 What was that? That was an evening greeting in my people's traditional Lakota language. I am absolutely delighted to be here and talk about a subject that I really care a lot about, and I really appreciate this invitation. And John, thank you also for that kind introduction. So what I'm going to talk about basically is this. The general outline is going to be, I'll start by talking about the relationship between mathematics and culture, because there is a theory out there that if there's one topic that you can think of that is totally culture free, it's got to be mathematics. That's not entirely true, and I want to convince you that there's evidence in native societies of that. Then the central part of my talk is going to be about indigenous western hemisphere mathematics, and that's going to be almost exclusively Mayan mathematics, but I'm going to mention that this PowerPoint slide or this PowerPoint presentation I have is actually, it has a lot more slides than I'm going to go through. There are points where you're going to see the little thing on the screen that says Skip. I'll click it and I'll go past those, but I also want it to be a document for people who might be interested in seeing some more about it, like what Aztec mathematics was all about, for example. So that'll be in there, and I will leave that here and anybody who wants to check with some of these people, they should be able to get it to you. I also want to mention in the central part of this in particular, there's going to be some number theory in there. I also understand that there are, for example, students here who are from courses that just want to hear about the cultural part of it. Don't worry too much about that because really what I want you to notice is that the mathematics itself that is being done there actually is quite impressive, even if you, and maybe sometimes I, don't understand everything that's going on there. The point is that this is mathematics that really was being done in the western hemisphere before European contact. I'm going to end the thing by showing why some people believe that the world was going to end four years ago because of some Mayan numerology that actually the Maya thought were nonsense, even if the people that were trying to purvey it did not. So that's quite a bit. Let's get on with it here. Why do I care about Native American mathematics? Again, it's this issue about the fact that mathematics can sometimes carry some culture in both directions. So does culture influence mathematics and conversely? I want to start with a couple of quick stories that perhaps might convince you that maybe Native people didn't really care much about mathematics, but then I'm going to try to counter that argument. So this one is from Copper, Inuit, Oral Tradition. And it goes like this. This may be a not so positive culture of reflection on pure mathematics, and right away I will apologize to any number theorists in the audience because they're going to think that you're talking about them, about you. So two hunters return, one with a wolf, the other with a caribou. They begin arguing as to which hide has the most hairs and in order to settle the argument, decide to have a contest, each pulling the hairs out one at a time. They count and count and become so engrossed in what they are doing that days pass and they die of hunger. This sort of reminds me of Andrew Wiles locking himself in his attic to prove for a month's last theorem. He didn't quite die of hunger, but the Inuit story teller added at the end, that is what happens when one starts to do useless and idle things that can never lead to anything. Well, there goes the interest in my geometry of bonnock spaces, what the heck. All right, another story, actually another fact. According to early 20th century accounts among many crew of that time, it was bad to try to count above a thousand since honest people have no use for higher numbers. So this book that I, from which I took these stories, really is an excellent read if you're interested in this kind of stuff. But actually native people did have a very close cultural involvement. Many native peoples had a very close cultural involvement in mathematics and since I am a Lakota guy who is standing on three fires ground, Anishinaabe ground in particular, Ojibwe being one of those peoples. I'm going to point out some stuff about a wonderful attitude that they had. I believe it's a very rich attitude that they had just toward number. Here was the idea and I actually heard other Algonquian language people, particularly Blackfoot elders in Canada tell me the same thing, that counting is okay but the thing is that numbers, if you tell somebody, I have 16 of these objects and they've never seen those objects, that raw count conveys something very sterile. They're these 16 things but you have no idea of what they are or what they're like. So their number words actually have a joint to them, suffixes that indicate all this rich other stuff about the things that you're describing. So the number words, the suffixes can also indicate dimensionality, manipulability, whether or not the things are organic. So if you have 10 hard inorganic stones and you see the word, that's the word for 10 in the Ojibwe language. Okay but you don't just say you have Midaswa Asinine, you have 10 stones. You say I have Midaswa Abih Asinine to also tell the person with that number word that you have 10 hard inorganic objects that you're calling stones. And then you get a better idea than just this raw naked count of what it is you're talking about. You have other suffixes, Minak, you can use for 3D organic solids. Well, blueberries certainly satisfy that. So if you want to talk about two blueberries, you say Nizho, two Minak, 3D organic Minan blueberries. And similarly you can describe four two-dimensional organic muskrat skins by saying you have four muskrat skins and the things that you're talking about are two-dimensional and they are organic. So all of that gets contained in the number words because culturally it just does not seem right to convey only the count. Now I very much appreciate the advance but I've got a point now to be able to skip because I have a number of other examples of Ojibwe words in there that anybody who's interested can look at the PowerPoint slide to see or PowerPoint file to see. This is not unique to western hemisphere, you know, native people to American Indians in both North and South America and Central America. The Maori also have this attitude which actually has been born out in their educational system about the importance of understanding a relationship between culture and mathematics. One of these authors of this statement, I know, Tony Trenick, he is a Maori fellow. I spent some very interesting time with him in New Zealand talking about this. That he worked on a project also with mathematicians at the University of Auckland and several other Maori elders and several other people interested in this to try to reconstruct a mathematical linguistic in Maori that made sense culturally for them. Now normally what happens, you know, particularly in portions of the world that have been subject to European contact where the people spoke an indigenous language, the words in English or French or Spanish just get ported into that language, basically verbatim with no understanding that there may be some cultural stuff that they carry with them that really doesn't work for that particular culture. Seems hard to believe, but here's the evidence that happens that way. First of all, this is Tony's statement about it. Cultural practices including ethnomathematical ones cannot be separated from the language in which they were developed. Changing the language or the linguistic register in which the practices are discussed will have an impact on how the practices are perceived by students. In particular, what you're doing is you're porting words into the language that really didn't originate there. This could result in a loss in the fundamental values that would normally accompany the practices, such as for the Ojibwe. The fundamental values that all that other information is supposed to carry in numbers. So what they did was they worked with elders to actually reconstruct if there ever had been one before, who knows, but to construct a mathematical linguistic that the elders and others said made sense within the culture of the Maori language. Now in the early grades of students in New Zealand, the Maori students in New Zealand get their education in Maori at a certain point and they shift to English. What happened was by using this reconstructed linguistic, they noticed a rather dramatic increase in the interest the students had in mathematics when they were taking those Maori courses and their performance later on when they got into the English language courses. So there is something there about the relationship between culture and mathematics. I always have to mention this in case any of you have ever heard of this because this is nonsense, but there are people a day that still believe it. There's a long-standing piece of folklore. I could have a stronger language to describe that claiming that there's something inherent in us or our cultures that makes us naturally bad at mathematics and the worst manifestation of this happened in the 1980s when there was a collection of articles that appeared in reputable neuroscience journals that said there was something genetically about us that made us bad at mathematics. This got all tangled up with what are called hemispheric dominance theories, the left and right brain stuff and the theory of the right brain to Indian. All of these results were either never replicable or they were actually genuinely proven to be wrong, but meanwhile we had to put up with some of the skull measures. You thought the phrenology went out in the 19th century. Well, what harm did this ultimately do? There's a generation of counselors in Indian schools that heard this and believed it. Some of them are still there and some of them are still counseling their students not to go into mathematics because they're doomed to be bad at it. Okay, so that's just demonstrable nonsense. So could we do mathematics? Okay. Now, whenever you start talking about genetics in American Indians, I and anybody else who does that is on kind of treacherous ground. I will say that the evidence is that genetically there is a lot of similarity between the people who came across the land bridge into this country in several ways, into this hemisphere in several ways. Enough so that if something genetically is making some of us bad at mathematics, some tribes are bad at mathematics, it really ought to be sort of universally affecting us and that sure is not the case as the Maya demonstrate. So here are, if you hear the words are, here are the homelands of the Maya. I didn't say were because they're still there. Okay, the area around the Yucatan Peninsula is where many Maya people still live. And many of you will, I suspect, have seen the basic Mayan barn dot system, which is really kind of interesting. It's a base 20 system with numerals, well, from 1 to 19, and another one we're going to talk about, that are formed this way. If you want to write the number one, there it is. Just one dot. Two dots would be the number two, four dots would be the number four. Now at this point in time, if you start building on that, you're going to end up having things disappear into a sea of dots. So what you do when you get to five, I'm going to replace five by a bar by smearing those five dots together. And then if I want to represent 13, five, ten, there's three dots for three more, there's 13. That's the numeral that would indicate 13. And to do that up to 19, and once you get past that, well, you have the Mayan system is a place value base 20 system. So if this were a Mayan number, let's say, I'm about to put something over on you, but I'm going to end up clarifying in a second, okay? So here's the unit's position, right? Okay, so you have 12 units. And then in the 10's position, oops, 20's position, right? Base 20. So in the 20's position, you have three 20's, that accounts for 60. And then the next position over, you go up by another factor of 20. This is the 400's position. So here you have 15, 400's. One more over by another factor of 20. That's 8,000's and there are six of those. You put that all together and that would be the way that the Maya would represent the number that we call 54,072. Except what I'm putting over on you, it's not the way that they represent them, because this is the way we do numbers in our Arabic system. Right to left with units on the right, the Maya tended to stack these things instead. Or put them on their side. There are several different versions that we see in stellies in these monuments that appear in Central America. But we can think of it this way. There's no harm in sort of thinking of it that way. But now here's the thing. You have a place value system here, numbering system. What do you have to have to make a place valuing numbering system work? This is audience participation. I saw this and I saw this. You've got to have a zero, right? The Maya did and here it is. And there's an interesting, totally false apocryphal story about this I'm going to tell you. Nowadays, not everything I'm saying is going to be a lie, I promise. I'll warn you when it is. Okay. That's clearly a picture of something, right? It's a pictograph. So what is it? Almost universally, and I agree I've seen the evidence, I believe, and ethnomathematicians, which I am not one, but they seem to be fairly universally agreed that that is a picture of a shell. A seashell or something like that. Reasonable, right? There's another thing that back in the middle, part of the 20th century, ethnomathematicians believed it could be instead. And wow, I would sure like to know if there's any chance that they're right. Okay, and so here's how this, here's how, but they're not, okay? It really is a shell. But let's suppose we project ourselves back to 1940 and we have two ethnomathematicians who are sitting here quarreling with each other about what that picture is. And one of them says, that's a shell. And the other one says, the Maya did not live on the shore. It's a leaf, shell, leaf, shell, leaf, right? And then it occurs to one of this flash of true genius and insight. How can we settle this? And in a moment of wonderful revelation in his head, he says, why don't we ask a Mayan? Okay, so they get on a boat and they go down to Yucatan Peninsula and they find a Mayan elder. By the way, this story is totally apocryphal. I've heard it, but it is, it just has all the ear marks of apocryphal, all right? So it is. But it's just still too good not to tell. So they go down and they find a Mayan elder who is totally versed in the number system and they say to him, we need to know what this is. And the one says, yeah, because it's obviously a shell. And the other one yells, leaf, shell, leaf. And the Mayan elder is sitting there thinking, you know what they say about these academics is right. The arguments are so fierce because the stakes are so small. All right, so he says, stop, stop, stop, stop. Isn't it obvious? Well, yeah, it's obvious. It's leaf, shell, leaf, shell, no, no, no. He says, okay, now this again is audience participation. I need you all to sort of speak up on this. How many? Five. Five, right? How many? Four. How many? How many? Five. How many? One. How many? Five. Do you see it? Okay. I would like that to be true. What difference does it make? It's a big one. Look, if you're doing sand reckoning on the shore with a stick and you're, you know, writing down a number and you say, wait a minute. I'm not going to put anything there and I've got to put something there. And you look around, you see a shell and you just grab it and shove it in there. That's one thing. But having something that represents a number that means one less than this, that means nothing at all, that is a much deeper thought. Okay. And I really wish that we could show that's true. You can't because there was way too much it was lost in the conquest. Okay. So we won't know. And yeah, the evidence is very strong that it really is a shell. But I could wish, can't I? Okay. So my point was showing you Yucatec words in a tone language. If there are any Ojibwe people in the audience, you know that I better apologize for my pronunciation of your words. Well, this would go a whole lot worse. It's a tone language and I can't do that. But I can at least sort of get the pronunciation right because I had a Mayan walk me through this once going like this. Okay. And there's a point to this because actually there is something here that also talks about how number and mathematics can convey culture. So just go through these, let me just quickly go through these words. Hun, ka, oh, kan, hu, hu, hua, hua, hua, hua, hua, hua, balon, la hun, baluch, la cha. Well, you see some kind of relationship between this number for 12 and the word two. But there doesn't seem to be any between 11 and one. Basically you have 12 unique numbers here. The way that the 12 unique names for numbers, okay, which apparently they had enough use for that they gave them unique names. So once you hit 13, look what happens. It gets real mechanical now. Oh, la hun, 13, oh, la hun, 14, same thing. You're just 13 through 19. All you do is you go over here, 19. Balon, la hun, you take the number, okay, 10 and you append to whatever the front of it is whatever you have to do to bring it up to that number. Then finally you get to ka, which is a different number. Okay, but somehow they saw a special need for numbers one through 12. Okay, and that might indicate, if we didn't already know it's true, that maybe the, my had a lot of interest in trade in merchandising because when you have unique numbers like that for one through 12, frequently that means that you're interested in selling things in batches of 12. Okay, or having that as your standard lot size and there's a real good reason for that that has to do with divisibility. Okay, my father-in-law was an egg farmer and he always sold eggs by the dozen, which is handy because if you have one standard batch of eggs you could sell people either the whole batch or half of it or a third of it or a fourth of it or a sixth of it or of course single eggs too. Okay, that's fine, that's handy. But on the other hand if your standard lot size is 10 and somebody wants a third of that you're going to have a real mess on your hands, won't you? So that could indicate that maybe there was some interest in trade and batch sizes. Okay, so there's an example where culture can convey something or well mathematics can convey something about a culture. Okay, if you've ever seen 18 rabbits granola it's some real high-class coffee shops carry that stuff. Actually if you look on the, if you look on the box frequently they'll have this little story about it that contains that number that we just saw for 18. That's a little bit different spelling but here's the Wikipedia explanation which is pretty close to what they tell you on the box. 18 rabbits was actually, 18 rabbit actually was a Mayan ruler. Okay, and this stuff is named after him for some reason. Okay, so that's just a barely outside chance you've seen that word before. The Maya had two calendrical systems that they ran simultaneously and that had a lot of interesting mathematical consequences. So they, now do you all have a copy of that sheet that I handed out? Okay, could you turn? I wasn't quite planning on doing it this way. Turn to the side, I'm going to. The side that says parsing Mayan dates. Okay, and at the top of that I have an example of something called a calendar round date. I won't define that quite yet but you see it says, for a hau eikum ku. Now, the Maya had a vague year okay that consisted, or hop in their language, in Yucatec they consisted of 18 months of, I'm putting months in quotes, it doesn't have anything to do with the moon, okay? But 18 months of 20 days each, that accounts for 360 days and then there was this little residual period of five days at the end of the year. It was called the vague year because of the lack of a leap year so it did not sink completely with the solar year. They had that little deficit there. The little deficit in the 365 days it has to be made up by a leap year. The Maya knew that. They knew that this didn't stay in perfect sync with the actual solar year but they didn't really care very much because over the course of one person's lifetime it wasn't going to get that far out of sync and besides getting out of sync with the seasons in Central America wasn't such a big deal when the seasons really aren't that much of a big deal themselves. By the way, occasionally you'll see claims that the Maya had computed the solar year to be 365.2422 days which is the correct number. That's generally considered to be nonsense. You see it, ignore it. That's one of the two calendars. There was also the second calendar called the Sacred Rounded Solcane Sequence of Days that that consisted of two cycles running simultaneously one of which was the whole thing ran simultaneously with the vague year cycle of 365 days and one of those two cycles in the Sacred Round had repeated every 13 days had a cycle length of 13 days and the other cycle repeated every 20 days. Now, but for the moment let's go back and take a look at this vague year. That's the one that's easier to think about because it kind of works a lot like our calendar does. So, here are the vague year month names. Remember I had 18 of, okay, so first I should point out on this sheet that says parsing Mayan dates. Ignore the calendar round part and just look at that date for Ahau 8 Kum Koo. And there the vague year date is the 8 Kum Koo part. Now, we have these 20, excuse me, we have these 18 months of 20 days each to account for 360 days and at the end we have this little five day stubby month that you call Y HIP, it only had five days in it. Okay, now you see the 8 Kum Koo just means that in the month of Kum Koo the 18th month, 18th, 20 day a month of a year okay, that was just the 8th day of that month. Okay, just like we would say the 8th day of March. No difference at all. So, yep, the vague year dates were named just as you would expect. So, pop was the first month of the year. So, you started with one pop, two pop, you went up to the 20th day of the year, was 20 pop, and then you started with the second month. And it was one whole, two whole, all the way up through 20 whole and so forth until you got to the end and those five days were one Hawaii, two Hawaii, three Hawaii, four Hawaii, five Hawaii. There were variations for the last day of the month that for a time might have excited some people into thinking that maybe the Maya had the meaning, had a different meaning for zero just besides a placeholder but it's kind of doubted that they really did. So, the last day of the month might be called 20 pop but you could also leave a number out and just say end of pop and mean the same thing or you could say seeding of wool. And if you said that this when some people got excited and said, whoa, they're talking about the 0th day of wool, they would not have thought of it that way. They just meant that it's coming right there next day. All right, now, here on the first slide you saw that little funny thing down here on the lower left corner. Remember that? Of course, remember everything was on every slide I've shown, right? Okay, I would go back and show it to you except it'd take me a long time to get back here. In the lower left corner, this was there and that actually is a vague year date. So, that fourth month in that list was called sots. And what this represents is och le hon, 13, the 13th day of the month of sots. This little head variant thing is what it's called. That little image indicates the month of sots. So, that's how they would likely represent that but that's how they would represent that. Now, that takes care of the kum ku part. The first part, the four a hao, is from the sacred cycle and that works a little bit differently. The sacred cycle has 20 day names that are listed like this, imikik and so forth, all the way up through koak and a hao. A hao will appear again. So, you might want to remember that is the 20th day name. Those were kind of like weekdays in the sense we might say, well, how do the days go when you're thinking about weekdays? Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Repeating cycle of size seven. This is the same thing except their week was 20 days long. So, they would say if today is imik, tomorrow be ik, the day after will be aakbao and 19 days from now will be a hao and then the 20th day from now would be imik again. So, that was a 20 cycle and running in parallel with that, we had day numbers sometimes called this kind of getting overwhelming. There's a way to fix this to make it less overwhelming and that's what we're going to do as soon as we get all of this stuff out of the way, respecting the fact that we should say the way that the Maya did it. We had this other cycle running in parallel, numbering the consecutive days except that repeated at 13. So, you had one imik, okay, if you started with one imik, you had the first day, the first number, the first day name. Then you went to the, all the way up to the 13th number, the 13th day name and the next day still had the 14th day name and continue on down to the 20th except now you've got to start this cycle over again at one. Okay, two cycles running in parallel. Who, we put all this together and you have calendar round dates now which is what I claim that you had on that sheet that I handed out and this is really the way that the Maya always did this. They would have not just one or the other but whenever they talked about a date they would talk about both the sacred date that 13 cycle and 20 cycle are running together and they would also talk about the day in the 365 day a year and they would write them side by side. So, on that sheet you have where it says, okay, at the top, four ahau, eight kumku, the four ahau, the four is your place in the 13 day number cycle, the ahau is your place in the 20 day, okay, day name cycle and the eight kumku tells you where you are in the year. Okay, now the whole thing seems to have disappeared into a collection of Mayan words with some numbers intermingled. So how about if we do this instead and this is what the modern convention is. Okay, down at the bottom of the page you see that ordered triple, four comma 20 comma 348. So what you can do is take this four ahau and eight kumku and say, okay, the sacred round date is four ahau. That means the fourth day in the 13 day cycle and ahau is the 20th day name. That's the 20th day in the 20 day cycle. So you represent that is just four comma 20, whole lot simpler. And it turns out that eight kumku, computation is done here for you, eight kumku is a 348th day of the year. So you just say, okay, we'll write 348 for that. We have a nice ordered triple that contains all that information and all of that Yucatec language and makes it a lot easier to think about. But that is a modern convention. The Maya just reveled in doing it the other way. So this is just what was on that slide, okay, that I just gave you. So problem one. Now I'm going to do, now I'm going to do a little bit of number theory here and I'm just going to ask you to believe the results. Okay, the mathematicians will know how I'm computing it. So the first question is this. What you really have here are three cycles running simultaneously. A 13 cycle, okay, a 20 cycle and a 365 cycle. So the question is, if you're at position, when you put all that together, the calendar round position, the position you are in those two calendars on a particular date, 420, 348, how long is it going to be before that date appears again? Which will be the same as asking how many different calendar round dates can there be? Okay, and it turns out that what you do is you just compute the least common multiple, the number that is the smallest positive multiple of those three different cycle lengths. And that is 18,980, which for some reason I wrote here as 52 times 365, which it is, right? That's 52 times 365 days. Now imagine this. Somebody is born on the date 4 Ahau, 8 Kumku, which we've just written as 4, 20, 348. What would that person consider to be their really significant birthday? It's going to be the day that 4 Ahau, 8 Kumku happens again. Right? So with that sort of leading introduction, what did the classical Maya consider to be the natural human lifespan? Let's see. That's how many different dates there are. 52 times 365. 52 years, right? And if somebody actually managed to hit their second birthday, there was just, which people did, there could be a whale of a celebration. I doubt if anybody ever made three of them. Okay, but people did live to be 104 years old, even considering that those years aren't exactly the years that we would think of. Now, distance numbers. We're actually closing in now on the end of the world. So, distance numbers. Okay, turn to the back of that sheet and you'll see there's distance numbers in the long count. Just go over the distance number part of it. The Maya loved to compute the difference between, in days, between two given dates. Every time they talked about this date and this date seemed like they always had on a stele that talked about this, on a monument that talked about this, they would have to tell you how many days are between those things and their computations, as far as we know, are always right, which is significant. So, for distance numbers, we're going to talk about the number of days between two dates. And one way we could talk about that, oh, we could write some notation down like this and say, well, there's days, there's 20-day units, 400-day units, 8,000-day units, and so forth. We do that, you know, just sort of like their base 20 numbering system. And to do that, that is what they did with one small modification. Okay, this is days. This is 20-day units. They call these kins. Okay. These are 20-day units, which they call winals. And these, well, are they 400-day units? That's awfully close to what they considered to be a year. So how about instead going from here to here, instead of multiplying by 20, just for that one case, multiply by 18 instead. You get 360. And that pseudo year is what they actually put in there. It turns out that there's mathematical reasons that that makes some computations really, really easy if you do that. Okay, if you think about adding two things about this, just think that you just carry, you know, you carry an addition. Instead of carrying 20s, at that one place, you carry 18 instead. Okay, it really makes it really slick and easy to add these things. And they no doubt took advantage of that. So if the Maya wanted to talk about the difference between two dates, they could say, well, it's, okay, 13 kins, 8 winals, 11 tunes, 3 katoons. Okay, so these are like days, months, 360-day pseudo years. Okay, these are 20-year pseudo-year periods. And these things over here are 400 pseudo-year periods. Okay? Now, by the way, they're called, so this is called a katun. This thing is called a boktun, except that that word is the one word here that really is just a modern invention. But a boktun is just to give you some idea of how big these things are. That's 144,000 days. Now, so here's an example from a Mayan inscription. And now, look, having sat through all of this, we can all actually read Mayans. So let's take a look at this. Well, actually, you can. You can at least see which pieces of this go with what's actually on this steli. This is a sketch of a real steli. Okay? On 8 ok, you see the 8. There's 8 ok. 13 yak. You see the 13. And that's the yak. It was born bird jaguar, lord of Yaxchilan. Now, they could have just said on then, at a later date, on 11 Ahau 8 sec, he was seated in his rulership of Yaxchilan. Except they don't do that. They also have got to tell you how many days apart those things are. It's just like a compulsion. Okay? So they said the distance number was then ten kins, five wieners, three tuns, and two katuns, which you can represent this way. Okay? After which on 11 Ahau 8 sec, he was seated in this rulership. At least these were a bunch of primitive people in Central America. Would they get this right? Always, as nearly as we can tell. And that kind of indicates maybe something was going on behind the scenes about the way they were doing these computations. And again, I stress that too much was lost during the conquest to be sure. But particularly for the folks who know how to do this computation, okay, I'm not inviting you to do it on the fly here. Okay? And for those who don't, I just like you to take a look and think about the fact that the mathematics is happening. It looks pretty impressive. Okay, so here's what you can do. You start with a date, giving as your place in the 13th cycle, your place in the 20th cycle, and your place in the 365th vaguer cycle. Start with that. And later you arrive at a later date. So start at T0, V0, Y0. Starting date, you arrive at a later date after a distance number that you can write like this. Okay? N5, N4, N3, N2, N1. Okay? So that happens. So what you would like to do, if you know these two things, okay, if you want to check this computation, you know the beginning date represented that way. You can figure that out from the language, and you know the distance number that's given to you also. So you might want to check to see if you really do arrive at that ending date that they claim. Here's what you have to do. Some number theory. For those who are familiar with clock arithmetic or modular arithmetic, that's what we're going to do. So one Bakhtun, that's 400 pseudo years, as I mentioned already, that's 144,000 days. Well, it turns out that that's equal mod 13 to negative 1, and it's equal mod 20 to 0. 20 clearly divides that number. Okay? And it's equal to 190 mod 365. So first, you've got to know that. Then you do the same thing for cartoons. Okay? Those 20 pseudo-year periods. That's 7,200 days. That's equal to negative 2 mod 13, 0 mod 20, and negative 100 mod 365. You do the same thing for tunes, those pseudo-years, and the winals, and the canes, and you get that whole collection of, you know, modular equivalences there. And now you have these formulas. Just take a look at these formulas. Okay? If you start at this date, T0, V0, Y0, so in particular, if you start at this starting position in the 13th cycle, and you want to know what the ending position is in the 13th cycle, here's what it is. T, the ending position, will be the starting position, minus 1 times N5, the rightmost position in that distance number, minus 2N4, minus 4N3, plus 7N2, plus 1 times N1. Where did the minus 1, minus 2, minus 4, 7, and 1 come from? Well, right here. There's the minus 1, minus 2, minus 4, 7, and 1. They came from those modular equivalences. And then your position in the 20th cycle, okay, is given by that formula. You can find your ending position in the vague year this way, and that's given by that formula. But you're interpreting mod 13 here by mod 20, and here you're interpreting that mod 365. Okay, so, you need to haul out your favorite junior-senior level number theory book, like, you know, Niven-Zuckerman and Montgomery or something. And you dig through it, you know, and you think, well, yeah, I guess I see why that's true. Would the Maya have known how to have read a book like that that we used in junior-senior mathematics? As some ethnomathematicians, I think many, including Michael Kloss, believes that they did these computations for, you know, it may have looked different the way that they did it, the process they went through, but they actually did them just exactly the way that we did them. That they knew how to do that number theory that I just put up there. Because they had brilliant people. They valued people in their societies. Okay, who were doing this kind of things for hundreds of years. They would have figured it out. Okay, rather than just sit there and say, I think I'll figure out what the distance number is here and I'm gonna sit down and scratch out 146,000 lines and hope I don't forget one of them. Okay, yeah, no, they would have, they almost certainly would have just known how to do this. But again, too much is lost to be sure. So let's check the computation and say that if you actually do the computation, Bert Jaguar was done on the sacred date, eight position in the 13th cycle. Okay, 20th day in the 20th cycle. 13 yacht, well anyway, 810 I guess that is actually. And then that's a 193rd day of the year. And that's the distance number. Here's where you arrive at the end. And the ending date is 11 Ahau'i, just exactly as claimed on that study. They got it right. Another, I'm just gonna leave this for any number theorists out there who'd like to do this old problem. So here's another one, P'Kal, perhaps the greatest of all Mayan kings, certainly the greatest known one, the best known one, Lord of Palenque. There's this problem on his, there's this thing that's written on his sarcophagus that sounds suspiciously like an algebra problem that he's inviting somebody to solve. Okay, on this date he was born, on this date he died, and he lived through parts of four cartoons. Parts of four of those 20 year periods, right? King P'Kal, Lord of Palenque. So the problem is, it's not stated, but you know, somebody can scratch your head and say, well how old was he when he died? The interesting part is without actually giving you the distance number, they've given you just exactly the clue that you need to solve this uniquely. That part they lived through four parts of, parts of four cartoons. So almost as they were inviting mathematicians of the future to figure out how old he was. Okay, anyway. So, the long count and the end of the world. So now you know everything about distance numbers, which is what you need to know to decide whether or not the world ended four years ago. You remember that other thing that was on the opening slide, that little list of numbers separated by dots? That was today's date, in the Mayan long count system. Did you notice that everything I've talked about concerning dates doesn't actually give you an absolute date because if today is for a hao e kum koo, that doesn't tell you exactly when something happened because for a hao e kum koo is going to occur again 52 years in the future. But they did have an absolute numbering system. And rather than look at this, let's take a look at the back of this sheet where it says the long count. Okay, these were the days computed as distance numbers, which we just talked about, from a starting date, way in the past, a particular starting date, way in the past. And here is the story, the Mayan creation story that leads to this. In their cosmology, they believed that it took the gods 13 bhaktuns, they didn't actually have that word, but 13 of those units, 13 bhaktuns, 13, 144,000 day periods to create a world that people could actually live in. And on the final day of that period, okay, so from the very start of time, 13 bhaktuns have passed. And they said, what happened is on that day actually the darkness lifted, does this sound like other stories you might have heard? Okay, let there be light. Okay, the darkness lifted and at that point the world was ready for habitation by humans. Okay, so that was the end of that period and 13 bhaktuns had passed. So the next day, you would say, see this thing, sort of every day of the past, this would have been one bigger until it reached 13 bhaktuns like this. So maybe you would say, well the next day should be called 13,0001, but they said no. A whole new era has started. So we're going to reset the odometer back to 0001. Okay, that'll be the first day of the new universe in which humans can be. So what turns out that if you know where they got this and you take a look at long count dates on Mayan monuments, you actually have enough information to nail down what their starting date would have been. So they run from about, you know, 8.12, 8 bhaktuns, 12 bhaktuns after the first day humans could be there. That was about 300 AD common era and it goes up to about 10,400 in 900 AD, 900 common era and there's enough information that actually ethnomathematicians are fairly confident that the Mayan base date was August 11, 3114 BC. You have to be careful about what calendar you're talking about. So we're talking about the Proleptic Gregorian calendar. Remember that funny thing about Julian and Gregorian calendars that happened a few centuries back where they had to correct the old calendar? Well, the Proleptic Gregorian calendar is the one in which you take the calendar that exists now assume it's always been there and project backward. Okay, so it's a Gregorian calendar projected backward into time. So please, is the Mayan base date there? And what happens when the odometer flips again? Okay, well, the gods had no use for a date in creation past 13,000. So once the odometer started rolling again we shouldn't expect them to have any use for a date past 13,000. And supposedly the wise maya completely understood the workings of the universe and knew that the modern world could not last longer than its creation. Thus, when the calendar would get to that number again on that day or the next, it's over. So when did that happen? Okay, here, you know, this is like using this system this is like 99999, right, with the calendar ready to flip. 12, 1919, 17, 19, which by the way incidentally hence be three co-opted conkin as the calendar round date. That was December 20th, 2012. The next day was 13,000. Okay, that's December 21st, 2012. So lots of people, you know, gave away everything they had went outside and waited for, you know, what's it called? The rapture, yes, that's it, thank you. Waited for the rapture. And there is evidence that it didn't happen. Now, by the way, today, assuming you can let this calendar keep going today would be 1304519, which is what that date was on the first page. Okay, I'm not making fun of the Mayan because the Mayans say when all this is going on they say, we never believed this. Okay, and you know the response is, yes, you did. All right, well, I'm not going to say much about the Aztecs because I'm within a few minutes of getting my hour through my hour. I would kind of like not to hold you longer than that. I do want to say something, though, about the Mexica people. Mistakenly called the Aztec. That was a misunderstanding and a conversation that occurred between one of the Spaniards and an Aztec and a Mexica person. Okay, the Mexica also had a really, really fine numbering system, which is described on the next 46 slides or something in great detail. But look, if you're interested in that, you can go look at it, okay, because it's on there. But you see that little thing there that says skip? We're going to click it. And I'll go down if I can get pointed to it, looking at this thing from an angle of about 30 degrees. Let's see. And we're going to go to the last few, very few slides. Ah, there we go. So all I'm going to say about the calendrical system of the Mexica people is, by the way, they're still there too. There's still five million people down there who speak Nahua, the traditional Aztec language as their first language. So the Aztec have not gone anywhere either. Okay, their calendrical system is similar to the Maya. It had, in fact, they got it from the Maya. Okay, it has a 13 cycle and a 20 cycle together, gives you the sacred cycle, and it has the 365-day vague year, but it has a special feature. A year is known by the sacred, they actually give the year's names. The year's meaning the vague year's names. They're known by the sacred day name of the 360th day of the year. The last day before that little stubby five-day month starts, a year is known by that date. So, and it turns out, of the 260 possible sacred day names, exactly 52 are used, and they occur in a cycle of 52. Okay, again, one person's lifetime, sort of. Now, here is a legend of Keakata Piltson that sometimes is conflated with a Mexica deity, Quezalcoatl of Feathered Serpent. They are actually different. Okay, but they're sometimes conflated. Okay, here is a legend. This was a legendary ruler of the Toltecs, the old ones, the builders of Teotihuacan, this amazing city, precursors the Mexica, the city was already ruined by the time the Mexica people got there. Okay, well, those were the Toltecs. Now, I'll mention that this all is disputed, it's controversial, but supposedly the legend was that Keakata Piltson, you know, the Arthurian legend, the once and future king, that Keakata Piltson would return to resume his rule in a year that was named Keakata. Hence his name, Keakata Piltson. He would return in such a year. One read. Akata is a 13 sacred day name. When does Cortez have to show up? Okay, now the thing was that this was considered to be a white god. And now, look, the Europeans who arrived there under Cortez, they had guns, and they had horses, but the Mexica empire was mighty. It really was. It was a very strong, very, very strong empire. And given the numbers of the Europeans, the Mexica could have shoved them just right back into the sea, that's the problem. Here's Cortez. It's Keakata. Cortez is a white guy. Am I about to try to kill Keakata Piltson? It does not go well with you if you kill off your gods. And so this actually paralyzed the emperor, Moktatsuma, paralyzed Moktatsuma, and he just could not do anything. Now, by the time that actually one of his generals put him back into shape and convinced him that something was seriously wrong, that gave Cortez the opportunity to go around and recruit the neighboring tribes, and the Mexica people were not very kind to them. They were, you know, taxing the daylights out of them and taking their people for sacrifices and everything. So he made alliances with them, and you couple that with the disease that was brought and sort of decimated the Mexica people, and the debt is what led to the conquest. So in one funny way, you might almost say, and I don't want this to be discouraged any of you from taking a number theory course, okay? But the conquest happened because you could argue that the Central American people knew too much number theory. Okay? I don't know if there's a moral in there. But if you're interested in this story, here's where I can point you to where you can learn more about it. For more on that story in the Mexica mathematics, including the symbol that's actually there for a cobbled reed, take a look at this poster, Math of the Aztecs, that you're going to find on the wall outside A2155 Mackinac Hall. Okay, it's all there. And in fact, it even extends on this, even expands on this a little bit. I will warn you, there's a couple of things I quarrel with. I think that they do make the equation, they try to equate Keapato-Pilson with Kessler-Quaddle, and that is not quite right. And also they don't point out that some of this is disputed. However, it has a ring of truth. It really does. And I do believe that there's a good chance that at least most of it is true. And it's laid out very nicely with a lot of nice pictographs on that particular poster. While you're at it, please also take a look at the other posters that are up there that talk to you, that tell you all about how mathematics and other cultures and what the cultural importance of all of that is. It's really a very nice display. And I'm going to actually finish with one last thing which is sort of bang, bang, hit that skip thing. Here. Okay, you probably figured out that stories about clondical systems have lots of apocryphal things and things that aren't sure about them. It's a way of having stories that are really sound, interesting and maybe aren't quite as true as you might think that they would be. How many people know that our months, September, October, November and December, which obviously got named after the Latin, September, October, and November because they're the 7th, 8th and 9th, 10th months. But wait a minute, they're not. They're the 9th, 10th, 11th and 12th. I won't ask you to raise your hand. But almost everybody knows that's because the Caesar shoved those two months right into the middle of the calendar that pushed things two months farther into the future, which for you, the future's that way, right? Okay, pushed it two months farther into the future. Okay. That is not the reason. Now, I'm going to let you do the research to see what the real reason was. And you can do it by answering this research problem, which just requires just a tiny amount of web research. Okay. George Washington was born February 11th, 1731, but that was by the Julian calendar, which the colonies were still using that. Now, the Julian calendar had fallen behind the Gregorian calendar at that point by 11 days due to the lack of leap years. Most of the rest of the world had already converted the Gregorian calendar, and the colonies finally did too. It actually took the colonies longer to do it than it took Britain to do it. But, so it was corrected to February 22nd, 1732. Well, there's the 11 days, but why did the year jump from 1731 to 1732? Once you know that, you'll know the real story about why those months are called September, October, November, and December. And with that, thank you for listening to me. And I hope the next time somebody tells you that mathematics is culture-free, you can stick out your tongue. I know better. Thank you.