The Paradox of Proof: Mochizuki and the ABC Conjecture

The Paradox of Proof: Mochizuki and the ABC Conjecture

On August 31, 2012, Japanese mathematician Shinichi Mochizuki published four papers on the internet.

The titles were incomprehensible. The volume was intimidating: 512 pages in total. The claim was audacious: he declared that he had proved the abc conjecture, a famous, tantalizingly simple number theory problem that had stumped mathematicians for decades.

Then Mochizuki simply walked away. He didn't submit his work to the Annals of Mathematics. He didn't post a message on any of the online forums frequented by mathematicians around the world. He simply published the papers 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 specified topics. On September 2nd, Google Scholar sent him Mochizuki's papers: "This might interest you."

"And I'm like: 'Yes, Google, I am kind of interested!'" Ellenberg recalls. "I posted them on Facebook and on my blog, noting: 'By the way, it looks like Mochizuki has proved the abc conjecture.'"

The internet exploded. Within days, even mainstream media outlets picked up the story. "World's most complex mathematical theory cracked," announced the Telegraph. "Possible breakthrough in abc conjecture," the New York Times reported somewhat more modestly.

On the mathematics forum MathOverflow, mathematicians from around the world began to dispute and discuss Mochizuki's claim. 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 might shed light on the abc conjecture?" asked Andy Putman, an assistant professor at Rice University. Or, to paraphrase: "I don't understand any of it. Does anyone?"

The problem that many mathematicians encountered when they rushed to Mochizuki's website was that the proof was impossible to read. The first paper, titled "Inter-universal Teichmuller Theory I: Construction of Hodge Theaters," begins with the statement that the goal of the work is "to develop an arithmetic version of Teichmuller theory for number fields bounded by an elliptic curve... by applying the theory of semi-graphs of anabelioids, Frobenioids, etale theta functions, and log-shells."

This reads like gibberish not just to a layperson. It was gibberish to the mathematical community as well.

"Looking at it, you feel like you're reading a paper from the future or from outer space," Ellenberg wrote on his blog.

"It's very, very strange," says Columbia University professor Johan de Jong, who works in closely related areas of mathematics.

Mochizuki had created so many mathematical tools and assembled so many disparate areas of mathematics that his paper ended up filled with language that nobody could understand. It was absolutely unfamiliar and absolutely intriguing.

As Professor Moon Duchin of Tufts University put it: "He truly created his own world."

It would be a long time before anyone could even begin to understand Mochizuki's work, let alone verify the proof. In the months that followed, the papers sat like a weight on the mathematical community's shoulders. A handful of people approached them and began to study. Others tried but quickly gave up. Some ignored them entirely, preferring to observe from a distance. As for the man who had caused all the commotion, the person who claimed to have solved one of mathematics' greatest problems — not a sound.

For centuries, mathematicians have strived toward a single goal: to understand how the universe works and describe it. For this purpose, mathematics itself is merely a tool — 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 conjectures. Simply put, a theorem is an observation that has been proven true. The Pythagorean theorem, for example, says that for all right triangles, the relationship between the three sides a, b, and c is expressed by the formula a² + b² = c². Conjectures are the precursors to theorems — they represent a claim to a theorem, observations that mathematicians believe are true but have not yet been proven. If a conjecture is proved, it becomes a theorem, and when this happens, mathematicians celebrate and add the new theorem to the account of the known universe.

"The point isn't to prove a theorem," Ellenberg explains. "The point is to understand how the universe works and explain what the hell is going on."

Ellenberg is washing dishes while talking to me on the phone, and I can hear a small child's voice somewhere in the background. Ellenberg is passionate about explaining mathematics to the world. He writes a mathematics column for Slate magazine and is working on a book called How Not to Be Wrong, which aims to help ordinary people apply mathematics in everyday life.

The sound of dishes goes quiet as Ellenberg explains what motivates him and other mathematicians. I picture him gesturing in the air with soapy hands: "We feel the existence of this vast dark region of ignorance, but together we're all pushing forward, taking steps to move the boundary."

The abc conjecture digs deep into the darkness, reaching the very foundations of mathematics. First proposed by David Masser and Joseph Oesterle in 1985, it makes an observation about the fundamental relationship between addition and multiplication. But the abc conjecture isn't famous for its deep consequences — it's famous because on the surface it seems rather straightforward.

It begins with a simple equation: a + b = c.

The variables a, b, and c, which give the conjecture its name, have constraints. They must be whole numbers, and a and b must not share common factors — that is, they must not be divisible by the same prime number. So, for example, if a were 64, which equals 2&sup6;, then b cannot be any number divisible by two. In this case, b could be 81, which is 3&sup4;. Now a and b share no common factors, and we can form the equation 64 + 81 = 145.

It's easy to come up with combinations of a and b that satisfy the conditions. You could take large numbers, such as 3,072 + 390,625 = 393,697 (3,072 = 2¹&sup0; × 3 and 390,625 = 5&sup8;, no overlapping factors), or very small ones like 3 + 125 = 128 (125 = 5 × 5 × 5).

What the abc conjecture then says is that the properties of a and b influence the properties of c. To understand this observation, it may help to first rewrite these a + b = c equations in versions consisting of prime factors.

Our first equation, 64 + 81 = 145, is equivalent to 2&sup6; + 3&sup4; = 5 × 29.

Our second example, 3,072 + 390,625 = 393,697, is equivalent to 2¹&sup0; × 3 + 5&sup8; = 393,697 (a prime number!).

Our last example, 3 + 125 = 128, is equivalent to 3 + 5³ = 2&sup7;.

The first two equations don't look like the third, because in the first two equations we have many prime factors on the left side of the equation and very few on the right. In the third example, it's the opposite — the right side of the equation has more prime factors (seven) than the left side (only four). It turns out that of all possible combinations of a, b, and c, the third situation is very rare. In essence, the abc conjecture says that when there are many prime factors on the left side, then usually there won't be very many 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 the abc conjecture, all of this is expressed in more precise mathematical terms. But even in this simplified version, you can appreciate the conjecture's implications. The equation is based on addition, but the conjecture's observations say more about multiplication.

"It's about something very, very basic, about the tight relationship that connects the properties of addition and multiplication of numbers," says Minhyong Kim, a professor at the University of Oxford. "If there's something new to be discovered in this direction, you can be sure it's very important."

This idea is not obvious. Although mathematicians invented addition and multiplication, based on our current understanding of mathematics, there's no reason to think that the additive properties of numbers can somehow influence or affect their multiplicative properties.

"There's very little evidence for it," says Peter Sarnak, a professor at Princeton University, who is skeptical about the abc conjecture. "I'll believe it when I see a proof."

But what if it's true? Mathematicians say it would reveal close relationships between addition and multiplication that nobody knew about before.

Even the skeptic Sarnak admits: "If it's true, it will be one of the most stunning achievements."

In fact, it would be so monumental that it would automatically unlock many legendary mathematical mysteries. One of them would be Fermat's Last Theorem, the famous mathematical problem proposed in 1637 and solved only recently in 1993 by Andrew Wiles. Wiles's proof earned him more than 100,000 German marks in prize money (the equivalent of roughly $50,000 in 1997), a prize that had been offered nearly a century earlier in 1908. Wiles didn't solve Fermat's Last Theorem using the abc conjecture — he chose a different path — but if the conjecture were true, then the proof of the theorem would be a simple consequence.

Thanks to its simplicity, the abc conjecture is well known to all mathematicians. Professor Lucien Szpiro of the City University of New York says that "every professional has at least once tried" to theorize about a proof. But few have seriously attempted to find one. Szpiro, whose eponymous conjecture is a precursor to the abc conjecture, proposed a proof in 2007, but problems were soon found in it. Since then, no one had dared to take on the search — until Mochizuki came along.

When Mochizuki published his papers, the mathematical community had many reasons for enthusiasm. They were excited not because someone had claimed to prove an important conjecture, but because of who that someone was.

Mochizuki was renowned for his outstanding intellect. Born in Tokyo, he 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, New Hampshire. There he finished early, in just two years, at age 16, with excellent grades in mathematics, physics, American and European history, and Latin.

Mochizuki then entered Princeton University, where he again finished ahead of schedule, earning his bachelor's degree in mathematics in three years and quickly moving toward a doctorate, which he received at 23. After two years of teaching at Harvard University, he returned to Japan, where he joined the Research Institute for Mathematical Sciences at Kyoto University. In 2002, he became a professor at the unusually young age of 33. His early papers were widely recognized as very good work.

Academic prowess isn't the only characteristic that sets Mochizuki apart. His friend, Oxford professor Minhyong Kim, says that Mochizuki's most outstanding quality is his complete focus on work.

"Even among the many mathematicians I know, he demonstrates incredible patience and the ability to just sit and do mathematics for long, long hours," says Kim.

Mochizuki and Kim met in the early 1990s, when Mochizuki was still an undergraduate at Princeton. Kim, who had come on exchange from Yale University, remembers how Mochizuki was studying the works of French mathematician Alexander Grothendieck, whose works on algebraic and arithmetic geometry are required reading for every mathematician in the field.

"Most of us gradually come to understand [Grothendieck's works] over many years, after several periodic deep dives," Kim said. "Add to that thousands and thousands of pages."

But not Mochizuki.

"Mochizuki... simply read them from beginning to end, sitting at his desk," Kim recalls. "He started this process when he was still a senior undergraduate, and within a couple of years he had already finished."

A few years after returning to Japan, Mochizuki turned his attention to the abc conjecture. In the years that followed, rumors surfaced about his confidence that he had cracked the puzzle, and Mochizuki himself said he expected results by 2012. So when the papers appeared, the mathematical community was already waiting with anticipation. But then the enthusiasm faded.

"His other works — they're readable, I can understand them, and they're stunning," says de Jong, who works in a related field. Pacing around his office at Columbia University, de Jong shakes his head, remembering his first impression of the new papers. They were different. They were unreadable. After working in isolation for more than ten years, Mochizuki had built a mathematical language that only he himself could understand. Just to begin parsing the four papers published in August 2012, one would need to read hundreds, perhaps thousands of pages of his previous works, none of which had been checked or peer-reviewed. It would take at least a year to read and understand everything. De Jong had been considering taking a sabbatical and was prepared to spend a year on Mochizuki's papers, but when he saw the height of the mountain, he balked.

"I decided there was no way I could do it. It would drive me insane."

Soon, frustration gave way to anger. Few professors were willing to openly criticize a fellow mathematician, but virtually everyone I interviewed immediately noted that Mochizuki hadn't followed the standards of the community. Typically, they say, mathematicians discuss their findings with colleagues. They usually publish preprints on reputable forums. Then they submit their work to the Annals of Mathematics, where papers are refereed by prominent mathematicians before publication. Mochizuki bucked the trend. He was, according to his colleagues, "unorthodox."

But the most outrageous thing was Mochizuki's refusal to give lectures. Normally after publication, a mathematician gives lectures, travels to various universities to explain their work and answer colleagues' questions. Mochizuki rejected numerous invitations.

"A prominent research university asked him: 'Come, tell us about your results,' and he replied: 'I can't do that in one lecture,'" says Cathy O'Neil, de Jong's wife, a former mathematics professor better known as the blogger "Mathbabe."

"And they said: 'Fine, stay for a week,' and he says: 'I can't do it in a week.'"

"So they offered: 'Stay for a month. Stay as long as you need,' but he still said no."

"The guy just doesn't want to do it."

Kim sympathizes with the frustrated colleagues but offers a different explanation for the resentment: "Reading other people's work is very painful. And that's all... We're just too lazy to read them."

Kim tries to defend his friend, saying that Mochizuki's reticence stems from his "somewhat shy personality" and devotion to work: "He works very hard and really just doesn't want to spend time on planes, hotels, and all that."

O'Neil, however, holds Mochizuki responsible, saying his refusal to collaborate puts his colleagues in an awkward position: "You can't say you've proved something until you've explained it," she says. "A proof is a social construct. If the community doesn't understand it, you haven't done your job."

Today the mathematical community faces a dilemma: the proof of a very important conjecture hangs in the air, but no one dares to touch it. For a brief moment in October, everyone turned to Yale graduate Vesselin Dimitrov, who pointed to a possible contradiction in the proof, but Mochizuki quickly responded that he had accounted for this issue. Dimitrov backed off and activity died down.

Months passed, and the prevailing silence began to call into question a fundamental rule of mathematical academia. Duchin puts it this way: "Proofs are right or wrong. The community renders its verdict."

This foundation is a point of pride for mathematicians. The community works together; they don't compete. Colleagues check each other's work, spending many hours verifying that everything is correct. They do this not merely out of altruism — it's necessary: unlike medicine, where you know you're right if the patient recovers, or engineering, where the rocket either launches or it doesn't, theoretical mathematics, better known as "pure" mathematics, has no physical or visible standard. It is based entirely on logic. To know you're right, you need someone else, preferably many others, to walk in your footsteps and confirm that each step was correct. A proof in a vacuum is no proof at all.

Even an incorrect proof is better than no proof at all, because if the ideas are novel, they can still be useful for other problems or can push another mathematician toward finding the correct answer. Thus, the most important question isn't whether Mochizuki is right — what matters far more is whether the mathematical community will fulfill its role and actually read the papers.

The prospects are murky. Szpiro is one of the few who has made attempts to understand fragments of the paper. He holds weekly seminars with scholars from the City University of New York to discuss the paper, but he says they are limited to "local" analysis and don't yet understand the big picture. The only remaining candidate is Go Yamashita, Mochizuki's colleague at Kyoto University. According to Kim, Mochizuki holds private seminars with Yamashita, and Kim hopes that Yamashita will then explain the work. If Yamashita can't manage, it's unclear who else could take on the task.

For now, all the mathematical community can do is wait. While they wait, they tell stories and recall great moments in mathematics — the year Wiles conquered Fermat's Last Theorem, how Perelman proved the Poincare conjecture. Columbia professor Dorian Goldfeld tells the story of Kurt Heegner, a high school teacher in Berlin who solved a classical problem proposed by Gauss: "Nobody believed it. All the famous mathematicians snorted and dismissed it." Heegner's paper gathered dust for more than a decade until finally, four years after his death, mathematicians realized Heegner had been right all along. Kim recalls the proof of Fermat's Last Theorem proposed by Yoichi Miyaoka in 1988, which received great media attention until serious flaws were discovered. "It was very embarrassing for him," Kim recalls.

While they recall all these stories, Mochizuki and his proofs hang in the air. Any of these stories could be the possible ending. The question is simply: which one?

Kim remains one of the few people who are optimistic about the future of this proof. He is planning a conference at Oxford University this November, and he hopes to invite Yamashita to come and share what he has learned from Mochizuki. Perhaps then more will be known.

As for Mochizuki, who has rejected all media requests, who so resists the dissemination of his own work — one can only wonder whether he is aware of the commotion he has caused.

On his website, one of the few photographs of Mochizuki available on the internet shows a middle-aged man with old-fashioned 1990s-style glasses, looking up and off to the side, somewhere above our heads. A self-proclaimed title hangs above him. It's not "mathematician" but "inter-universal geometer."

What does that mean? The website gives no clues. There are only his thousand-page papers, piles of dense mathematics. His CV is modest and formal. He lists his marital status as "single (never married)." There is also a page titled "Thoughts of Shinichi Mochizuki," containing just 17 entries. "I would like to share my progress," he writes in February 2009. "Let me tell you about my progress," October 2009. "Let me tell you about my progress," April 2010, June 2011, January 2012. Then comes mathematical speech. It's hard to tell whether he is excited, depressed, frustrated, or inspired.

Mochizuki has been reporting on his progress for years, but where is he heading? This "inter-universal geometer," this probable genius, may have found something that will overturn number theory as we know it. He may have discovered a new path into the dark unknown of mathematics. But for now, his steps cannot be traced. Wherever he is going, he appears to be going alone.