Paradox evidence
On August 31, 2012, Japanese mathematician Shiniti Mochizuki published four articles on the Internet.
Headlines were incomprehensible. The volume was intimidating: 512 pages in total. The promise was insolent: he declared that he had proved abc- hypothesis, the famous, seductively light numerical theory, which for decades led mathematicians to a dead end.
Then Mochizuki just left. He did not send his work to Annals of Mathematics. He did not leave a message on any online forum that is frequently visited by mathematicians from all over the world. He simply published the articles and waited.
')
Two days later, Jordan Ellenberg, a professor of mathematics at the University of Wisconsin-Madison, received an email alert from Google Scholar, a service that scans the Internet for articles on the topics indicated. On September 2, Google Scholar sent him Mochizuki’s articles: “It might interest you.”
“I’m like this:“ Yes, Google, I’m kind of interested in it! ”,” Recalled Ellenberg, “I posted them on Facebook and in my blog, with the note:“ By the way, it seems that Mochizuki proved the abc hypothesis ” ".
Internet exploded. Within days, even the distant from mathematics media picked up the story. "The world's most complicated mathematical theory has been solved," the Telegraph announced. “A possible breakthrough in abc- hypothesis,” wrote the New York Times a bit more modestly.
At the MathOverflow math forum, mathematicians from around the world began to challenge and discuss the Mochizuki statement. The question that quickly became the most popular on the forum was simple: “Can anyone explain the philosophy of his work and comment on why it can shed light on the abc hypothesis?” Asked Andy Putman, an assistant professor at Rice University. Or, to paraphrase: “I did not understand anything. Did anyone understand? ”
The problem faced by many mathematicians who had fled to the Mochizuki site was that the proof was impossible to read. The first article, entitled “Teichmüller’s Inter-Universal Theory 1: Building Hodge Theaters,” begins with the statement that the goal of the work is to “develop an arithmetic version of Teichmüller’s theory for digital fields bounded by an elliptic curve ... functions and log shells.
This is similar to gibberish not only for the average man. It was gibberish and for the mathematical community.
“Looking at it, you feel as if you are reading an article from a future or distant cosmos,” wrote Ellenberg in his blog.
“She is very, very strange,” says Johan de Jong, a professor at Columbia University who works in related areas of mathematics.
Mochizuki created so many mathematical tools and collected so many incompatible areas of mathematics that his article was filled with a language that no one could understand. She was completely unusual and completely intriguing.
As Professor Mun Duchin of Tufts University, expressed it: "He truly created his own world."
It must be a long time before anyone will be able to understand the work of Mochizuki, much less appreciate the fidelity of the proof. In the following months, the articles were a stone on the shoulders of the mathematical community. A handful of people crept up to them and began to study. Others tried, but quickly surrendered. Some completely ignored them, preferring to watch from afar. As for the culprit of anxiety, the man who declared that he had solved one of the greatest problems of mathematics - there was not a sound from him.
For centuries, mathematicians strived for one goal: to understand how the universe works and to describe it. For this purpose, mathematics itself is only a tool — it is a language that mathematicians invented to help describe the known and explore the unknown.
The history of mathematical research is marked by milestones in the form of theorems and hypotheses. Simply put, a theorem is an observation that is considered true. The Pythagorean theorem, for example, says that for all right triangles the ratio between the three sides a , b and c is expressed by the formula a 2 + b 2 = c 2 . Hypotheses are the forerunners of theorems - they are an application for a theorem, observations that mathematicians consider to be true, but not yet proven. If the hypothesis is proven, it becomes a theorem, and when this happens, the mathematicians celebrate and add a new theorem to the known universe.
“The point is not to prove the theorem,” explains Ellenberg. - "The point is to understand the work of the universe and explain what the hell is going on."
Ellenberg is washing the dishes while talking to me on the phone, and I can hear the voice of a small child somewhere in the background. Ellenberg is eager to explain mathematics to the whole world. He maintains a math column for Slate magazine and is working on the book “How not to be wrong”, which should help ordinary people to use mathematics in everyday life.
The sound of the dishes freezes when Ellenberg explains what motivates him and other mathematicians. I imagine him gesticulating in the air with soapy hands: “We feel the existence of a huge dark area of ​​ignorance, but we are all pushing forward, taking steps to move the border.”
The abc- hypothesis digs deep into darkness, reaching the very foundations of mathematics. First proposed by David Masser and Joseph Esterle in 1980, she makes observations regarding the fundamental relationship between addition and multiplication. But abc- hypothesis is not known because of its profound consequences, but because on the surface it seems rather uncomplicated.
It starts with a simple equation: a + b = c .
The variables a , b , and c , which give the hypothesis its name, have limitations. They must be integers, and a and b should not have common factors, that is, they should not be divisible by the same prime number. So, for example, if a was 64, which equals 2 6 , then b cannot be any number that is divisible by two. In this case, b can be 81, which is 3 4 . Now a and b do not share common factors, and we can get the equation 64 + 81 = 145.
It is easy to come up with a combination of a and b , which satisfy the conditions. You can take large numbers, such as 3072 + 390625 = 393697 (3.072 = 2 10 x 3 and 390.625 = 5 8 , there are no overlapping multipliers), or very small ones, such as 3 + 125 = 128 (125 = 5 x 5 x 5 ).
What abc- hypothesis then says is that the properties of a and b affect the properties of c . To understand this observation, it can help to rewrite these equations a + b = c in a version consisting of simple factors.
Our first equation, 64 + 81 = 145, is equivalent to 2 6 + 3 4 = 5 x 29.
Our second example, 3072 + 390625 = 393697 is equivalent to 2 10 x 3 + 5 8 = 393697 (prime!)
Our last example 3 + 125 = 128 is equivalent to 3 + 5 3 = 2 7 .
The first two equations are not similar to the third, because in the first two equations we have many simple factors on the left side of the equation and very few on the right side of the equation. In the third example, on the contrary, on the right side of the equation there are more prime numbers (seven) than on the left (only four). It turns out that of all the possible combinations of a , b and c , the third situation is very rare. In essence, abc- hypothesis says that when there are a lot of simple factors on the left side, then, usually, there will not be many of them on the right side of the equation.
Of course, “many”, “not very many” and “usually” are very vague words and in the formal version of abc- hypothesis, all this is expressed in more precise mathematical terms. But even in this simplified version, the consequences of the hypothesis can be assessed. The equation is based on addition, but the observations of the hypothesis speak more about multiplication.
“It’s about something very, very basic, about a close relationship that relates the properties of addition and multiplication of numbers,” says Minhion Kim, a professor at Oxford University. “If there is something new that can be opened in this direction, then you can be sure that this is very important.”
This idea is not obvious. Although mathematicians invented addition and multiplication, based on the current understanding of mathematics, there is no reason to think that the properties of the addition of numbers can somehow influence or affect their properties of multiplication.
“There is very little evidence of this,” says Peter Sarnak, a professor at Princeton University, skeptical of the abc hypothesis. "I will believe only when I see the evidence."
But if this is true? Mathematicians say that it will open the close relationship between addition and multiplication, which no one knew before.
Even the skeptic Sarnak admits this: "If this is true, then this will be the greatest achievement."
In fact, it will be so great that it will automatically reveal many legendary mathematical puzzles. One of them will be the Great Fermat theorem, a well-known mathematical problem that was proposed in 1637 and solved only recently in 1993 by Andrew Wiles. Proof Wiles brought him more than 100,000 German marks in prize money (the equivalent of about $ 50,000 in 1997), an award that was offered almost a century earlier in 1908. Wiles did not solve the last Fermat's theorem with the help of abc- hypotheses, he chose another way, but if the hypothesis were true, then the proof of the theorem would be a simple consequence.
Because of its simplicity, abc- hypothesis is well known to all mathematicians. Lucien Spiro, a professor at the City University of New York, says that “every professional at least once tried” to theorize on evidence. But few people seriously tried to find it. Shpiro, whose hypothesis of the same name is a precursor of abc- hypothesis, offered evidence in 2007, but it soon revealed problems. Since then, no one dared to take up his search until Mochizuki.
When Mochizuki published his articles, the mathematical community had many reasons for enthusiasm. They were excited not because someone had declared evidence of an important hypothesis, but because who this person was.
Mochizuki was famous for his outstanding intelligence. Born in Tokyo, then moved to New York with his parents, Kiichi and Anne Mochizuki, when he was 5 years old. He left home to study at Phillips Academy in Exeter, in New Hampshire. There, he completed his studies as an external student two years later, at the age of 16, with excellent grades in mathematics, physics, American and European history, and Latin.
Then Mochizuki entered Princeton University, where he again completed his studies before the others, received a bachelor's degree in mathematics in three years, and quickly moved towards the candidate’s, which he received at 23. After two years of teaching at Harvard University, he returned to Japan, where he joined the research institute of mathematical sciences at Kyoto University. In 2002, he became a professor at an unusually young age of 33 years. His early articles were widely recognized as very good works.
Academic prowess is not the only characteristic that distinguishes Mochizuki from the rest. His friend, an Oxford professor, Minhyeong Kim, says that Mochizuki's most outstanding quality is his total focus on work.
“Even among many of my fellow mathematicians, he demonstrates incredible patience and the ability to just sit and do math for long, long hours,” says Kim.
Mochizuki and Kim met in the early 90s, when Mochizuki was still studying for a bachelor’s degree in Princeton. Kim, who arrived at the exchange from Yale University, recalls how Mochizuki studied the works of French mathematician Alexander Grothendieck, whose works on algebraic and arithmetic geometry are required for each mathematician to read.
"Most of us gradually come to understand [the works of Grothendieck] over many years, after several periodic dives," said Kim. - "Add to this thousands and thousands of pages."
But not Mochizuki.
"Mochizuki ... just read them from beginning to end sitting at his desk," recalls Kim. “He started this process when he was still a recent year student, and in a couple of years he had already finished.”
A couple of years after returning to Japan, Mochizuki turned his attention to abc- hypothesis. In subsequent years, rumors about his confidence that he solved the puzzle, and Mochizuki himself said that he expects results by 2012, appeared. Therefore, when articles appeared, the mathematical community was already looking forward. But then the enthusiasm was gone.
“His other works are readable, I can understand them and they are amazing,” says De Jong, who works in a similar field. Strolling through his office at Columbia University, De Jong shakes his head, recalling the first impression of new articles. They were different. They were unreadable. After working in isolation for more than ten years, Mochizuki built a mathematical language that only he himself can understand. In order to just begin to analyze the four articles published in August 2012, you need to read hundreds, maybe thousands of pages of his previous works, none of which have been reviewed or reviewed. It would take at least a year to read and understand everything. De Jong was already thinking of taking a vacation and was going to spend a year on Mochizuki articles, but when he saw the height of this mountain, he would pass.
“I decided that I could not do it in my life. It will drive me crazy. ”
Soon, disappointment was replaced by anger. Few professors were willing to openly criticize fellow mathematicians, but almost every person I interviewed immediately noted that Mochizuki did not follow community standards. As a rule, they say, mathematicians discuss their findings with colleagues. Usually they publish preprints in reputable forums. Then they send their work to Annals of Mathematics, where articles are reviewed by prominent mathematicians before publication. Mochizuki resisted the trend. He was, according to his colleagues, "wrong".
But the most outrageous was Mochizuki’s refusal to lecture. After publication, the mathematician usually gives lectures, goes to various universities explaining his work and answering the questions of his colleagues. Mochizuki rejected many offers.
“A prominent research university asked him:“ Come, tell us about your results, ”and he replied:“ I can’t do it in one lecture, ”says Katie ONil, De Jong's wife, a former mathematics professor better known as the blogger“ Mathbabe ".
"And they said:" Well, stay for a week, "and he answers:" I can not do it in a week. "
“Then they suggested:“ Stay for a month. Stay as long as you need, “but he still said no.”
"The guy just doesn't want to do this."
Kim sympathizes with disappointed colleagues, but offers a different explanation for the insult: “Reading other people's work is very painful. And everything ... We are just too lazy to read them. "
Kim is trying to protect his friend, he says that Mochizuki’s reticence is caused by his “slightly shy temper” and diligence in his work: “He works a lot and really doesn’t want to spend time on airplanes, hotels and the like.”
O'Neill, however, considers Mochizuki to be in charge, saying that his refusal to cooperate puts his colleagues in an awkward position: “You cannot say that you proved something until you explained it,” she says. “Proof is a social construct.” If the community does not understand, you have not done your job. "
Today, the mathematical community is in a dilemma: the proof of a very important hypothesis is in the air, but no one dares to touch it. For a brief moment in October, everyone turned to Yale University graduate, Vesselinu Dimitrov, who pointed out a possible contradiction in the proof, but Mochizuki quickly replied that he had taken this problem into account. Dimitrov retreated and activity subsided.
Months passed, and general silence began to question the basic rule of the mathematical scientific community. Duchin explains it this way: “The evidence is true or not true. Society renders a verdict. ”
This foundation is the pride of mathematicians. The community works together, they do not compete. Colleagues check each other's work, spend many hours checking that everything is correct. They do it not simply from altruism, it is necessary: ​​unlike medicine, where you know that you are right if the patient has recovered, or in a technique where the rocket either takes off or does not. Theoretical mathematics, better known as “pure” mathematics, does not have a physical or visible standard. It is entirely based on logic. To know that you are right, someone else is needed, preferably many other people, who would follow in your footsteps and confirm that every step was right. Proof in vacuum is not proof.
Even the wrong proof is better than its absence, because if ideas are new, they can still be useful for other problems or may push another mathematician to find the right answer. Thus, the most important question is not about the rightness of Mochizuki, much more important, will the mathematical community fulfill its role and read the articles?
Prospects are foggy. Shpiro is one of the few who attempted to understand the passages from the article. He conducts weekly seminars with scientists from the City University of New York to discuss the article, but he says that they are limited to “local” analysis and do not yet understand the big picture. The only candidate remains Guo Yamashita, Motizuki's colleague at Kyoto University. According to Kim, Mochizuki holds private workshops with Yamashita, and Kim hopes that Yamashita will then explain the work. If Yamashita fails, it is unclear who else will be able to master the task.
For now, everything that the mathematical community can do is wait. While they are waiting, they tell stories and recall great moments in mathematics — the year when Wiles defeated the Great Fermat theorem, as Perelman proved the Poincare conjecture. Colombian professor Dorian Goldfeld tells the story of Kurt Hegner, a high school teacher in Berlin who solved the classic problem proposed by Gauss: “No one believed it. All the famous mathematicians snorted and rejected it. ” Hegner’s article had been collecting dust for more than ten years until finally, four years after his death, mathematicians realized that Hegner had been right all this time. Kim recalls the proof of Fermat's Great Theorem, which was proposed by Yoichi Miyoka in 1988, which received a lot of attention from the media until it showed serious flaws. “He was very embarrassed,” Kim recalls.
As long as they remember all these stories, Mochizuki and his evidence are hanging in the air. All these stories may be possible endings. The only question is what?
Kim remains one of the few people who are optimistic about the future of this evidence. He is planning a conference at Oxford University this November, and he hopes to invite Yamashita to come and tell him what he learned from Mochizuki. Perhaps then more will be known.
As for Mochizuki, who rejected all media inquiries, who so resists the spread of his own work, one can only guess whether he is aware of the hype he has raised.
On his website, one of the few photos of Mochizuki available on the Internet shows a middle-aged man with old-fashioned glasses in the 90s style, looking up and off to the side, above our heads. The self-proclaimed title hangs over his head. This is not a "mathematician", but an "inter-universal geometer."
What does it mean? The site does not give tips. There are only his articles a thousand pages long, piles of dense mathematics. His resume is modest and formal. He indicates his marital status as “single (not married)”. There is also a page called “Thoughts of Shiniti Mochizuki”, on which there are only 17 notes. “I would like to share my progress,” he writes in February 2009. “Let me talk about my progress,” October 2009. “Let me talk about my progress,” April 2010, June 2011, January 2012. Then comes a mathematical speech. It is difficult to say whether he is excited, depressed, disappointed or inspired.
Mochizuki talked about his progress over the years, but where is he going? This “inter-universal geometer,” this probable genius, may have found something that will reverse the known theory of numbers. He may have opened a new path to the dark unknown of mathematics. But while his steps do not track down. Wherever he goes, he seems to go alone.
According to rumors , behind the pseudonym Satoshi Nakamoto, the creator of Bitcoin, is still the same Mochizuki. Mochizuki article links: 1 , 2 , 3 , 4
Headlines were incomprehensible. The volume was intimidating: 512 pages in total. The promise was insolent: he declared that he had proved abc- hypothesis, the famous, seductively light numerical theory, which for decades led mathematicians to a dead end.
Then Mochizuki just left. He did not send his work to Annals of Mathematics. He did not leave a message on any online forum that is frequently visited by mathematicians from all over the world. He simply published the articles and waited.
')
Two days later, Jordan Ellenberg, a professor of mathematics at the University of Wisconsin-Madison, received an email alert from Google Scholar, a service that scans the Internet for articles on the topics indicated. On September 2, Google Scholar sent him Mochizuki’s articles: “It might interest you.”
“I’m like this:“ Yes, Google, I’m kind of interested in it! ”,” Recalled Ellenberg, “I posted them on Facebook and in my blog, with the note:“ By the way, it seems that Mochizuki proved the abc hypothesis ” ".
Internet exploded. Within days, even the distant from mathematics media picked up the story. "The world's most complicated mathematical theory has been solved," the Telegraph announced. “A possible breakthrough in abc- hypothesis,” wrote the New York Times a bit more modestly.
At the MathOverflow math forum, mathematicians from around the world began to challenge and discuss the Mochizuki statement. The question that quickly became the most popular on the forum was simple: “Can anyone explain the philosophy of his work and comment on why it can shed light on the abc hypothesis?” Asked Andy Putman, an assistant professor at Rice University. Or, to paraphrase: “I did not understand anything. Did anyone understand? ”
The problem faced by many mathematicians who had fled to the Mochizuki site was that the proof was impossible to read. The first article, entitled “Teichmüller’s Inter-Universal Theory 1: Building Hodge Theaters,” begins with the statement that the goal of the work is to “develop an arithmetic version of Teichmüller’s theory for digital fields bounded by an elliptic curve ... functions and log shells.
This is similar to gibberish not only for the average man. It was gibberish and for the mathematical community.
“Looking at it, you feel as if you are reading an article from a future or distant cosmos,” wrote Ellenberg in his blog.
“She is very, very strange,” says Johan de Jong, a professor at Columbia University who works in related areas of mathematics.
Mochizuki created so many mathematical tools and collected so many incompatible areas of mathematics that his article was filled with a language that no one could understand. She was completely unusual and completely intriguing.
As Professor Mun Duchin of Tufts University, expressed it: "He truly created his own world."
It must be a long time before anyone will be able to understand the work of Mochizuki, much less appreciate the fidelity of the proof. In the following months, the articles were a stone on the shoulders of the mathematical community. A handful of people crept up to them and began to study. Others tried, but quickly surrendered. Some completely ignored them, preferring to watch from afar. As for the culprit of anxiety, the man who declared that he had solved one of the greatest problems of mathematics - there was not a sound from him.
For centuries, mathematicians strived for one goal: to understand how the universe works and to describe it. For this purpose, mathematics itself is only a tool — it is a language that mathematicians invented to help describe the known and explore the unknown.
The history of mathematical research is marked by milestones in the form of theorems and hypotheses. Simply put, a theorem is an observation that is considered true. The Pythagorean theorem, for example, says that for all right triangles the ratio between the three sides a , b and c is expressed by the formula a 2 + b 2 = c 2 . Hypotheses are the forerunners of theorems - they are an application for a theorem, observations that mathematicians consider to be true, but not yet proven. If the hypothesis is proven, it becomes a theorem, and when this happens, the mathematicians celebrate and add a new theorem to the known universe.
“The point is not to prove the theorem,” explains Ellenberg. - "The point is to understand the work of the universe and explain what the hell is going on."
Ellenberg is washing the dishes while talking to me on the phone, and I can hear the voice of a small child somewhere in the background. Ellenberg is eager to explain mathematics to the whole world. He maintains a math column for Slate magazine and is working on the book “How not to be wrong”, which should help ordinary people to use mathematics in everyday life.
The sound of the dishes freezes when Ellenberg explains what motivates him and other mathematicians. I imagine him gesticulating in the air with soapy hands: “We feel the existence of a huge dark area of ​​ignorance, but we are all pushing forward, taking steps to move the border.”
The abc- hypothesis digs deep into darkness, reaching the very foundations of mathematics. First proposed by David Masser and Joseph Esterle in 1980, she makes observations regarding the fundamental relationship between addition and multiplication. But abc- hypothesis is not known because of its profound consequences, but because on the surface it seems rather uncomplicated.
It starts with a simple equation: a + b = c .
The variables a , b , and c , which give the hypothesis its name, have limitations. They must be integers, and a and b should not have common factors, that is, they should not be divisible by the same prime number. So, for example, if a was 64, which equals 2 6 , then b cannot be any number that is divisible by two. In this case, b can be 81, which is 3 4 . Now a and b do not share common factors, and we can get the equation 64 + 81 = 145.
It is easy to come up with a combination of a and b , which satisfy the conditions. You can take large numbers, such as 3072 + 390625 = 393697 (3.072 = 2 10 x 3 and 390.625 = 5 8 , there are no overlapping multipliers), or very small ones, such as 3 + 125 = 128 (125 = 5 x 5 x 5 ).
What abc- hypothesis then says is that the properties of a and b affect the properties of c . To understand this observation, it can help to rewrite these equations a + b = c in a version consisting of simple factors.
Our first equation, 64 + 81 = 145, is equivalent to 2 6 + 3 4 = 5 x 29.
Our second example, 3072 + 390625 = 393697 is equivalent to 2 10 x 3 + 5 8 = 393697 (prime!)
Our last example 3 + 125 = 128 is equivalent to 3 + 5 3 = 2 7 .
The first two equations are not similar to the third, because in the first two equations we have many simple factors on the left side of the equation and very few on the right side of the equation. In the third example, on the contrary, on the right side of the equation there are more prime numbers (seven) than on the left (only four). It turns out that of all the possible combinations of a , b and c , the third situation is very rare. In essence, abc- hypothesis says that when there are a lot of simple factors on the left side, then, usually, there will not be many of them on the right side of the equation.
Of course, “many”, “not very many” and “usually” are very vague words and in the formal version of abc- hypothesis, all this is expressed in more precise mathematical terms. But even in this simplified version, the consequences of the hypothesis can be assessed. The equation is based on addition, but the observations of the hypothesis speak more about multiplication.
“It’s about something very, very basic, about a close relationship that relates the properties of addition and multiplication of numbers,” says Minhion Kim, a professor at Oxford University. “If there is something new that can be opened in this direction, then you can be sure that this is very important.”
This idea is not obvious. Although mathematicians invented addition and multiplication, based on the current understanding of mathematics, there is no reason to think that the properties of the addition of numbers can somehow influence or affect their properties of multiplication.
“There is very little evidence of this,” says Peter Sarnak, a professor at Princeton University, skeptical of the abc hypothesis. "I will believe only when I see the evidence."
But if this is true? Mathematicians say that it will open the close relationship between addition and multiplication, which no one knew before.
Even the skeptic Sarnak admits this: "If this is true, then this will be the greatest achievement."
In fact, it will be so great that it will automatically reveal many legendary mathematical puzzles. One of them will be the Great Fermat theorem, a well-known mathematical problem that was proposed in 1637 and solved only recently in 1993 by Andrew Wiles. Proof Wiles brought him more than 100,000 German marks in prize money (the equivalent of about $ 50,000 in 1997), an award that was offered almost a century earlier in 1908. Wiles did not solve the last Fermat's theorem with the help of abc- hypotheses, he chose another way, but if the hypothesis were true, then the proof of the theorem would be a simple consequence.
Because of its simplicity, abc- hypothesis is well known to all mathematicians. Lucien Spiro, a professor at the City University of New York, says that “every professional at least once tried” to theorize on evidence. But few people seriously tried to find it. Shpiro, whose hypothesis of the same name is a precursor of abc- hypothesis, offered evidence in 2007, but it soon revealed problems. Since then, no one dared to take up his search until Mochizuki.
When Mochizuki published his articles, the mathematical community had many reasons for enthusiasm. They were excited not because someone had declared evidence of an important hypothesis, but because who this person was.
Mochizuki was famous for his outstanding intelligence. Born in Tokyo, then moved to New York with his parents, Kiichi and Anne Mochizuki, when he was 5 years old. He left home to study at Phillips Academy in Exeter, in New Hampshire. There, he completed his studies as an external student two years later, at the age of 16, with excellent grades in mathematics, physics, American and European history, and Latin.
Then Mochizuki entered Princeton University, where he again completed his studies before the others, received a bachelor's degree in mathematics in three years, and quickly moved towards the candidate’s, which he received at 23. After two years of teaching at Harvard University, he returned to Japan, where he joined the research institute of mathematical sciences at Kyoto University. In 2002, he became a professor at an unusually young age of 33 years. His early articles were widely recognized as very good works.
Academic prowess is not the only characteristic that distinguishes Mochizuki from the rest. His friend, an Oxford professor, Minhyeong Kim, says that Mochizuki's most outstanding quality is his total focus on work.
“Even among many of my fellow mathematicians, he demonstrates incredible patience and the ability to just sit and do math for long, long hours,” says Kim.
Mochizuki and Kim met in the early 90s, when Mochizuki was still studying for a bachelor’s degree in Princeton. Kim, who arrived at the exchange from Yale University, recalls how Mochizuki studied the works of French mathematician Alexander Grothendieck, whose works on algebraic and arithmetic geometry are required for each mathematician to read.
"Most of us gradually come to understand [the works of Grothendieck] over many years, after several periodic dives," said Kim. - "Add to this thousands and thousands of pages."
But not Mochizuki.
"Mochizuki ... just read them from beginning to end sitting at his desk," recalls Kim. “He started this process when he was still a recent year student, and in a couple of years he had already finished.”
A couple of years after returning to Japan, Mochizuki turned his attention to abc- hypothesis. In subsequent years, rumors about his confidence that he solved the puzzle, and Mochizuki himself said that he expects results by 2012, appeared. Therefore, when articles appeared, the mathematical community was already looking forward. But then the enthusiasm was gone.
“His other works are readable, I can understand them and they are amazing,” says De Jong, who works in a similar field. Strolling through his office at Columbia University, De Jong shakes his head, recalling the first impression of new articles. They were different. They were unreadable. After working in isolation for more than ten years, Mochizuki built a mathematical language that only he himself can understand. In order to just begin to analyze the four articles published in August 2012, you need to read hundreds, maybe thousands of pages of his previous works, none of which have been reviewed or reviewed. It would take at least a year to read and understand everything. De Jong was already thinking of taking a vacation and was going to spend a year on Mochizuki articles, but when he saw the height of this mountain, he would pass.
“I decided that I could not do it in my life. It will drive me crazy. ”
Soon, disappointment was replaced by anger. Few professors were willing to openly criticize fellow mathematicians, but almost every person I interviewed immediately noted that Mochizuki did not follow community standards. As a rule, they say, mathematicians discuss their findings with colleagues. Usually they publish preprints in reputable forums. Then they send their work to Annals of Mathematics, where articles are reviewed by prominent mathematicians before publication. Mochizuki resisted the trend. He was, according to his colleagues, "wrong".
But the most outrageous was Mochizuki’s refusal to lecture. After publication, the mathematician usually gives lectures, goes to various universities explaining his work and answering the questions of his colleagues. Mochizuki rejected many offers.
“A prominent research university asked him:“ Come, tell us about your results, ”and he replied:“ I can’t do it in one lecture, ”says Katie ONil, De Jong's wife, a former mathematics professor better known as the blogger“ Mathbabe ".
"And they said:" Well, stay for a week, "and he answers:" I can not do it in a week. "
“Then they suggested:“ Stay for a month. Stay as long as you need, “but he still said no.”
"The guy just doesn't want to do this."
Kim sympathizes with disappointed colleagues, but offers a different explanation for the insult: “Reading other people's work is very painful. And everything ... We are just too lazy to read them. "
Kim is trying to protect his friend, he says that Mochizuki’s reticence is caused by his “slightly shy temper” and diligence in his work: “He works a lot and really doesn’t want to spend time on airplanes, hotels and the like.”
O'Neill, however, considers Mochizuki to be in charge, saying that his refusal to cooperate puts his colleagues in an awkward position: “You cannot say that you proved something until you explained it,” she says. “Proof is a social construct.” If the community does not understand, you have not done your job. "
Today, the mathematical community is in a dilemma: the proof of a very important hypothesis is in the air, but no one dares to touch it. For a brief moment in October, everyone turned to Yale University graduate, Vesselinu Dimitrov, who pointed out a possible contradiction in the proof, but Mochizuki quickly replied that he had taken this problem into account. Dimitrov retreated and activity subsided.
Months passed, and general silence began to question the basic rule of the mathematical scientific community. Duchin explains it this way: “The evidence is true or not true. Society renders a verdict. ”
This foundation is the pride of mathematicians. The community works together, they do not compete. Colleagues check each other's work, spend many hours checking that everything is correct. They do it not simply from altruism, it is necessary: ​​unlike medicine, where you know that you are right if the patient has recovered, or in a technique where the rocket either takes off or does not. Theoretical mathematics, better known as “pure” mathematics, does not have a physical or visible standard. It is entirely based on logic. To know that you are right, someone else is needed, preferably many other people, who would follow in your footsteps and confirm that every step was right. Proof in vacuum is not proof.
Even the wrong proof is better than its absence, because if ideas are new, they can still be useful for other problems or may push another mathematician to find the right answer. Thus, the most important question is not about the rightness of Mochizuki, much more important, will the mathematical community fulfill its role and read the articles?
Prospects are foggy. Shpiro is one of the few who attempted to understand the passages from the article. He conducts weekly seminars with scientists from the City University of New York to discuss the article, but he says that they are limited to “local” analysis and do not yet understand the big picture. The only candidate remains Guo Yamashita, Motizuki's colleague at Kyoto University. According to Kim, Mochizuki holds private workshops with Yamashita, and Kim hopes that Yamashita will then explain the work. If Yamashita fails, it is unclear who else will be able to master the task.
For now, everything that the mathematical community can do is wait. While they are waiting, they tell stories and recall great moments in mathematics — the year when Wiles defeated the Great Fermat theorem, as Perelman proved the Poincare conjecture. Colombian professor Dorian Goldfeld tells the story of Kurt Hegner, a high school teacher in Berlin who solved the classic problem proposed by Gauss: “No one believed it. All the famous mathematicians snorted and rejected it. ” Hegner’s article had been collecting dust for more than ten years until finally, four years after his death, mathematicians realized that Hegner had been right all this time. Kim recalls the proof of Fermat's Great Theorem, which was proposed by Yoichi Miyoka in 1988, which received a lot of attention from the media until it showed serious flaws. “He was very embarrassed,” Kim recalls.
As long as they remember all these stories, Mochizuki and his evidence are hanging in the air. All these stories may be possible endings. The only question is what?
Kim remains one of the few people who are optimistic about the future of this evidence. He is planning a conference at Oxford University this November, and he hopes to invite Yamashita to come and tell him what he learned from Mochizuki. Perhaps then more will be known.
As for Mochizuki, who rejected all media inquiries, who so resists the spread of his own work, one can only guess whether he is aware of the hype he has raised.
On his website, one of the few photos of Mochizuki available on the Internet shows a middle-aged man with old-fashioned glasses in the 90s style, looking up and off to the side, above our heads. The self-proclaimed title hangs over his head. This is not a "mathematician", but an "inter-universal geometer."
What does it mean? The site does not give tips. There are only his articles a thousand pages long, piles of dense mathematics. His resume is modest and formal. He indicates his marital status as “single (not married)”. There is also a page called “Thoughts of Shiniti Mochizuki”, on which there are only 17 notes. “I would like to share my progress,” he writes in February 2009. “Let me talk about my progress,” October 2009. “Let me talk about my progress,” April 2010, June 2011, January 2012. Then comes a mathematical speech. It is difficult to say whether he is excited, depressed, disappointed or inspired.
Mochizuki talked about his progress over the years, but where is he going? This “inter-universal geometer,” this probable genius, may have found something that will reverse the known theory of numbers. He may have opened a new path to the dark unknown of mathematics. But while his steps do not track down. Wherever he goes, he seems to go alone.
According to rumors , behind the pseudonym Satoshi Nakamoto, the creator of Bitcoin, is still the same Mochizuki. Mochizuki article links: 1 , 2 , 3 , 4
Source: https://habr.com/ru/post/183374/
All Articles