NOTE: Another Western Journalist reporting on the fundamental
"multicultural" nature of mathematics and, therefore, the centrality
of peoples of color being the founders of mathematical thinking
The Quipu (or Khipu) has come to Western mathematical academic
circles as "discovery" back in the 70's and 80's by progressive
math and science educators challenging the Eurocentric nature
of Western Education... particularly in the mathematical and
scientific fields of study. Maurice Bazin's pioneering pedagogical
work at the Exploritorium <www.exploratoriumstore.com/msac.html>
culminating in a book: "Math and Science Across Cultures" and
George Gheverghese Joseph's "The Crest of the Peacock" (Princeton
University Press) books have placed the Incan mathematical pioneering
work in their proper historical and cultural perspective.
These books, exchanges and trainings have directly contributed
to a new wave of research by indigenous scholars into the profundity
of mathematical thinking among their respctive pre-Columbian/pre-slavery
It also shows that a computer does not have to be electrical
Inca may have used knot computer code ======================================
The Independent (UK)
Inca may have used knot computer code to bind empire
By Steve Connor, Science Editor
They ran the biggest empire of their age, with a vast network
of roads, granaries, warehouses and a complex system of government.
Yet the Inca, founded in about AD1200 by Manco Capac, were unique
for such a significant civilisation: they had no written language.
This has been the conventional view of the Inca, whose dominions
at their height covered almost all of the Andean region, from
Colombia to Chile, until they were defeated in the Spanish conquest
But a leading scholar of South American antiquity believes the
Inca did have a form of non-verbal communication written in an
encoded language similar to the binary code of today's computers.
Gary Urton, professor of anthropology at Harvard University,
has re-analysed the complicated knotted strings of the Inca -
decorative objects called khipu - and found they contain a seven-bit
binary code capable of conveying more than 1,500 separate units
In the search for definitive proof of his discovery, which will
be detailed in a book, Professor Urton believes he is close to
finding the "Rosetta stone" of South America, a khipu story that
was translated into Spanish more than 400 years ago.
"We need something like a Rosetta khipu and I'm optimistic that
we will find one," said Professor Urton, referring to the basalt
slab found at Rosetta, near Alexandria in Egypt, which allowed
scholars to decipher a text written in Egyptian hieroglyphics
from its demotic and Greek translations.
It has long been acknowledged that the khipu of the Inca were
more than just decorative. In the 1920s, historians demonstrated
that the knots on the strings of some khipu were arranged in
such a way that they were a store of calculations, a textile
version of an abacus.
Khipu can be immensely elaborate, composed of a main or primary
cord to which are attached several pendant strings. Each pendant
can have secondary or subsidiary strings which may in turn carry
further subsidiary or tertiary strings, arranged like the branches
of a tree. Khipu can be made of cotton or wool, cross-weaved
or spun into strings. Different knots tied at various points
along the strings give the khipu their distinctive appearance.
Professor Urton's study found there are, theoretically, seven
points in the making of a khipu where the maker could make a
simple choice between two possibilities, a seven-bit binary code.
For instance, he or she could choose between weaving a string
made of cotton or of wool, or they could weave in a "spin" or
"ply" direction, or hang the pendant from the front of the primary
string or from the back. In a strict seven-bit code this would
give 128 permutations (two to the power of seven) but Professor
Urton said because there were 24 possible colours that could
be used in khipu construction, the actual permutations are 1,536
(or two to the power of six, multiplied by 24).
This could mean the code used by the makers allowed them to convey
some 1,536 separate units of information, comparable to the estimated
1,000 to 1,500 Sumerian cuneiform signs, and double the number
of signs in the hieroglyphs of the ancient Egyptians and the
Maya of Central America.
If Professor Urton is right, it means the Inca not only invented
a form of binary code more than 500 years before the invention
of the computer, but they used it as part of the only three-dimensional
written language. "They could have used it to represent a lot
of information," he says. "Each element could have been a name,
an identity or an activity as part of telling a story or a myth.
It had considerable flexibility. I think a skilled khipu-keeper
would have recognised the language. They would have looked and
felt and used their store of knowledge in much the way we do
when reading words."
There is also some anecdotal evidence that khipu were more than
mere knots on a string used for storing calculations. The Spanish
recorded capturing one Inca native trying to conceal a khipu
which, he said, recorded everything done in his homeland "both
the good and the evil".
Unfortunately, in this as in many other encounters, the Spanish
burnt the khipu and punished the native for having it, a typical
response that did not engender an understanding of how the Inca
used their khipu.
But Professor Urton said he had discovered a collection of 32
khipu in a burial site in northern Peru with Incan mummies dating
from the time of the Spanish conquest. He hopes to find a khipu
that can be matched in some way with a document written in Spanish,
a khipu translation. He is working with documents from the same
period, indicating that the Spanish worked closely with at least
one khipu-keeper. "We have for the first time a set of khipu
from a well-preserved and dated archaeological site, and documents
that were being drawn up at the same time."
Without a "khipu Rosetta" it will be hard to convince the sceptics
who insist that, at most, the knotted strings may be complicated
mnemonic devices to help oral storytellers to remember their
lines. If they are simple memory machines, khipu would not constitute
a form of written language because they would have been understood
only by their makers, or someone trained to recall the same story.
Professor Urton has little sympathy with this idea. "It is just
not logical that they were making them for memory purposes,"
he said. "Tying a knot is simply a cue; it should have no information
content in itself other than being a reminder." Khipu had layers
of complexity that would be unnecessary if they were straightforward
mnemonic devices, he said.
Translating the secrets of the ages SUMERIAN CUNEIFORM The world's
first written language was created more than 5,000 years ago,
based on pictograms, or simplified drawings representing actual
objects or activities. The earliest cuneiform pictograms were
etched into wet clay in vertical columns and, later, more symbolic
signs were arranged in horizontal lines, much like modern writing.
Cuneiform was adapted by several civilisations, such as the Akkadians,
Babylonians and Assyrians, to write their own languages, and
used for 3,000 years. Many of the clay tablets, and the occasional
reed stylus used to etch cuneiform on them, have survived. Knowledge
of cuneiform was lost until 1835 when a British Army officer,
Henry Rawlinson, found inscriptions on a cliff at Behistun in
Persia. They were identical texts written in three languages
- Old Persian, Babylonian and Elamite - which allowed Rawlinson
to make the first translation for many hundreds of years.
EGYPTIAN HIEROGLYPHICS The original hieroglyphs, dating from
about 5,000 years ago, were etched on stone and were elaborate
and time-consuming to make, which meant they were reserved for
buildings and royal tombs. A simplified version, called hieratic,
was eventually developed for everyday bureaucracy, written on
Later still, hieratic was replaced by demotic writing, the everyday
language of Egypt, which appeared on the Rosetta stone with Greek
and hieroglyphic script, allowing scholars to translate the original
MAYAN HIEROGLYPHICS The Maya used about 800 individual signs
or glyphs, paired in columns that read from left to right and
top to bottom. The glyphs could be combined to form any word
or concept in the Mayan language and inscriptions were carved
in stone and wood on monuments or painted on paper, walls or
pottery. Some glyphs were also painted as codices made of deer
hide or bleached fig-tree paper covered by a thin layer of plaster
and folded like an accordion. The complete deciphering of the
Mayan writing is only 85 per cent complete, although it has been
made easier with the help of computers.
Only highly trained Mayan scribes used and understood the glyphs,
and they jealously guarded their knowledge in the belief that
only they should act as intermediates between the gods and the
common people. © 2003 Independent Digital (UK) Ltd