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Tuesday, August 16, 2005

Kipus = Libros contables.

Los quipus son cordeles coloreados, de longitud variable, anudados cada cierto trecho. De la cuerda superior (cuerda primaria), se desprenden otros (c. colgantes), a las que se adhieren otros laterales (c. subsidiarias). Las primeras descripcciones de su uso provienen de Garcilaso de la Vega -1606.Comentarios Reales de los Incas-: "...de acuerdo a su posición, los nudos significaban unidades, cientos, mil, diez mil, excepcionalmente cien mil... y ellas estan ahí bién alineadas en sus diferentes cordeles al igual como ordena sus cuentas un contador, columna por columna, en su libros..." y Guamán Poma de Ayala -El Primer Nueva Corónica y Buen Gobierno.1615-16-:("...por los colores sacaban lo que contenía...:oro al amarillo..plata para el blanco..colorado para la gente de guerra...los sin colores puestos en orden significaban de más, a menos calidad...cada cosa en su género, como en mieses y legumbres...los hilitos delgados, eran para los casados, las viudas...las cuentas eran anuales..."), quienes sustentaron que eran registros contables (no, calculadoras), donde se almacenaban datos numéricos relacionados con estadísticas vitales ("...el método de hacer estos reportes es por medio de los nudos, hechos de varios y tales y tales colores, para tales y tales faltas y sus castigos...cordeles más pequeños adheridos a cordeles más gruesos de diferente color, significaban la naturaleza precisa del castigo inflingido..."-Garcilaso de la Vega.1606-), armas, alimentos, etc. Los trabajos de Gary Urton y C. Brezine, publicados esta semana en la revista Science, acaban de confirmar los asertos anteriores.
Para ello, ambos investigadores crearon un sistema informático, que simulaba con exactitud meridiana, las características físicas de los quipus. Ingresando a ella cuanto quipu estuvo a su alcance (de entre 2 a 1500 nudos), establecieron relaciones numéricas -para los nudos-, conexiones y relaciones para el color y, los espacios vacíos de los cordeles. Recuérdese que los incas desconocieron el cero.
No obstante por la forma dispuesta de las estadísticas, se ha inferido que los incas empleaban el sistema decimal, para el alamacenaje de objetos. Así, el número 586, estaría representado de la siguiente forma (ver figura de la derecha). De otro lado, Urton y Brezine, tras revisar y analizar 21 quipus procedentes de Puruchuco, concluyeron que la mitad superior de los quipus, expresaba un incremento en el número de objetos almacenados (simulando multiplicaciones), en tanto la inferior, las disminuía (simulando divisiones).
No obstante, una cosa son las cifras numéricas empleadas en el almacenaje o, archivo y otra el cálculo previo. Las matemáticas previas continúan siendo un problema irresuelto. Es decir : ¿como se calculaban datos gigantescos : suma, resta y multiplicación de millones de granos de maiz?. Un solo cronista Joseph de Acosta (1590.Historia Natural y Moral de las Indias), relata que calculaban en cuadrados especiales (yupanas), empleando granos de maiz. "...hay que verlos a ellos, empleando una suerte de calculador...Así para ejecutar cálculos dificiles al modo como lo hace un calculista, empleando lápiz y papel, estos indios hacen uso de granos de maíz. Colocan 1 grano aqui, 3 allá y 8 en cualquier lugar. Entonces mueven 1 grano aquí y allí y, el hecho es que ellos son capaces de completar el cálculo, sin cometer el mínimo error. En el considerar a personas así, como salvajes o, poco ingeniosos, dejemos a cada cual, con su juicio. Personalmente considero que su modo de calcular es superior al nuestro...".
Ha contribuído a alimentar la sospecha de que los incas calculaban empleando granos y yupanas, la clásica recreación (angulo inferior derecho-arriba-, de un dibujo realizado por Guamán Poma de Ayala, en el que se aprecian círculos negros y claros (granos de maiz), insertos en un tablero (posible yupana). Sin embargo, no todos aceptan que los incas hallan utilizado estas yupanas (cuasi, ábacos), para calcular. Algunos opinan que el dibujo (1615), representa tan solo una analogía entre ábacos transportados al Perú por los españoles y los quipus.
Empero, hay al menos un cientifico ingeniero -matemático: Nicolino de Pasquale, que ha logrado establecer un sistema numérico de base 40 (ver cuadro de la derecha), para realizar calculos con granos de maíz. Pasquale, agrega que no hay desencuentros con el sistema decimal empleado en el almacenaje. Los números calculados en las yupanas (s. numérico de base 40), eran transformados a base decimal, a fin de ser almacenados en los quipus.
Kipus = countable Books.
Quipus are colored cords, of variable length, knotted every certain stretch. The superior cord (c. primary), others are come off (hanging c.), to which lateral ones adhere other (subsidiary c.). The first descriptions of their use come from Garcilaso de la Vega -1606-. Real Comments of the Incas-: "... according to their position, the knots meant units, hundreds, thousands, ten thousands, exceptionally one hundred thousand... and they are aligned in their different cords from the equal one as an accountant orders his accounts, column by column, in their books..." and Guamán Poma de Ayala -the First New Corónica and Good Government-1615-16-:("... by the colors we know that yellow is to gold as white to silver...red for military people... without colors the things were put in order from more, to less quality... each thing in their sort, like ripe grain and vegetables... the thin little threads, were according to the married ones, the widows... the accounts were performed annually..."), that they sustained were countable registries (not, computers), where were stored numeric data related to vital statistics ("... the method to make these reports is by means of the knots, facts of several and such and such colors, for such and such faults and their punishments... smaller cords adhered to thick cords of different color, meant the precise nature of the applied punishment... "-Garcilaso of la Vega.1606-), arms, foods, etc. The works of Gary Urton and C.S. Brezine, published this week in the Science magazine, finish confirming the previous assertions.
For it, both investigators created a computer science system, that simulated with dazzling exactitude, the physical characteristics of quipus. Carryng to the computer system whatever quipu characteristic (of between 2 to 1500 knots), settled down numerical relations, for the knots, connections and relations for the color and, the empty spaces of cords. Remember that the Incas did not know the zero.
Owing to the form of the statistics, tey has inferred that the Incas used the decimal system, for the storage of objects. Thus, number 586, would be represented in the following form (See figure of the right). On another side, Urton and Brezine, after reviewing and analyzing 21 quipus coming from Puruchuco, concluded that superior half of quipus, expressed an increase in the number of stored objects (simulating multiplications), in as much the inferior one, diminished them (simulating divisions).
However, a thing is the used numerical numbers in the storage or file, and another one the previous calculation. The previous mathematics continue being a problem. It is to say ¿ how w ere gigantic data calculated?: In which way they sum, reduces and multiplicate million grains. A single cronista Joseph de Acosta (1590, Natural and Moral History of the Indians), relates that they calculated in special squares (yupanas), using grain of maiz. "...it is necessary to see them, using a computer type... Thus to execute dificcults calculations to the way since a planner does, using pencil and paper, these Indians make use of maize grains. Here 1, 3 there and 8 in any place. Then they move 1 grain here and there and, the fact is that they are able to complete the calculation, without committing the minimum error. In considering to people thus, like savages or, little ingenious, we leave everyone, with its judgment. Personally I consider that its way to calculate is superior to ours... ".
The previous has contributed to feed the suspicion that the Incas calculated using grain and yupanas, according to the classic recreation (angle inferior right), of a drawing made by Guamán Poma of Ayala, in whom black and clear circles are appraised (grain of maiz), inserted in a board (possible yupana). Nevertheless, all do not accept that the Incas used these yupanas (cuasi, abacuses), to calculate. Some think that the drawing (1615), represents so single an analogy between transported abacuses Peru by the Spaniards and quipus.
However, there is at least an scientific engineer and mathematician : Nicolino de Pasquale, that has managed to establish a numerical system of base 40 (See picture of the right), to make calculos with maize grains. Pasquale, adds that there is no contradiction with the decimal system used in the storage. The numbers calculated in the yupanas (numerical system of base 40), were transformed to base decimal, in order to be stored in quipus.
Referencias :1-Gary Urton and Carrie J Brezine.Khipu Accounting in ancient Peru.Science.309(5737);1065-67. 2-http://www.Kb.dk/elib/mss/poma/

22 Comments:

Blogger Django said...

Hola Victor, un post digno de resaltar como las otras, pero esto me ha hecho recordar que el Museo de la Nación dicta curso de Quipu aqui el enlace http://inc.perucultural.org.pe/mnacion.shtml , acaba de culminar un curso, pero me dicen que cada cierto tiempo dictan este curso, por un costo de S/. 50.00 con certificación, la verdad para un legado tan valioso dejado por los Inkas quizas valga la pena, yo ya separé mi inscripcion para el proximo curso.

Slds amigo Victor

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Blogger Jenny said...

Gracias por este gran aporte!!!
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Anonymous Anonymous said...

Hola somos alumnos de la I.E Jose Granda Gracias por el aporte :D
Quisieramos saber que significaban los colores para los quipus

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