Amir Aczel

Transcription

MALE SPEAKER: Good morning, and welcome to the first Talks at Google of the year here in Cambridge, Massachusetts. Today, it's my great pleasure to introduce Amir Aczel. Dr. Aczel is here to discuss his new book, "Finding Zero-- A Mathematician's Odyssey to Uncover the Origins of Numbers." This is the amazing story of his lifelong obsession with finding the origin of the numbers we use every day and have been using for centuries and the journey that took him around the world to answer his questions. Did I mention the story is amazing? You are absolutely going to love this book, absolutely. Dr. Aczel has degrees from Berkeley and the University of Oregon. He's the author of over 20 general books and textbooks including "Fermat's Last Theorem." He's been a Guggenheim Fellow and a visiting scholar in the History of Science at Harvard. Since 2003, he's been a research fellow with the BU Center for Philosophy and the History of Science. And as of 2011, he's been teaching math courses at UMass Boston. Please join me in welcoming Amir Aczel. [APPLAUSE] AMIR ACZEL: Thank you. Thank you very much. It's really a great pleasure and a great honor for me to be here at Google in Cambridge. As you mentioned, I've written 20 books, and they're all about other people, interesting people. "Fermat's Last Theorem" got me to write about Andrew Wiles and other amazing mathematicians. I wrote a book about Descartes, and a couple of books about Einstein. And this book is about me. So it's a very, very different story. And the reason I had to sort of-- it's not about me, but I had to write in the first person, which is very unusual for me. And it took a lot of time to get used to. It's almost like you're writing about yourself. And the reason was that I had this obsession about trying to trace the numerals, the Hindu Arabic numerals, and especially the zero. And I wrote a book proposal. When you're a writer, you have to write a proposal. The book doesn't write itself. And you need a publisher to be interested in it. And this is the first book in 20 that no publisher was interested in. I went to all the publishers in New York, and they all turned it down. And the reason is that the main character-- one of the reasons, the main character in this book is not like Einstein or Andrew Wiles or Descartes. It's a very boring guy. He's this French academic by the name of Coedes. And he was so boring that every publisher would just read the proposal, and they throw it away. It's boring. So I was very depressed. I didn't know what to do. I mean, this is the topic I care a lot about, and I can't write a book. So I just got worse and worse. And I got physically ill at some point. And I was about to give up on everything. And then, I got this grant from the Sloan Foundation, and I thought, wow. They trust me to write this book. So I have to do it. And the grant was to try to rediscover the earliest zero. Now, the zero is discovered by the Maya, and the Maya calendar has a glyph for zero which is like a crescent moon. But that didn't go anywhere. That remained in Mesoamerica. But there is a zero that became our zero. And it was studied by this guy that I mentioned, Georges Coedes, this French scholar. And it disappeared. And I wanted to find it again to bring it to the attention of the world of science. And it was also an excuse to travel. So if I can have the first slide, that would be great. So everybody knows the numbers come from India. Every Indian you talk to will say, oh, we invented the numbers. And we invented the zero. And so I sort of took it by faith. This odyssey of looking for the first zero was really an adventure to try to find the first zero. And it got me talking to a lot of people, emailing, phone calling, writing letters to some people who are less accessible. And at the conference in Australia, I met an Indian researcher by the name of C. K. Raju who gave a talk at this international conference at the University of New South Wales. And he said, people, you don't really know this. But the Indians invented everything. Einstein's relativity came from India. And Euclid's theorems come from India. The Pythagorean theorem was invented in India. And he went on and on. The audience got very upset. They would have pelted him with rotten eggs, but they're slightly more polite, and there were no eggs around. At any rate, everybody got very irritated. But when I got the grant, I thought, well, he seems to know about India and the things that happened there. And so I went to India to meet him. So India has a lot of temples. And there are markets where you can buy manuscripts. And I was sort of looking for things, like a fishing expedition. Let's see what I can find. And I came into contact with a Japanese mathematician who had traveled, by the name of Hiyashi, Takao Hayashi, who had traveled to this part of India called Khajuraho which has these really amazing statues on temples. And they get more and more graphic, let's say. But I took out all the slides that were more explicit because there's really no end to how graphic they become. And people go there from all over the world to see these statues. There's some hints of them. I don't know, maybe not. Maybe I was censoring this too much. And then, there are all these erotic statues. And then, there are numbers. And this is a magic square that was known since the 1800s. But after a while, Hayashi found it. But he forgot where it was. So he told me it was somewhere in Khajuraho. So I spent a day looking for it, and I rediscovered it. If you Google Khajuraho, you will find that magic square. But there's no photograph of it because nobody knew where it was. And it's rewritten in our numerals. But since you're all brilliant people, I'm sure you can figure out these numerals. So they're different from ours. And don't be fooled if something looks like our numeral. It's not. They are very different. Some of them are similar and almost the same. But they're really mostly very different. My wife, who is not a mathematician, figured it out in about 10 minutes. So I'm sure some of you maybe already figured it out. So you know about a magic square. A four by four magic square, a normal magic square, a regular one, it has the numbers from 1 to 16. Has to have a sum of 34. It's an easy proof, too. You can probably prove it to yourselves. So knowing that the sum of everything is 34, you can do it mathematically by setting up equations. But you can also do it the way my wife did, just by guessing, I guess. And it's a cool exercise to try and do it. I went there because I was fascinated by the magic square. But I also wanted to see early numerals. I said 11th century. We don't really know. On Wikipedia, it says 10th century. It's either 10th or 11th century. It's very early. Four by four, three by threes have been known in Persian and in China and different parts of the world. A four by four is more complicated. And if you set the sums, what's cool about this magic square is it's not only the two full diagonals, all the rows, and all the columns, but also every four by four in a corner, like the-- oh, two by two in a corner. The one on top, to the right and to the left, and the central one. So this is a very sophisticated magic square that has many possibilities of adding to the sum of 34. Which, I think, is magical. Maybe that's why they put it next to all these very strange, sensuous statues and phrases all over. And nobody knew where it was. I had to ask maybe 30 people. And then, there was a French tourist who heard me ask his guide. And he said yes, I've seen it. And it's in a different part of Khajuraho, less visited. So the main part is the western group of temples. And this is in the eastern group, which is across town. There's nobody there except dogs and a few children begging for money. For some reason, nobody goes there. And this is at the Parshvanath Temple, which is what that piece of information was lost. And here's a better view of it. So I forgot to tell you the best part. So I did go there. And I did trace also the earliest zero in India to a place called Gwalior, and that zero comes from the mid 9th century, 850 something. And there was a big controversy in the '30s and '20s about who invented the zero. There was a British scholar by the name of G. R. Kaye, K-A-Y-E, who was actually an expert on India, but was very racist. And he claimed that the Indians didn't invent anything, sort of the opposite of C. K. Raju today. And he was the first person to study the Bakhshali manuscript, which is very important in the history of mathematics, but undated. If they could date it-- it's made of bark, tree bark. If they could just take a tiny bit of it and radiocarbon date it, the whole history of mathematics would come into much, much clearer view. And we would know what the sequence, the chronology, was like. But the Brits don't allow you to do that. It's in Oxford, at the library there, at the Bodleian Library. And so we don't know. It could be first few centuries, even BC or AD, around that time. Or it could be 1200. So he claimed that it was 1200s, 1300s, something like that, Kaye himself. And he dissed the Indian chronology by saying, well, the Indians claim that their history goes back several million years. So of course they don't know what they're talking about. And he claimed, his theory was that the zero was invented either in the West, in Europe, and brought through Arab trade to the east, or the Arabs themselves invented it. It's unlikely that the Indians had anything to do with the zero. So there is this guy by the name of Georges Coedes who is originally a Hungarian Jew who immigrated to France and changed his name so it looks very French with the C-O-E-D-E-S and an accent grave, and the O and the E are ligated, and so it looks very, very French. And he just had this natural ability-- I'm very fascinated by people who have a natural ability to do things. Ramanujan comes to mind, as a mathematician, just out of the blue can come up with these identities. And nobody understands where they come from. And he can just invent or discover them. And he said a goddess, I guess, from one of these statues, the Hindu goddess Parvati gave him all this information. And you all know the story about Hardy, great mathematician, British mathematician, brought him to England and tried to figure out how he got this knowledge. And Coedes had a similar ability of understanding dead languages, like old Khmer. He could read a stele, and within a short period of time, translate it. Just an incredible gift. And he ended up translating into French 2,000 steles in Cambodia and Thailand. And one of these steles that he found had an early zero. The problem with the Indian zero in Gwalior is that it's 800s. And that's coincidental with the caliphate in Baghdad, the big, very extensive Arab trade. So if you have a zero in India from that time, it could've come there from Europe or from the Arabs, or the other way. There's no conclusive proof. But if you could-- what Coedes knew was that if he could find a zero that's earlier than Gwalior, the earliest Indian zero, he could prove that the zero was an eastern invention because it predates that Arab trade. And that's exactly what he found. He found a zero in Cambodia. And basically was able to completely destroy the theory of Kaye, and to establish that the zero is Eastern. But everything that he was basing this theory on disappeared. The reason it disappeared was because of the Khmer Rouge. So I knew that, and that's what I got the grant from the Sloan Foundation, to try and find, to rediscover the stele that may or may not still exist. We didn't know. But I guess they had faith that maybe I'll be able to find it. And once I got the grant, it's like, I had to do it. I had to find it. So my first stop after going to India was to go to Bangkok. And I kind of used this as just a great, fun city to go to and an excuse to show pictures of places I like. This is one port and temple there, and all these Buddhas. And it doesn't want to go. And of course, all the pickpockets are not Thai. They're foreign, foreign. And so this is why I went there. I couldn't write-- the software wasn't written by Google, so it doesn't work very well. It doesn't allow you to write the name correctly. The font is-- I wasn't able to add all the French. OK, so you say, why did you go to all these places? So I need to explain this. Once I started researching it, I sort of became obsessed with the philosophical idea of, why the zero? Why would the zero be invented in the east and not in the west? And it was, I think, we know conclusively that it's an eastern invention. I call it an invention. Of course, mathematicians would argue is it an invention or discovery? Most mathematicians I've polled, including some people we've talked about earlier, most mathematicians think, of course, that numbers exist. They're the Platonic point of view. The numbers exist regardless of us. But the ability to write something is sort of an invention. You invent the zero. And the zero that I'm interested in here is not the zero, just an idea of emptiness, even though it's very interesting. But the zero as a placeholder which allowed our number system to work so well. If you try to construct a multiplication using Roman numerals, then you can see why our numerals are so much better. So what happens algebraically, when you add the zero, you turn it into a ring. And you can do things. And these are stacked rings for each decimal, for each position. And you couldn't do that without something like a zero because you need to hold an empty space for the further ones. So you can write 2 is just 2. But 2 followed by 0 is 20 or 200 with two 0's, and so on. So it's also an intellectual idea for creating a number system-- and it's independent of the base, of course. It doesn't matter whether the base is 8 or 10 or 2, the way a computer would work. So the idea of doing that is really what gives our numbers their efficiency and power. But the idea of writing something that means nothing, there's so many, of course, people play with language. Nothing means nothing, or nothing means everything, and so on. So nothing, the idea of nothing, is a sophisticated idea. And I became convinced, maybe for no reason, that it's an eastern idea for a reason. And that it may have come from Buddhism, the void in Buddhism, you know? When you meditate, if you're a Buddhist, I guess. You imagine the void. And so I went to Luang Prabang in Laos, because that's the one town in the world that has more Buddhist monks per square foot than anywhere in the world. And it's also a very interesting place. Has anybody been there? I would go there soon because it's a gem that's going to be destroyed. They're building a high speed rail line from China and a new airport. And it's a very authentic town that's both French colonial and Asian at the same time. You can see their art is very intricate and different from what you see in Thailand. So I was talking to all these monks. And I became convinced that yes, the zero does come because they even tell you, when we meditate, we think of the void. And then, we start counting. So from emptiness, there's a one, two, and we count when we breathe in and we breathe out. And so I think it's logical that the void does come from some eastern idea. Maybe not Buddhism, but the eastern mind of the early centuries was able to conceive of nothingness. And the Mekong goes through this place. And 200 kilometers southeast in that direction is where the earliest zero was found, in a place that looks just like this, but it is just further south. There was a temple there dated to 683. So it's 200 years before Gwalior. And that's where Coedes found his zero. There's the Mekong again. So I finally got serious about this search. And I knew I had to go to Cambodia to try to find this lost stele with the first zero. It's not necessarily the first zero. In the 80 years that passed since Coedes analyzed it, nothing else was found that's earlier. So it's the earliest because of that, because we don't have anything earlier than that. But likely, the zero was invented before that. Coedes defined these civilizations as Indianized because they were Hindu and later Buddhist and back to Hindu. There's a back and forth with religions. So if you've been to Angkor Wat, it was built as a Hindu temple to Vishnu, and then became Buddhist, and Hindu, and Buddhist, and depending on the kings which ruled. But this is Phnom Penh, and I went there to meet the Deputy Minister of Culture and Fine Arts of Cambodia because after about two years of emailing, writing, phone calling, I got to this person who, for some reason, agreed to talk to me. And I needed doors opened to me. Otherwise, I couldn't find anything. And I don't know why, this is a guy-- Hab Touch is his name. They called him His Excellency, Hab Touch. And he goes around the world. He comes to New York often to try to get back to Cambodia all the lost art that was sold out of the country during the years of the Khmer Rouge rule. Maybe you've read about it in the "New York Times." There are stories about it all the time. And he goes to all the museums in the world and tries to negotiate the return of the art. For some reason, he thought that maybe this was interesting. And I met him. And then, he sent me to this place, which is near Angkor Wat. This is called Bayon, which is outside Angkor Wat, built by Jayavarman VII. And the zero is there. But I didn't know that. He just sent me there and told me to look. These are the faces of Jayavarman VII, the great Buddhist King who built Angkor Wat. So there's a place near Angkor Wat in a clearing in the jungle that's called Angkor Conservation. And after all these years and months of searching, I came there. And a phone call from this man opened the doors to me because this is closed to the public. Nobody can go there. And what it is is a repository for all the art that was destroyed by the Khmer Rouge. They tried to restore it, but mostly they do nothing. It's just a storehouse. This is in a shed that's about twice the size of this room. And that's what you see there. And the great statues, they were just destroyed. At some point-- I don't have a slide of that, but an archaeologist there showed me a statue where you can see clear marks of destruction. So the Khmer Rouge were a little bit like the Taliban. I don't know if you remember before 9/11, maybe a few years before, they blew up those two large Buddha statues in Afghanistan for no reason. And the Khmer Rouge did the same thing. They weren't Muslim. I guess they were Communists, the Khmer Rouge. They tortured and killed 1.7 million of their own people, Cambodians. And then, they destroyed all the art. So I'm not sure why they would want to do that. But so the place is full of this just discarded art. And I had to look through it. They told me, if it exists, then maybe it's here. But they may have destroyed it. So there it is. I found it. So it took a while, maybe half a day going through. And I finally found it. This is K-127, the number Coedes gave it. This is from behind. And this is from the front. And here are two pictures of the front of this stele. And the zero is dead center on the right photograph, right there. And this is what it says. It says, the Caka 605 year began on the fifth day of the waning moon. And 605, this is better. So you can see the zero is the dot. So why the dot? A zero discovered sometime later in Indonesia is a year younger than this one. So this is the oldest zero we know. And apparently, the Khmer Rouge didn't think this was important. And that's why it survived destruction. There is damage to it, but maybe it was the Khmer Rouge-- I have no idea-- but not the part that's important. You can see the damage on the last slide, on the top there. We don't know whether this damage was done by the Khmer. It doesn't look like a statue, so they wouldn't have wanted to destroy it necessarily. But it does have damage in the top and on the side here. And you can see the zero on the second line from the bottom. So why is it a dot? The other zero from Indonesia, which is-- so I need to explain the dating, though. So Caka began in 78 AD. And this is the year 605 of the Caka era. And so 605 plus 78 is 683, which is a prime number, by the way. So 683 is the date. And therefore, it's two centuries earlier than Gwalior. And there was no extensive Arab trade at the time, and this is further east. So that is very strong proof of the hypothesis that the zero is indeed an eastern invention. The Arabs today use a dot for zero, a circle, an uneven circle that's kind of fatter on the bottom is a five. And the dot is a zero. And we use a circle as a zero. So it is a good question. Why was it a dot or a zero? But apparently the two forms still co-exist and evolved at the same time. The zero, circular one in Indonesia, and this dot in Cambodia. And these cultures were Indianized in the sense that they were Hindu at the time. And this comes from a Hindu temple. And they used Sanskrit in addition to old Khmer. So the story would end now, but it didn't. Because I was stupid. I'm standing there, looking at it. And I was euphoric. I thought, I finally found-- took me really five years to find it. I didn't know what to do. Suddenly, these two women walked into this shed. Nobody ever comes into this shed. And they're wearing lab coats and speaking Italian. And as I described in the book, I was raised on a cruise ship in the Mediterranean. I speak French and Italian fairly well. And I'd been traveling for months and didn't talk to anybody other than these phone calls and trying to beg people for information. And usually somebody says, oh, I know somebody who might know somebody, and that kind of thing. It was just this kind of research, like trying to find somebody who knows somebody who might know something or might talk to you about it. And I got a lot of nos. The director of the museum right there just refused to talk to me. And the head of this place, Angkor Conservation, just hung up the phone on me and, in an email, told me, I can't talk to you. And that's why I had to go to the government to go over their heads. And then, of course, they were there. But so these two women walk in. They speak Italian. So I started to speak Italian with them. And I said, you know, this is the oldest zero in history? And they said, really? Thank you so much for telling us. We'll take it. And they took it. So it weighs a couple of tons. They didn't take it right away. But they said, we run a lab here. And we teach the Cambodian students how to restore archaeological artifacts. And since you told us it's important, we'll use that, all these stones. And it looks fine to me. I hate to have somebody restore it. This is crazy. And I tried to say, there's nothing. No, no, we'll take it. And they took it. So this archaeologist from Palermo who runs this place has control of all the artifacts because the Cambodians can't handle it themselves. So they got somebody from the University of Palermo to come and teach them how to do archaeology. And she's in control. And so I told her it was important. She took it, and she went to write articles about it. And she took it into her lab. And I thought, OK, it's lost now. I went back to Bangkok, and an art dealer who helped me on my way said, it's lost now. The mafia is going to get its hands on it. She's from Palermo. And it will be sold somewhere in the world for millions of dollars. It's the first zero. You told me it's important. It's gone. There's nothing you can do. I came back to Boston after everything. I found it, and now it's gone. I thought it should be in a museum. And then I thought, the grant had run out long ago. And so I paid my own way back to Cambodia. I went back to Phnom Penh. I stayed in the same hotel from which that picture was taken that I showed you. And I asked for a meeting with Hab Touch, his excellency, the Deputy Minister of Culture and Fine Arts. And he came to the hotel to meet me. And I thought, I'll wait for him. He said, no, wait in your room. I'll come, and I'll call you. And he came, and we had dinner in the hotel. And I took my PC, and I showed it. And I said, here, this is the oldest zero. And you helped me find it. If it hadn't have been for your phone calls, I wouldn't have been allowed there. And he said, that's wonderful. It belongs in a museum. And I said, that's where I want it to go. And he says, I want you to write the description of it. And I wrote one right away. And I emailed it to him. And he said, don't worry. It's going to go to the museum. This was in 2013. I rediscovered this stone on January 2, 2013. And nothing happened. I emailed him again. Not a word. I thought, maybe my friend in the gallery in Bangkok was right. It's probably sold at auction somewhere. It's gone. They won't talk about it anymore. And then, my sister died. So it seemed like it would have nothing to do with this, but it did. My sister was interested in Cambodia and the Khmer Rouge era in particular. It's because of her interest in it that I got interested in Southeast Asia. And that's what got me to do this research even before I started to do the research. And when she died, she left some amount of money. And I thought, I can commemorate her by paying for this piece to be moved to the museum. So I emailed them, and they hadn't been answering me for months, maybe over a year. And the minute I mentioned money, not only His Excellency responded, but the Minister of Culture and Fine Arts herself wrote me an email saying, thank you so much for wanting to support this important find that you made of the first zero. And we'll place it in a museum and everything. So I'm in the process of going to Cambodia and trying to arrange-- I haven't seen it yet, though. So I'm not sure that it hasn't been stolen. You always have this fear in the back of your mind. Maybe they're saying yes, yes, because they want the money. But they haven't talked about this for a long time. And maybe the Sicilian archaeologists did spirit it out of the country. I wondered how a piece of stone like that weighing a few tons can be taken out of Cambodia. But the art dealers told me, no problem. They just put it on a crate, the crate, they put it on a ship, and it's gone. Who knows? But they promised me that they have it under control, and as soon as they get the money, they'll place it in a museum. So I think it's important because there's one thing you read in a book or in an article about some kind of a discovery, or an artifact, and it's gone. That's not the same. When you are able to see it and touch it, it's in a museum, and study it further, then that is, I think, of value. And I even found a place in a museum, the National Cambodian Museum in Phnom Penh, where it should go. There's a little room in the corner of it. It's sort of a square display like that that's dedicated to the seventh century. And that's where it should go. And there's some art from Sambor in Mekong, where this comes from. So that's the end of the story at this minute. Thank you very much. If you have any questions, I'll be-- [APPLAUSE] AMIR ACZEL: I'll be glad to try and answer them. Yes? AUDIENCE: So as you mentioned, the importance of zeroes for place value, and I guess just a little more context, someone reading this stele, were they already familiar with the idea of place value, but had just never seen a numeral yet that had a zero in the middle of it? Or is place value itself a novel thing as of the writing of this that the reader would have had to have some education or training to expect? Does that make sense? AMIR ACZEL: Yes. I worked as a statistician for many years. And one of the problems we do with estimation is suppose you want to estimate the left point of support of a distribution. So the minimum point. And so you use the smallest order statistic. If you look at data, that's the last one towards the left. But there's always bias. If you know about bias, this is a biased estimator. It's always biased because you can't-- this is the smallest value you could have. You can't always-- if it's a continuous distribution, probability of 0 of hitting it exactly. So we don't know how early this is. So we have to see it in context. So my guess is because of that bias inherent in an estimation, this is a kind of an estimation process. You dig up new steles and new archaeological things. And each one of them is a data point. So the Gwalior one in India is one. And then, you go 200 years earlier. In the last 80 years, nobody has found anything older than this. So we don't know. But seeing it in context, you may imagine maybe this or an earlier piece of stone like that, maybe 200 years earlier than that, where somebody has this idea. I need to write the year 605. So the Babylonians just left a blank. They had a very sophisticated number system based on 60. So you'd have the 0 power would be the units, the same as for us. And the power one for us would be the tens for them. It's 60s, right? So two 60s or three 60s, like we have 30 or 20? And then the next power would be 3,600. That was a huge base, and therefore it was not as efficient. But they didn't have a zero. But they'd leave a blank. So one can imagine that maybe when they wanted to write 505, 100 years earlier, the year 505 of the Caka era, they'd write 5 and leave a blank, or just 55, and sometimes when somebody sees it, he or she will say, well, it has to be 505 because we know it's not 55. And that's what the Babylonians did. The example with the Babylonians, they used things in context. Like you say, in a modern example, say something costs $6.95. We're talking about a magazine. It would be $6.95. If I said $695 for an airline ticket, you know it's $695. So that's-- the way I understand it-- what the Babylonians did. There was a blank sometimes. There was a blank, and you could understand from the context. At some point, some person, then or earlier, decided to chisel a dot there to say the 6 is for hundreds and the 5 is for units, and there's nothing for the 10s. So that's my guess. I hope I answered your question. AUDIENCE: I just visited Chichen Itza a couple weeks ago, and I'm totally impressed by the Mayan culture. But did you have a date on the Mayan zero? AMIR ACZEL: I think it's 37. AUDIENCE: 37? AMIR ACZEL: Yeah, AD 37, just from memory. But I know very little about it. Something like that. The number 37 pops into mind. So I think it's 37. AUDIENCE: So thank you for your previous answer about the place value represented by the blank. How far back does that go? AMIR ACZEL: How far back does that go? Well, the Babylonian culture is very old. So 2000 BC, something like that. So Plimpton 322, the famous clay tablet at Yale, is 1650 something like that BC, BCE depending on how you want to call it. So it's something like 3,600 years. AUDIENCE: The concept of zero goes back to then, then. It's just the symbol of zero? AMIR ACZEL: Right. I think so. That's a good point. Maybe the Babylonians didn't need the Buddhist void. The Buddha was born much later. There goes my theory. Good point. AUDIENCE: The other question is you were talking about things like 695. Do we know where the final zero was first used? So this is not a medial zero. It's possible that it was not contemporaneous. I don't know. AMIR ACZEL: A year later, there's a zero from Panin Bank in Indonesia, a stele that you can still see in a museum there in Panin Bank. And it is a circle. So they co-existed apparently. The oldest one is a dot, but-- AUDIENCE: What I meant was it's possible that you might write 605 as 6, 0, 5, but you write 600 as just 6. So when is the first evidence that you have a zero at the end of a number? AMIR ACZEL: That's a good point. AUDIENCE: Is that a different issue? I don't know. Maybe it happened at the same time, maybe it didn't. I don't know. AMIR ACZEL: Right. I see what you're saying, right. I don't know. I have no answer to that. Sorry. AUDIENCE: If I understand correctly, you dated this to 683 because of the text 605. But do you have evidence that that text wasn't written years later? AMIR ACZEL: Right, so that's a very good question. The temple there was built around that time. Around 683, and we know that from stylistic forms on the door frames, lintels, and all these things there. So that site was studied extensively by Coedes and by others. So we know when it was built. So for example, yeah, that's how you know. So we know when the temple was built. So maybe it's-- well, 683, they wouldn't write it before because they were actually describing an event that happened at that time. They were saying, the era began on the fifth day of the waning moon. It seems like something that's localized in time. Unless, hm, could you write it at a different time, year 605 of the Caka era began on the fifth day of the waning moon? Maybe they're talking like we would be reading the Bible. Something happened on a certain-- yeah, it could be. But the temple was built around that time. It's a seventh century temple. The styles are seventh century. And Hab Touch himself is actually an expert on Cambodian art. And he actually has classified the Cambodian art styles into five different styles. And the style of Sambor-- there are two Sambors, by the way. And for several months, I was going to the wrong Sambor trying to look for it years before. There's a Sambor that's near Siem Reap, where Angkor Wat is. And this Sambor is all the way on the other side. This is northeast of Phnom Penh. And the other one is northwest. But they're both called Sambor. One is Sambor something else, and this is Sambor on the Mekong. The other one is Sambor Prei Kuk, and this is Sambor Mekong. And they're both seventh century. So the style is very well known. The artistic style of everything, so we know it's seventh century. AUDIENCE: Thank you. AMIR ACZEL: Sure. AUDIENCE: I had originally had the same question about this, and that other inscription in India, right? How do you know somebody didn't scribble it into the temple walls later, right? But the other question I had was is there any use of it in arithmetic at some point in old Khmer? AMIR ACZEL: We don't know. In terms of the old Khmer, this is all we have. So we have no knowledge about their arithmetic. It would be very interesting to find out. This is all we have. That's just one number. Well, we know some things, which are actually very interesting. So Coedes was an amazing guy. He wasn't a mathematician at all, but he was fascinated by mathematics. I don't remember the exact way it works. But numbers in the east were often given names. So you'd say, the veda, and there are four vedas. So the number four is veda. You say veda, it means four. Eyes means two, obviously. And census, I think, is five, or something like that. And flavors is six. I may be confusing them. But every number of the first few numbers has a word. And it's like a code. And you'd say, I'll meet you at-- I don't know-- a particular street, veda eyes census, and you know exactly where it is. And he traces that, how they do arithmetic with that. That would be a good question, whether eyes plus eyes would be whatever four is. Well, veda. Veda is eyes plus eyes, or eyes square. So I don't know if they did arithmetic that way. Coedes started and made a lot of progress. But it stopped. We know nothing. There are thousands of steles in Cambodia. You go to museums, and they have them in their store rooms. And he was a genius. He was just able to translate. He did 2,000 of them. And that's it. Nobody knows. Nobody has consistently studied these further. Why? I have no idea. Maybe they just have no resources. I guess I do know why. You don't want to justify colonialism. But France says resources, and Coedes wasn't just by himself. The was the Ecole Francais d'Extreme Orient, which is the French School of the Far East, I guess, the extreme Orient. And they had a lot of money, and they could do things. That's why they were able to do it. Now, the Cambodians go to Palermo to get somebody to teach them how to restore artifacts. So I think that's where the problem lies. Somebody needs to-- there are tons of these things that need to be translated. This is one of only 2,000, 3,000 inscriptions that have been translated. And there are probably-- well, I don't know how many there are in Cambodia. In India, there are millions. And they still need to be translated. The problem with the dating, again, a few of you alluded to is how do we know that it's the date? That's a huge problem in India. So in India, the dates are very, very murky. And the reason is that many of the inscriptions in India are on copper plates. And the date is always on the side, or often on the side. And you don't know if it was added five centuries later or whatever. So the dates are very fishy. And that allows somebody like Raju to claim, we invented everything. On the other hand, they probably did have a lot of science, astronomy in particular, and a lot of mathematics that they did develop. But it's all hidden. Nobody has translated these things. And nobody knows the dates with precision. Here, we are lucky. The good thing is that it's in stone. It looks like it hasn't been-- doesn't look to me like somebody could have come in 1932 and chiseled that 605. There are forgeries. I studied one of them, a very interesting one. Maybe you know that story about the Drake Plate in California. Many years-- it was discovered by a Professor at Berkeley. And it was in the trunk of his car for several years. And this is Francis Drake put this plate here on the California coast, commemorating his arrival there by Point Reyes Peninsula. And they found it was fake. The copper was made using modern techniques. And the script wasn't Elizabethan at all. It was forgery. So there are forgeries. There are lots of people who have an incentive to forge things. And so this, you never know. I don't think this is fake. And the cool thing is that we have that date. And it happens to have a zero. And the same, in India, the Gwalior zero is 270. And it looks just like our numerals, even though it's ninth century. The two looks like that. It's a very overdone two. And the zero is a circle. I'm sorry, the seven actually goes back very far. The seven is one of the first numerals that remained the same from its inception, whatever that means. I've looked at a lot of these scripts in India. There's Brahmi numerals, and Ashoka was a Buddhist king in India, first few centuries. And he had pillars all over India-- maybe you know-- they're very famous pillars. And there are numerals on them, and the numerals are called Ashoka numerals. Both Ashoka numerals, and there's a Naneghat cave in the Western Ghats, southeast of Mumbai. And you can go up there. It's a long hike to the top. There are numerals there called the Naneghat inscription. So there are these three scripts that go back the first few centuries, two, three centuries AD. They're all around the same time. I think Ashoka is about the third century, or maybe it's 300 or so. So all three of them have a seven that's just like ours, without the-- just a seven the way we write it, not the way European write it, crossing it. So the seven is one of the first numerals that just traveled through time to us exactly as it is. So you have the seven, and then you have a zero in India, the Gwalior zero. Now, there's another way-- there are many ways of dating things that can authenticate them quite well. I'll get to you in a second-- that can authenticate them quite well. This is one example. It's not unique, but it's very clear. You have a date. It has a zero in it. And you know when that dynasty started. That dynasty started in 78, so we know the date. When you go earlier in history, there's something very interesting that can be done, and that is early civilizations like the Babylonians were very careful in recording total solar eclipses. They were very obsessed with Venus and not the phases, they couldn't see them, but the motions of Venus when it rises with the sun and so on. And they also were interested in total solar eclipses. So they tell you something happened. The king died, whatever king died on the day of, or five days after the total solar eclipse. You can find it exactly because you use modern astronomy. You go back in time roughly, and you know where the eclipse path past on a certain time on a certain day in a certain year. And that has been used, actually, to establish a lot of Assyrian, Babylonian chronology. So that's another very nice modern method. And of course, radiocarbon is unbeatable. I've worked a little, devised a statistical routine for making radiocarbon more precise using the bootstrap. And the radiocarbon is now, I don't know, 50 years or within a few centuries. So it's very accurate. AUDIENCE: Would you speak to the difference between the numeral zero and the number zero? In particular, I'm thinking imagining a record of an inventory that might say, number of bushels of wheat, zero. Number of jars of wine, zero, which is distinct from the use of the numeral as a placeholder. But the numeral to represent the number zero. AMIR ACZEL: Right. So the way I understand it, that is a sophisticated idea, too. In the west, at least, it was difficult for people to say, we have zero bottles of wine left for the holiday, or something like that. And my understanding is that the double-- what do you call it? The bookkeeping method, double entry bookkeeping evolved because of that inability to say there's zero. So you have numbers of one kind here and numbers of the other kind there. What you owe, what you are owed, and so on. Yeah, it takes a little sophistication to say, we have zero of something. Negative numbers are even worse. What do negative numbers mean? So when you study mathematics and you look at the numbers, how they evolved, the idea of numbers, how we perceive them, the positive, the natural numbers and natural to us. One, two, three, four, five, even if they go on to infinity. In order to define the negative numbers, you need a zero. Because the zero is the point of symmetry where the natural numbers go to the right, and then the negative ones go to the left. And you can have a group under addition where you have the inverse is the minute you have the negative numbers, and you have a zero. So the zero is the additive identity. The multiplicative identity would be one. So then you can have a field. So yes, it's a sophisticated idea. It was slow in the west. I don't know, in the east, it seems to have been more natural, maybe. I feel, and I have no strong proof of that, and infinity also is an idea that was maybe easier to evolve in the east, just having read a lot of stuff about the east. In Hinduism, there's an infinite sea, and the sea turtle on which Vishnu lies when Brahma springs out of his navel. It's called Ananta, which means infinity. So there are ideas of infinity. They didn't understand them. But they could talk about them maybe more than in the west. Because Augustine writes about the infinitude of God in certain ways. So infinity as well-- yeah? AUDIENCE: In the system where the designation is just a blank space, like 60 is 6, or 601 is 6, blank space, 1, how would you judge when there's more than one space? And if there's more than one, is it two or three? In order to really do that with any accuracy, you need something in there, a little jot of some kind. AUDIENCE: Or it could be clear columns. AUDIENCE: Yes. AMIR ACZEL: Good point, good point. That's why the zero is so important. You don't know how many blanks are there. And the blanks also, remember, they didn't have anything standardized like you'd print on a page, and a blank is an exact width. This was written by hand, or chiseled. And how do you know if there's a blank there, or just the guy couldn't judge distances very well? Especially if the distances vary, there's no way to tell where there's a blank unless it's really a significant width. So that's a good point. But you could-- going back to your previous question, you could assume-- and what somebody else asked, too, and commented on-- a blank could be quite rationally a void. And maybe it's not that Buddhist. Maybe it's an idea that came other ways. Going back to infinity, infinity really was understood with the work of Cantor in the 1800s in Germany. So infinity really becomes a mathematical concept in the late 1800s. But maybe what I'm trying to say is that eastern people talked more about infinity than people in the west. Yeah? AUDIENCE: Just on the subject of multiple zeroes in a row, I'm trying to remember the tables of chords in Ptolemy's Almagest, which is second century AD, right? Which has trigonometric functions of every half degree value between 0 and whatever they went up to, 30 degrees or something like that, for trigonometric calculations. There's calculations in the Almagest that certainly go out to like five places in sexidecimal or something like that. So certainly, there must been calculations that they were doing at that time, all the trigonometric calculations that they were doing that involved multiple zeroes in a row. It seems very unlikely that they would never have hit two adjacent base 60 slots that were empty. But of course, all their calculations were done in very strict tables, these tables of values of the chord function had the firsts, and the seconds, and thirds, and the fourths, and the fifths, which are fractional powers of 60, all lined up with each other. So I don't know how Ptolemy actually wrote it, though. AMIR ACZEL: You're right. If you're careful with writing a table and there are five zeroes that have to be there, so there are five blank columns, or whatever, however they did it. That makes sense. Yes. So they must have had a technology. So maybe ours is more advanced just in the sense it's standardized. You have a zero, so you don't have to worry about making a table and columns. So the ideas must have been there early on. But maybe they don't agree with our sense of neatness and keeping things much more universal. So you can write a number here and read it in Japan, and it makes sense that it's 1,005 and not something else. So good point, good point. They did have very extensive tables, right? Good. Any other questions? Well, thank you very much for having me come here. [APPLAUSE]