Hexbyte  Hacker News  Computers Mathematics as thought

Hexbyte Hacker News Computers Mathematics as thought

Hexbyte Hacker News Computers

‘These foundations [of mathematics] are, moreover, a perpetual source of reflection and discovery for the masters of the science. Even in number system you will find material for long reflection. Remember that Leibniz did not disdain to occupy himself with it.’ – Paul Valéry, letter to Pierre Honnorat, 1942

There are almost too many examples of the power and pervasiveness of mathematical ideas. For instance, this essay was written on a computer. The software of the computer, its mind and spirit, if you will, is a compilation of code that is based on the ideas of Claude Shannon, the father of information theory, and his article ‘A Mathematical Theory of Communication’ (1948). But perhaps this is both too obvious and too slight an example. The personal computer is hardly an essential part of human existence, even if most of us have structured our lives around it today. Let’s take something more basic and widespread, something most of us probably already apprehend, even if only dimly – like the idea of regression to the mean.  

As the Oxford English Dictionary explains, regression to the mean is ‘the tendency for the values of any distributed variable to move towards the mean over repeated independent trials’. In other words, the more trials, the less random or mistaken the measure. For instance, say you’re in a race at school. You do surprisingly well and beat most of your classmates. All things being equal, the next time around, you’re actually not likely to do as well, relative to the other runners. Obviously, one’s actual rank depends on skill and talent – if you did well the first time, you probably are pretty fast – but each result also depends on luck as well as a host of other circumstances. Therefore, in order to mitigate against any selection effect, one has to run the experiment multiple times. In order to be able to see just where you actually place or rank, you have to be able to know the shape or form of the distribution of outcomes. The notion of regression to the mean informs how we think about a wide range of things, from the design of clinical trials, to gambling, to, well, the prosaic pep-talks we give ourselves after coming up short by saying: ‘OK, next time will be better.’ Actually, it probably will be.

The Victorian polymath (and eugenicist) Sir Francis Galton first discovered and formulated the idea of regression to the mean in his studies of heredity. The ‘regression toward mediocrity’, in Galton’s memorable phrase, grew out of his studies of why genius seemed to run through families (think of the Bachs), but then also how and why it dissipated over time. Genius and talent themselves being impossible to measure, Galton studied other more quantifiable phenomena, for instance, the height differences between parents and children, and the spatial distribution of falling objects. He devised a contraption called the Quincunx, and found that when uniformly dropped objects encountered small deviations (think of a pinball machine with lots of stoppers), they nevertheless distributed themselves at the bottom along a normal distribution – what we generally call a bell curve. The bell curve, of course, is just one example of the various forms used to plot the relationship among independent and dependent variables. Galton’s conceptual innovation, his idea of regression to the mean, made certain features of the world newly visible. By doing so, he helped us count and describe other things we can’t see, or test, but that shape our lives.

The modern separation among scholars between intellectual history and the history of mathematics is untenable as mathematics might be the ultimate intellectual endeavour. In the words of the 19th-century German mathematician Carl Friedrich Gauss: ‘mathematics is the queen of the sciences’; like literacy, widespread numeracy is one of the defining features of modernity. In fact, one of the great shifts of modernity has been how mathematicians changed their view of mathematics, transforming the focus of their work from the study of the natural world to the study of ideas and concepts. Perhaps more than any other subject, mathematics is about the study of ideas. Yet, when people invoke the history of ideas, you are unlikely to hear about Dedekind’s cut (that is, the technique by which the real numbers are rigorously defined from the rational numbers), or L E J Brouwer’s rejection of Aristotle’s ‘law of excluded middle’, which states that any proposition is either true or that its negation is (put technically: for all propositions p, either p or not p). Nor are you likely to hear about the contested history of these ideas. Generally, when they talk about ideas, intellectual historians today mean political thought, cultural analysis, and maybe a sprinkling of economic and religious concepts, too.

Nor has this divide been one-sided. As the historian John Tresch recently noted: ‘For most historians of science trained in the past 30 years, doing history of science has meant avoiding the history of ideas.’ Even though most historians of science would surely consider the history of mathematics part of their field, in fact the subject is more often conspicuous by its absence than its presence. However, the larger problem among historians of science is that, while the consolidation of the field over the previous century has been an institutional and economic success, it has segregated the field. Today, it primarily takes place within separate departments or committees, with separate training. This reinforces, in practice and effect, a division between the study of science and the study of society – something its own literature repeatedly criticises.

‘Mathematics, like any literature, is created by human beings for their own amusement’ 

One of the many damaging results of this intellectual division is that most of us – that is, those of us who are not mathematicians, physicists or engineers – adopt a view of mathematics that is primarily the product of our encounter with it in grade school. For most people, unfortunately, mathematics is a set of confusing, repetitive, formalistic and abstract techniques. Yet this is exactly the opposite of how mathematicians see their own work. Rather, what attracts them, in the words of The Mathematics Lover’s Companion (2017) by the graph theorist Ed Scheinerman, are ‘joyful, beautiful’ theorems and proofs, which they arrive at through the sweat of intellectual play. Akin to the best poems, they contain truths about the world perfectly expressed.

Perhaps the most impassioned remonstration against our bifurcated view of mathematics is the pamphlet A Mathematician’s Lament (2009) by the American private-school teacher Paul Lockhart. Written in 2002 and circulated for years among mathematicians and educators, Lockhart’s essay ruthlessly criticised the simplistic nature of most mandatory mathematics education. Mathematics, Lockhart wrote, is almost always taught in a way to obscure the actual insights and reasoning, the grandeur and insight, the excitement and frustration, that drive mathematicians. ‘At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement,’ Lockhart writes, attempting to humanise the subject.

Continuing with his literature analogy, Lockhart emphasises that: ‘A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?’ And like most pieces of art, truly appreciating it is not a rote exercise: ‘works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste’. By teaching mathematics as a series of techniques, the appeal and nature of mathematics is masked: ‘Of course there’s no criticism going on in school – there’s no art being done to criticise!’ The notable English mathematician G H Hardy also compared mathematics to poetry in his own work A Mathematician’s Apology (1940): ‘A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.’

What then might we all learn from better integrating the two? What might be gained if, to quote Egon Spengler in the film Ghostbusters (1984), ‘we cross the streams’ of the history of mathematics and the history of ideas?

We should not understand the history of ideas as one besotted by speculations about politics. With that, hopefully, we can gain a wider sense of the diverse ideas and visions that, deep down (for better and for worse), motivate and inform any of our endeavours. Intellectual history today is dominated by elite political ideas. There are many factors that have contributed to the shortchanging of the history of mathematics and its ideas. High among them is how narrowly political the work of leading intellectual historians has been.

The British scholar Quentin Skinner and the French philosopher Michel Foucault, for example, are arguably the most influential intellectual historians of the past 50 years. Superficially, they are unlike each other in almost every way, from their prose style, to their subjects. Yet both portray intellectual history mostly in terms of political languages and expressions of power. The discourses and mentalities of individuals, for both of them, describe much of the social and political world, and account for when and why change happens. But one of the most important reasons for the rise of the modern state is the increasing possible reach of politicians and bureaucrats, changes powerfully facilitated by improvements in mathematics and technology, and that are absent in both of their accounts.

Recognising the role of mathematics in the history of how we think would actually make us more sophisticated in our political judgment. Those wishing to lend certainty and authority to their ideologies use mathematics, but they also abuse it. As the epitome of formal knowledge and the premier example of abstract reason, the spectre of mathematics has haunted European thought for centuries. From theologians to politicians and economists, many have naively – even manipulatively – adopted mathematical methods to develop, and too often implement, overly simplistic theories of the world. Despite the repeated problems with this approach, the impulse has been seminal, formative and comparatively neglected.

The discipline of economics offers a recent example, but this is also true for areas as widely separate as architecture and philosophy, as the political scientist and anthropologist James C Scott shows in his book Seeing Like a State: How Certain Schemes to Improve the Human Condition Have Failed (1999). The allure of formal schemes has turned people’s attention to the results of mathematics, rather than to the creative ways that mathematicians work and think. Commenting on the aftermath of the late 19th- and early 20th-century controversy over the foundations of his subject, the influential mathematician John von Neumann wrote in the essay The Mathematician (1947) that ‘[T]he very concept of “absolute” mathematical rigour is not immutable. The variability of the concept of rigour shows that something else besides mathematical abstraction must enter into the makeup of mathematics.’ While the reason that mathematics is authoritative is due of its supposed rigour and certainty, what we mean by rigour has changed throughout history, based on novel ideas and realisations. How mathematics works rests on no absolute timeless standard, despite what many assume today, given its precision and efficacy.

Taking mathematics seriously might lead thinking back to questions of aesthetics and beauty

In some ways, mathematics is the hardest, and most realistic, art. The German philosopher Martin Heidegger, for instance, offers an example of an influential thinker who misrepresented mathematics as a kind of thinking divorced from the natural world. In the essay ‘Modern Science, Metaphysics, and Mathematics’ (1962), he writes: ‘In the mathematical project develops an obligation to principles demanded by the mathematical itself.’ In Heidegger’s view, the embrace of mathematics and technology brings a kind of repetitious, routinised, alienating world in its wake. The paradox, however, is that a desire to bypass repetitious exercises is precisely one of the things that motivates mathematicians. Elementary proofs are nothing if not ways to think about something in all applicable cases so that repetitive testing on each possibility is not necessary. Nothing could be more elegant. Describing technology and mathematical modes of thought as mechanistic and alienating mistakes the result for the process.

Finally, taking mathematical ideas seriously might lead to discovering that technical ideas are as important as political or religious ones. Taking mathematics seriously might also, counter to stereotypes, lead thinking away from current preoccupations with culture and power, and back to questions of aesthetics and beauty. Aesthetics and beauty are ever-present concerns in art and in mathematics, though seemingly small matters to historians and humanists today, preoccupied as they are with power.

Take, for example, the case of the Weils: Simone Weil was an important literary and political writer, regarded as one of the more insightful religious philosophers and social critics of the first half of the 20th century; her brother André, a mathematician, made important contributions to a variety of areas of mathematics, but is much less widely known, unless one is a mathematician. There are many colourful anecdotes about André Weil’s life – he escaped war-torn Europe and lived all around the world – but perhaps most importantly, and amusingly, he was a founding member the Bourbaki group. This was a loose association of mostly French mathematicians in the first half of the 20th century who tried to put mathematics on a new and self-contained basis (a recurring goal in the history of mathematics and, more generally, the history of thought). One of the main ways they tried to achieve this goal was by collectively writing textbooks under the name of a fictitious character, one Nicolas Bourbaki. Bourbaki doesn’t exist, but he published not a few books, and came up with some of the terms and symbols modern mathematicians use, such as Ø to represent the empty set. André Weil led the first meeting of the Bourbaki group.

And Weil is just the beginning. There’s Shannon and his insights into information theory, along with his teacher, the prodigy Norbert Wiener, who coined the term ‘cybernetics’ (and wrote a scintillating two-volume autobiography). There’s the debates between Brouwer and David Hilbert about the basis of mathematics. And there’s the extraordinary career of Emmy Noether, who made contributions across many fields of mathematics and physics, and helped to apply abstract algebra to topology (which deals with the preservation of spatial properties during transformations of shape and size, and is one of the main ways topology is done today). In other words, it would mean beginning to be as interested in the creators of our technological civilisation as history has been in those who lament it.

The history of ideas can, should and, historically, has often been precisely about the limitations of politics and the rewards of other ideas; it need not primarily be the study of thinkers who, more often than not, never even directly held power. The ideas that most deeply shape us, and our societies, are just as often not part of the history of political thought, but come from many other areas. This is not an argument for a neo-Platonic worshipful examination of ideas, or for a move away from considerations of politics. It is rather a call to recognise the ways in which humans actually encounter ideas, and the unique vitality of mathematical ideas in the world.

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Hexbyte  Hacker News  Computers Titans of Mathematics Clash Over Epic Proof of ABC Conjecture | Quanta Magazine

Hexbyte Hacker News Computers Titans of Mathematics Clash Over Epic Proof of ABC Conjecture | Quanta Magazine

Hexbyte Hacker News Computers

number theory
ByErica Klarreich

September 20, 2018

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

In a report posted online today, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a “serious, unfixable gap” within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory.

Despite multiple conferences dedicated to explicating Mochizuki’s proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called “a sense of infinite regress.”

Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct, wrote Ivan Fesenko of the University of Nottingham in an email. But only mathematicians in “Mochizuki’s orbit” have vouched for the proof’s correctness, Conrad commented in a blog discussion last December. “There is nobody else out there who has been willing to say even off the record that they are confident the proof is complete.”

Nevertheless, wrote Frank Calegari of the University of Chicago in a December blog post, “mathematicians are very loath to claim that there is a problem with Mochizuki’s argument because they can’t point to any definitive error.”

That has now changed. In their report, Scholze and Stix argue that a line of reasoning near the end of the proof of “Corollary 3.12” in Mochizuki’s third of four papers is fundamentally flawed. The corollary is central to Mochizuki’s proposed abc proof.

“I think the abc conjecture is still open,” Scholze said. “Anybody has a chance of proving it.”

Scholze and Stix’s conclusions are based not only on their own study of the papers but also on a weeklong visit they paid to Mochizuki and his colleague

Yuichiro Hoshi

in March at Kyoto University to discuss the proof. That visit helped enormously, Scholze said, in distilling his and Stix’s objections down to their essence. The pair “came to the conclusion that there is no proof,” they wrote in their report.

But the meeting led to an oddly unsatisfying conclusion: Mochizuki couldn’t convince Scholze and Stix that his argument was sound, but they couldn’t convince him that it was unsound. Mochizuki has now posted Scholze’s and Stix’s report on his website, along with several reports of his own in rebuttal. (Mochizuki and Hoshi did not respond to requests for comments for this article.)

In his rebuttal, Mochizuki attributes Scholze and Stix’s criticism to “certain fundamental misunderstandings” about his work. Their “negative position,” he wrote, “does not imply the existence of any flaws whatsoever” in his theory.

Just as Mochizuki’s high reputation made mathematicians view his work as a serious attempt on the abc conjecture, Scholze and Stix’s stature guarantees that mathematicians will pay attention to what they have to say. Though only 30, Scholze has risen quickly to the top of his field. He was awarded the Fields Medal, mathematics’ highest honor, in August. Stix, meanwhile, is an expert in Mochizuki’s particular area of research, a field known as anabelian geometry.

“Peter and Jakob are extremely careful and thoughtful mathematicians,” Conrad said. “Any concerns that they have … definitely merit being cleared up.”

Hexbyte Hacker News Computers The Sticking Point

The abc conjecture, which Conrad has called “one of the outstanding conjectures in number theory,” starts with one of the simplest equations imaginable: a + b = c. The three numbers a, b and c are supposed to be positive integers, and they are not allowed to share any common prime factors — so, for example, we could consider the equation 8 + 9 = 17, or 5 + 16 = 21, but not 6 + 9 = 15, since 6, 9 and 15 are all divisible by 3.

Given such an equation, we can look at all the primes that divide any of the three numbers — so, for instance, for the equation 5 + 16 = 21, our primes are 5, 2, 3 and 7. Multiplying these together produces 210, a much larger number than any of the numbers in the original equation. By contrast, for the equation 5 + 27 = 32, whose primes are 5, 3 and 2, the prime product is 30 — a smaller number than the 32 in the original equation. The product comes out so small because 27 and 32 have only small prime factors (3 and 2, respectively) that get repeated many times to make them.

If you start playing around with other abc triples, you’ll find that this second scenario is extremely rare. For example, among the 3,044 different triples you can make in which a and b are between 1 and 100, there are only seven in which the product of primes is smaller than c. The abc conjecture, which was first formulated in the 1980s, codifies the intuition that this kind of triple hardly ever happens.

More specifically, coming back to the 5 + 27 = 32 example, 32 is larger than 30, but only by a little. It’s smaller than 302, or 301.5, or even 301.02, which is about 32.11. The abc conjecture says that if you pick any exponent bigger than 1, then there are only finitely many abc triples in which c is larger than the product of the prime factors raised to your chosen exponent.

“The abc conjecture is a very elementary statement about multiplication and addition,” said Minhyong Kim of the University of Oxford. It’s the kind of statement, he said, where “you feel like you’re revealing some kind of very fundamental structure about number systems in general that you hadn’t seen before.”

And the simplicity of the a + b = c equation means that a wide range of other problems fall under the conjecture’s sway. For instance, Fermat’s Last Theorem is about equations of the form xn + yn = zn, and Catalan’s Conjecture, which says that 8 and 9 are the only two consecutive perfect powers (since 8 = 23 and 9 = 32), is about the equation xm + 1 = yn. The abc conjecture (in certain forms) would offer new proofs of these two theorems and solve a host of related open problems.

The conjecture “always seems to lie on the boundary of what is known and what is unknown,”

Dorian Goldfeld

of Columbia University

has written


The wealth of consequences that would spring from a proof of the abc conjecture had convinced number theorists that proving the conjecture was likely to be very hard. So when word spread in 2012 that Mochizuki had presented a proof, many number theorists dived enthusiastically into his work — only to be stymied by the unfamiliar language and unusual presentation. Definitions went on for pages, followed by theorems whose statements were similarly long, but whose proofs only said, essentially, “this follows immediately from the definitions.”

“Each time I hear of an analysis of Mochizuki’s papers by an expert (off the record) the report is disturbingly familiar: vast fields of trivialities followed by an enormous cliff of unjustified conclusions,” Calegari wrote in his December blog post.

Scholze was one of the paper’s early readers. Known for his ability to absorb mathematics quickly and deeply, he got further than many number theorists, completing what he called a “rough reading” of the four main papers shortly after they came out. Scholze was bemused by the long theorems with their short proofs, which struck him as valid but insubstantial. In the two middle papers, he later wrote, “very little seems to happen.”

Then Scholze got to Corollary 3.12 in the third paper. Mathematicians usually use the word “corollary” to denote a theorem that is a secondary consequence of a previous, more important theorem. But in the case of Mochizuki’s Corollary 3.12, mathematicians agree that it is at the core of the proof of abc. Without it, “there is no proof at all,” Calegari wrote. “It is a critical step.”

This corollary is the only theorem in the two middle papers whose proof is longer than a few lines — it fills nine pages. As Scholze read through them, he reached a point where he couldn’t follow the logic at all.

Scholze, who was only 24 at the time, believed the proof was flawed. But he mostly stayed out of discussions about the papers, except when asked directly for his thoughts. After all, he thought, perhaps other mathematicians would find significant ideas in the paper that he had missed. Or, perhaps, they would eventually come to the same conclusion as he had. One way or the other, he thought, the mathematics community would surely be able to sort things out.

Hexbyte Hacker News Computers Escher’s Staircase

Meanwhile, other mathematicians were grappling with the densely written papers. Many had high hopes for a meeting dedicated to Mochizuki’s work in late 2015 at the University of Oxford. But as several of Mochizuki’s close associates tried to describe the key ideas of the proof, a “cloud of fog” seemed to descend over the listeners, Conrad wrote in a report shortly after the meeting. “Those who understand the work need to be more successful at communicating to arithmetic geometers what makes it tick,” he wrote.

Within days of Conrad’s post, he received unsolicited emails from three different mathematicians (one of them Scholze), all with the same story: They had been able to read and understand the papers until they hit a particular part. “For each of these people, the proof that had stumped them was for 3.12,” Conrad later wrote.

Kim heard similar concerns about Corollary 3.12 from another mathematician, Teruhisa Koshikawa, currently at Kyoto University. And Stix, too, got perplexed in the same spot. Gradually, various number theorists became aware that this corollary was a sticking point, but it wasn’t clear whether the argument had a hole or Mochizuki simply needed to explain his reasoning better.

Then in late 2017 a rumor spread, to the consternation of many number theorists, that Mochizuki’s papers had been accepted for publication. Mochizuki himself was the editor-in-chief of the journal in question, Publications of the Research Institute for Mathematical Sciences, an arrangement that Calegari called “poor optics” (though editors generally recuse themselves in such situations). But much more concerning to many number theorists was the fact that the papers were still, as far as they were concerned, unreadable.

“No expert who claims to understand the arguments has succeeded in explaining them to any of the (very many) experts who remain mystified,”

Matthew Emerton

of the University of Chicago



Calegari wrote a blog post decrying the situation as “a complete disaster,” to a chorus of amens from prominent number theorists. “We do now have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else,” Calegari wrote.

PRIMS soon responded to press inquiries with a statement that the papers had not, in fact, been accepted. Before they had done so, however, Scholze resolved to state publicly what he had been saying privately to number theorists for some time. The whole discussion surrounding the proof had gotten “too sociological,” he decided. “Everybody was talking just about how this feels like it isn’t a proof, but nobody was actually saying, ‘Actually there is this point where nobody understands the proof.’”

So in the comments section below Calegari’s blog post, Scholze wrote that he was “entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12.” He added that mathematicians “who do claim to understand the proof are unwilling to acknowledge that more must be said there.”

Shigefumi Mori, Mochizuki’s colleague at Kyoto University and a winner of the Fields Medal, wrote to Scholze offering to facilitate a meeting between him and Mochizuki. Scholze in turn reached out to Stix, and in March the pair traveled to Kyoto to discuss the sticky proof with Mochizuki and Hoshi.

Mochizuki’s approach to the abc conjecture translates the problem into a question about elliptic curves, a special type of cubic equation in two variables, x and y. The translation, which was well known before Mochizuki’s work, is simple — you associate each abc equation with the elliptic curve whose graph crosses the x-axis at a, b and the origin — but it allows mathematicians to exploit the rich structure of elliptic curves, which connect number theory to geometry, calculus and other subjects. (This same translation is at the heart of Andrew Wiles’ 1994 proof of Fermat’s Last Theorem.)



conjecture then boils down to proving a certain inequality between two quantities associated with the elliptic curve. Mochizuki’s work translates this inequality into yet another form, which, Stix said, can be thought of as comparing the volumes of two sets. Corollary 3.12 is where Mochizuki presents his proof of this new inequality, which, if true, would prove the



The proof, as Scholze and Stix describe it, involves viewing the volumes of the two sets as living inside two different copies of the real numbers, which are then represented as part of a circle of six different copies of the real numbers, together with mappings that explain how each copy relates to its neighbors along the circle. To keep track of how the volumes of sets relate to one another, it’s necessary to understand how volume measurements in one copy relate to measurements in the other copies, Stix said.

“If you have an inequality of two things but the measuring stick is sort of shrunk by a factor which you don’t control, then you lose control over what the inequality actually means,” Stix said.

It is at this crucial spot in the argument that things go wrong, Scholze and Stix believe. In Mochizuki’s mappings, the measuring sticks are locally compatible with one another. But when you go around the circle, Stix said, you end up with a measuring stick that looks different from if you had gone around the other way. The situation, he said, is akin to Escher’s famous winding staircase, which climbs and climbs only to somehow end up below where it started.

This incompatibility in the volume measurements means that the resulting inequality is between the wrong quantities, Scholze and Stix assert. And if you adjust things so the volume measurements are globally compatible, then the inequality becomes meaningless, they say.

Scholze and Stix have “identified a way that the argument can’t possibly work,” said Kiran Kedlaya, a mathematician at the University of California, San Diego, who has studied Mochizuki’s papers in depth. “So if the argument is to be correct, it has to do something different, and something a lot more subtle” than what Scholze and Stix describe.

Something more subtle is exactly what the proof does, Mochizuki contends. Scholze and Stix err, he wrote, in making arbitrary identifications between mathematical objects that should be regarded as distinct. When he told colleagues the nature of Scholze and Stix’s objections, he wrote, his descriptions “were met with a remarkably unanimous response of utter astonishment and even disbelief (at times accompanied by bouts of laughter!) that such manifestly erroneous misunderstandings could have occurred.”

Mathematicians will now have to absorb Scholze and Stix’s argument and Mochizuki’s response. But Scholze hopes that, in contrast with the situation for Mochizuki’s original series of papers, this should not be a protracted process, since the gist of his and Stix’s objection is not highly technical. Other number theorists “would have totally been able to follow the discussions that we had had this week with Mochizuki,” he said.

Mochizuki sees things very differently. In his view, Scholze and Stix’s criticism stems from a “lack of sufficient time to reflect deeply on the mathematics under discussion,” perhaps coupled with “a deep sense of discomfort, or unfamiliarity, with new ways of thinking about familiar mathematical objects.”

Mathematicians who are already skeptical of Mochizuki’s abc proof may well consider Scholze and Stix’s report the end of the story, said Kim. Others will want to study the new reports for themselves, an activity that Kim himself has commenced. “I don’t think I can completely avoid the need to check more carefully for myself before making up my mind,” he wrote in an email.

In the past couple of years, many number theorists have given up on trying to understand Mochizuki’s papers. But if Mochizuki or his followers can provide a thorough and coherent explanation for why Scholze and Stix’s picture is too simplistic (assuming that it is), “this might go a long way towards relieving some of the fatigue and maybe giving people more willingness to look into this thing again,” Kedlaya said.

In the meantime, Scholze said, “I think this should not be considered a proof until Mochizuki does some very substantial revisions and explains this key step much better.” Personally, he said, “I didn’t really see a key idea that would get us closer to the proof of the abc conjecture.”

Regardless of the eventual outcome of this discussion, the pinpointing of such a specific part of Mochizuki’s argument should lead to greater clarity, Kim said. “What Jakob and Peter have done is an important service to the community,” he said. “Whatever happens, I’m pretty confident that the reports will be progress of a definite sort.”

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