Bamana Sand Divination: Recursion in Ethnomathematics
Author(s): Ron Eglash
Source: American Anthropologist, New Series, Vol. 99, No. 1 (Mar., 1997), pp. 112-122
Published by: Blackwell Publishing on behalf of the American Anthropological Association
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RONEGLASH/
0 HIO S TATE U NIVERS ITY
Bamana Sand
Recursion
in
Diuination
Ethnomathematics
NEWdiscipline of ethnomathematTHERELATIVELY
ics has been motivatedby its straightforwardepistemological opposition to primitivismas well as its potential
application to education and development (here entering into the more general "indigenous knowledge"
framework). But because of these potent and direct
applications, there has been less emphasis on theoretical development. Inparticular,there is little application
of the move toward reflexive anthropology in ethnomathematics.This is not surprising,given that reflexive anthropology attempts to turn the Western gaze
back on itself, disruptingrealist claims throughits portrait of ethnographic representation as a social construction. Whatpossible point could there be to making
the reflexive move in ethnomathematics?It is already
hard enough to get anyone to believe accounts of topological theory woven in palm fronds; so why bother
disrupting it? And since mathematics is never subjective, but ratherthe unvaryingresult of pure logic, what
could possibly be aconstructed"about it? Nevertheless,
there are indeed good reasons for using reflexivity in
ethnomathematics.In particular,there is reflexivity already present in many mathematical systems in the
form of recursion. Through the example of Bamana
sand divination, this essay will attempt to show how
reflexive cultural analysis and recursive mathematics
can be brought together.
Theoretical Background
Ethnomathematicsis primarilythe child of anonWestern mathematicsband "mathematicalanthropology." Non-Western mathematics (e.g., Raum 1938)
traces its genealogy to the same reports of traders and
missionaries that provided the origins of cultural anthropology, but rather than changing into an analytic
Studiesin the
is a lecturerin the Divisionof Comparative
RONEGLASH
Columbus,OH43210.
OhioStateUniversity,
Humanities,
methodology, it maintainedits descriptive emphasis in
the transition to scholarly rigor. Its discursive trope is
typically one of translation, with each example framed
as a non-Western "version"of Western mathematics
(such as the use of base-five counting systems). Mathematical anthropology,generally defined as mathematical modeling of social and material culture, could be
said to begin with the early classificatory systems for
kinship(e.g., Morgan1871).The anthropologicalsignificance of this approachwas soon opposed, however, by
functionalists such as Bronistaw Malinowski, who insisted that kinship is the result of 'sahost of personal
intimateinterests"and could not be zreducedto formulae" (1930:19).
Malinowskiwas quite willing to grantthe epistemological status of science for certain types of indigenous
knowledge (e.g., Trobriandoutrigger technology) but
carefully separated these analytic narratives from the
real stuff of anthropological inquiry, warning that
"thereis no more fallacious guide to knowledge than
language" (1925:78). At first one might think that
structuralismwould provide direct opposition to this
notion, but here too there was a curious reticence to
consider indigenous mathematical knowledge. LeviStrauss,for example, used indigenous botanical classificatory systems as illustrations of the epistemological
equivalence between West and non-West,but reserved
the more complex algebraicanalysis of kinshipsystems
as an anthropologicalunderstanding.Laterrefinements
of mathematical anthropology (such as Kay 1971) expanded this analytic modeling to a variety of social
phenomenaand to very complex mathematicalsystems
but maintainedits location in the mind of the anthropologist. Aside from anthropologicaltradition,the reasons for this distancing may also be related to the
Platonic realism of the mathematics subculture. For
mathematiciansin the Euro-Americantradition,truthis
embeddedin an abstract realm, and these transcendental objects are inaccessible outside of a particularsymbolic analysis.
Association.
AmericanAnthropologist99(1):112-122. Copyright 1997, AmericanAnthropological
BAMANASAND DIVINATION/
Thusthe two parent disciplines of ethnomathematics showed a sort of symmetrical lack. Non-Western
mathematicsdid not (with the exception of the empires
of ancient Chinese and Hindu civilizations) have the
abstractcomplexity of mathematicalanthropology,and
mathematical anthropology could not credit its subjects of inquirywith its own complex discoveries. The
two were brought together through the device of cultural relativism.
In Claudia Zaslavsky's seminal Afrtca Counts (a
title that conveys the oppositional stance of ethnomathematics), a somewhat vague collection of patterns art, games, economics, and so forth took on a
new epistemological status. They were not framedas a
'sfirststep"in a universalhistorical progression,norwas
the complexity revealed by analysis implied to be the
sole property of the mathematician-anthropologist.
Thereremaineda question of balance between the complexity of the analysis and the attributionof intentionality on the part of the Africans, but this was a useful
tension that opened the possibilities for indigenous
mathsystems withoutdissolvingthem into subjectivity.l
Similarideas have been developed independently
by several African scientists and mathematicians. In
Senegal, for example, physicist Christian Sina Diatta
has lectured on the use of Jola concepts in mathematical modeling,and SakirXiamhas explored mathematics
pedagogy in Wolof (see also Njock 1979). Workin various areas of the world has expanded this synthesis,
providingcomplex mathematicalanalyses for a variety
of indigenous patterns and abstractions while pushing
the location of mathematicalthought toward the local
culture.2
At the same time, however, researchers in science
and technology studies (STS) have been looking in the
directionof Westernmathematicsas a possible location
of cultural thought. STS has been quite successful in
demonstratingthe culturalinfluences in a varietyof the
"soft"sciences (see, for example, Gould's [1981]history
of racism and sexism in biology), but the task has been
increasingly difficult as we look toward the "hard"science end (thus mathematics signified the extreme in
difficulty). This problem was somewhat mitigated by
the move from an analysis of cultural influence in science to portraits of cultural construction (the "strong
programme"of Bloor [1976]). Culturalconstruction no
longer maintainedan inner core of science as a neutral
or value-freeinstitution whose outer edges were biased
by social influence. Rather,it held that both failureand
success in science were the result of social constructions of knowledge, and that logical certaintycould still
be multiple(as Bloor [1976]showed in his discussion of
mappingthe historical alternatives for definitions of a
polyhedron).In additionto these new social analyses of
mathematics (see Restivo et al. 1993), the mathemati-
RON EGLASH
113
cians themselves have recently been an active force in
consideringthe social aspects of their subculture,probably because the widespread assumption that mathematics is culture-freeallows them to relax a guardthat
must be defended in other sciences.3
Thus the anthropology of mathematics which,
ideally, would not be using the ethno- prefix to designate non-Westernsocieties (especially given the undertheorized status of white ethnicity) Emdsitself split
between its non-Westernsubject in ethnomathematics
and its Western subject in STS. Ethnomathematics
looks at a society previously framed as distant from
science, and shows that this culture does indeed have
mathematical content. STS analyzes a mathematical
practice previously framedas culture-freeandshows its
basis in social process and cultural meaning. To fully
btidge this gap is beyond the scope of this essay, but the
following example may be helpful in thinking about
how a recursive exchange between the two approaches
might be beneficial to a broader understandingof the
relationshipbetween culture and mathematics.
AfricanEpistemologyand Divination
Comparisons of Westerntechnoscience and traditional knowledge systems are nothing new in the discourse of Africanepistemology.AnthonyAppiah(1992)
provides an extensive discussion of this intersection,
starting with ethnophilosophy.His analysis weaves between the positions of Kwasi Wiredu(1979), who critiques the focus on comparison to Western science
rather than religion (noting that it leaves the superstitions and folk philosophies of the West unexamined),
and Paulin Hountondji (1983), who argues against any
mimetic comparison, suggesting that ethnophilosophy
and its allies are dressing European motivations in
autochthonous garb. Like V. Y. Mudimbe's(1988) Foucaultian discourse analysis and Paul Gilroy's (1993)
fractal history, Appiah'sdialectical contour maps African epistemology as a historical process ratherthan an
object of strictly pre- or post-Westernpresence.
Divination enters Appiah's analysis through Edward E. Evans-Pritchard'sclassic Azande study (1937),
showing that the supposedly self-limitingsystem of explanation for failures in Azande magic are quite similar
to the theory-laden observation and resistance to new
paradigmsdescribed in STS.Appiahdoes not, however
allow either technoscience or the Azande knowledge
system to be reduced to a closed feedback loop, citing
BarryHallen's (1977) evidence for satisfaction of Karl
Popper's(1962) "criticalreflection"critieriain the work
of a Yoruba diviner. A similar rejection of the uclosed
world"portrait underlies the recent collection on African divinationstudies edited by Philip Peek, who notes
114
* VOL. 99, NO. 1 * MARCH1997
AMERICANANTHROPOLOGIST
Figure 1
Baoule door carving.Vogel (1977) recordsthis carvingas an iconic
representationof social forces in balance;thus it can be viewed as
analogous in both structureand applicationto the Western negative
feedback loop diagram. Photo courtesy IFAN,Dakar.
that Evans-Pritchardhimself was interested in possibilities for a more reflexive comparison of knowledge
systems (Peek 1991:7-8).Inthe case of ethnomathematics, the issue is not the applicabilityof negative feedback as a model for the traditionalknowledge system,
but rather the ways that traditional knowledge might
create feedback models (Figure 1).
Bamana Divination Pedagogy
My study took place in Dakar,Senegal, where the
local Islamic culture credits the Bamana (also known
as Bambara) with a potent pagan mysticism. As in
many other areas of Africa, the clash between Islamic
economic hegemony and animist spiritual authority is
a complex dynamic (see Masquelier 1993). There is a
more subdued syncretism with Islam within Bamana
culture itself, which organized the states of Segou and
Kaartaunder animist rule from 1712to 1862, and even
after political defeat maintained strong resistance to
Islam in many areas of the Mande diaspora.4
The strategy of "otheringXranges from repression
(Fanon 1963) to resistance (Taussig 1993);here it was
an important part of the professional identity of the
diviner. Rudolph Blier (1991) and Elizabeth Colson
(1966) suggest that an alien status allows diviners to
be seen as more impartial. Individualsfrom the Wolof
ethnic majorstyseemed to frame the outsider status
as indicating powers from outside the norm (and
hence outside the natural). Almost all diviners had
some kind of physical deformity "the price paid for
their power"- and these were displayed rather than
hidden.5 Dress and mannerisms also served to distance them from the mainstream Wolof culture. (One
woman had hands and feet dyed with indigo.) They
were quick to show me their Malianpassports, which
were presented as official proof of their Bamana identity.
At the site of the study there were six diviners,
usually with no more than four present at any one
time. All were located at the edge of the very urban
MarcheSandaga,on a quiet street between a dumpfor
construction materials and some shipping companies. The construction material was put to good use
by the sand diviners. Both men and women used
cowrie-shell divination, which concentrated on
iconic patterns discerned in the tossed shells, allowing complex narratives to build up around a putative
future. I was a bit of a disappointment to the cowrieshell diviners, who found my mathematical questions
to be a distraction from their efforts to entice me to
pay them for extra services to guarantee good fortune. One pointed to his deformed foot as an indication of the potential dangers.
The sand diviners were somewhat more flexible,
particularly when using the palm liqueur and marijuana, which improve their occult vision. Marijuana
is illegal in Senegal, but the police officers' fear of
being cursed allows diviners to smoke with impunity.
They were also much more interactive than the
cowrie-shell diviners, often recording and discussing
the results of their work on scraps of paper.6One sand
diviner, who was always accompanied by a friend in
urban dress, seemed quite willing to teach me his
system, suggesting that it "wouldbe just like school."
James Clifford (1988) mentions the relationship
of student-teacher as one of the many possible
choices for ethnographic interactions. It did not necessarily seem optimal to me, but it was indeed the
relation of choice for the diviner (perhaps because it
helped cover my status as an economic resource).
The friend in urban dress did not do divination himself, but he was introduced to me as "a professor of
his people" and held authority over the entire group.
The first few sessions went smoothly, with the diviner
showing me a symbolic code in which each sign,
represented by a set of four vertical dashed lines
drawn in the sand, stood for some archetypical concept (such as traveling, desire, or health) with which
they assembled narratives about the future. But when
I finally asked how they derived the symbols in particular the meaning of some patterns drawn prior to
the symbol writing-they just laughed at me and
shook their heads. UThat'sthe secret!" My offers of
increasingly high payments were met with disinterest. Finally, I tried to explain the social significance
of cross-cultural mathematics. I happened to have a
copy of LindaGarcia'sFractal Explorer (1991) on me
and began by showing a graph of the Cantor set,
explaining its recursive construction. The head diviner suddenly stopped me, snapped the book shut,
and said, "Showhim what he wants!"
BAMANA
SANDDIVINATION
/ RONEGLASH 115
Recursionand RosicrucianJews
For the mathematics subculture of 1877, Georg
Cantor'stransfiniteset theory was as much an intrusion
of the supernaturalas any Bamanapresence is in Dakar.
Infinityhad been banished by Aristotle on the basis that
it was self-annihilating" (infinity + infinity = infinity)
and therefore could only be a potential. Although the
introduction of calculus in the second half of the 17th
century broughtattention to the concept of the "inElnitesimal"(revived from its Greekbanishmentin 1615by
Johannes Kepler's Stereometr?a) and to the convergence to a limit as infinity is approached, infinity as a
unitary mathematical object was strictly forbidden.
Cantor'srigorousfoundation for an infinite set, classes
of infinity, and their relation to the real-numbercontinuum made possible the impossible, destroyingthe Aristotelian distinction as a difference between legitimate
and illegitimate mathematics (Maor 1987).
The Cantorset (Figure 2) was his visualization of
transfinite numbertheory. It shows the interval of zero
to one on the real number line and indicates that the
number of points are not denumerable, that is, the
number is greater than infinity. The set (which has a
positive measure but zero dimension) was a prototype
for other recursive set constructions; in the late 20th
century these would become the basis for the computational modeling of natural self-organizing systems in
Benoit Mandelbrot'sfractal geometry (1977). But at the
time, applied mathematicswas far from Cantor'smind.
l
-
-
n
*
|l
l
*
ll ll ll ll
ll ll ll ll
118111
IIR111
0111111
11111111
IIR
1 111 IIR
_
_
There is no more vicious academic hatred than that of one
Jew for another when they disagree on purely scientific
matters. Whentwo intellectual Jews fall out they disagree
all over, throw reserve to the dogs, and do everything in
their power to cut one another'sthroat or stab one another
in the back. [Bell 1937:562-563]
In a scholarly masterpiece on Cantor, biographer
Joseph Dauben flatly declares that since Cantor's
mother was Roman Catholic, "in fact, Cantor was not
Jewish"(1979:i).Nazi scholars solved their own worries
by spreading a story that Cantorwas found abandoned
on a ship bound for St. Petersburg (Grattan-Guinness
1971:352).
Actually Cantor's Jewish identity was quite complex. His family had indeed converted to Christianity,
but he was well aware of his heritage.He referredto his
grandmother as uthe Israelite" and wrote a religious
tract attemptingto show that there was no Virginbirth
and that the real father of Jesus Christ was Joseph of
Arimathea.Cantor eventuallyjoined the Rosicrucians,
whose mystical-scientific approach to a supposed
Egyptian origin for all religions probably appealed not
only to his intellectual interests but also to his syncretic
ethnicity. Cantor chose a Hebrew letter as his new
symbol. The aleph, the beginning of the alphabet, was
used to represent the beginningof the nondenumerable
sets. While his biographers arguedJew or not-Jew, off
or on, zero or one, Cantor himself proved that the
continuumfrom zero to one cannot be delimitedby any
subdivision process, no matter how long its arguments.
Recursionand BamanaSandDivination
*
*
*
"
| |
| I
| |
| |
11111111
light of his remarks on Cantor'sarchrival,the Jewish
mathematicianLeopold Kroneker:
As Figure 3 indicates it is not surprisingthat the
diviners reacted so strongly to the Cantorset. The divination begins with four horizontal dashed lines, drawn
11111111
veryrapidly, so thatthere is some randomvariation in
the number of dashes in each. The dashes are then
011111
11118 110
01 11111S connected in pairs, such that each of the four lines are
left with either one single dash (in the case of an odd
11N
Rl 11"11111181N
211111 number) or no dashes (all pairs, in the case of an even
Figure 2
The Cantorset. FromMandellbrot1977.
His real fascination was in theologis balimplications:the
increasing classes of infinity he diC,covered seemed to
point toward a religious transcende Zntal.Cantors biographers differ greatly on the cultura]1signiElcanceof this
point. Eric T. Bell felt that Cantor's t Jewishethnicorigin
ruled his life and made several rema]rks aboutthe inheritance of personality traits particl ularly disturbing in
number).The narrativesymbol is then constructed as a
column of four vertical marks,with doublevertical lines
representing even dashes and single representing odd.
At this point the system is very similarto the famous Ifa
divination:there are two possible marks in four positions and so 16 possible symbols. Unlike the process in
Ifa divination,however, the randomsymbol production
is repeated four times rather than two. The difference
is quite significant. Each of the Ifa symbol pairs is
interpreted as one of 256 possible Odu, or verses. The
Ifa diviner must memorizethe Odu;hence four symbols
* VOL. 99, NO. 1 * MARCH1997
AMERICANANTHROPOLOGIST
116
1) Four sets of random dashes are drawn:
_
_
_
2) Each of the dashes are paired, and the odd/even results recorded:
l l
l
l l
3) The process is repeated four times, resulting in four symbols. Each row of
the first two symbols and the last two symbols are paired off to generate
two new symbols
l
+
l
l
+
Q
+
l l
l
l l
l
l
l
>
l
l
l
l
l
l
l
+
+
+
l l
l l
4) The two newly generated symbols, now placed below the original four, are again
paired off to generate a seventh symbol. Then the four are read sideways to create
four more symbols.
I
l l
1|ll
l
l l
l l
l
l
l
l l
l l
l
l l
\
l l
l
A I II II I
JJ
=,',4<
..
..
l
l l
.
5) The four new symbols are used to generate another three, which are placed underneath them,
creating a second set of seven.
l l
l
l l
l l
l
l l
l
l l
l
l
l l
l l
l
l l
l
l
l l
l
l
l l
l l
l l
l l
l
l
l l
l
l
l
l
l
l l
l l
l
lll
l
l
l
l
Figure 3
Bamanasand divination.
would be too cumbersome (65,536possible verses). But
the Bamana divination does not require any verse
memorization;as we will see, its use of recursionallows
for verse self-assembly.
Recursion is generally deElned as any iterative
mathematical function in which the output of each iteration is used as the input for the next iteration.In this
case the function is addition modulo 2 ("mod 2"), the
same simple even-odd distinction in the parity-bitoperation that contemporary computer systems use to
check for errors.There is nothing particularlycomplex
about mod 2; in fact, I was quite disappointed at first
because its reapplicationdestroyed the potential for a
binary placeholder representation in the Bamana divination. Rather than interpret each position in the column as having some meaning (as would our binary
number 1001, which means one 1, zero 2s, zero 4s, and
one 8), the divinersreappliedmod 2 to each row of the
first two symbols and each row of the last two symbols.
The results were then assembled into two new symbols,
and mod 2 was applied again to generate a third. Another four symbols were created by readingthe rows of
the original four as columns, and mod 2 was again
recursively applied to generate another three symbols.
The use of an iterative loop, passing outputs of an
operationback as inputs for the next stage, was striking
to me; I was at least as taken aback by the sand symbols
as they had been by the Cantorset. It would be naive to
claim that this was somehow a leap outside of our
culturalbarriersand power differences in fact, that is
just the sort of pretension that reflexive anthropology
has been dedicated against but it would also be ethnocentric to rule out those aspects that would be attributed to mathematical collaboration elsewhere in the
world: the mutual delight of two recursion fanatics
discovering each other. Andthe appearance of the symbols laid out in two groups of seven the Rosicrucian's
mystic number (not to mention the respective publication dates of Cantor and Mandelbrot in 1877 and
1977) added some numerologicalicing on the cake.
The following day I found that the presentation had
not been complete. There were an additional two symbols that were left out; these were also generated by
mod 2 recursionusing the two bottom symbols to create
a 15th, and using that last symbol with the first symbol
to create a 16th (bringingthe total depth of recursion to
5). The 15th symbol is called "thisworld,"and the 16th
is "the next world";so there was good reason to separate them fromthe others. But it maybe that the emphasis was partlydone for my benefit, as a bit of mathematical translationto better fit the Cantorset model.
The final part of the system creating a narrative
from the symbols was still unclear, but I was assured
that it could be learned if I carefully followed their
instructions. I was to give seven coins to seven lepers,
place a kola nut on a pile of sand next to my bed at night,
and in the morning bring a white cock, which would
have to be sacrificed to compensate for the harmful
energy released in the telling of the secret. I followed
all the instructionsand the next morningwas told to eat
the bitter kola nut as they preparedthe chicken, marking divinationsymbols on its feet with a blue Bic pen. A
little sand was thrown in its mouth, and then I was told
to hold it. Therewas no action on the part of the diviner;
the chicken simply died in my hands.
While I was still a bit shaken by the chicken's
demise (as well as a respectable buzz from the kola nut),
BAMANASAND DIVINATION/
they explained the remainingmystery.Each symbol has
a house" in which it belongs the position of the 16th
symbol is "thenext world" but in any given divination
most symbols will not be located in their own house.
Thus the symbol for desire in the house of travel indicates a desire for travel, and so on. Obviouslythis still
leaves room for creative narration on the part of the
diviner, but the beauty of the system is that no verses
need to be memorized or books consulted; the system
creates its own complex variety.
The most elegant part of the method is that it only
requires four random drawings; after that the entire
symbolic array is quickly self-generated (a timesaving
device that allows more clients; see Meyer 1991 on
client overload). A similar system for self-generated
variety was developed as a model for the "chaos"of
nonlinear dynamics by Marston Morse (1892-1977).
Morsebegins by counting from zero in binarynotation:
000, 001, 010, 011, 100, and so on. He then takes the sum
of the digits in each number + 0 + 0 = 0, 0 + 0 + 1 = 1,
and so forth and finally mod 2 of each sum. The result
is a sequence with many recursive properties but also
endless variety. Morse did the same "misreading"of the
binarynumberas did the Bamana althoughhe did not
have an anthropologist scowling at him for ignoring
place value and he did it for the same reason: combined with the mod 2 operation, it maximizes variety.
Geomancy
In Westernculture the dichotomy between "hard,"
or quantitative,science and "soft,"or qualitative,science has created a spectrum of status based on claims
to objectivity. When I have described Haraway'sstudy
of primatologyto Westernphysicists, for example, they
usually reply, "Wellof course, I've always thoughtbiology is too subjective; that's why I became a physicist."
But mathematicsoccupies a special position at the end
of this spectrum; in many ways it is a "closed world"
operating only by its own axioms, and perhaps that is
why divination receives less attention as scientific
knowledge.This is particularlyunfortunatein the crossculturalcomparison of ideas such as chance and determinism,since the recent discovery of deterministicaperiodicity as framed by nonlinear dynamics maps
quite well onto the traditional African conceptions of
tricksters and related forms of causal unpredictability.
Evans-Pritchard (1937) noted that the Azande
rankedthe validity of divinationmethods in proportion
to what he saw as their probabilistic variation, and
similar observations are made by Rene Devisch (1991).
Peek (1991) notes that Ifa diviners vary in their use of
correlations between the two sides of the divination
chain, which would also introduce control over prob-
RON EGLASH
117
abilistic variation.Variationsbetween chaos and order
within individualdivinationsessions are also well documented. Rosalind Shaw (1991), for example, shows an
intricate combination of mod 2 and mod 4 calculation
with random casts in Temme divination,providingthe
semantic process with variations in both periodic-aperiodic and chance-deterministicoppostitions.
In my reading of divinationliterature,I eventually
came across the duplicate of the Bamanatechnique in
Malagasysikidy (Sussman and Sussman 1977) and the
historical debate on its diffusion. The strong similarity
of both symbolic technique and semantic categories to
what Europeans termed geomancy was first noted by
Etienne Flacourt (1661),but it was not until Rene Trautmann (1939) that a serious claim was made for a diffusion fromthe Arabicilm al-raml ("thescience of sand")
to European,West African,and East Africandivination
techniques. This was supported in a detailed formal
analysis by Robert Jaulin (1966). Stephen Skinner
(1980) provides a well-documentedhistory of the diffusion evidence from the first specific written record, a
ninth-centuryJewish commentary,to its modernuse in
Aleister Crowley'sLiber 777. Skinner'smost intriguing
connection is the similarity between the geomancy of
RaymondLulland the design of Lull's"logicmachines."7
But his orientalist perspective (a "lethargic"Africa is
Uwokenby Islam") makes the ultimate attribution to
Arabicinvention suspect.
The oldest Arabic documents (those of az-Zantiin
the 13th century) claim geomancy's origin through the
Egyptian god Idris (the Arabic name for Hermes Trismegistus), and while we need not take that as anything
more than a claim to antiquity,a Nilotic influence is not
unreasonable.WallisBudge (1961) attempts to connect
the use of sand in ancient Egyptian rituals to African
geomancy, but it is hard to see this as unique. Mathematically,however, geomancy is strikinglyout of place
in non-Africansystems.
Like other linguistic codes, number bases tend to
have an extremely long historical persistence. Even
under Platonic rationalism,the ancient Greeks held ten
to be the most sacred of all numbers; the Kabbalah's
Ayin Sof emanates through ten Sefirot, and the Chrisdecimal notatian West counts on its "Hindu-Arabic"
tion. In ancient Egypt, on the other hand, base-2 calculation was ubiquitous, even for multiplication and
division, and Claudia Zaslavsky (1973) notes archaeological evidence linkingit to the use of doublingin the
counting systems of sub-SaharanAfrica. T. Kautzsch
(1912) notes that both Diodorus Siculus and Ailian reported that the ancient Egyptianpriests had a method
of seeldng truth through division by two. Doublingis a
frequent theme in African divination and many other
Africanknowledge systems, connecting the sacredness
of twins, spirit doubles, and double vision with material
118
* VOL. 99, NO. 1 * MARCH1 997
AMERICANANTHROPOLOGIST
objects, such as the blacksmith's twin bellows and the
double iron hoe given in bridewealth.
Moreover,the use of the additionmodulooperation
has an independent origin in Africa with the game that
is variously termed ayo, bao, giuthi, lela, mancala,
omweso, owart, and soro (among many other names).
The game is played by sequentiallyplacing counters in
twin (or double twin) rows of cups (sometimes referred
to as uhouses"),and for large counter-cupratios, addition modulo is required to calculate winning moves.
Zaslavsky notes that it can be played as a game of
chance by beginners, underscoring the relation between deterministic aperiodicity and our intuitive notions of randomness. Boards cut into stones, some of
extreme antiquity,have been found from Zimbabweto
Ethopia (see Zaslavsky 1973:Elgure 11-6). That the
game, while of Africanorigin is knownthroughoutEast
Africa under its Arabicname of mancala suggests that
Skinner's linguistic basis for an Arabic origin of geomancy is less certain than it might at Elrstseem.
The recursive aspects of Bamana divination can
also be illuminatedby comparisonto geomancy'scrosscultural history. European geomancers like Raymond
Lull, Robert Fludd, de Peruchio, and Henry de Pisis
persistently replaced the deterministic aspects of the
system with chance. By mounting the 16 figures on a
wheel and spinning it, they maintainedtheir society's
exclusion of any connections between determinismand
unpredictability(see Porter 1986).The Bamana,on the
other hand, seem to have emphasized such connections. On a video recordingthat I made of the Bamana
divination,I later noticed that they had used a shortcut
method in some demonstrations. (This may have been
a parting gift, as the video was shot on my last day.) As
first taught to me, when they count off the pairs of
randomdashes, they link them by drawingshort curves.
The shortcut methodthen links those curves with larger
curves, and those below with even larger curves. This
upside-down Cantorset shows that they are not simply
applying mod 2 again and again in a mindless fashion.
The self-similar physical structure of the shortcut
method vividly illustrates a recursive process.
African divination can be elsewhere linked to recursion, as in Devisch's (1991) description of the Yaka
diviners'Uself-generative"
initiationanduterinesymbolism. But the Bamana represent recursion in other domains as well. Figure 4 shows a chi wara sculpture
visualizing the cyclic iteration of living generations.
Their cultural neighbors, the Dogon, are famous for a
cosmology based on recursive nesting of the human
form, and Bamanalamps and merenkunpuppets sometimes feature a self-similar cascade of human shapes
(see Figure 5). The architecture of the Sudanese area
also makes use of self-similarstructures(Eglash 1995a,
1995b; Eglash and Broadwell 1989), and Alexander
Badaway (1965) found that a recursive numeric sequence was used to create such scaling in the construction of ancient Egyptian temples (such as the one at
Karnak).Icons linking this architecturalself-similarity
to a self-generating cosmology, also represented by
nested human formsXwere used in ancient Nilotic civilizations (Figure 6).
Cantor and the Bamana
Figure 4
Recursioninthe chi wara.Thecycles of livinggenerationsare depicted
as an iterativestructure in the chi wara headdress symbolism.
Before the 1970s,the standardanalyticapproachto
Cantorand the Bamanawould have been a mathematical portrait of Cantor'swork and an ethnographicportrait of the Bamana. By including ethnomathematics
and STS perspectives, we find a new array of causal
explanations and meanings. In the STS view, Cantor's
work cannot simplybe the discovery of new mathematical objects, because its universal truths are also the
result of his local cultural meanings. Conversely, an
ethnomathematicsview of the Bamanadiviners would
focus not on their local social semantics but on their
work as mathematicians,as theorists of the universal.
Whereasthe pre-1970sapproachset up a mathematicsversus-culture division, the more recent alternatives
show that this division exists within each side of the
divide. But there is no reason for stoppingafter only two
iterations; if we allow for recursive subdivisions, then
BAMANASAND DIVINATION/
RON EGLASH 119
Figure 6
Recursivecosmology in ancient Egypt.FromDescriptionde l'Egypt,
Paris, 1820.
Figure 5
Scaling cascade in Bamanamerenkunpuppet representingmultiple
spirits (see Arnoldi 1977). The puppet is worn on the head, thus
adding self-referentialimageryto this scaling cascade. Photo courtesy the IndianaUniversityArtMuseum, Bloomington.
the two sides may begin to show some strong similarities.
The Cantorset and the double-seven configuration
of the initial 14 divinationsynlbols may have a superElcial visual sinzilarity,but the comparisononly becomes
mathematicallysignificantif we hold one upside down.
Cantor'sproblem was in takingthe finite-a line of unit
length and demonstratingthat it could be expanded
beyond infinity. The diviners are faced with the infinity
of possible futures and must show how they can be
narrowed down to a predicted unity. It is clear that,
while we can consider the diviners as theoreticians,
their mathematicsis drivenby the performativerequirements of their work. But European mathematicians
must also gather clients and perform their theories; it
was quite some time before Cantorwas acknowledged
as a legitimate mathematical actor. And while both
Figure 7
The Cantor set in Egyptiancapitals. This capital from an ancient
Egyptiantemple represents the lotus, symbol of the self-generating
origins of life. FromDescription de l'EgyptJParis, 1820.
120
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AMERICANANTHROPOLOGIST
mathematicalsystems are unified in their use of recursion-united through the universality of mathematical
process they are culturally linked as well. Indeed,
given Cantor'sRosicruciantheology and the proximity
of his cousin Moritz Cantor at that time a leading
expert in the geometry of Egyptian art (M. Cantor
1880) it may be that an African concept of self-generated fecundity (as visualized in Figure 7, an ancient
Egyptianrepresentation of the lotus creation myth) is
the shared origin of both the Bamana divination and
transfiniteset theory. Neither mathematicsnor culture
should be viewed as firmlyfixed on the universal-local
divide, for there are divisions within divisions never
ending.
Notes
1. Significantly,the most direct statement in Zaslavsky's
text concerning math and cultural relativism is in a quote
from F. E. Chapman, an African American historian sentenced to life imprisonmentfor a robbery and murderwhen
he was 19. A more diasporic view of mathematicsin African
cultures-for example, MalcomX on the numericcapabilities
of his mentor in the numbers racket or on his own oppositional adoption of the generic symbol for the mathematical
unknown-might reveal some wider implications for the
power-knowledgerelations of mathematics and society.
2. Ascher 1990;Closs 1986;Crump1990;D'Ambrosio1990;
Gerdes 1994; Washburnand Crowe 1988; and others. See
Crowe 1987and Fisher 1992for reviews.
3. Davis 1988;McClearyand McKinney1986;Wilder1981.
4. Imperato 1983;McNaughton1988;Zahan1974.
5. Aftergivinga lecture on Bamanadivinationin the United
States, I was approached by a mathematics faculty member
who was quite taken by this phrase. That's just like us!"he
exclaimed. We get the power of mathematicsonly at the cost
of our social deformityas nerds.'
6. They typically used computer printouts. This recycled
paper was generally available but may have had particular
significance for diviners due to the symbolics of computers
in Africa;see Jules-Rosette 1990.
7. Since Lull's "logic machiner inspired Leibniz (about
1670) in his development of the modern binary code, Skinner's theory about the influence of geomancy on Lull would
mean that the streams of ones and zeros running through
every digital circuit, from alarm clocks to supercomputers,
can trace their origins back to Africandivination.
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