Good afternoon, everyone. Welcome to the Carnegie Endowment for International Peace. My name is Doug Paul. I think virtually everyone here knows we all know each other. But my name is Doug Paul. I'm vice president for studies in the Asia program. And it's my great privilege this afternoon to welcome a very old friend and colleague from Korea who has done a great many things in his career. Chung Min Lee has been a RAND Corporation researcher and PhD out of, well, you see it all in the biography, I think, but PhD out of Fletcher School, the master's there as well. And that'll account for his loquacious English that really stands out. And when you read the book, you'll be pleased to see some even more interesting verbal dexterity. He's a really accomplished writer. One of my favorites was trying to run the world so he ran into the world. He's written a book that has really done a lot to counter the triumphalism you get from a lot of the writing about Asia. I think a really bad example would be when China rules the world by Jax. But Kishore Mopobani in Singapore writes always about how we Westerners think we're so superior, but the Asians really have got it all done. And what I think you'll learn today from Chung Min is that there has been, there's a paradox in Asia. And he's captured it very well in the volume. Before I ask him to say a few words about what he's written and why he thinks it's worth all of you giving up such a beautiful afternoon to come inside, I'll mention this book, Fault Lines in a Rising Asia is now available on Amazon. So you can't get it easier than that. And it's a really solid and well-informed read. Chung Min, do you want to give us an introduction? It's really embarrassing because he's basically plugging my book in front of my very old friends. So it's a real pleasure, Doug. Thank you so much. I'd like to thank all the folks at Carnegie, especially their publications and communications department. I see many familiar faces in this audience. The reason why I wrote this book, which came out a couple of days ago, is basically a three-fold. And then I hope I can do Q&A with you as well as questions from Doug. And the first is, if you look at almost all of the airport books in airports in Barnes-Novel and whatever, 90% of the narrative on Asia's rise is what I would call the Horatio-Elzer story, the rags to riches, Japan rising, China rising, India rising, Taiwan rising, South Korea rising, everybody's rising. So it's a very big picture of Asia's economic success, which is undoubtedly true. And yet, I thought that something was missing. And so one of the first things that I looked into was, what is the other side of the Asian story? Well, the other side of the Asian story, unfortunately, is that it is the depository of the world's largest political, military, and security challenges. And so if you take all the good things about the world, technologically innovation, political stability, economic trade, it's all in Asia. But all the world's biggest problems in magnitude are also in Asia. And that's a facet of Asia that people just weren't talking about. So I thought, well, there must be an Asian voice that corrects the narrative. And the second reason why the second major victim in the book is I don't buy the argument that Asia's rise is synonymous with the demise of the West, the death of the West. Obviously, because of the US elections now and a certain candidate who shall remain nameless, people are having a great time mocking the American democracy, how it's falling on its face, it's sinking, it's horrible. But I think Americans are smarter than you think. But the key issue here is you cannot blame the West for all of Asia's ills. And I think that's a shortcut to intellectual, I would argue, very shallow thinking. And it is also politically wrong, because although it is expedient to blame everything on the West and the US, if you look at all the problems that Asia faces, they're all indigenous, whether it's historical, I guess, legacies, territorial disputes, political deficits, and the list goes on. So that's what I really wanted to say was that, I'm a huge believer in Asia's success and I'm very proud to be Korean, but I'm also absolutely convinced that the greatest gift I have as a Korean citizen is that I have the ability to say what I want to, without fear of persecution. Living in a democracy is a great thing. And as we know from last week, the ruling party lost elections in Korea, but that's the way the chips fall. And so democracy is not something that only quote unquote Westerners practice. Japan, Korea, Taiwan, they have shown that you can be rich and free in Asia. It is not something that somehow Asian values creeps in to the discussion. So that was my second major theme. The third major theme is this whole idea that Asia's rise is almost relentless. It's going to go on almost forever. Well, let's assume that's to be the case. Then what? What happens after Asia rises? Do you build more cars? Do you build more factories? And I would argue that after Asia rises, it's time for Asia to really look into global issues. How much money does Asia pay for climate change? What are the great big ideas that Asians have come up with that have resolved outstanding global issues? What contributions have Asians made to the ongoing Syrian debacle and genocide? And the list goes on. Now, obviously you have countries like Japan who really pay up to the UN and membership and so forth, but the rest of Asia, in my opinion, has remained terribly silent on a host of global tragedies. And this must stop. So if Asia wants to rule the world, quote, unquote, you must first make contributions to the world. So those are the three broad themes that I have in my book. And as Doug said, I think it's on Amazon, as we speak. Also available at Brookings Store, right? At Brookings Store, if you want. You can't wait for Amazon to deliver it. But the more I've, and when someone asks me, what background do you have that enables you to say things about Asia? Well, I'm Korean, of course, but since the age of four, I've lived in 10 countries. My father was a foreign service officer, so we ended up living in Indonesia, Singapore when I was working at the Lee Kwan Yew School for a couple of years. Obviously, in Japan, I've lived there twice, although as my students know, my Japanese is what I call Starbucks Japanese. I can order coffee in Japanese, but that's basically it. I've lived in the U.S., I've lived in Germany and France, and then Mexico, and twice in Africa. So I think I've been able to see Asia from a much more objective, I guess, perspective than most other people who are in my field. And so from that perspective, let me just stop now, and then- Don't stop, you've got all the time you want to expatiate on all this. But I think rather than telling you how great the book is, which is basically up to you, if you have questions on my argument, on the pitfalls of the argument as you see them, or to explain in more depth what my book is about. If I just go through the chapters one more time, I have an introduction. And then the first chapter, I look into the introduction, basically says, if you stood in Asia in 1950, no one could have said that Beijing, Seoul, Tokyo, Taipei, et cetera, Shanghai, would be world cities in 30, 40, 50 years. So that's the good side of Asia's rise. But the bad side of Asia's rise is all in those cities as well. And so then on chapter one, I look into, for example, the predicaments of strong states. And here I focus on Asia's political deficits. We have the world's largest one-party state, China, that is struggling to come up with some type of a modus vivendi, but I believe that we are reaching a very critical tipping point in the PRC. You simply cannot control the way people think when you've never had more educated, globalized, and outgoing Chinese people. You cannot monitor 1.4 billion people, even with all of their monitoring and surveillance equipment. Then you have a number of outliers in Asia, whether it's Pakistan or North Korea, with nuclear weapons, and democracies in transition, i.e., Thailand. You've had two military coups in 10 years' time. And you have two very successful, but aging democracies, Japan and South Korea. So there are huge political problems, even as we speak. The third chapter I focus primarily on what I call the new, I guess, game in Asia, which is a competition for supremacy. No one power is going to dominate Asia, as the Chinese did back in the good old days, or the U.S. right after World War II. So you'll see some type of a mismatch between the great powers and the major powers, and the middle powers. Then I focus on what I call a virtual arms race. Since 2010, Asia spends more money on defense than NATO. And so despite the fact that China's defense budget may go down a little bit, owing to a slowing growth rate, I foresee the fact that the next 20, 30 years, Asia is going to become the most hotbed for arms transfers and military modernization. Then the fifth chapter, I look into what I call the second nuclear age, which all of you know, Paul Bracken coined the phrase about 10, 15 years ago. And here I focus primarily on the Indian, Pakistani nuclear quagmire. And my final chapter, which I call the bomb next door, apropos of South Korea, is a nuclearized Korea. And in particular, how the Americans, the Japanese, and the Koreans have responded, or not in some cases, to a nuclearized North Korea. And then I end very shortly with a conclusion. So that's basically a walk of the land of my book. Truman, I forget who was the originator of the phrase, the Asian paradox, but it meant to capture the notion that on the one hand you have all this economic success and educational attainment and social development, and on the other you have this grinding military and nationalist conflicts in the region. I wonder if you'll like to talk a little bit about that. Right, that's a phrase that I use also in my book. I think I first read about this in the early 1990s. So the paradox simply is this, how can a region that is so successful economically minus outliers such as North Korea and Myanmar before she began to reform, whereas politically and strategically and otherwise, you have the world's most pronounced dichotomies and problems all at the same time. In other words, if you look at every single bilateral relationship within Asia, between China and Japan, Korea and China, Korea and Japan, and then the list goes on, there is a fairly deep repository of mistrust. And you have huge political deficits in all those regions. And so if you go from, if you take Singapore as the middle point, and the right side of Singapore is what I would call the right vector, that right vector is the Asia that we're all familiar with. The rising Asia, the megalopolis, technological wonders, Samsung cell phones and whatnot. But the left vector from Singapore all the way up to the subcontinent and on to, for example, Afghanistan, that's the very weak link of Asia. And that's an Asia that people simply don't pay attention to within East Asia proper. And so if you look at that, that paradox is something that Asians really haven't come to terms with. So in that sense, you know, we can talk about the good sides of Asia, but unless you understand the paradox of Asia, you simply will not be able to come up with regional or even global solutions. So that's basically my argument in a nutshell. And so then my daughter asked me that, if you could give us the elevator speech of your book, what would it be? So I said, well, Jean, I would say Asia has many problems. So she said, well, it took you 367 pages to say this. I said, well, that's what academics do. But seriously, I really wanted to show a different facet of the Asian narrative. And I think coming from someone who is Korean and Asian and was lived abroad, I believe, the lens credibility to the extent that it does on showing the other side of Asia's rise. I'm glad in your response to my question that you mentioned Myanmar because when you rattled off the names of all the cities that nobody would have predicted in 1950 would have great success. You left out the fact that everybody in 1950 thought Rangoon was going to be the future for Asia and or Yangon and Myanmar. My first visit to brought all this home for me in a funny way. It caused me to recall it now.  What year was this? 1982. 1982, okay. The Korean team came in to blow up the Korean cabinet who were there to visit Nae Woonch and will Nae Woonch in his government. And the great, wonderful Korean diplomat, Ibum Suk, was killed in that. The foreign minister, that's right, yes. And all of that came to bear at one place. And I think when people ask me, well, can Asians be both wealthy and free? Well, I say, well, look at Korea. I was a university student at Yonsei University for my undergraduate days. And I remember that at that particular time you could not speak about politics openly.  And today Korea has one of the most vociferous, open, political, I guess, environments. Some might say we're too noisy. But in a democracy you really can't choose who says what and which party loses or not. And that's a great thing about living in a democracy. And so it's inefficient. And there are other countries that really don't like that way of democracy. But I would say that that is the biggest lesson that I've learned growing up in Asia. So whether it's Japan or Korea or Taiwan or Indonesia even today or the Philippines for that matter, you can be economically successful and also have a liberal society. And if you look at what's happening in Hong Kong over the last several years or what's happening within China on the university structure, this really is a big step backwards for academic freedom. Well, it's a big choice. Clearly the most recent rulers in China have decided that their choice is to prefer control over the benefits of unleashing greater energy among the people, among intellectuals, political opposition even. It seems to be far away. They can't really allow people who've got this high level of development, middle class of over 300 million people and then tell them they're not gonna get information. I think if there's one emblematic event in the last few years that has colored the policies and actions of the Xi Jinping government, it was the arrest of Boasi Lai and the allegations back and forth of corruption within the leadership, which were all a proxy fight for real power. You're corrupt and therefore you're bad, but that's not really the point. The point is you oppose me and I want you out. And we see now that this Pentagon Papers, excuse me, Panama Papers. Panama Papers has seen it. Same thing, same thing in some respects. The Panama Papers have been published and implicated members of the families of seven of China's recent leaders. And none of this has been reproduced in many Chinese media, but everybody I know from China seems to be well aware of it. You can't stop this stuff. You know, in my book, I'm not a China specialist and I'm sure there are many people in this audience who are infinitely more informed about China than I am. But as someone who lives next door to China, what happens in China has a huge impact on my life and that of my daughter and hopefully in the future, grandkids like Chris. But the key issue here is this. As I was writing my book, I found out that there are 87 million party members of the CCP as of July 9, 2015. That's three million less than the entire population of Vietnam, which is 90 million. Well, I ask myself, if you have 87 million people in a party, how many people do you need to watch those 87 million? So you take a rough figure, 10%, we're talking 8.7 million, which is larger than most European countries. And you ask yourself, who watches those 8.7% that comes out to maybe about what I'd say, at least a good 870,000. And of the 870,000, who do you really pick and choose to run and give orders to the 870,000? Well, so on down the line. And it's come to the point where as you know, ever since 2008 or 2009, China spends more money on domestic security than on external defense. And that tells you that they're definitely concerned, not only about social upheaval and so forth, but the fact that the very model the CCP has produced and strengthened, I don't know how resilient this is going into the next 20, 30 years. I'm not saying that the PRC is going to do another Gorbachev type of meltdown. I don't see that happening for many reasons, but nor do I believe that the CCP will be in power forever. Nothing lasts forever, but they're at this crucial tipping point where you simply cannot control how people think. And regardless of how many people watch the internet, as I was researching my book, Doug, there are 2 million people who monitor the internet in the PRC, 2 million. And very soon that's going to be impossible because with Twitter and Instagram or whatever versions you have in China it's going to be virtually impossible to trace micro blogs and to say exactly what words or concepts or phrases you cannot basically find on Google or whatever it may be. Especially when you've got a language that's just heavy with homonyms. That's right. You can always find something to express what it is you're saying that sounds like, what was the frog? They had a frog phrase a while back that covered a whole bunch of suits. And I quote extensively a good friend of mine who just happened to write an article in Foreign Affairs but I won't mention his name, but it's the same name as me. I didn't get his first name. And he is one of those really great, I guess, flag bearers for the CCP. And I know him personally very well. And I think it's gotten to the point where I think he's very happy that the CCP has turned out good leaders as he says and for a country of that size, you cannot simply have a rambunctious democracy like India and so you need a strong state that basically churns out good leaders that are responsible to the state. But at some point in time I told him, what do you do for a 14 year old who doesn't believe in that dream for China? What do you do to a 16 year old who doesn't believe that his or her life depends on the imprimatur of the party? And that's something that people simply don't have an answer for. Now, if you ask countries in Asia that turned democratic before when they were authoritarian, whether in South Korea or Taiwan or whatever, you probably would have gotten the same answer. We need a strong state with great growing nationalism and so forth, but at some point in time, you simply cannot maintain that depth of control. And that's one of the big things I think I wanted to get out of the book, which is there's nothing wrong with Asians who want freedom, who want to live in a free society. And that's a very powerful message. And one other thing, Doug, that if you look at Asia, one of the greatest maps I have in my wall is a map of China. I think China shares borders with 14 countries, as I recall, 14 countries. 14 is right, right? Then the Maritimes are separate. And of those 14 countries since 49, not one is a natural enemy of China, which is a great feat of Chinese foreign policy. But at the same time, not one of those 14 states is a natural friend of the PRC. And if you look at what the US has done over the last 50, 60 years, despite all the ups and downs and pivot or no pivot or new balance or new pivot, if you look at all the treaty allies, all your core partners, people who really want to engage with the US, India, Vietnam, et cetera, that's a huge reservoir of soft power and goodwill that China at this particular moment in history in Asia doesn't have. And that's another message that I wanted to tell to the book that despite Asia's rise, how many Asians really like fellow Asians? How many Asians will call each other and say, look, you know what, I really need you to send troops to defend me because country X is going to invade me? And almost all of the Asian states, except as I said for outliers, will rely on the US if there is a huge, I guess, security problem. Now that should change, obviously. And I think one message that Donald Trump has said, I think the way he articulated it is very wrong and I totally disagree with him, but I do agree in some respects that Asians must assume much greater share of the common defense burden. And that's basically a fact of life. But if you look at the overall disposition of the Chinese way of friendship and the American way of friendship, I would choose the American way of friendship anytime. As I would tell my Chinese friends, and they're not very happy, and I would tell them if your best friends in Asia are North Koreans and Burmese and Pakistanis, that's absolutely fine with us. But on the other hand, if you look at America, despite all the ups and downs, you have Japan, South Korea, Australia, New Zealand, Southeast Asia, India, that's an amazing, I guess, collection of friends and people who really share in many respects similar interests and goals. You just made a point which I think is really profound and I try to do my own Johnny Appleseed kind of work on this to spread it around, but it's really underappreciated. And that is we tend to, in our thinking, our journalistic narratives, tend to write about bilateral relationships. It's always us against China, or maybe a Japan in the US against China. But we don't take the whole picture and see that the 14 neighbors of China never really want to get into a hostile relationship with China, but don't ever want to be subject to overweening Chinese pressure. And somewhere between those desires, there's a lot of partners, people who will, if China's behavior really does become offensive to its neighbors, the neighbors will be glad to work with us and others. So we have partners to do the kind of countervally work that may occasionally be necessary as China migrates through its future. And we don't conceptualize this broadly enough. And I think if we do take that conceptual step, then we have to ask ourselves, are we doing the right thing with Russia? Are we paying enough attention to Central Asia and a few other of the actors on the scene? And what's interesting is China has treated Burma with a very high-handed approach and they got the reaction that we saw when the government shifted and looked west.  They've looked and they say, well, you know, we did score our own goals here. And so they invited Aung San Suu Kyi up to Beijing last year. I don't underestimate the point or the fact that China has made enormous strides compared to pre-49 days. And one of the most remarkable, I guess, developments in the last 30, 40 years is this. China produces incentives for the rest of the planet, whereas the Soviet Union never did. And so every single country in Asia, especially that are allied with the U.S., we're running into the so-called China dilemma. And that's becoming progress of the worse because almost every single U.S. treaty ally trades more with China than with the U.S. And so that means China has huge leverage which is economic or trade or commercial, technological, et cetera, a leverage that the U.S. has never had. And so if you want to craft a viable U.S. strategy in the next 30, 40 years, you've got to understand that China is a very different major great power than, for example, the Soviet Union. And that's something that not only the U.S., but every single major Asian country is now facing. How do you calibrate your China policy so that you will not really make them antagonistic to your goals, but at the same time, you want to also make sure that your ties with Uncle Sam remains robust, but maintaining that balance is very difficult. There's no real field manual that says how do you coexist between these two large humongous powers? And I think if you look at the map of Asia, there were two countries who really all are affected by this, but my personal view is that Vietnam and Korea are the two most countries on the opposite sides of the Chinese continent who really feel this pressure. How do you operationalize your fine-tuning, fine-tuned China U.S. policy? And I, as I was about to say a moment ago, the Chinese were scoring goals against themselves for a few years there. And you could argue they're still doing that the South China Sea, but they're Southeast Asian neighbors, but they're not passive. They have, in the last couple of years, had two major conferences on their peripheral relationships where they talked about how we gotta improve our game for China. And then secondly, they've operationalized it with the One Belt, One Road, with huge promises of infrastructure investments, market access, and then on top of that, the financial means by which to do it, the Asian Infrastructure Investment Bank. So we can sit back and observe this and stick to military activity, but if we're not engaged across the spectrum, then the competition's gonna be very tough for us to beat. You know, I'm sure all of you in this audience recall this. When the Chinese announced the AIIB, many people in Asia wanted to join, but they didn't want to become the first country to join because they knew that Washington was not really cheering the AIIB at that particular time. So I think if I'm not mistaken, Korea raised her hand after Britain said, I'm gonna join the AIIB. So after Britain said, okay, she broke the ice and the Koreans came on board and others. And yes, this is a new reality. That's a new normal that all Asians have to deal with. But that does not negate the fact, and that's one of the key lessons that I hope that I can deliver to this audience. It doesn't negate the fact that Asia has immense problems of its own, that China has immense developmental challenges, that India has enormous economic and social problems, that if you look at how Korea is going to reunify over the next period of time, there really is no road map in that sense. How does Japan cope with a shrinking population? Let me just give you one huge number. By 2100, according to current UNDP estimates, population estimates at the current fertility rate, Japan's population, which is about 127 million today, is going to decrease to about 85 million. But if you look at how it could even shrink even further, depending on what, so this is why I tell my students, it is absolutely nonsensical to believe that Japan is the next big threat. You know why? Grandpas and grandmas are not gonna drive tanks. You know, it's not gonna happen. And so all the rich countries of Asia, i.e. Korea, Japan, Taiwan, Singapore, Hong Kong, and so on on the line, are becoming what? Rapidly aging, shrinking in population with ballooning welfare costs. So the issue today is national security versus social security. And that's a dilemma that even China faces. Every single Asian country dug, you're going to see exponential increases in social welfare spending. And that would have huge political repercussions. Why? Every single political party is going to tell a lie, which is I will give you all the benefits without raising taxes. And I'll also give you all the other benefits. And that is simply not doable. It's not doable in Korea or in any of the other major democracies in Asia. And that's a problem that you have to really, you know, head up. Otherwise, you will find yourself in a huge financial and fiscal hole that you simply cannot dig yourself out of. But that's a problem that all the three big major players in Asia faces. I guess you can draw your lessons from European capitals on how to turn yourself into a gilded retirement home as these changes occur. I wanted to ask you about something closer to home being Korea for you. Right. We've got now a fairly well-developing security dilemma in the South China Sea, where China does something, they say, to defend their sovereign interests. And we and others do things in response because we're defending peace, truth, justice in the American way. And we're getting into this escalation process. We've got a potential security dilemma. I don't think the new Taiwan government is seeking to provoke one, but given the gap between the positions of that authority and the Chinese government on cross-strait relations, we could have a very rapidly escalating security dilemma there, too. And then now we've got one beginning in the Korean Peninsula, where the North Koreans have tested military systems and weapons. And the natural response for any democratic leader, Korean or American, is to take advantage of the technologies we have, in this case, high-altitude anti-missile defense, and we're now, the two governments are talking about deploying that in South Korea. And then China sees that as a threat to its second-strike capability because of the anti-missile character being so close to Chinese territory. Are we heading into another security dilemma on the Korean Peninsula? You know, I think there's a security dilemma in almost all facets of Asia because, and this is something that China really has to understand. The more she becomes a bit more aggressive in her power projection, whether it's in the East China Sea or the South China Sea. And if you look at the map of Asia, China today has the ability to constrain activities in the so-called first island chains. And their objective, I would argue, in the next 10, 20 years, is to have the ability to reach out into the second island chains. And if that happens, the American fleet, the southern fleet in the Indo-Pacific will face huge problems. Then the question is, which other country will be able to fill that gap?  He is looking at the US for more security guarantees and increasing however possible it is its own military footprint. Going back to your question about that, this was the $64,000 question. And for President Park and her predecessors, every time North Korea tested a nuclear weapon or a long-range missile, we did two things, which is we want you to go back to the six-party talks and we will also apply sanctions. And none of those really worked all that well. So this time around, after the January nuclear test, the Korean government said enough is enough. We will now talk with the Americans on deploying the THAAD at some point in time, although a decision has not yet been made. And the Chinese have become really vociferously against such a deployment. They said anything about THAAD is negative. We do not want you to do it. And that's the Chinese position. And diplomats aren't supposed to intervene in each other's affairs, but the Chinese envoy in Seoul has spoken out publicly on many occasions why THAAD is detrimental to its own security. But from our perspective, short of going nuclear, what option do we have? If Kim Jong-un does a fifth nuclear test, if he miniaturized nuclear warheads, which he already basically has, if he develops SLBN capabilities, the only guarantee we have of any type of defense is, for example, a THAAD system. So that's something that South Korea has to really make a decision. It's a political call, it's a difficult call, but it's better than doing nothing. And so that type of political calculation is going to become much, it's going to become progressively more difficult because the Chinese can become overbearing on these issues. It was extraordinary to see Foreign Minister Wang Yi call your ministry and publicize that he called and opposed this. I'd never seen that in the history of modern Chinese diplomacy. It's quite overt pressure. I want to, before we open the floor to the audience for their questions and comments, I'd like to ask you, push you one more area, which is how do you see Russia in Asia? What's Russia's role? What's Russia's role? I have a short section on all the major powers in Asia and much has been written about the so-called Russian pivot to Asia. But if you look at almost all of the empirical evidence, it just isn't there. First of all, the Russian population east of the Urals is nothing compared to the Chinese. And so if you look at the longer term projections of white Russians in the Siberian part of Russia, it is not really a place for economic growth. Number two, Vladivostok was a major, I guess, seaport and naval base in the Cold War, it no longer is. And if you look at Russia's trade with Japan, Korea and the rest of Asia, it is very minimal. And so despite the rhetoric, Russia is a one commodity economy, natural gas and oil. And the largest importer of that is China. But outside of that, there really is no, I would argue, a bilateral or even a multilateral engagement between Russia and the other parts of Asia. So in a nutshell, Mr. Putin has done this for political reasons. They have tilted towards Beijing on a number of issues. But if you look at the long-term projections, Russians are getting very, very, very anxious because China will basically overwhelm them economically. And that is a prognosis that is very bad for Moscow's longer-term strategic interests. Every time you talk to a Russian, you get yes, but when it comes to this part of the world, one of the emblematic facts for me is that there's about, it's about a 2,800 kilometer riverine border between Russia and China. And how many bridges cross that river? I don't know. None, okay. But bear this in mind because you'll soon be seeing a reporting on Prime Minister Shinzo Abe's upcoming visit with Putin and Sochi. And he has long had an ambition to regain the lost Northern territories from Russia. And the Russians are suggesting there might be some give in this, and so I think you're gonna be hearing more about Russia's involvement in a construct, attempting to portray it as heavily involved and constructive in the next few weeks as this story plays itself out. You know, one final point, note on this, on Russia's tilt to Asia, Doug, in the book I mentioned the fact that perhaps, strategically speaking, it makes sense for Asians to actually support Russian economic development as a countermeasure to the rising Chinese economic behemoth that it is. But the problem is, if you look at the political discussions across Asia, it is not, Russia is no longer a key security factor. She is important in the context that she is, of course, a member of the Security Council, but if you look at their footprints, military and otherwise, it really is very weak compared to the Chinese, and of course, nothing in comparison to what the Americans have. The Russian defense budget in 1990 was $212 billion, and in 2000 it was 20 billion. That's right, yeah. So you can see where the trend lines went. Now ladies and gentlemen, I'm happy to take questions and comments from the floor. Mike Mozetic, in the fourth row. Just wait for the microphone please. So that the Chinese censors can hear what you have to say. Mike Mozetic, PBS online news hour. First, a couple points that you had made in the prelude. Ying, Ming Pao, the major independent Chinese language newspaper in Hong Kong, yesterday published a whole front page piece on the tycoons involved in the Panama paper. This morning the editor got fired. This morning. This morning. And secondly, one usually doesn't go to the magazine Foreign Affairs for Happy Talk, but there's a co-authored article with two unusual co-authors, Kishore Mabubani, whom you mentioned in Larry Summers, predicting, Kishore changes his line quite a bit. A nice fusion between East and West and the world is going to be better off. Sorry, now to my question, why do you put Singapore in the left vector rather than in the right vector? Is that geographic or something else? It sits there. It's just because it's right in the middle. So of course Singapore is part of the successful part of Asia, she's the financial hub and so forth. But if you draw a line West of Singapore, that's where I think the problem, more problematic like it's real estate of Asia lies. But Singapore is not. No, no, no, no, no, no, no, certainly not, certainly not. No. Microphone up front, please. It's over here, Chris, yeah. You got an intern from Asan Institute and so on. Oh, great, I'll see you next week. Thanks, Chris Nelson, Nelson Report. Attorney, thank you so much. I look forward to reading the book. I'm going to read it on the plane on the way out. So that'll keep me awake, because it's obviously so very stimulating. Or put you to sleep. Well, that helps too, yeah. That mellotone in two glasses of red wine. As a professor trying to help people organize how they think about these things, what's your advice to the upcoming G7 or for that matter the upcoming G20, leaders, how should they be ranking dealing with China in a positive way if possible to bring it into the world community as a, you know, as a Bobbizelic stakeholder? I think we all agree if China fails, we're all in deep trouble. So no matter how angry we get, we've really got to help these guys do it the right way. What things are we collectively not doing that you think we should be doing? What is the role of Korea here as a possible bridge on this kind of thing? I'd love to hear your thoughts on that. You know, that requires an entire new book, which I do not plan to write. Well, how about the elevator speech? Yeah, but the elevator speech is this. I think China's way past models that could take it out of these problems. In other words, nobody has a solution for what China faces. And nobody has a pamphlet that tells China this is how you should act to become a greater stakeholder in the world. And the reason is this, it is only their leadership who can basically tell themselves, you know, if you look at the world of 200 odd countries, how many are prospering politically and economically and socially? And they must understand that you cannot have this strange bifurcated model of a one-party authoritarian dictatorship of the proletariat who existing with a flourishing market economy. They think this can last forever, but again, Chris, that's the dilemma that only the Chinese can basically resolve. What role can we outsiders play for a small country like Korea? Not all that much. I don't think China considers South Korea as a bridge. She considers Korea as more of a province who turns out to be an independent state. So, but it is the US where the Chinese leadership is paying the greatest attention. And so I would argue that over the next 20, 30 years, much like the so-called strategic consensus America had after the Cold War, you need a new strategic consensus bipartisan on China. And that's a homework that only Americans can do. And that's something that I think the Chinese will understand and accept as a co-equal. But other countries, whether it's to EU or even Japan or even India, the Chinese will say, no, no, no, we don't want to hear from you. We want to hear exactly what the Americans will do in the context of this emerging new order. About 30 years ago, Jonathan Spencer wrote a really good book on China called Two Changed China, which chronicled all the efforts of all the types of people over the centuries who failed trying to bring their particular version of change to China. Larry Nicks in the back, in the second row, from the rear. Good to see you, Dr. Lee. Larry Nicks from CSIS and George Washington. Your reference to this issue of the Asian countries trying to balance how they deal with China on the one hand and the US on the other hand, civically as this debate politically in the US has emerged over the US economic relationship with China, especially what Donald Trump has been talking about, but also what 60 minutes featured in late January about Chinese theft of American trade secrets. This issue is not gonna go away, regardless of whether Trump gets nominated or not. It's going to go into the next administration. The relationship with China. As US dissatisfaction with this relationship seems to be mounting, is there any thought being given in the other Asian countries how to, well, I guess I would describe it, how to advertise themselves and their own economic relationship with the US? Get their input into the US debate. Make the point that, look, our economic relationship with you Americans is not the same as the Chinese economic relationship is. Is there any thinking going on in Korea or any of the other Asian countries about how to deal with this rising debate about US-China economic relations? Now I noticed Prime Minister Abe did have an op-ed piece addressing this to some degree, and I guess reading that leads me to ask this question, but it seems to me it's gonna be a dilemma for other Asian governments as we go ahead into the next US administration. Well, I think the short answer, Larry, is this, that TPP is obviously what President Obama wanted before he left office, but the Republican-controlled Congress will not give that to him. And I'm not really sure whether the next incoming administration will pass TPP as soon as the government comes in. In other words, that's a reality. You cannot trade with China on Monday and with the US on Tuesday and go back to India on Wednesday and then back to China on Thursday. That's not gonna be possible. So all Asian states have to trade with China and the US, but the key issue is this, for those countries that overwhelmingly trade with China, whether it's Korea, Taiwan, and other parts of Southeast Asia, the bad news is as China's growth rate dips below 6%, and if it goes down to, let's say, 4% or 5%, who will feel the repercussions? It's the South Koreans and others who depend on exports to the PRC market. So we've got to change our own economic strategy. That's much more important for Asians than somehow sending messages to the US. So the TPP is one major vehicle. The Koreans didn't get on board on the first round, but we're hoping to get on board the TPP on the second round. But despite all of the problems with the Chinese economy with slower growth, the fact remains that all Asian states are so intertwined with the Chinese economy that it is impossible to divorce yourself completely from the PRC economically. And that's a reality that I think even America realizes over the next 10, 20, 30 years. Larry, you mentioned at the beginning of your question, Donald Trump, in today's issue of foreign affairs, there's an article from a very unusual source, Eric X. Lee, who's a Stanford-trained capitalist in Shanghai, who's written controversially about the authoritarian lessons China can teach the world, where democracy might be making us weak. In this article, he dissects the Trump campaign as perhaps, maybe not rightly, as perhaps setting the basis for a new and more constructive long-term relationship between the US and China by being honest about some of the problems and directly addressing them rather than skirting around them in the name of protecting the elites. This is picking up on the campaign theme of Sanders. I haven't read that, but does Eric mention the fact that about at least in China, whether the transparency should also apply to the Chinese leadership in the article? Probably not. No, no, no, no, that's a different topic. You can be sure of that. Okay, third row here, please. Mark, did you already have that? Hi, my name is Kunio Kikuchi, and I'm with Washington Research and Analysis and formerly of the World Bank. Now, it seems to me, in ancient Greece, there was Sparta, which is a militaristic, unrelenting country, and Athens, which was, let's say, democratic and arts and everything else that's good about society. And sitting in Washington discussing China these days, everybody is focused on the Spartan aspect of China. The increase in military might and how to contain it, how nasty they could be with internet hacking and this and that. But if you go to Japan, you're overwhelmed by this sudden increase in Chinese tourists who are enjoying the heck out of the place. There are now, a few years ago, it was only three million a year. Now we're looking at six million. And by the way, in January, there were over a million Chinese, not counting the Hong Kong and Taiwan people, but there were only 900 Russians, by the way. So that's the kind of influx of Chinese people and identification. And it's not only that they're coming in, the Japanese are welcoming them. And when I went to New York the other day, there were lots of Chinese tourists.  So it seems to me that there is almost a schizophrenic aspect of China. And we are focused so much on the Spartan aspect of China when we should be also paying us attention to the Athenian democratic sort of wish to become democratic thing. In fact, Xi Jinping sent his daughter to Harvard. I don't think she went there to spy on Harvard. I think she's going to carry some information from there. So I wondered if you had any comments on that. Well, obviously, China is a great civilization. And as Lucian Pai once said, China is a great civilization trying to act like a nation state. So the benefits that we have gained, the world and Asia from Chinese civilization is, of course, something that we all know. And as one Chinese academic told me, he said, you know, Cheng Min, if we had to, if you guys, the world, had to pay intellectual property rights on paper, on gunpowder, and whatever else we've invented, you'd all be bankrupt. So obviously, yes, we were all beneficiaries of Chinese civilization. But the key issue is you've got the differentiate with the Chinese regime and the Chinese people. And so whether it's in Korea, obviously we've got tons of Chinese coming in daily. They're buying things, they're engaging. Well, you have thousands of Chinese students studying all over the world here in the US in particular. And when they go back home to the PRC, I hope they'll bring back not just how great American burgers are, but the fact that despite all the, I guess, noise and disrespect in some ways and discomfort, this large 300 million society keeps on trucking. And there's got to be a reason why this happens. And it's because despite all the trumps of this world, there is a budding consensus that builds over time. And that's, I think, a great aspect of American society that I hope our Chinese students can bring back. So whether they're studying overseas or wherever else, they will compare the world as they see it and the world as a regime tells them as it sees it. And that's a dichotomy which the Chinese government cannot control. And that's a dilemma that I think will deepen of the PRC. Two things, Kukuchi-san. One, remember Athens made Socrates drink poison. There's some. No, no, no, no. Yeah. And secondly, I'm reminded that at the spring holiday this year, Japanese took back about a million heated Japanese, excuse me, Chinese tourists took back from Japan about a million heated Japanese built toilet seats. And there was some comments about why can't we make good, but I approve of that. I lived in a traditional Japanese house for a time. And these are wonderful inventions. Mark, the second row here. Mark Wall, former State Department now affiliated with the Global Studies Center at the University of Wyoming. Could you comment on China's relationship with its other big land neighbor, namely India? How do you see that playing out? And also, how does the U.S. fit into that? You know, throughout the Cold War, India was one of Russia's key strategic partners. Despite the monolite movement, Russia was a major supplier of arms to New Delhi. Moscow supported New Delhi on key foreign policy initiatives. And particularly under the Gandhi regime, in Delhi, American foreign policy interests were not always aligned with, of course, with India's because of U.S. support for Pakistan. But after the Cold War, and particularly today, I think the Russian influence on India is marginal. The defense relationship has basically tanked over the last 20 years, although the Russians are still supplying spare parts for the aircraft and so forth, no new orders have come in, really, from New Delhi. And so if you look at where India is planning to build its next, I guess, greatly forward in terms of military projection, it's naval capacity, naval capabilities, which the Russians are not supplying, although they have supplied submarines to Vietnam. And from the Indian perspective, it took them a long time, but finally I think they realize that they have more in common strategically with the U.S. than with Russia. And that paradigm shift took about 20 years. And I think Indian leaders today from Prime Minister Modi on down can say this quite publicly, without the backlash that existed even 20, 30 years back. You could not say 30 years ago, oh, you know, there are certain aspects of America that actually works, and it's beneficial to Delhi's interests. And so when you meet with, for example, the interlocutors in Delhi, I had a very good dinner with Jai Shankar, who is the permanent secretary, the foreign secretary, and one of Prime Minister Modi's right-hand men on foreign policy issues. He was very erudite. He said, look, we do not want to go into an alliance with the U.S., but as long as our interests are aligned, we'll work with America. And one of the real things that I mentioned in my book is this, that there's an office in the Pentagon that focus exclusively on defense partnerships with India. Can you imagine this happening 30, 40 years ago? That would have never happened. And that, I think, tells the entire story of how much you've seen a shift in India's foreign policy thinking over the last couple of decades. You'll learn a lot about that, Mark. If you dial into the newly opened Carnegie office in Delhi, where we've got, we've got an absolutely superb... Yeah, headed by a great guy, C. Rajamohan. Rajamohan is spreading that, and we would actually tell us from here as it's contributing a lot, especially on the security side of the debates. About two years ago, Shyam Saran came here, and I've always thought of him as about one of the most fiercely articulate people I've ever met. And he always stuck to the non-alive. Two years ago, he came here, had a visit next door, and said, I'm here to tell you that's changed. What you couldn't say for 20 years is now something that they're quite straightforward about. I can roll in the back. Hi, then. Thank you for being over here and greets to you guys for your presence. I'm at the American University right now, and I'm doing my research on Pax Asiana. So is the Pax Asiana possible? And my question to you in this regard was, first of all, is a cohesive Asian identity possible to achieve that Pax Asiana period from Pax Americana? Right. And the second question I had was, if that is to be possible, we also need the United States out of the region. So is a cohesive, not only a cohesive Asian identity, but is it possible for Asian countries to get together and manage themselves in a security manner without the supervision of the United States, especially when it comes to strategic triangles like India, China, Japan, or Asian countries with China and Japan, South Korea, Japan, China? So other than the outliers like North Korea and Pakistan, can these powers come together and form an identity in economics, in security struggles, in human security, in environment? So that was my question. Thank you. My short answer is no, no, no, no, no. In the sense that, again, Asian has made great strides, it's over 30 years, there is an Asian identity in that sense, but that is very different from whether their interests actually converge. I spent two years in Singapore, traveled all over Southeast Asia. I can tell you that the feelings between Malaysia and Singapore are not really all of that in unison on a range of issues. And so I don't buy the argument that you can have one Asia, one big happy Asia, because Asia is a figment of the imagination. You cannot tell me that somehow the Pakistani mindset, however one defines it, coincides with what's happening in Taiwan. There are very, very different regions even within one particular country. So Asia has made progress in certain issues there's a greater acceptance of norms on proliferation on human rights, et cetera. But this is why I believe Asia's future lies in accepting and adapting and strengthening universal norms. I reject the argument that Asian values is the future of Asia. That is simply not true. And there are people who shall remain nameless, who go around selling this wonderful medicine called Asian values, but they're all from countries that are either authoritarian, semi-democratic or don't want to be democratic. And so as someone who lived in a country that was authoritarian, but is now a robust democracy, I can assure you that living in a democracy is a lot more fun. There was a question. The other question. So I'm Anna Concom with USAID. Relatedly I was gonna ask what the role is in regional organizations like ASEAN, our regional initiatives like NAPKI and addressing these fault lines. The more you have alphabet soups, it's good because it constrains behavior and it also increases norms. But at the margins, they make key contributions, but on the big, big, big issues on political stability, political development, the future of North Korea, nuclear proliferation and the list goes on, the future of South China Sea. I doubt highly whether these multilateral institutions, including for example the ARF, will be all the defective. So it's a talk shop, it's a necessary talk shop, but if you really have to butt heads and resolve outstanding issues, those issues calls for major power diplomacy and that is something that these so-called multilateral organizations simply aren't equipped to do at the present time. Very perceptive observation. This is a persistent problem. To every year we have a wonderful, one of the Washington-based NGO sponsors, two students from each of the best universities around Asia, Korea, Japan, Vietnam, Singapore, Philippines, you name it. And they come in, these are really presentable, competitive people. And what takes over the conversation, not 15 minutes into it every time, is all the animosity among the different nationalities. Now you'd think these kids would all say, well, we're all interested in this kind of computer, let's talk about Apple and what's, very quickly this stuff comes to the surface and it's gonna be, if it's those young people now, it's gonna be around quite a long time. Any more questions or comments? Please send them a fourth row. Hi, Rudy Dahl, Global Holding Copers. We do a lot of international infrastructure development and you were talking earlier about how the Chinese were influencing people with finance and infrastructure development. That's turning. The reality is hit, Africa, Bangladesh, Philippines. And they find out that the money isn't always there and they're bringing in thousands of Chinese laborers and merchant people. And now it's turned back and people aren't putting the new, I wonder what impact that's gonna have back on China. As opposed to Korea, I'm introducing some Korean companies into these. And by teaming XM and K-Shure and K-XM, we put together stuff and they're not interested in bringing other than the management in there. They just want it because you don't have enough local labor in Korea. So what do you think's gonna happen with this? My take on that is that many people say, well, South Korea should be much more forthcoming in exporting its so-called successful development model. And there's a certain truth to that, but it will be highly presumptuous of Koreans to go to country X in Africa or Latin America or South Asia and say, well, this is how we did it. And so you follow our path. That simply is not going to work. I think you've got to be a lot more, there has to be a lot less hubris. This is something that the Chinese are now learning.  And so every country in Asia benefited from two things. One is its ties with the global market really helped it to what export itself out of poverty. And that's the model that the Japanese, the Koreans, the Taiwanese and even the Chinese basically adopted. And the second thing which people don't really think about all that much in Asia is the reason why Asia was able to do this was because writ large, you had security, umbrellas and ties with the US. And the US did this for its own intrinsic, non altruistic reasons, but the end result is Asians could focus on development because you knew that Uncle Sam was behind you in certain respects. And that is one of the key conditions for Asia's super growth from the 60s onwards. That cannot duplicate I believe in other regions. But going back to your China question, yes, China has to learn that it's not enough to go in there with money. And although technology is a cultural carrier, you've got to really impress the local people that you're looking up for their interests, not necessarily your own. I think this point about learning is important. I think the Chinese are learning from their mistakes. And there's a history to this. The United States in the 50s suddenly found itself all over the world in ways it'd never been before.  And we had to learn a lot about local cultures and how to do business overseas. The Japanese followed us in a lot of clumsy ways in the 70s. And Korea did it in its own way in the 90s with the expansion of construction industries because you had built everything in Korea so you had to go overseas to keep building. And China's doing that now. My rule of thumb with China is if they say 40 billion, it's 40% of that. And maybe it won't even be 40% of that when we get down to it. But there's, my final point on this subject is 40% of 40 billion is a whole lot more than the 15 million we've got here and there. I'm not saying we have to go out and spend a lot of money, but we gotta get a lot more active in our representing our interests in these peripheral nations if we want them to pay enough attention to us when the time comes. You know, this has been a wonderful session for me, and all the more proud that we asked you to be a non-resident senior associate of the Carnegie Endowment and that you've undertaken this book when you first presented the pitch to me. I thought it was really onto something but the book itself delivers. Ladies and gentlemen, please find the book and read it and thank you. Thank you. Thank you. Can I just say one final word? Thank you so much. And to all my friends here in the audience from the US and from elsewhere, as someone who has benefited from Asia's rise, I think I should leave you with one final word and that is my relationship or my country's relationship with the US has really been a game changer and that's something that I'd like to really remind my American friends that the reason why Korea, Japan, other parts of Asia have been successful in many respects, despite the hard work and everything else is because of the fact that we have been linked with the US for the last 70 years and that's been a huge boon in the way that we have changed our thinking on key global issues so thank you for being there for the last 70 years.


Welcome back to our lecture series Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misledine. So we finally come to it, Lecture 25, what's so significant about Lecture 25? Lecture 25 is when we're really going to open up the cage and let calculus run free in our geometric system. So this lecture, if you've been following along with our lecture series here, is going to be the most calculus-based, the most analytic lecture of the entire lecture series, where we're really going to use the full strength of our continuity axiom for neutral geometry. Beforehand, we've talked about neutral geometry, we've talked about continuity, but we've never really used it, right? We use the Archimedean principle here or there, maybe a circular continuity, but those axioms do not require the full-blown deacon axiom. It doesn't require the full-blown least upper bound property of the real numbers. For example, the rational plane, which doesn't satisfy deacon's axioms, it has the Archimedean principle. The constructible plane, that is all the points and lines you can create in a plane using a straight-edge encompass, that'll have the Archimedean principle, that'll have circular continuity, and thus it'll have elementary continuity as well, but it doesn't have the full-blown deacon axiom. What do we need it for? Well, now we've arrived upon that issue. While we were studying the topics of Zachary and Lambert quadrilaterals, we have seen that the summit and fourth angles respectively of those quadrilaterals must be less than or equal to 90 degrees. I made some comments about, well, if the fourth angles or summit angles were equal to 90 degrees, that was equivalent to the Euclidean parallel postulate, but if they were properly less than 90 degrees, they were authentically at the acute angles, then we were in hyperbolic geometry. It begs the question, how small can they get? How much smaller than 90 degrees can these angles get? It turns out that even in hyperbolic geometry, we can get the summit angles of the secure quadrilateral to get really, really close to 90 degrees, in particular if it's a very small secure quadrilateral, then the measure of the summit angles will be very, very close to 90 degrees. Therefore, if we as creatures living in a hyperbolic universe, if we don't know we're hyperbolic, but we suspect we are, we could try to measure the summit angles of a secure quadrilateral. But if the quadrilateral is small relative to the size of the universe, then it turns out their angle measures will be so close to a right angle that we might not be able to measure them. That is, our instruments might not be able to detect the difference because the difference between 90 degrees and the actual angle measure might fall within the margin of error of our instruments. But on the other extreme, if our quadrilateral is large compared to the size of the geometry, then it turns out the summit angles can get arbitrarily small. These summit angles can in fact, well, we know by the Sicari-Legioner theorem that they're bounded above by 90 degrees. But we actually want to argue in this situation that the smallest they can get is zero. And zero is never actually going to be obtained. 90 degrees is only obtained in Euclidean geometry. But in hyperbolic geometry, we can get arbitrarily close to 90 degrees and zero degrees if we want to. And it depends on basically how big the quadrilateral is. And so in this lecture, I want to make explicit what we mean by how big it is. And so we're going to talk about this notion of an angle of parallelism. So let me explain this, this definition is a lot more cumbersome than other ones we've seen before. So let's explain what's going on here. So we have some line in our geometry for which we'll place two points on that line A and B. And so let's call this line L. L, of course, is the line determined by A and B. Then we are going to take a line T, which is the unique perpendicular line to L that passes out of the point A like so. So this is our point T. And we're going to select a point D on the perpendicular line and so I should indicate that this is in fact a perpendicular line like so. So we're going to select some point D that it's on the perpendicular line T, but it's not itself A. And then we're going to pick some point C that's on the same side of the line T that B is on. And so we get some point over here, call it C. And so if we take the line determined by D and C, you might get something like this, we're going to call this line M like so. And we ask ourselves the question, are the lines M and L parallel to each other? Now, in a Euclidean sense, if I were to extend this picture on, beyond the edge of the screen right here, you would think that these lines would eventually intersect each other, but that's because I've been drawing Euclidean lines in this diagram. If we're in a hyperbolic setting, the hyperbolic lines could look very, very different. And because of their curvature, they actually might avoid each other. Because remember, parallel doesn't mean anything to do with equidistance, it doesn't have anything to do with slope, because those are notions we have not introduced into our geometric framework yet. Parallel just means do they intersect each other? And so with this framework, we can ask, do the lines L and M intersect each other? Or are they parallel? So in that direction, consider the following set of numbers M, which this is gonna be a subset of the real numbers. And what we're gonna do is we are going to gather together all of the possible angle measures ADC, where we're treating C as a variable. So A, we don't even care about B so much in this conversation. Well, I mean, we do, we do, we do, because we have to talk about the angle ABD. With C is selected to be a point interior to the angle BAD, that's what I meant to say. And so we are going to gather all the points C, all the points in the interior of the angle BAD, such that the lines AB, which is L, and the lines CD, intersect each other, all right? So such an example would be something like this, right? If this was our choice of C, this line, DC would intersect the line L. So the measure of this angle would then be something we throw inside this set, okay? What's another choice? We could take a point C over here, take the line intersects. We could take some point over here, take the line, oh, intersects off the screen, but there's that. And we then measure what are the angle measures of these angles? And we gather all of those together, all of the angle measures into this set M. Now, the first thing to mention is that the set M is, it's a set of real numbers, of course, but it's a non-empty set of real numbers. Why is that? Well, because if you choose C to equal B, which after all, I said interior earlier, but I guess actually, if I correct myself the way I said it, you just have to be in the angle, which includes the boundary as well. But even if you don't like that, that there's an easy workaround of that, I'll say in a second. If you just take C to be B and you take the line that connects them together, well, that line clearly intersects L because the point B is on there. Now again, if you want an interior point, it's not such a big deal. Just take some point between D and B, call that point C, it'll be interior to the angle and so that C will do it. So again, if you want an interior point, like I originally said, not a big deal. But in particular, there do exist lines, there do exist, excuse me, I should say there do exist points so that the line imitating that passes through D, I should say, it will intersect L. So this isn't just an artifact of my diagram right here. No matter what the geometry is, as long as it's a neutral geometry, in every neutral geometry, there will be at least some point that causes the line DC to intersect the line L and therefore if you take the measurement of that angle, whatever it is, there's something there, it'll be non-intersecting. Excuse me, it will be intersecting, it won't be parallel. And so this set is non-empty, but I want us also to consider the case, what if M was perpendicular to T itself? And we're again, I'm treating the point C as a variable in this situation. There does exist, for example, the perpendicular that erects out of the point D that is perpendicular to the line T in that situation. So what if C was such a point where you get this right angle? Well, by the alternate interior angle theorem, this line D determined by this point C right here, that version of M would in fact be parallel to T in that situation. This is what we call the guaranteed parallel line. Such a line always existed in neutral geometry here. And so because there does exist a guaranteed parallel line and the guaranteed parallel line occurs when the angle in question here, ADC, has measure equal to 90 degrees, this gives us an upper bound to the set M. So I want us to consider that for a moment. The set M is not empty, it's a set of real numbers and it has an upper bound, 90 degrees. So the least upper bound property of the real numbers comes into play and describes this set M right here. It has to have a least upper bound, a real number for which anything strictly less than that number will make the lines be intersecting. And so we call such a number D naught. And we say that this value D naught is the so-called angle of parallelism with respect to the line L and with respect to the point D. Now I want to make it very clear in this situation that when we talk about an angle of parallelism, the angle depends on the line L and it just depends on the point D in this situation. Different lines and different points will give you different angles of parallelism potentially, right? In particular, even if you keep the same line L, if you choose different points D, those can give you different angles of parallelism, okay? So in particular, if you start moving D closer and closer and closer to A, that actually will enlarge the angle of parallelism. And if we take the limit as D approaches A, that actually will then obtain the upper bound of 90 degrees. So when A equals D, you would actually get the angle of parallelism there as 90 degrees. That doesn't really say so much because if D equals A, you're no longer a point off the line. But the idea is these angles of parallelism converge towards 90 degrees when D converges towards A. But it also goes the other way around that if we take points farther and farther and farther and farther away from A, in that situation, the angle of parallelism is going to converge towards zero degrees. That is when D diverges away from A, then you could say that D is D approaches infinity. Then in that situation, the angle of parallelism will converge towards zero degrees. And this is why we can get essentially any angle measure, any acute angle we want, because it depends how far apart the point in the line are. So when it comes to this discussion about angle of parallelism, it's important that we keep in mind that the segment AD is paramount in that discussion. It doesn't just depend on the line, it depends on the segment. In fact, in a neutral geometry, all lines are congruent to each other. So really the defining factor for the angle of parallelism is going to be how far off the line L is the point D. So since 90 degrees is an upper bound for any line and point, and this is a consequence of the alternate to your angle theorem, we see that we get D is bounded above by 90 degrees. And I mentioned as we converge, as D converges towards A, we can actually, we can get closer and closer. We can get arbitrary close to that 90 degrees as well. And as, like I said before, as D diverges away from A, we can get that this thing is going to sit between 90 degrees and zero degrees, like so. So the reason we care about the angle of parallelism is because we care about how other angles interact with it. So let's draw our picture again. We have L, we have T, like so. And we pick some, we pick some ray that obtains the angle of parallelism. So I'm going to add my points to the screen here. We have A, we have B, we have D. We're going to have then here in this angle, we'll call this angle the angle of parallelism. Okay. So since D naught is the least upper bound to our set M, what that means is the following. If we take, if we take some ray emanating out of D inside the angle D AB right here, in this case, if our angle is smaller than the angle of parallelism here, so we have some point C in consideration here. So the measure of ADC is less than D naught. Then what this tells us, because it's the angle of parallelism that's eventually this ray will intersect the line and that these lines have to intersect themselves somewhere. So this line L and M are not parallel to intersect because we are less than the angle of parallelism in this situation. Now, conversely, if we choose some point C so that when you look at the ray DC, if you look at the angle in this situation, we actually are getting an angle measure for ADC that is greater than the angle of parallelism and that situation that lines L and M are gonna in fact be parallel. So if you're larger than the angle of parallelism, you're gonna be parallel. If you're smaller than the angle of parallelism, then you're gonna be intersecting.  But it does beg the question, what happens if your angle is equal to the angle of parallelism? Does it intersect or are they parallel? It's the boundary point. It's the least upper bound but Dedekin's action doesn't tell us which one it belongs to. We have a Dedekin cut here. So the boundary point exists but it could belong to either the parallel side or the intersecting side. So the good news is we can actually prove that if you obtain the angle of parallelism, so you have a line AB and some point D and suppose that the angle of parallelism relative to this line and point is D naught in that situation, then if you have an angle measure ADC that's equal to the angle of parallelism, then the two lines are gonna be parallel in that situation. And so we can do this by contradiction of course. So considering our picture again, here's our lines T and L like so with their usual labels A, B and D. And it's important to note here that T and L are always right angles in this consideration and suppose that we have some line determined by the points D and C and suppose that the angle between these two lines here that is the angle between the angle between line T and line M is in fact the angle of parallelism. I'm gonna label my line L or excuse me, my line M in this situation. So suppose for the sake of contradiction that the line AB and the line M actually intersect each other. And so in such a situation we can actually without the loss of generality assume that of course the point of intersection turns out to be B in that situation. I'm actually gonna extend. No, no, this is fine right here. We'll just put that there. So we can actually take C to equal B and suppose that we have the angle of parallelism right here. So basically what we're saying is yep, we've obtained the angle of parallelism but the line turns out to be intersecting in that situation. What happens there? Well, by extension there's gonna exist some point E over here so that B is between A and E like so. Now because of that you can then consider the angle ADC right? Which of course C I didn't draw the second time but the angle ADC is equal to the angle ADB. And so if we take the line segment that connects D to E here, sorry about the crudeness there. Let me try that one more time. If we take the line segment that goes from D to E right there I should say it's the line determined by D and E there. This angle because B and C are interior points to the angle ADE. This is gonna be an angle that's bigger. It's gonna be bigger than the angle of parallelism. So D naught is less than the angle ADE. And as such that means the lines AB and DE are parallel to each other. But that's not the case, they contradict. That the contradiction because these lines do in fact intersect L contains E so does the line M here. And so we do get a contradiction. So if you obtain the angle of parallelism then that actually means that you are parallel itself. And so this line that obtains the angle of parallelism is essentially the smallest line that would be parallel to L passing through the point D. Now this concept deserves a definition here. So we have our line CD it's parallel to AB. We call this a limiting parallel. Some people like to call it an asymptotic parallel if the angle ADC obtains the angle of parallelism. So the limiting parallel is a parallel line that obtains the minimum boundary. So if you make that angle just a teeny bit smaller by any margin, any level of smaller it actually would intersect. So the idea with the limiting parallel is that these lines intersect each other like at infinity. So if you think of like the hyperbolic disk model here we could have for example a line that looks something like this. I'm actually gonna draw one with some semicircles. You might get semicircles that look something like this. It's possible and despite the crudeness of my drawing here it's possible that you have these semicircles which they share a point on the boundary but they don't share any points in the geometry. And as such, these are what we call limiting parallels. Another example of we use the half plane model for hyperbolic geometry. A vertical line is example of a hyperbolic line. We can also take a semicircle which contains that same point, something like this. These line and circle intersect on the x-axis but the x-axis doesn't belong to the geometry. These are both examples of limiting parallels. They touch each other at infinity but infinity doesn't belong to the geometry. And so they're asymptotic to each other. Notice they converge towards the same location but they never actually touch each other much like a rational function which is converging towards its vertical asymptote. They never touch each other but as you go towards the extreme they are both approaching the same value. So hyperbolic geometry does have these limiting parallels. Euclidean geometry has them too. They just don't look so weird. In Euclidean geometry, your all lines are gonna be limiting parallels. And that situation, your angle of parallelism is actually a right angle. But as you go from Euclidean geometry to elliptic geometry, that is you add points at infinity, these points intersect at infinity. I should say these lines intersect at infinity. It doesn't happen in the affine geometry but it does happen in the covering projective geometry. A topic we'll talk about a little bit later. All right, so the of a parallel line obtains the angle of parallelism we call it a limiting parallel. And yeah, in Euclidean geometry the angle of parallelism is always 90 degrees. We'll talk about that in just a second. Now, I wanna also clarify something before we go much farther here. In this situation, when we talked about parallel lines we really were talking about a line being parallel to a ray. And so if we were to draw our picture again, we have L, we have T. I'm actually gonna move T to put a little bit more in the middle right here. Whoops, one more time. So we have T like so. L, we have A, we have B. We have our point D, say like so. When we talk about the point C, it's always as we're talking about our ray, something like this, for which we do know that the angle of parallelism has to be an acute number or right. It can't be obtuse because after all the angle of parallelism was the least upper bound to the set M that we described earlier where M was the set of all angle measures that make the two lines intersect each other. Once you have 90 degrees, those rays aren't gonna intersect anymore. If you have some of the two-sangle going on like this, those aren't gonna, they're not gonna intersect each other. Now, mentally, if you go on the other side of the ray, then maybe you get some intersection going on there. But I want you to be aware that while we're talking about rays here, it's not such a big deal. The other side isn't going to intersect, but in particular, the concern I have is something like the following. What if we pick, let me distort my picture a little bit. What if we have something like the following? What if we have two rays on opposite sides of the line right here? What if in that situation, we have an angle over here that's smaller? Okay? So what I'm saying is the following, let's take some point C over here on this side of the line. Let's take some point C prime over here, and let's consider the angle D naught, which let's call that the angle of parallelism for D on the right side of T. And let's take this one over here, D naught prime, which we'll say that's the angle of parallelism on the left side of the line T. And again, for the sake, without the loss of generality, let's suppose that D naught is greater than D naught prime. Could you have something like that? Could it be unbalanced? That one side, it's easier to converge on than the other? That other one has maybe a bigger angle of parallelism, so that's actually harder? I don't know. Well, I do know. I'm just trying to speak hypothetically, of course, right here. Could it be possible that such a thing could happen? Well, I claim the answer is no. And why is that? So under the assumption that D naught is greater than D naught prime, then consider the point E that lives inside of the half plane AB that contains C. So we have some point over here. Some point over here, we're gonna call it E, such that the angle ADE, something like that, ADE, then is the angle D naught prime. Now, this is going to imply, since D naught prime is smaller than D naught, this implies that our line, DE actually must intersect the line L, and without the loss of generality, we can assume that point of intersection is gonna be B, no big deal there. In fact, we can say that B and E are the exact same point for all we care about, the same point in this consideration. I want you to then consider the triangle ADE, or ADB, whichever you prefer. It doesn't matter what you call the point there. So you have this triangle over here. Well, by triangle translation, we can translate this onto the other side of the line, for which case we're then gonna get a triangle over here, right, and so there's gonna be some point E prime over here so that ADE prime is congruent to ADE, and as this is a right angle, this is gonna be a right angle. So when we translate it over, we actually do get these supplementary angles at EAD and E prime AD like so. So these are triangles are intersecting, but then this, excuse me, these triangles are congruent, and that then forces an intersection between the ray, C prime D with the line L, but that refutes the fact that D not prime was in fact the angle of parallelism. So this tells us that D not prime cannot be smaller than D not, but by the same argument, D not cannot be smaller than D not prime. And so because of this argument here, we see that the angle of parallelism has nothing to do with the ray itself. It has entirely to do with the point D, right? Because whether you're on whichever side of the line you're on, it doesn't matter. You're gonna get the same angle of parallelism. So our definition of angle of parallelism is in fact a well-defined concept. All right, another thing we need to mention here is what about security quadrilaterals? We've seen previously that if you take a security quadrilateral, let me draw one for us right here. So security quadrilateral, remember the thing we know about security quadrilaterals is that we have these two right angles. The summit here, maybe it's potentially longer than the base. That's the possibility. But we do know that if we call the vertices A, B, C and D, we know that angle A and B are right angles thus congruent to each other. We know that the legs A, D and B, C are congruent to each other. That's what it means to be a security quadrilateral. We've discussed previously the summit angles of a security quadrilateral. They're congruent to each other, but in particular, they cannot be obtuse angles. They're either both acute angles or they're both right angles, all right? What we can then say because of the angle of parallelism is that these angles, these summit angles actually have to be greater than the angle of parallelism because of a security quadrilateral, as we also discussed, if we take the midpoint of the summit and the base and connect them together, you get the so-called altitude of the security quadrilateral. This shows us that the base and summit of the security quadrilateral are in fact parallel to each other because they have a common perpendicular, all right? I want you to look at this picture right here. This kind of looks like the picture we've been drawing over and over and over again, right? So we have the point B over here, the point A. I'm gonna throw these midpoints in there for the altitude as well. But we have the line L, which is determined by A and B. We have the line T, there's some point D right here. There's some point C, right? That determines this thing here. The summit is the line D, C. It is parallel to the base AB. And as such, this angle must be larger than the angle of parallelism. So for every security quadrilateral, the summit angles have to be greater than the angle of parallelism determined by the length of this. Of course, if the legs have different lengths that could change the angle of parallelism. But with whatever this length is given here and that's what determines the angle of parallelism, that length, how far apart are the points A and D? That length determines this. But for a security quadrilateral, this angle has to be, the summit angle has to be larger than the angle of parallelism. Now a nice little trick about security quadrilateral is if you introduce the altitude, this will cut the security quadrilateral into two congruent Lambert quadrilaterals because we actually now have three right angles in this situation. But nonetheless, when you look at these Lambert quadrilateral right here, this angle D, which was a summit angle for the security quadrilateral, this is the fourth angle for a Lambert quadrilateral. The same argument applies right here. The fourth angle of a Lambert quadrilateral has to be greater than or equal to the angle of parallelism because these two sides of the Lambert quadrilateral are parallel and it has to do with this distance here. So whatever this length is, we'll determine what the angle of parallelism is. And for a Lambert quadrilateral, those things have to be bigger than the angle of parallelism. So this is very much related to this topic of security and Lambert quadrilaterals that we had introduced in the previous lecture videos. All right, so let me mention two theorems about the angle of parallelism that are extremely important. The first one, theorem 3611 in our lecture series, I'm gonna leave this as a exercise to the viewer here. It's a very important theorem, don't get me wrong, but it's one that I believe the viewer should be able to prove. If the angle of parallelism with respect to a line L and with respect to a point P is acute, then there exists more than one line through P, which is parallel to L. And as such, this means that the hyperbolic parallel posture has to hold. And there's a couple of ways you could think about this. If your angle of parallelism is acute, one way to construct multiple parallel lines would be to consider a ray that obtains the angle of parallelism on one side and a ray that obtains the angle of parallelism on the other side. Those can't be the same lines because they're acute. That's gonna give us two different parallel lines, right? But we can also use the guaranteed parallel line. Let me use a different color there. The guaranteed parallel line is going to use a right angle, right? By the ultimate here, angle theorem, this gives the guaranteed parallel line. Since this is a right angle and these are acute angles, those lines can't be the same. So I've actually constructed three and then you could even make the argument that any ray that sits between the limiting parallel and the guaranteed parallel is also gonna be a parallel ray for which on the other side, that gives you then parallel lines. So in fact, if your angle of parallelism is acute, then you're gonna have an infinitude of parallel lines that live in sort of a spectrum. But again, I've given you the basic idea here. I'm gonna leave the viewer the opportunity to provide the necessary details there, but a pretty good hint has then been already executed here on the screen. So the next theorem here, this one I'm gonna prove this one in our video right now. In neutral geometry, the Euclidean parallel posh is equivalent to the following statements. The measure of the angle of parallelism is 90 degrees for all lines and all points. Very similar to the theorem 3, 6, 11 that we just talked about a moment ago. So if your angle of parallelism is ever acute, then you actually have the hyperbolic parallel posh in neutral geometry. But if your angle of parallelism is 90 degrees, then you get the Euclidean parallel posh. And so this measurement, this angle of parallelism really determines the parallel posh. And so looking at angles of parallelism, which is why we call it the angle of parallelism gives us a parallel posh for our neutral geometry. So there's two directions we need to go here. It's logically equivalent. So there's an if and only if statement. We can assume the Euclidean parallel posh and then prove that the limiting parallel has to be a right angle. And then we have to go the other direction as well. So we will assume that the angles of parallelism are always 90 degrees and then argue that we have the Euclidean parallel posh. So let's first assume that we have the Euclidean parallel posh. Let L be a line that we have, we have the same picture here and let P be a point not on L. Usually we call it in that D, but to be consistent with my notes here, let's call that point now P and let T be the unique perpendicular line to L through P. So that's this picture we have so far. Let D not be the angle of parallelism with respect to the line L and with respect to the point P. So we wanna argue that this angle of parallelism has to be 90 degrees. Let M be a limiting parallel through the point P with respect to the line L. Such a limiting parallel exists by D to consaxium. So we're gonna get something like this. This is our line M and let's claim that the angle between T and M is in fact this angle of parallelism D. Now by the Euclidean parallel posh, M is the only line parallel to L through P. That's what EPP gives us. On the other hand, by the alternator angle theorem, the unique line perpendicular to T that passes through L that's parallel to L is gonna be that guaranteed perpendicular line. So let's use a different color for that one. Because we do have our guaranteed perpendicular that if we form a right angle between T and this line we just directed out of P, that's gonna be parallel by the alternate interior angle theorem. But because of the Euclidean parallel posh, there can only be one parallel line to L that passes through M. So it turns out that passes through P, excuse me. So it turns out that these lines have to be one and the same thing. And since the angle between M and T was a right angle, this then gives us that the angle of parallelism was 90 degrees. Now given that the line L and the point P were arbitrary, this shows us that the angle of parallelism is always 90 degrees in Euclidean geometry. Now let's go the other direction here. So this time, let's suppose that the angle of parallelism is always going to be 90 degrees. Let's then argue that we have a unique parallel line. So to prove the Euclidean parallel posh, what we're gonna do is we always start off with a line L and we then take some point P that is off of the line. We then wanna show there's only one parallel line that passes through P in that situation. Well, let's take the perpendicular. It's perpendicular to L and goes through the point P. Such a construction exists in neutral geometry. We know that by the alter interior angle theorem that there's at least one parallel line, the so-called guaranteed parallel that will erect out of the point P here. I never gave this line a name. It's T of course. So we have this guaranteed line M. That it's gonna be a right angle in that situation. Now this line of course forms a right angle with T that we see right here. And hence it is a limiting parallel since it obtains the angle of parallelism. We're assuming right now the angle of parallelism is 90 degrees. Therefore no other parallel line can form an angle with T less than 90 degrees. But the two angles forms by such a parallel line either side, right? If we go on the other side as well, this one's also a right angle, right? Right angles are their own complements here. So either side, the angles are gonna be both right angles. You don't have one acute and one right angle either. So if we had some other line that passed through P, whoops, we had some other line that passed through P. Could it be possible to L? Well, one of the angles associated to that line would have to be an acute angle because we're thinking of this as a distinct line. One of those angles would have to be acute and therefore it's smaller than the angle of parallelism which then means that somewhere the lines are gonna intersect each other which means it's not in fact a parallel line. So we get a unique parallel line passing through P that's parallel to L and that then proves the Euclidean parallel postulate. So in neutral geometry, these two notions of the 90 degree angle of parallelism and the Euclidean parallel postulate they are logically equivalent to each other. Let me provide one last result about the angle of parallelism which then is going to connect the notions of hyperbolic geometry to the angle of parallelism. In neutral geometry, the hyperbolic parallel postulate is equivalent to the following statement. The measure of the angle of parallelism is acute for all lines and all points. I'm not saying they're equal to each other but if you have the hyperbolic parallel postulate all angles of parallelism will be acute for any line with any point and the direction is then reversible. If you have ever an angle of parallelism that is acute then actually implies the hyperbolic parallel postulate. Now I want to point out that of course theorem 3611 that we had mentioned on the previous slide but did not prove, it gives us the first direction we had mentioned in that situation that if your angle of parallelism is acute then you're gonna get multiple parallel lines and therefore that gives us the hyperbolic parallel postulate and given that this is true for all lines and all points that construction is arbitrary and therefore you always get multiple parallel lines for any line and any point. What about the other direction? So the direction we're trying to go now is assuming this statement, excuse me, no. The direction we need to go this time is that we're gonna assume the hyperbolic parallel postulate and argue that the angle of parallelism has to be acute. Well, let the geometry be hyperbolic. That is, it satisfies the hyperbolic parallel postulate. So there exists more than one line parallel to L through some point P. Thus the angle of parallelism cannot be 90 degrees. If the angle of parallelism was 90 degrees by the previous theorem we just did then there would only be one parallel line and therefore that's not hyperbolic. So we get that the angle of parallelism can't be 90 degrees. Now, since the angle of parallelism is less than or equal to 90 degrees, we know this in general, that's the only possibility. It can't be 90 degrees. It can't be obtuse and therefore it's gotta be acute and that's the last possibility. And so that brings us to the end of course of lecture 25 about the angle of parallelism. Like I had mentioned at the beginning of this video we definitely let our calculus run wild in this lecture but it was very necessary to grapple these notions of angle of parallelism and what it has to do with the parallel postulates inside of neutral geometry. The fruit of course of this will bear very quickly in the next couple lectures as we talk more about our search for a rectangle in neutral geometry and then ultimately we lead to the schism between Euclidean geometry and hyperbolic geometry. If you learned anything about the angle of parallelism in this video, please like it, subscribe to the channel to see some more mathematical videos like this in the future. And if you have any questions about angles of parallelism or anything you see in any of these lecture videos feel free to post your comments below and I'll be glad to answer them.