It is not often that major conjectures in number theory are proved—much less conjectures stated in elementary terms. Any time a well-respected mathematician claims to have done so, the mathematical world pauses until the experts have interrogated the argument. It is always possible that a subtle mistake may be hiding inside a long proof; there is no lasting shame in such a thing when it is handled with grace. All mathematicians have had the experience of convincing themselves that a result is true only to have it collapse or wobble under later inspection. That said, experts have highly trained intuitions, and can quickly recognize and separate novel ideas from routine procedures, and powerful techniques from symbol shuffling. When Andrew Wiles first revealed his first, and ultimately flawed, proof of Fermat’s Last Theorem, number theorists could still confidently say that his work was groundbreaking. Before Grigori Perelman’s proof of the Poincaré conjecture was vetted, it was clear his ideas were incredibly novel.
Shinichi Mochizuki’s purported proof of the abc conjecture is otherwise. The proof itself is embedded in a framework that Mochizuki calls inter-universal Teichmüller theory (IUT).1 There is disagreement about the proof between leading number theorists and some of Mochizuki’s few, but very confident, supporters. In December 2017, a rumor spread that Mochizuki’s papers were to appear in a journal of which he is the editor in chief.2 Much of his work had already been made accessible to mathematicians. Publication would have served to give his results the authority of peer review. Number theorists started expressing their concerns online. “Since shortly after the papers were out,” wrote a mathematician going by the initials PS, “I am pointing out that I am entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12.”3 He was not alone. “I assumed the referee process,” Brian Conrad wrote, “would ultimately lead to a revision that completely clarified the proof of 3.12.”4 The rumor turned out to be false, but Mochizuki did little to clarify Corollary 3.12. “Corrected a misprint,” he wrote in his change log, deleting a parenthesis.5 This does not seem the sort of revision that Conrad was anticipating.
PS turned out to be Peter Scholze, a Fields medalist for his work on arithmetic geometry. In March of 2018, Scholze and Jakob Stix visited Mochizuki in Kyoto to discuss their concerns. The meeting was known only to a few insiders, but became common knowledge a few months later.6 Neither side was able to convince the other.
The discussions had broken down.
ABC
The abc conjecture describes a mysterious link between how numbers behave under addition and the size and number of their distinct prime divisors. There is a very distant family resemblance between the abc conjecture and Fermat’s Last Theorem. Both involve simple equations. In the case of Fermat’s Last Theorem, the equation is an + bn = cn, and solutions are subject to the condition that n > 2 and abc ≠ 0. There are no such whole number solutions. The abc conjecture involves an even simpler equation: a + b = c; and affirms that for positive integers a, b, and c with no common prime divisors, if ε > 0 and c > rad(abc)1+ε, then a + b = c has only finitely many solutions. The radical, rad(abc), denotes the product of the distinct prime divisors of the number abc. What is mysterious about this conjecture is the connection drawn between two very simple arithmetical operations: addition, a + b, and multiplication, abc.
To attack any large problem, mathematicians must, to some extent, build their own toolkit. In trying to prove the Weil conjectures, Alexander Grothendieck needed first to construct his own version of cohomology theory. The four Weil conjectures serve the grand effort of algebraic geometry, which is to count the points on those shapes cut out by polynomial equations. Grothendieck envisaged a galaxy of new geometric objects called schemes, each with its own attendant world, its topos. Grothendieck’s grand plan was carried out during the 1960s in the Séminaire de Géométrie Algébrique du Bois-Marie (SGA).7 By 1964, Grothendieck had proved three of the four Weil conjectures. Ten years later, Pierre Deligne conquered the last and most difficult of them, using a mixed approach. The structures, techniques, and terminology that Grothendieck introduced revolutionized algebraic geometry. Schemes have been generalized to stacks, derived schemes, and spectral schemes. Topos theory has escaped its origins, and found uses in mathematical logic as well as other parts of geometry.
Mochizuki did not begin his proof by counting the solutions to the abc equation. Should he have been interested, a distributed computer program could have done it for him.8 These searches must come to an end; the abc conjecture goes beyond every enumeration of examples. Mochizuki began by placing the abc conjecture into the broader, richer framework of elliptic curves. There, geometric tools were available, or could be developed.
A part of Mochizuki’s journey to the 2012 release of his IUT papers involved a detailed study of structures he called Frobenioids. Although conceptually similar to a Grothendieck topos, a Frobenioid is richer than a topos and comes in an array of different flavors. All this happened under the radar, receiving little attention from others who might have been interested. While Mochizuki’s work on Frobenioids is given as a prerequisite for IUT, much of it is general theory, and not really needed for his proof.9 This makes extracting a clear path to the matter of most interest difficult. In ordinary cases, the background material would have been pounced on and reexplained by other mathematicians, even simplified, and applied to other targets. This is largely what happened with Grothendieck’s revolution. Mochizuki’s work went largely unnoticed, and he pressed on without apparently engaging others in his growing theory.
It is entirely possible that cultural and geographic isolation have placed Mochizuki at a disadvantage. Mochizuki is completely fluent in English, but he is reluctant to leave Kyoto. Efforts have been ongoing in order to figure out what can be done. Several conferences have recently been devoted to Mochizuki’s IUT papers and their precursors, although not to great effect. Attendees have said that too much material was crammed into too short a time, allowing little flexibility for minute interrogation by experts. Some mathematicians have made an attempt at exposition. The arithmetic geometer Taylor Dupuy started a video blog in 2015 in order to cover the background material.10 In six months, he produced almost three dozen videos without quite getting to the heart of the matter. Dupuy has recently started up again, focusing on the inequality from the contentious Corollary 3.12.11 The meaning of Mochizuki’s inequality required an hour-long video using standard number-theoretic terminology; the background assumptions, another hour.
Serving Two Masters
Mathematical exposition, if it is to be effective, must give both an intuitive picture of difficult ideas and a precise technical description of their meaning. The best writers do this seamlessly. Mochizuki’s papers are very technical, and they also contain a good many analogies, evocative imagery, and metaphors. There are times when standard mathematical terminology gives way to neologisms. “Two mutually alien copies of conventional scheme theory,” he writes, “are glued together.”12 He describes his technique as, “dismantling the two underlying combinatorial dimensions of a ring [emphasis original].”13 Even to a mathematician, these are not illuminating phrases.
Such ever-present metaphors lead to a pervasive sense of unfamiliarity.
Scholze has also introduced new objects and structures into mathematics: the most famous and influential were perfectoid spaces.14 These were immediately used by mathematicians to solve long-standing problems, simplify complicated proofs, and extend the scope of important theorems. Perfectoid spaces were studied in seminars around the world; by the end of 2018, MathSciNet listed 47 papers published in a wide range of top-tier journals using the word “perfectoid.” At the time of writing, MathSciNet lists three published papers mentioning Frobenioids—all by Mochizuki.15
“What I care most about,” Scholze wrote, “are definitions.”16 Arriving at a perfect and suggestive name, and the correct definition of the concept it denotes, is an art. When the right nomenclature and notation arise, it can be surprisingly productive. Mochizuki and Scholze have both introduced new definitions and concepts. Each is in a sense following Grothendieck’s counsel: rather than cracking a hard mathematical nut with a hammer, immerse it in water until it is soft enough to peel. Whether Mochizuki’s work will have the desired effect is yet to be seen. If he is successful, the theory of Frobenioids and Hodge theaters will be dissected and examined in far closer detail than it has been. But, as Go Yamashita has pointed out, the constructions in the first three of his IUT papers constitute a single algorithm—the multiradial algorithm of Theorem 3.11.17 It is not yet clear what results can be achieved short of the full abc conjecture, since Theorem 3.11 leads directly to Corollary 3.12, which is the heart of the matter.
As a formal object, a proof is either correct or not. But most mathematicians do not deal with formal proofs.18 Formal proofs at the professional level are produced using specialized computer languages and software. After the release of Mochizuki’s papers, some mathematicians wondered whether a formal computer proof of his results could be developed.19 Not any time soon. And for the most obvious of reasons. A formal computer proof would require mathematicians first to understand Mochizuki’s work.20 A team of mathematicians—Kevin Buzzard, Johan Commelin, and Patrick Massot—is attempting to formalize Scholze’s definition of a perfectoid space, an undertaking that still requires building part of the theory of adic spaces; it is work that involves concepts and definitions developed over decades.21 Formalizing Mochizuki’s definition of a Hodge theater is a more daunting task.
Formalizing Theorem 3.11 of IUT, whose statement runs to more than five pages, is Herculean.
In the absence of a formal proof, the scruples expressed by Scholze and Stix gave nonexperts something to hold on to. “I received unsolicited emails from people whom I knew in quite distant parts of the world,” Conrad remarked, and “[e]ach of them told me that they had worked through the IUT papers on their own and were able to more or less understand things up to a specific proof where they had become rather stumped.”22 The specific proof was, of course, that of Corollary 3.12.
For all that, there are a small number of mathematicians who have intensely studied Mochizuki’s work, and affirm quite emphatically that it is correct.23 Mochizuki himself remarked that
IUTch has been checked, verified, read and reread, and orally exposed in detail in seminars in its entirety countless times since the release of preprints on IUTch in August 2012 by a collection of mathematicians (not including myself) involved in this line of research [emphasis original].24
No one wishes to see the story of Kurt Heegner repeated. A private scholar in Germany, Heegner published a paper in 1952 claiming to provide a solution to the class number problem in number theory.25 His proof was regarded as fatally flawed. “The arguments were sufficiently obscurely written,” Alf van der Poorten remarked, “to leave considerable doubt about their completeness, even in essence.”26 As it turned out, Heegner was correct. His real contribution, van der Poorten adds, “is by now well recognized.” A poignant question remains. “[W]as it a disgraceful scandal that his contribution was not recognized in his lifetime?”27 Van der Poorten thinks not.
[A] recognized mathematician, had best have clear arguments written in the language of the majority—the language expected by other mathematicians—if her surprising arguments are to get a proper hearing. That’s not unfair; it’s our playing the odds.28
It is one thing for a proof to be correct; quite another, for a proof to be comprehensible. In 1976, Kenneth Appel and Wolfgang Haken published a proof of the four color theorem. Their proof was essentially a giant case-checking computation; they relied on the computer to do the checking.29 Their proof did not explain why their theorem was true. No one then knew, and no one now knows. It just is so. In writing about the ensuing controversy, William Thurston remarked in 1994 that it had “little to do with doubt people had as to the veracity of the theorem or the correctness of the proof.” What Appel and Haken had failed to satisfy was the “continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true [emphasis original].”30 We are in the current case at a double disadvantage, in that those few who say they understand Mochizuki’s proof and believe it correct, do not seem to be able to pass on that human understanding.
Structural Thinking
In the report containing his reactions to their March meeting, Mochizuki discussed a few of the points that Scholze and Stix had apparently failed to appreciate.31 The report makes for interesting reading because it shows Mochizuki in some measure rejecting a trend in mathematics that has been gathering steam for over a century. David Hilbert once dismissed the objects of Euclidean geometry as incidentals. “At any time,” Hilbert said, “one must be able to replace ‘points, lines, planes’ by ‘tables, chairs, tankards.’”32 These remarks have become a part of mathematical folklore. Hilbert had himself composed an axiomatic system for geometry, one that filled the subtle gaps in Euclid’s system of definitions, postulates and common notions. Euclid had relied on his real-world intuitions to justify certain operations; Hilbert wished everything to be specified by his axioms. The axioms given, the precise identity of the objects they govern is irrelevant. This kind of structural thinking was on the rise at the end of the nineteenth century. An early and famous example dates to 1888, when Richard Dedekind published Was sind und was sollen die Zahlen? (What Are Numbers and What Should They Be?). Like Giuseppe Peano at roughly the same time, Dedekind offered an axiomatic definition of the natural numbers. He gave a careful account of infinite sets and arbitrary mappings, and defined what he called “simply infinite systems.” The natural numbers followed:
If in the consideration of a simply infinite system N set in order by a transformation φ we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation φ, then are these elements called natural numbers [emphasis original].33
The precise identity of the elements is not important: what is important is that N satisfy the properties required of natural numbers. This is no trivial matter. Any pair (N, φ) satisfying Dedekind’s axioms is essentially unique. Given two pairs (N1, φ1), (N2, φ2) satisfying the axioms, there is precisely one function N1 → N2 compatible with φ1 and φ2, and this function is one-to-one and onto. Any natural number object is unique up to unique isomorphism.
Henri Poincaré put the point well in his 1908 International Congress address: “Mathematics is the art of giving the same name to different things. … It is enough that these things, though differing in matter, should be similar in form.”34
A similar result is true for the real numbers: there is only one Dedekind-complete ordered field, up to a unique isomorphism of fields. There are other constructions with numbers that do not have this rigidity, and as a result, one needs to be careful when specifying which construction one uses. The algebraic closure of a field F is not unique in this way. Proofs depending on a choice of algebraic closure must indicate when one is singled out.
Category Theory
The modern way to make such structures precise is category theory.35 A category C is determined by the objects that it contains, and the morphisms that connect pairs of objects in C. Morphisms may themselves be composed by a binary associative operation; and for each X in C there is an identity morphism X → X. In many cases of interest, but not all, the objects of a category are sets equipped with extra structure, and the morphisms are functions compatible with that structure.36
A category does not and cannot distinguish between isomorphic objects. Anything that can be specified about a given object, using only the language and structure of a category, is true of any other isomorphic object. This idea is immensely powerful, but it comes with a caveat: to transfer information from one object X to another object Y one needs to choose an isomorphism. The information subsequently attached to Y can depend on the choice. Y can be X itself, but even in this case, the information transferred by the isomorphism is not necessarily identical with the original information attached to X.
Since a category consists primarily of objects and morphisms between them, the information attached to X almost always consists of relationships with other objects via specified morphisms. The algebraic closure F stands in relation to the field F by a specified morphism F → F of fields. Given an isomorphism of fields a: F → K, it follows that K is an algebraic closure of F via the morphism F → F → K. A different isomorphism b: F → K generally gives a different way to think of K as an algebraic closure of F.
Consider the abstract case where X and Y are objects of some category, and {φi: X → Y} is a set of isomorphisms from X to Y. Such a set is called a polymorphism by Mochizuki; we can consider it as information attached to Y, in the sense of a collection of ways in which X and Y can be regarded as the same. Given some reference isomorphism c: X → Y, replacing Y by X under c yields {c–1φi: X → X}, a collection of isomorphisms of X with itself. In the special case that our starting set consisted only of c, the final set would be just {idX: X → X}, where idX is the identity morphism. But otherwise, the result can be highly nontrivial.
This restricted fungibility is what is meant by structural thinking, but there are still more subtleties to consider. Quite often, mathematical objects, informally construed, give rise to objects in several different categories related by forgetful functors. If the objects of a category C are sets with some additional structure, and the morphisms are structure-compatible functions, then there exists a forgetful functor U: C → Set, sending an object to its underlying set, and a morphism to the resulting function between sets. Many other examples are possible between various categories. For the case of fields, given a field F and an isomorphism of sets U(F) → S, there is a unique field whose underlying set is S and which is isomorphic to F as a field via the given function.
It is important to remark that given two fields F1 and F2, say, functions U(F1) → U(F2) need not arise from morphisms F1 → F2 between the fields. One immediate obstruction is that field morphisms are always one-to-one. If the set U(F1) is larger than the set U(F2), there is no morphism F1 → F2 of fields.
This setup with a category of structured sets and the associated forgetful functor can, with sufficiently rich structures, be broken into several stages. A given forgetful functor C → Set may be the composite C → D1 → D2 → … → Set for various forgetful functors. Sometimes we do not wish to consider only the underlying set of some object, but its underlying Dn-structure, for one of the categories in the chain. A typical case considered in Mochizuki’s work is the purely multiplicative structure attached to a field. This is an example of what is called a monoid. So we have a chain of functors Field → Monoid → Set, where the first forgets the additive structure and the second forgets everything but the underlying set.
Categories themselves are examples of mathematical objects, and so can be gathered together to form the objects of a category. Equivalent categories can be markedly different in size. The category of Dedekind-complete ordered fields, properly speaking, has more objects than fit inside any set, but it is equivalent to a category with only one object—any particular complete ordered field. Since complete ordered fields are rigid, there is really only one way to do this. But for other categories, there are many choices to be made, and the choices must be made in a compatible way. Sometimes it is best to avoid making such choices, but it is possible to do if desired. After all, a pair of equivalent categories cannot distinguish between themselves using only categorical properties. It can thus be a deep theorem to establish such an equivalence, and highly nonobvious.
This way of thinking is becoming more and more entrenched in certain disciplines of mathematics, especially those where category theory has been used extensively. Algebraic geometry is one such discipline. One can, with care, sometimes work as if isomorphic objects are identical. When Mochizuki insists that the isomorphic objects he describes must be distinguished at all costs, and so labelled to keep them distinct, it feels like prohibiting a boxer the use of his fists.
The 10-page note in which Scholze and Stix express their concerns reads like a detailed (if preliminary) referee’s report.37 In their critique, they appeal to “certain radical simplifications” that seem to get the heart of the matter, but they are also aware that “such simplifications [might] strip away all the interesting mathematics that forms the core of Mochizuki’s proof.”38 They have allowed themselves to identify isomorphic objects for the purpose of simplifying an argument. In places where Mochizuki supplies two distinct but isomorphic mathematical objects, Scholze and Stix see only one on the grounds that they are isomorphic. It is this that Mochizuki condemns as illicit, and in his own support, he offers a number of examples that, he claims, lead to incorrect results if so treated.39 But Mochizuki, in defending himself, again uses some idiosyncratic definitions for common constructions in category theory, while still using standard terminology.
Not all of Mochizuki’s complaints are of a structural nature. Category theory
does not itself solve hard problems in topology or algebra. It clears away tangled multitudes of individually trivial problems. It puts the hard problems in clear relief and makes their solution possible.40
There are still subtle aspects of anabelian geometry at work, an area of which Mochizuki is a recognized expert. It remains entirely possible that those radical simplifications engineered by Scholze and Stix identified objects that are isomorphic only after some stage of a tower of forgetful functors, but not at the earlier stage at which they were meant to be considered. A system of objects may have been identified with a different, simpler system of objects unnaturally, various necessary compatibility conditions violated. Even if Scholze and Stix’s analysis is flawed, and Mochizuki’s categorical foibles are harmless, his papers may still have a gap, some innocuous assumption unchecked, some existence statement unjustified—an abc-sized gap deep in the proof of Theorem 3.11 or Corollary 3.12.41 This will only be found by careful study and ideally a rewriting of Mochizuki’s papers into more standard language. Or, like Heegner, Mochizuki may have just written a proof in a way that the rest of us just cannot parse, and time will prove him right.
