuity. Third, essentialist portraits of African culture, particularly dualistic claims around holism vs. reductionism, intuition vs. logic, etc. must be challenged by more detailed accounts of the highly heterogeneous makeup of traditional African societies, as well as the diversity engendered by the African diaspora. This essay will briefly discuss these theoretical challenges, outline some of the author's work in traditional African mathematics, and review some possibilities for researching these cultural influences in American mathematical and information sciences.
1) Cultural Studies of Science
Social studies of science have only recently begun to establish the tools needed to describe how science is culturally constructed. Early portraits by Merton and Popper saw science as an essentially culture-free process, and even Kuhn, while establishing the existence of an internal scientific community, did little to allow for any external influence. But changes in the 1960s -- the need for technology transfer to the third world under cold war politics, the civil rights movement, the publication of "Silent Spring," critiques of the "military-industrial establishment," etc. -- inspired new outlooks. Gould's 1984 "Mismeasure of Man" was perhaps the most influential in its portrait of the racist and sexist history of biological determinism. But Gould made a strong distinction between this ideologically influenced "bad science" and what he saw as culture-free "good science."
A similar dichotomy existed in the accounts of African Americans in the history of science. The under-representation of Blacks in science was under discussion in times predating the existence of America (having served as a frequent illustration for early pro-slavery arguments), and continues today as one of the most controversial issues in education. The majority of this discourse has, for reasons to be discussed, constructed an opposition between Black culture and participation in science; particularly in its narratives of successful African American scientists. The frequent separation of African American scientists from Black culture in historical accounts is not due to a lack of opposition to racism; but rather because of the particular oppositional strategies employed.
In Bedini's (1971) meticulous "Life of Benjamin Banneker," for example, we are persistently provided with excerpts from historical descriptions suggesting innate intelligence: Banneker's African grandfather was "a man of bright intelligence" (pg 17), his Black mother was "a woman of uncommon intelligence" (pg 22), and as a youth Banneker had a "latent mechanical skill" and "natural mathematical skill" that awaited the proper stimulus to be "aroused" (pg 42). Bedini suggests that because Banneker saw few people other than his parents, sister, and some relatives, he "became more and more withdrawn into himself, and his senses became all the more alert to the world around him" (pg 42). This implication of the self-birthing of a scientist, with inherited talent, is in direct opposition to the racist accusations that Banneker was incapable of his mathematical achievements, and had stolen them from a white man. There is nothing either inaccurate or unethical about Bedini's emphasis; it is just that the strategy of emphasizing genetics and independent thought results in a de-emphasis of cultural contributions -- including Black culture.
A second reason for the separation of African American scientists from Black culture in historical accounts is due to fears that African American youth are discouraged from science careers because they realize the odds against leaving their socioeconomic position (Beane 1985, Lockheed et al 1985). Hence Carwell's (1977) youth-oriented "Blacks in Science" emphasizes the successful struggle against poverty and racism in the early careers of individual African American scientists. This theme of a break with social position also results in a de-emphasis of cultural continuity. The only mention of a Black cultural connection in a portion of the narrative concerning scientific work is a sentence on M. Rabb's sickle cell anemia studies (which merely states that the disease is most frequent in Blacks).
This conflict is not unique to the connections of science and Black culture. In the study of slavery, for example, past descriptions emphasized the oppressive deprivation and emptiness of slave life. More recent approaches, however, have found a complex "slave culture," in which subtle modes of resistance, inventive coding practices, and cultural continuities to Africa can be revealed (c.f. Sobel 1979). What is unique to science connections is the fact that science itself has often been defined in terms of the transcendence of culture. As previously noted, researchers such as Gould had only portrayed culture as a contamination, a block or blinder to scientific truth, rather than a positive force. But even before Gould, a "strong program" in social studies of science (now typically referred to as Science and Technology Studies or "STS") had begun to emerge (Bloor 1976). Recent work (Latour and Woolgar 1979, Gilbert and Mulkay 1984, Latour 1987, Traweek 1990) has demonstrated that social meanings are infused throughout scientific practice and technological productions, and are as much a part of scientific success as they are contributors to error. Thus the recent theories supporting both Black cultural studies and STS indicate that the connections between these two areas would be potent and beneficial locations for investigation.
2) African cultural studies
In contrast to a static, singular conception of Africanness, many researchers have begun to emphasize the diversity and dynamism with which both traditional Africa and the African diaspora has organized its cultural flows. Paul Gilroy (1993), for example, quotes James Brown on his visit to hear Fela Kuti in Nigeria:
"Some of the ideas my band was getting from that band had come from me in the first place, but that was okay with me. It made the music that much stronger" (pg 199).
Gilroy cites the impact of the Virginia Jubilee Singers on tour in South Africa in 1890, the return of slaves from Brazil to Nigeria, the Rastafari culture in Zimbabwe, and other examples of "mutations produced during its contingent loops and fractal trajectories." Perhaps his most radical move is a claim for diasporic mixing with Jewish culture -- W.E.B. Du Bois passing for a Jew to maintain safety in Eastern Europe, the use of the Exodus theme in M.L. King and Marcus Garvey, and E.W. Blyden's childhood in a Jewish community.
Other writers, such as A. Appiah and V.Y. Mudimbe, have concentrated on exploding the myth of Africa as a natural, self-contained, historically stable social object. Cornel West, bell hooks, Houston Baker, and Hazel Carby have made interventions in the African American cultural debates in similar ways. The most explicit example for my own purposes has been J.E. Phillips' "The African Heritage of White America," in which the survival of African cultural traits (e.g. the banjo, baton twirling, etc.) is completely dislocated from its usual racial boundaries.
3) Research in traditional African mathematics
Studies of traditional African mathematics are relatively scarce previous to Zaslavsky (1973), whose beautifully comprehensive survey opened the field to a wide variety of investigations. The studies of this author began in an investigation of the fractal patterns which can be seen in aerial photos of traditional African settlements. Spectral analysis of digitized photos showed that these did indeed have nonlinear scaling (Eglash and Broadwell 1989). Subsequent review of anthropological literature, along with a year of field work in west and central Africa, has shown that these architectural fractals result from intentional designs, not simply unconscious social dynamics, and that recursive scaling structures can be found in other areas of African material culture (art, religions icons, indigenous engineering, and games). In the design rationales and cultural semantics of many of these geometric features, as well as in quantitative and symbolic systems, there are abstract ideas and formal structures which parallel some of the fundamental aspects of fractal geometry (Eglash 1992).
In addition to (and sometimes accompanying) these formal utilizations of recursion, non-symbolic or "analog" representation techniques are also common in traditional African societies, often in relation to animist religious conceptions. These include time series and frequency domain representations of both physical movement and energy (cf. Zaslavsky 1973 figure 23-2, Badaway 1959). Quilting patterns, for example, provide visual representations of the phase relationships in syncopated music (Thompson 1983). Diagrams representing stability through negative feedback and the inverse relationship between frequency and wavelength also appear in various artifacts (Eglash 1994).
If these studies are correct, then a good case can be made for indigenous versions of both nonlinear mathematics and cybernetics in traditional Africa. It is important to caution that these are rather simple versions in comparison to their highly technologized western counter-parts. There is not, for example, any aspect of African fractals that specify the notion of fractional dimension, and very little symbolic notation in general. Nonetheless, this work suggests that we should do away with the unilineal model of mathematical progress. The history of mathematics is not a ladder in which we climb from primitive counting to advanced recursive functions and frequency transforms. Rather, it is a branching structure in which different cultures may take different paths, and what came only recently for some may have been the first steps for others.
A second caution concerns the essentialist interpretations that might arise from this work. Analog representation was mistakenly assumed to be a less concrete form of communication by many researchers in the 1960s. The counter-culture radicals of the cybernetics community -- Norbert Wiener, Gregory Bateson, Hazel Henderson, Paul Goodman, Kenneth Boulding, Barry Commoner, Margaret Mead, and others -- made the erroneous claim that analog systems were more "real" or "natural," and (according to this romantic cybernetics) therefore ethically superior. In social domains, this converged with Rousseau's legacy of the moral superiority of oral vs. literate cultures. It was not until recently, however, that mathematicians have shown that analog systems are capable of the flexible representation required to perform complex (Turing Machine-equivalent) computations, as demonstrated in both theory and experiment (Wolfram 1984, Touretzky 1986, Rubel 1989, Blum et al 1989). In particular, a new appreciation for analog systems was fundamental to the rise of fractal geometry, nonlinear dynamics, and other branches of chaos theory (Gleick pg 258, see also Dewdney 1985, Pagels 1988). By viewing physical systems as forms of computation, rather than merely inert structures, researchers became opened to the possibility of having infinite variation in deterministic physical dynamics. Analog systems can achieve the same levels of recursive abstraction as digital systems; the two are epistemological equals.
Thus the assumptions that fractals and analog systems are "more natural" are entirely unfounded; these are simply cultural associations serving to symbolize the self-other division. In fact, this presumed naturalism was a major stumbling block for certain mathematical developments in the west, and it is here that an important role of African cultural influence will be discussed in the following section.
Finally, it should be noted that comparisons of African vs. European cultures cannot be reduced to dichotomies of fractal vs. euclidian or analog vs. digital. In part this is because both African and European societies are simply too heterogeneous; there are euclidian aspects of African knowledge systems throughout the continent, and vice-versa for Europe. In addition, these are technologies, not cultural characteristics, and so even in cases where many different African societies are all using fractal structures, the cultural semantics associated with fractals will tend to vary greatly from one society to the next.
4) Africanisms in American mathematical and information sciences
Although it might be possible to simply survey, through a text such as "Black Mathematicians and their Works" (Newell et al 1980), the instances of work by Black mathematicians dealing with scaling, recursion, etc., such an approach would miss the cultural concept I have been attempting to outline in this essay. It is not as though mathematical style is a kind of psychological trait; it is an active invention, and we need to examine how the construction of individuals' identities -- their inspirations, personal semiotics, envisioned futures and imagined pasts -- interact with their work as scientists.
The earliest African American influence in mathematical sciences is the work of Benjamin Banneker, and there are two items in the historical record which indicate possible African influences. The first, which might be due to any African cultural origin, occurs in a mathematical riddle written by Banneker, in which he used doubling to generate an estimate for the method of false position. Although the false position method was common for European society of that era, doubling sequences are more often associated with African than European arithmetic procedures.
The second item is more specific to Banneker's particular cultural heritage. Banneker's grandfather Bannaka, who was brought to the U.S. as a slave, was the most direct route for possible African influences in his life. Although the legend that his grandfather taught him African irrigation techniques is fictitious, Bedini notes that conversations between Bannaka and young Benjamin were certain to have occurred. Research by this author indicates that Bannaka was of Woolof descent, and there is one geometric symbol, common in Woolof society, which appears in a description by Benjamin Banneker. This is in a journal passage which records a dream in which he discovers that, after death, our soul takes the shape of a "quincunx." It is possibly a coincidence, but the quincunx is a religious symbol appearing on floor tiles, prayer mats, iron and leather work, and other material designs in both ancient and contemporary Woolof society (including protective amulets for children). It is not unreasonable to suggest that Bannaka had drawn the quincunx for Benjamin and explained its religious significance to him, and that the experience had been recalled in the dream.
It is important to note that while the doubling sequence draws on the nonlinear tradition discussed earlier, the quincunx is a distinctly euclidian geometric form (I have seen two Woolof fractal designs that use the quincunx, but these were not of any importance to the cultural semantics of the design). Again, this suggests that attempting to essentialize African mathematics and search for its presence will lead us astray.
African slave influences in American science also include contributions in biological knowledge and metalwork; the biological (e.g. botanical) is particularly significant for cybernetics due to its involvement in models of information coding. While the romantic accounts of cultural difference would use botanical expertise to emphasize the "naturalness" of African traditions, this is certainly not the only interpretation. George Washington Carver, for example, declared that not only did God create the Kingdom of Plants and the Kingdom of Animals, but that He also had a "Kingdom of the Synthetic." This spiritual legitimization of the artificial fits well into the African religious traditions of analog representation discussed previously. Animism typically concerns a transfer of energy or information through culturally mediated activities. Rather than the orthodox Judeo-Christian opposition between inert material and abstract symbolics, animism allows for a view of physical dynamics as a representation form.
Non-orthodox Judaism, on the other hand, also includes some animistic ideas in its traditions, and here lies another possible connection. As in many African cultures, the ancient Egyptians used sculptures as animist energy forms, and some of them were capable of mechanical movement through cords and lever systems. These automata provided the substance for the golem legend of Jewish mysticism. Norbert Wiener, the Jewish founder of cybernetics, was quite influenced by this concept of information embedded in physical dynamics (Heims 1984). He made several influences to the golem in his writing, and his religious identity was closely tied to "gashmuit," the non-orthodox spirituality of material substance.
Another case for African influences in analog cybernetics can be seen in the work of E.E. Just, who used music as both a conceptual model for decentralized biological morphogenesis, and as a cultural basis for understanding his African heritage (Manning pg 203, 261). It is important to note this is not simply intergenerational transmission. Just was actively examining the idea of an African cultural tradition; it was as much his future heritage as that of his past which he drew upon in this moment of pan-African identity (see Appiah for an excellent examination of pan-Africanist intellectual history).
Just's scientific work, particularly on information encoded in non-symbolic representation (in part his rebellion against the suggestions that the only intracellular information is that of a "master code" in the cell nucleus), was taken up by Ross G. Henderson, who was an important influence in the General Systems Theory (GST) community (Haraway 1976), which in turn influenced the origins of cybernetics through their studies of aggregate self-organizing phenomena and positive feedback loops.
The lived experience of African-Americans' interactions between these African diasporic innovations and their survival of American racism is particularly apparent in the work of African American women. As Nakano Glenn (1992) argues for the case of service workers, gender and race cannot be reduced to "additive oppressions," and must be seen as the site of an interlocking or relational dynamic. For example, both the traditional work of African women (Hay and Sticher 1984), and specific labor locations for women of all ethnicities in America have contributed to the frequency of their involvement in biomedically related fields. From 1876 to 1969, over half of the Black women science PhDs have been in bio-sciences (Jay 1971), and the first Black American women inventor, Clara Fry, worked in health care tools (James 1989 p. 80). The most relevant example in cybernetics is the work of Patricia Cowings, who makes cyborgs for NASA. In a recent interview (Eglash in Gray 1995), Cowings discussed her use of analog biofeedback as a method for reducing motion sickness in space, and noted several complex interactions between her identity as a Black woman and her successful career in cybernetics. Yet she has distanced herself from the claims for any simple mimesis of "African Culture" in her construction of cybernetics. The contributions of African American women to what has become modern cybernetics should be seen as a form of resistance that cannot be reduced to either the restoration of tradition or a relocation to universalism.
Related to these systems of analog recursion are studies on computational self reference; these too have possible African influences. For example, Seymour Papert, a white computer scientist who championed hierarchical, non-recursive computing in the 1960s, made a dramatic conversion to decentralized computation following his U.N. work in Africa in the mid-70s. Another white engineer, N. Negroponte, developed his conceptions for self-organized computing following his study of "vernacular architecture," most of which was African.
The most recent example is that of Phillip Emeagwali, a Nigerian researcher currently at the graduate school of computer science in the University of Michigan. Emeagwali's work not only concerns parallel processing for nonlinear dynamics, but his research takes a strongly historical approach, utilizing the history of mathematics and its interaction with computer hardware to improve his perspective on the equations. It is, perhaps, this awareness of how cultural traditions -- be they mathematical or social -- are actively constructed that marks the greatest insight that W.E.B. Du Bois' "double vision" can bring into focus.
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Dr. Ron Eglash:
Department of Science and Technology Studies
Rensselaer Polytechnic Institute (RPI)
Troy, NY 12180-3590
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