 Gary is a member of this institute for longer than any of us can remember. He has been involved in Indian studies as one of its primary players for a very long time. He got his degree at the University of Illinois, taught for many years at Colgate, and of course now he's at Harvard and is currently the chair of the apology there. And with some help from my friends, we've tried to reduce Gary down to one cent. And we'll see if I can get this right. And the idea is that Gary has made some of the greatest contributions to understanding how the Indian mind sees the universe. Something we could all use a really big dose of and especially from the brilliant mind of Gary. So Gary can you take us to keep him alive? Thank you, John. Goodness, I have no idea what to say now. There's nothing to say really. Except to say in the first place, thank you so much to John and to all of the officers of the Institute of Indian Studies. I am aware that this pushes the bounds of propriety to be able to speak before the audience in the keynote lecture twice in three or four years or something of that order. So I am deeply grateful to you for this privilege and honor. I'm also painfully aware that I'm the only thing that stands between you and the good beer and wine and food of the president's reception. So I'll do this with as much alacrity and good humor as possible. So again, thank you so much to all of you for coming out this evening. For those who are coming in sort of from the outside and are not intimately familiar with the subject matter we've been dealing with in this meeting over the past two days, we're talking about the continent of South America and more specifically we're talking about that portion of the central Andes on the west central part of the continent of South America. So this includes the Pacific Coastal Desert, the Andes Mountains, the central portion of the mountains that run, of course, from Columbia and the Gulf of Mexico down to Tierra del Fuego. And then in the north the headwaters of the Amazon and in the southeast the headwaters of the Paraná River that flow down through Argentina. So this is the region that we're interested in. And we, in this talk, I'm interested specifically in that area insofar as it was the territory of the Inca Empire, Tawantinsuyu, the four parts intimately bound together that had their northern boundary more or less on the present-day boundary between Ecuador and Columbia, running down through Peru, through Bolivia, northwestern Argentina, and Chile down to a couple of hundred kilometers or so south of Santiago de Chile. So this is the area that we're concerned about. This was the great empire, the most extensive empire of the pre-Columbian New World. The Inca Empire existed only for a very short period of time. It was the end product of a long sequence of cultural evolution in the central Andes. It existed only from around 1450 or so, maybe slightly earlier than you see here, until its conquest by Francisco Pizarro and its troops in 1532. And so that's the period that we'll be interested in. And so this was the time of the Inca Empire in the central Andes. And since we have to talk about something, the thing about that we're going to talk about is about record-keeping and accounting. So we could talk about ceramics or we could talk about textiles or Yama herding, but it happens that this evening we're going to be talking about record-keeping and accounting in the Inca Empire. So you see here a drawing on the bottom left from the work of an indigenous chronicler, Guamampoma de Ayala. He wrote a thousand-page letter to the king of Spain near the end of the 16th century, in which he protested the conquest of the Andes in that work, in that letter of protest. He included drawings of life in Peru before, during, and after the time of the Spanish conquest. And this is one of our best sources for visual representations of the world of the Inca. Again, his idealization before, during, and after the time of the Spanish conquest. You'll see many images from the work of Guamampoma de Ayala. As I'm talking to you about the Kippus or record-keeping in the Inca Empire today, I will have in the back of my mind a comparison between what was going on in the Andes in terms of record-keeping and accounting and what was going on at the same time in Europe. And it so happens that this was just the time, 1494, so just before the conquest of the Inca Empire in 1532, of the production of what was the first book of double-entry bookkeeping. This was a mathematical text by Luca Pacioli, which was written and published in 1494. And it contained the first description of double-entry bookkeeping, which for any accountants in the audience, in the first place, why are you here? But thank you for coming. But this, of course, you will know, is the crown jewel of accounting in Western Europe and still is the principle method used in accounting practice. And I want to have this development in global accounting history conscious in the back of our minds as we go about looking at what was going on in terms of Inca accounting at this time. And the principle device for accounting was the Kippu, this knotted string, generally colorful device that was used for recording various kinds of information. On the right, we see another drawing from Mamma Poma de Ayala. On the left, a drawing from the Chronicle of Martín de Marúa, a priest who wrote a Chronicle in the 1590s, and we think that the drawings that are included in his work, in fact, were produced by the indigenous chronicler Mamma Poma de Ayala. So we'll see some images from Martín de Marúa's Chronicle as well. But so we see images of the holding and the manipulation of Kippus in these two colonial visual source works. And then we also have now almost 900 Kippus that exist in museum collections around the world. And I'll be talking to you about a set of those today. So basically, when we're talking about Kippus, there were essentially two different types of Kippus. So there was one type of Kippu for administrative work, and that's what I'm going to be talking to you about tonight. There was another type, though, that, as we're told by the Spanish chroniclers, recorded historical narratives, recorded poems, recorded songs, all sorts of narratives telling the life history of the Incas, etc. These were a very complex form of Kippu that I, in truth, haven't the foggiest idea how to interpret now. They're very complex. They are knotted string devices and knots are spread all over the surface of the Kippu, quite unlike the Kippus that I'll be talking to you about this evening. So here are two examples of what I think are probably narrative Kippus, ones that were we to understand how to interpret them. We could read, we could take information from them, and we could tell the life history of perhaps this or that Incas. The kinds of Kippus that I'm going to be talking to you about this evening were ones that had to do with the administration of the Inca Empire. So for administrative purposes, the Incas used a decimal system of administration. Here it's important to point out that the main form of tribute in the Inca Empire took the form of a demand for labor tribute. So the Incas did not take anything from the larder or from the storehouse of any individual, but rather they required that every subject of the Empire work a certain number of days each month for the Empire. And these work groups were formed in groups of ten, and five groups of ten could form a larger work group of fifty, two groups of fifty could form a group of a hundred, and you see how these groups could incrementally by fives and twos ultimately formed very large work groups up to the largest named group, the Hunu, which was a group of ten thousand workers. Information about the requirement for labor time was kept by the Kippus, so here they were recording time, time owed, time paid, and we think that was an important part of the recording of information in the Kippus. Each one of the eighty provinces of the Inca Empire had what was called a Tokrikuk, the one who oversaw, the one who oversaw the business of the Inca in the province, and Tokrikuk was attended in the business of state administration by a group of Kippu keepers, so we see one of them here, an administrador de provincias and administrator of the provinces, who has a couple of Kippus here, he's holding in his hands, which he presumably then has information that pertains to this business of keeping track of tribute. There were other uses of the Kippus, so they recorded information on the periodicities and the cycles of celestial phenomena of the sun and the moon and the planets and the stars, and so the calendar specialists were specialists in Kippu keeping, and then also Kippu's information about the business of state moved around the empire on the Inca road system carried by Chaskis, an example of which we see here in one of the drawings of Guama Poma, in which he just emphasizes what this is, he has a little cartoon sign here that says karta or letter that's attached to the Kippu that he's carrying. And another function of the Kippu keepers was to keep track of the goods that were in the Inca storehouses, and here it'll be useful to just say right now that all territory in Tawantinsuyu in the Inca state belong to the Inca, everything in fact belong to the Inca, and in terms of land, land was divided into three parts, one third was the land of the state, one third was the land of the gods, and one third belonged to the commoners. So those were the lands that were for the sustenance of the tributaries of the Inca. On the lands of the state, these were the lands that were planted and harvested by work groups, and the harvest then went into the state storehouses. And what we're going to be talking about then today, this evening, is storehouse accounting. So the goods that we're talking about insofar as they involve food products, plants, were products of the fields of the Inca that had been tended by the state workers. So just a short introduction here to the Kippus for those who don't really know anything about them. So the Kippus built on the framework of one fairly thick cord that's called the primary cord or the main cord. Often these cords are quite colorful, so they're wrapped cords. These are, I should just say here, the Kippus are spun and plied either cotton, fibers, or camelid, that is yama or alpaca. The majority of Kippus that we have that survive are of cotton. They come from the coast, and we think that the tradition of making camelid or yama and alpaca Kippus was the primary form of Kippu making in the highlands. And conditions for preservation there are very poor, so we have very few camelid fiber Kippus that exist from the highlands, most come from the coast. So you have the main cord or the primary cord to which are attached a variety of what we call pendant strings. So these are tied or these are knotted onto the primary cord. And in many cases we have subsidiary cords, second level cords that are tied onto the pendant strings. And you have hierarchical arrangements of subsidiary cords attached to subsidiary cords, which are attached then to the pendant string. We have Kippus with up to six levels of hierarchies of information recorded on the Kippus. We also have some Kippu cords that come off from the top, whose attachments bind together groups of these pendant strings. These are particularly important in terms of mathematics. Turns out the Kippus we're talking about tonight don't have top cords, so we're not going to talk about that structural feature. We can talk about it more if you have any questions. And then some Kippus have what I call loop pendants, and these are pendant strings that are attached to cords that are tied onto the main string and that loop down below the main cord. Also the Kippus we're talking about don't have those, but I just thought you should have the full sort of run of the structural features of the Kippus. On your sort of ordinary garden variety Kippus, or the majority of the Kippus, when you look at them you see the cords are knotted and the knots are not like the ones that I showed you on those earlier Kippus that are spread all over the surface of the Kippu, but rather that are in ranked files up the length of the Kippu cords as you see here. And this relates to the recording of numerical information in the base 10 decimal system of the Kippus. So like us the Incas had a base 10 system of numeration. You see the names for the numbers here. And the Kippu cords were knotted in files or in tiers that were linked to the place value of the numerical information that was being recorded on the Kippu. So that toward the bottom of the strings you have the ones, then the tens, the hundreds, the thousands, the tens of thousands and we have some there are few Kippus that record up to hundreds of thousands, but theoretically you could record into the millions. I've not encountered a Kippu that records the millions, but you could do it quite easily just with a very long string. So in these different place values they tied a number of different types of knots. So they had several different types of knots to record the values from one to nine at the ones level. They used the simple overhand or what I call granny knot at the tens, hundreds, thousands level. And then they, yeah, so those are the principle knot types. You have what we call a figure eight knot that was tied for the value one and then eight different varieties of what we call long knots to record the values two to nine. On these knots you can run your fingernail over the knot and you can count the number of turns between two to nine. So there you can actually read the numbers. So we can read the numbers on these Kippus. The numerical Kippus make up about two thirds of this almost 900 Kippus that I've inventoried now. So it's only about a third of them that are those in narrative Kippus. The two thirds of them, or the great majority, are Kippus that record numerical values. So they're quantitative. So that on these Kippus, the quantitative ones, you can read down. You can look at the placement of knots and here we see one knot in hundreds level, one in the tens level and a knot of three in the ones level. So that's 100 plus 10 plus three or 113. It's important to point out here since at this time in Western Europe basically the Hindu Arabic numerals were being introduced were being used for the first time. They've been introduced by about a thousand AD or so but it took four or five hundred years to actually be incorporated into the mathematical system and the accounting system. But there they had a symbol for zero for the absence of value and in the Inka Kippu they had a way of signifying the absence of value which was not by a sign like the zero but rather by the absence of a knot in the place of value. So if you want to record the number 102 you just leave the tens placed empty and then you read 100 plus two is 102. So the numbers were quite good and I should just point out here that they didn't do calculations with the Kippus. The Kippus were knotted with numerical values that were calculated elsewhere. We think they use these devices called upanas that are counting boards. I'll talk about this a little bit later. So they did all the calculations over there with kernels of corn or with pebbles in these counting devices and then when they had the values they recorded them on the Kippus. So what we're looking at with the Kippu is the results of calculations. So when we have a string with a certain value so we can be pleased with ourselves we know this string says 102 what we don't know is 102 what. And that's actually sort of a fundamental problem and that's the fundamental problem of Kippu studies is we still don't know how to consistently interpret what kinds of information has been recorded. We think that partially it was done using color so many of the Kippus are quite colorful from the dyeing of cords or from the natural colors of camel and fibers that come in all shades of browns and blacks and greys etc. And so we think that color coding was an important part of it. I've been recording the colors in Kippus. We recorded about 650 of the 900 Kippus now in a database that I started working on some 14 years ago or so. So we think color was important. Color was used not only probably to signify certain identities certain colors had certain associations with objects or products or statuses and also color was used to organize information on a Kippu that you would have cords of one color followed by cords of another color followed by cords of another color and those color differences we think were the ways they organized different categories of information on the Kippus. So that's all important to know but just in the background for us now we're not going to be worrying about that too much but just so you have some sense of the way these things operate. They were actually operated by a cadre of Kippu keepers those who make or organize or animate the knots. So the term Kippu itself means knot and so we have various representations here from the Chronicles of Marua and Wama Poma de Ayala showing Kippu keepers holding and manipulating and presenting Kippus in various ways. So there was a hierarchy of Kippu keepers with the principal ones being in the capital Kusko and mid-level Kippu keepers being out in the provincial centers and then local Kippu keepers who were keeping their local records and passing the information on that chain of hierarchy. We can talk about this but keep that in the back of your mind because that's important. There were also archives of these things we're told by the Chronicles for instance here Martin de Marua says that the accountants had great heaps of these cords in the manner of registries like our scribes had written documents and they kept their archives in such a manner that if they needed to know something they had only to go to one of these Kippu Kamiacs. From our studies of the Kippus that are extant in museum collections so this is in Europe, North America and South America so I say here they're 875 but they're close to 900 now that I've inventoried and about 250 of them or so have reasonably good proveniences. A lot of these came from graves and were plundered by grade robbers and so we don't have good provenience data but we have reasonably good provenience data and about 250 of them and these form what I call archives that were found in certain places certain named places or regions in a river valley we have some general confidence that they sort of pertain to each other and what you see over here is a general representation a sort of summary icon of the characteristics of Kippus from that archive so we find for instance a lot of these loop pendants in this collection in the far north we have six cord groups here in the Santa Valley four cord groups in one part of the Remont Valley three cord groups here et cetera a lot of subsidiaries in the group of Kippus down in Avika so there are these differences and I think that they represent something like species of a genre like they're all of the genus Kippu but there are these various differences and I don't frankly know now how to account for why there are those differences you know a Kippu when you see it but one may be a little bit different from the other there may be regional differences ethnic differences, something on that order so those are the Kippus that were known up until 2013 and then in 2013 there was a fairly large archive of 34 Kippus it was added from the excavation of an archaeological site on the south coast of Peru the site of Inkawasi which we'll talk about today this was the work of Dr. Alejandro Chu an archaeologist trained at San Marcos who's now teaching at Catolica University in Lima so this is the Kippu archive that I'll talk to you about this evening so the site of Inkawasi then is located in the Canete Valley it's one of these rich river valleys bordered on either side by the very dry coastal desert the river here is running from our left to our right water coming down out of the Andes and running to the Pacific Ocean the site of Inkawasi is located down from where that last photo which was the town of Luna Juana which is the nearest town of Inkawasi here now we're looking to the north and the river's running from the right down to the left and this is the edge of the river valley here and Inkawasi sits above the river valley stretches a little less than a kilometer along the valley there are various parts of it I'm sorry, a residential area down here a palace, the storehouses we're going to talk about here and administrative buildings there the site of Inkawasi we have some information on it from the Spanish Chronicles and we're told that it was built as a staging area for the Inkakon quest of the south coast and especially when the Inkas went down to conquer these very delicose war-like people the Warko people, they were the sort of bad guys on the south coast and so the Inkas came down, built Inkawasi and that was their site for provisioning the troops in the conquest of the south coast one of our chroniclers, Ciesa de Leon writing in 1553 tells us that the Inkas built a new city to which he gave the name New Cusco the same as his main seat or the Inca capital they also tell that he ordered that the districts of the city and the hills should have the same names as those of Cusco so they built this city and they named everything the same apparently as in Cusco one of the students of Inkawasi, John Hizlop the anthropologist who did the great study of the Inca road system worked from 1980 to 1984 at Inkawasi, Map the Side did a fantastic job there and he felt that Inkawasi could be divided into four quadrants just as the Inca empire itself was divided into four quadrants that you have the big division of the upper part or Hanan on one half of the site and the lower part of Hurin on the other side and the two parts were then subdivided making the four quarters of the side of Inkawasi I don't know whether to think about that but there you have his particular interpretation of it Ciesa tells us that after they defeated Diwarko and other tribes on the south coast and peace having been restored to the valley the Inca ordered the new Cusco he had built raised and with all the army he returned to the city of Cusco so supposedly the site was completely wiped off the face of the earth burned destroyed completely but that's not the case so the site still exists today in a fairly reasonable condition at the core of the side are these two facilities so the one on the left includes a palace and an ushnu or a ceremonial center and to the right is a temple of the sun that borders against the northern edge of the storage facility that we'll be talking about here so here's a balloon photograph of the storage facility an oblique angle view that John Hislop made showing us the temple of the sun over here a narrow little corridor that divides the temple of the sun from this great complex of the storage facility here you see that structure here this is overhead as a balloon photograph that John made in the 80s showing then the core of the storage complex these rectangular structures in the center of which there's a platform to probably was an administrative place or facility overseeing the movement of goods into and out of the storage complex small square storage deposits that surround the facility on three sides and then up here here's the corridor dividing this from the palace of the sun and four kayankas these rectangular buildings and closed buildings and two open storage spaces here I call these sorting spaces or drying spaces we'll get to those in a moment but so the kipu finds were from this upper part of the storage facility here so as Alejandro began excavating it he began excavating first in this corridor and he found that the walls of the corridor had been pulled down so as though perhaps from the purposeful destruction of the walls of the storage facility and there then beneath the stones and the mortar he could see like streams coming out so he started excavating that part of the site and then he subsequently began excavating in the kayankas and found various kipus just a few centimeters below the surface of the kayanka floors and of those floors of the open sorting spaces and in all then in 2013 and 14 he excavated 34 kipus from that part of the storage facility so here's Alejandro work with his wife Rocio and so here this was in 2014 he was excavating some of the kipus at the time we had present here on the site the vice minister of culture of Peru who blessed the kipus as they were found now actually this is Luis Jaime Castillo butters so many of you know him and Luis Jaime was demonstrating how high off the kipu you should hold your camera to make a 3D photographic model of it right so they were photographed of course after they were excavated and then they were collected and then after being stored in the storehouse at Incoasi they were moved to a conservatory in Lima so this is the house of Patricia Landa and so there she received the kipus she worked on them she would slightly lightly miss them and straighten the strings out and pin them down and let them sit for a day or two and then after she had finished that then I was able to study them so she did great work of storing out all these great masses, great tangled mass of the kipus so when we start looking at the kipus where they came from so just to be reminded there are four of these cayancas the enclosed rectangular buildings there are two of these sorting spaces or drying spaces and then there are actually 36 of these big rectangular storage spaces and 209 of those small square storage deposits around the three sides of the building and he actually then in terms of where kipus came from they all came from this upper portion of the storage facility from the corridor here and from the four cayancas and the two storage spaces so from the corridor here where the walls had been pulled down there's a huge pile about that big around and about that tall of just a great tangle of kipus some about a dozen eleven to twelve kipus they were all in a big pile buried under the walls that had been tumbled down then from this sorting space this open sorting space there was a low depression and a basket had been placed there and two kipus were tied together and were enrolled they were put in spiral form placed in the basket and covered with chili peppers the cool thing about these kipus is that a lot of them are covered with plant products so these two are covered with chili peppers then in the corner here of this cayanca fourteen kipus were placed in the corner and they were covered with peanuts so they were just like peanuts were dumped over the top of them and then from the center of this cayanca there was one kipu placed in a depression and covered with a couple of handfuls of black beans so I mean I think that these things are found with the products that they were accounting I mean that's my hypothesis could be wrong but and also I'll just say there's nothing about any one of these kipus like this one for instance screams out beans at you or at least at me I mean it may be that there's some features but I cannot yet identify unambiguously let's say a bean kipu and distinguish it from a chili pepper kipu but it's extraordinary that we I mean we've never had this before the discovery of kipus with the products that we think they were being used to account for now talking about accounting and beans and peanuts and chili peppers of course nobody accounts beans and chili peppers individually I mean we caricature accountants as bean counters but not even accountants count beans and one by one so how do you so in fact how do you count uncountable things and what you probably do is you create standardized units of measure and we think that there's a way they were doing that at Incawasi if you look at the two open spaces as Alejandro began excavating the floors of those spaces he found that they had been marked off so that the one that you see here on the left that as he excavated down on that he came down on these rows of panels of squares so you see here a set of them in an overhead photograph you see sorry that's what it looks like it's pretty cool but they're these panels of squares like three squares and then 39 squares long and then a little walkway between this panel and the next panel and then all the way across so that in reconstruction this is the way we think it looked this sort of central area is not preserved it was worn down, weathered and so he has this part of the layout of the squares and this other side and so but this is the reconstruction of how we think this thing looked and so these are each of these grid panels I call these grid panels was three squares by 39 or 117 squares just like the Inca they never do anything like regular right you know like 120 squares or something like that we think there were about 30 of them laid out across here so making about 3,510 total squares each one of these is a pretty standard 23 centimeters by 23 centimeters so we think that what they were doing was that they would bring it produce into the site they would be maybe offloaded into those caliancas the rectangular buildings the products would be brought out into these drying spaces or accounting spaces and the uncountable produce would be spread across the panels of squares and then you take up a unit here's a unit of beans two units of beans three units of beans right so account for them in that way so in terms of the accounting of these produce we think that they're accounting for standardized units of measure of these goods and so this accounted this amounted to a system of standardization surveillance and control and the control of accounting at the site in terms of Inca Wassi accounting methods so we have found several different methods that are evidenced in the study of the Kippu chords themselves so this is like sitting and looking chord by chord and recording the numerical information and all structural information on each one of the chords so for instance we see a kind of summing within chords so that here's one of the 14 Kippus covered with peanuts here's chord 1 and it has this value 13,328 then it has another value here I'm pretty sure this is 208 you have 208 in that position on the other chords and then other smaller values totaling 13,328 so it's taking this value and breaking it down into smaller sub-values or having a number of values here that add up to that sum so I think that there's a some kind of process going on here of the summing of values or of laying out a single value maybe a certain quantity of goods that are brought in and then they get redistributed they get broken down into smaller values some get stored here some get stored there etc one thing I want to point out is that you have here a fixed value a repeating value in each one of these these accounting sets so I'm calling an accounting set a big value plus a smaller sum the total of that big value each one has a 208 208, 208, 208 so this is a fixed value that's represented here so keep that in mind here's another kipu one of the ones covered with peanuts and this one in fact has a fixed value as well but it's 47 so 3317 47114 498, 370, 2287 comes to 3316 so close and you can see here there's some sort of difference between these sums and the large sum that's at the beginning of the series but they're pretty close in general so I think this is accounting it's like you know so much of this stuff's coming in we got a big batch of like peanuts coming in 1876 units something happens with 47 regularly 47, 47, 47, 47 and then the remainder gets broken down into different smaller units so I think there's something here going on in terms of identifying a certain value one of these fixed repeating values and then breaking down the remainder into other values I suspect it's indicating how many of the units of that batch gets stored here or there or some other place so one question is why if these are both peanut kipus you have 208 in one kipu and 47 in the other kipu don't everybody answer it once because I don't know either, no but that's one of our problems here's a really interesting pair or two pairs of kipus so here are those two kipus that were inside the basket that were tied together there's the knot that ties the main cords together here's a pair of kipus that were found out in the corridor that had the walls pulled down on top of it they're tied together and there's the knot tying them together and now the interesting thing about these two pairs is that they're two pairs of matching kipus so that in fact when you read the numerical values on the different kipus these numbers match those numbers and these numbers match those numbers so it's sort of as though these were inside the storehouse maybe they're used for active accounting and maybe this pair was out in the corridor maybe they're a record to compare with what's being accounted for here in this batch but the further really interesting thing is that they record the same numerical values but not the same way and I think what they're doing is that they're so that you can see for instance in that chart over on the right that these are a set of numbers from a group of cords here of that kipu these are of that kipu so here's cord 44, 45, 46, etc there's cord 50, 51 I'm having a hard time reading over there and I have small numbers here so in the first cord of those two it's 141, then it's 15 then over here it's 126 and then over here it's 126 and it's 15 so 141 minus 15 is 126 or 141 minus 126 is 15 so you have the same numbers but they're organized differently in one you're subtracting the 15 to get the remainder and in the other you're subtracting what is the remainder of the other calculation and getting the 15 so the 15 is fixed again it's like our 47 and 208 the 15 is fixed they're interested in the 15 get the 15 but get it in one case by adding it and in the other case by subtracting it so if you go down you see that the fixed value is 15 but that it gets placed in a different place in the three number calculations there yeah so I think it's like a check a check on the math of it right and so here from this other set these two keepers they match but they and they have the same difference but now they're fixed values 10 these guys they're fixed values 15 this one's 10 so 394 minus 10 is 384 394 minus 384 is 10 so again you get the same three numbers but they're placed differently to provide a sort of check on the math of those numbers yeah so what they're doing then is that they're in these two keepers I think they're employing to what I call arithmetic paradigms or arithmetic formulas so one is a value and a fixed number and the remainder of that value and the other is a large value and a smaller value what will be the remainder of the subtraction of this number over here from the original larger number so they're working with two different paradigms but I think essentially what they're doing is they're interested in the problem of how do we ensure good math here and so they're moving those different numbers around putting them in different places in the arithmetic calculations that they're doing so I mean they're checking their numbers here we see a pair of keepers so the one on the right is the one we saw earlier that records the fixed value 47 turns out the one on the left also has the fixed value 47 and but now we're going to look at another aspect of the matching between these two and the two columns of numbers in the center just record the numbers above 1,000 that are recorded on those two keepers and if you look at those you'll see that in fact those they're matches of those so I can't really see it from there but 337 matches 3317 this pair sums that value 2287, 2089, 2089, 1271, 1271 and you go down and you can see that they're very close there and this is where the matches not exact but it's close so I mean I think that they're looking at similar values checking and balancing similar values at these large numbers so above the level of the 47 that matches in the two and now what I think is interesting about this is that again this is at the same time the double entry in western Europe is being established and it's a system in which you have values that are recorded in two columns and in one column is the debits entry and the other column is the credits entry and those values are supposed to match between the debits and the credits so in the two on the left you have the debit cash in Simone's name here and that debit cash in Simone's name you credit to Francesco and the amount is indicated here and the same amount is then indicated as a credit to Francesco and debit cash in Simone's name the same amount so debit and credit amounts in two columns and those are supposed to match up those are supposed to balance and that's the sort of core principle of the double entry bookkeeping system and I just think we're seeing something like at least the structure of the duality and the matching of pairs of accounts that is if it's not double entry which I don't imagine the ideas but I imagine it's something that's on its way to a double entry like accounting debits and credits of pairs of matching Kippus that we have here so just I'll get toward the end here now so one thing that I just want to reflect on is this business of these fixed values or the repeating values so for chili peppers we have the fixed values 10 and 15 for the peanuts we have fixed values 17, 30, 47, 208 so these are repeated over and over again in those Kippus linked to those products and I again I'm assuming but maybe I'm wrong that the association of those Kippus with those products means that in some way those Kippus are recording information about those products could be wrong but in any case we have this interesting phenomenon of the fixed values that are repeated in those Kippus and my suggestion for this is I think this is for a sort of nascent form of taxation I think what they were doing is a certain amount of goods comes into the facility the facility has to be maintained, has to be supported you have a certain number of workers who are there the Kippukomaioks and other people who are moving stuff into and out of the storage facilities so you have to support that so these are the king's crops that are coming in but I think that the accountants are taking out certain quantities and they're holding it out for the support of the facility I could be 100% wrong but I think something like this my suspicion is that we see something here like the introduction of the concept of the levy or the tax which I think if left to play itself out could have had profound consequences for the financing of the Inca state but it's really interesting but in terms of the kind of information that we have we have all of this accounting information and we always bemoan the fact that we don't have history we can't read those historical documents but I think there is history that's embedded in these there's history about the making of those panels of squares there's history about the tying of two Kippus together there's history then about the recording of matching bits of information and in fact there's often you could say a lot of history in these recorded administrative forms and in fact I just found this as I was putting the talk together by Herman Waugh who notes that income tax returns are the most imaginative fiction being written today which for myself that's not true of course I don't know about you but it is the case that a lot of fudging of numbers gets done and there's a story there and I think that we can say that there is a real historical writing that can be done from the information that we get in these administrative documents a couple of other general things I want to say we've been looking at the numbers when we look at those complicated numbers of like 394 10 384 384 384 10 etc we're looking at the numbers that we're familiar with so we can follow this those did not exist for the Kippu Kamayak the Kippu Kamayak saw only the knots and all of that math was being done with the knots right? so here you have for instance seven knots in the tens place one in the tens here and five down there that's 70 minus 15 equals 55 so we can say all that we can write it down in the number 55 but they never translated this transferred it into a graphemic form it was always in this three dimensional knotted form so I think that there's something going on in the minds of these characters that was just extraordinary that we haven't even begun to approach haven't even begun to think about the complexities of thinking in three dimensional knot forms of the level of mathematical complexity that we've seen here these are very sharp characters at the same time in fact we have this famous engraving of the contest between the Abbasist and the Algarist so the Abbasist this is a poor character he looks pretty like in a state of consternation here he's being challenged to do calculations with the Abbasist and is being opposed to the fellow the bright fellow who has the Hindu Arabic numerals and he's doing the math with the algebra there and so demonstrating here at this moment in western Europe about the advantages of the adoption of Hindu Arabic numerals and algebra that comes in then from Arabic from Islamic sciences and here we have then in our case in terms of the recording of values we have the knots a vastly different system very different, highly complex I don't even know how to compare these this is the true original case of apples and oranges but in terms of the calculation again here these the calculations that recorded here get done in these constructions called upanas and we're told in some of the Spanish chroniclers that they were moving around corn kernels or small pebbles, small stones doing their calculations and those then would get recorded on the kippu and here's a drawing by from a poma that shows one of these calculators and the kippu and no I don't think by the way that's those panels of squares just to say because it looks a lot like those panels of squares but I think this is truly a calculator and that those panels of squares are something else so this all of course and I'll be through in five minutes or less but this all comes to a close of course with the conquest so Francisco Pizarro and his 126 troops come in to the Andes in September wasn't it November of 1532 and then they meet in Cajamarca and Francisco's troops utterly defeat the more than 10,000 troops of the Inca and capture Arawalpa and this brings about then the beginning of the conquest of the Andes and the colonization of the Andes under Spanish colonial rule and Guamampoma then shows a number of images of the abuse of native people under Spanish colonial rule one form of this abuse that was represented not so much by Guamampoma as by other subsequent writers was in fact the mathematics and accounting of the system of administration that comes in at the time of the Spanish conquest and here in Bishop's book western mathematics the secret weapon of cultural imperialism he talks about the significance of the impact and the domination of non-western cultures by western mathematics so mathematics with this clear rationalism and cold logic its precision is so-called objective facts its lack of human frailty its power to predict and control its encouragement to challenge and the question is thrust towards yet more secure knowledge was the most powerful weapon indeed and in fact this is then in terms of the imposition of administrative forms and administrative procedures that mathematics was central to the administrative and colonial imposition on indigenous peoples so that the document written in alphanumeric form became the standard administrative document but then it's important to point out that indigenous people the Andean people did learn to read and write Blazvalera who himself was a mestizo who was the source of a lot of information in Garcelazo de la Vega says that we are slower in understanding their books than that is their kippus than they in following hours for we have been dealing with them for more than 70 years without ever learning the theory and rules of their knots and accounts whereas they have very soon picked up not only our writing but also our figures which has proved for their great skill so you did get a whole class of look and very often these were the former kippucumaios who learned to read and write and manipulate the numbers and keep records at the local level at least and in many cases as well they continued to keep the kippu records as well so at this time at the time of the conquest just before Francisco Pizarro shows up you have this accounting tradition in the Andes that's going on and in western Europe you have a highly complex system that is developing my notion here is that if we think that we have enough now we can begin to see something about the real complexity in the contours of inka accounting from the material from inka wasi by the way much of what I've shown you this evening we've never seen before I've been working on kippus for like 25 years and I've never seen fixed values repeated across kippus I've never seen those like formulas of like 394, 10, 384 394, 384, 10 this is completely new and most of what we see with the inka wasi kippus is completely new so we're getting really profound insights into the level of complexity of the accounting system of the inka empire and it was quite complex I think indeed I just think this is a wonderful image here the comparison of the accountants and also the sort of poignancy of the finger pointing at the accounting books in those two images which is I think quite interesting and in terms of the European side of this this is the moment of the development of what was known as the venetian method so first used by venetian methods in Italy in the 15th century this gets codified by Luca Pacioli who is known as the father of double entry bookkeeping I don't know if we have double entry or not it's not a big deal to me it would be pretty darned interesting if it was the case if we could establish that what we do have is highly complex state accounting that involves internal summing calculation with fixed values it might be some form of taxation we have paired accounts so a duality principle which is rampant in the inka empire of course with matching or closely matching values it may be something like nascent double entry like system alternate arithmetic formulas for checking sums and methods for producing standardized units of accounting for uncountable products so I think we have a lot of information to go on to begin to like study seriously inka mathematics in accounting thank you very much still the thing is it's a part of the arithmetic calculation to get to the total so if it is persons then in some way you are counting in the persons adding them to the products if those other numbers are products are about products right so you have to add them in so that doesn't make a whole lot of sense to me could be but but yeah thanks to alize inka wassia to the rest of the kanyepi valley in our survey we found several thousand storage structures on the same side of the hill 10 to 15 kilometers higher than the site I'm wondering how does how you would take all those different storage structures presumably something has to be going on to move things from those individual structures into here and how you would be accounting for individual structures or locations on the sides of the hill or who is taking care of particular structures and whether that would be tied in in some way to some of these values I suppose to get down to the truth of the matter here we are going to have to just get down on the ground and see how many storage facilities here and how many storage facilities there and count them and mount them and do GIS and begin to hypothesize then that so many storage facilities here were used in this or that manner in the provisioning of INCOACI so I think it's as big as task is my biggest dread is I don't want to start having to count beans but I feel like I have to count all of these damn like chili peppers we have a lot of that data we have a lot of that data in our survey you're welcome oh you do you've been my best friend for years now and then is that Peter? it just seems to me that INCOACI is a facility that's taking care of the army in a combat situation and the product might be coming in and going out faster than in normal INCOACI I think of state warehouses where the tax relations coming in and accumulating maybe later getting distributed so it may be that the accountants have to have a faster means of recording what's coming in and what's coming out and I thought when you're tied together people that might have something to do with that some sort of solution to the faster movement of the product how would that move in faster if you keep track of the faster I would think that in the state warehouses they would have separate key groups for incoming and out going well presumably there's a relationship between people coming in with llamas or carrying bulldoze and coming in and they bring the stuff in and I don't think the accounting gets done before the stuff comes in and I think gets put into accountable units so I mean there's a sequence of this it's a linear sequence isn't it the stuff comes in and then it gets accounted it now gets counted and then it gets accounted and so I'm not sure how you speed that up especially in the fact in the circumstance where you're not doing the accounting it's not an abacus right you're not doing the accounting with the accounting device you're doing the accounting with the accounting device and then you're recording the results of the accounting on the key group so there's still a linear time like process that you have to go through in order to get the information in the key group that's a fascinating thing I've heard in years I was very competitive when he told me about the ability to suggest something and he told me it might make some sort of sense one of the things that I do is the budget protection for the non-profit organization that's very incorporated and with large complicated numbers like this unless there's some means of self-checking you don't know that you have the wrong numbers there until it's too late that's what I'm basically assuming the payers are I think it's possible that recurring numbers could simultaneously solve 3 2-3 things and with that I may be completely wrong I think the one that I think the overhead idea is really intriguing the cost of the story which is integral to any sort of enterprise system a cost-efficient business overhead so they might represent an overhead value but one of the things in spreadsheeting that you want to calculate a set of different ways from the same or at least two different ways from the same set of numbers is the overhead and you know that everything is correct when those two pallies match if they're off there's some error in that spreadsheet that you have to go through one and one's possible and we saw things that were off that's okay but if you're a food budget for field schools off of $10,000 you're in trouble I think I'm under by a different problem I mean I understand but also it really sounds from what you said that it's a way of checking and you would use that overhead number and either saw it and had it made and then I can suggest maybe a third use of those is it the valid is that you would want a carton of coffee so these may be actually you do two of these and you calculate it in a different way and you see it plus or minus a few sacks of peanuts if they agree and then you establish some calculations are correct and you have to say the carton of coffee should be sent off to the expo for planning purposes and be sure that when the big man comes through with his army that there isn't even enough food and the two keepers agree so I don't know if any of that makes sense I mean the interesting thing about that is that and you're saying it like that makes me think of it is that the only thing that's really fixed is the overhead value I mean because we saw in those numbers that there's leakage and sometimes they're counting more than the big sum and sometimes less so you know some's getting lost some's draining out maybe there's a mis-copulation here or there but it's certainly close within the ballpark but the the overhead value is fixed I mean it's like our taxes are always like our expenses are not fixed but our taxes are always fixed and the amount so like the state wants it it's like slice of the pie and it wants it to be fixed and that comes out first and then you jigger the figures there and hopefully they're close if they're not well nobody got in trouble but you know how much how much difference was allowable before you had to like stop and really fix things and you can even have a sort of stress that these people got out there because they think of the numbers well one presumes yeah exactly so they would want to have to stop it oh absolutely you haven't seen any other cycles what Carol Mackie excavated at Manchana Manchana excavated a room excavated a room that had some markings on the floor and she said that in that and there were squares but she saw depressions in the centers of the squares and she thought there they were actually to like a giant calculator but that's the only other example I know of there is the side of Kebrada de la Barca which is not far from Pagnete of course and there there's that rectangular storehouse that has to water polished cobbles embedded in the plaza that make lines like layout lines in the plaza that's a little bit like this but those are the only examples I know of I would just suggest that if there's another way to roughly get the standard heat and the measures that people actually are producing in the front you might be dealing with receding in person inventory or something like that if people show up and say I have three hundred and eight other units you know like okay hold that with how young I am but with someone bringing in like Tim Yamalod's of deans know how many units of standardized units of deans they have I'm assuming no they just fell off the path and they come in and they spread it out and the accountant says oh you got 42 units of deans there and so you know I don't know they had it for whatever but then also after it gets stored all stored together in one of these deposits then the general down on the coast says we need a hundred units of deans down here so they come and you know I mean we talk like this this is the way we talk you're doing business here right I mean I suppose you're doing business here you're moving deans around back there yeah something he was talking about this issue is that as far as I understand the keepers are tied this isn't the most editable medium that they have it's not terribly editable although it is editable you can untie knots but it's more trouble it's easier just to take the deans string off and tie them to string it's something of an investment because you do have to get these strings and tie them and it seems like there is some level of investment in the creation of this keep boot which makes me wonder is this something that would be very convenient for the counting of an actual inventory or does it seem more like this would be a ratio like a fixed ratio this grouped down here in this valley brings up 394 coca loads of beans every year if we keep 10 and 384 sent along maybe these are like paradigmatic this is the standard that you're expecting to meet and you need the standard or you're expected to meet the standard and maybe one is the standard and the other is the degree to which you like need the expectation I mean that would be like a checks and balances sort of thing and something has to account for the dates as well is that what you were saying yes absolutely there must be time here somewhere as well and the keepers must record the time and when you figure out I mean they must be recording the time because that's critical because you have income and you have output like it must be that you can must be that you keep track of that in a relative sense income output if not in an absolute sense you know the fourth year of the Inca Rural Project would be a funky record that's the question from the last set okay I feel sure you must have noticed this but you didn't mention it so I want to say about your fixed values PNAS your numbers were 17, 30, 47, 208 17 plus 30 is 47 47 plus 47 is 108 108 excuse me 104 10 plus 2 is 208 so there's a mathematical progression there 47 no I didn't thank you now that you know to tell you the truth I never even thought that 17 and 30 was 47 it's just you just can't think of everything but that's pretty low on the totem pole I'm thinking about I'm just never struck things great this is obviously fascinating and what's encouraging to me is it shows that this crowd has a hole actually has capacity for some sort of productive work with numbers we probably could entertain alternative professions to you thank you so much for the functions of this meeting these two million studies are now finished and for you who would like to continue we encourage this the president's reception is in the archaeological research facility