Category Archives: Number theory

It’s official: the author of MWA is not a philosopher

For anyone who read MWA it’s obvious that its author would never be mistaken for, nor would want to be mistaken for, a philosopher.  Still, three referees for the journal Synthèse were kind enough to put to rest any lingering doubts on that score (there were none) in their reviews of the article I have just posted on arXiv under the title Virtues of Priority.

In the fall of 2018 I received a message inviting me to contribute to “a special issue of the philosophy journal Synthèse on virtues and mathematics.”  The prospective guest editors wrote, “We would be delighted to be able to list you as a prospective contributor. This would, of course, be in no way binding on either party.”  This looked like the perfect excuse to write on a topic that had been much on my mind ever since the controversy broke out among number theorists on the correct attribution of the conjecture on modularity of elliptic curves.  From the very beginning this controversy was marked by the contrast between the acrimony with which certain colleagues addressed one another and the superficiality of the analysis with which they justified their positions.  Even while Wiles labored in secret in his attic, I was already wishing for a philosophical umpire to require the warring parties to base their arguments on general principles rather than on raw chronology and personal affinities.

The story is interesting, well written and the conjecture as well as the main actors play important roles in 20th century mathematics. In that sense the paper was a good read, but a good read is not enough to make a good philosophical paper.… The paper does not connect to the current debates on value ethics in mathematics at all; in fact, none of the papers on the reference list belongs to the philosophy of mathematics, and the most recent reference is from 2001. The paper should be situated more firmly in the recent literature on mathematical values … Along the same line, the paper does not make a clear and substantial contribution to current philosophical debates on virtues in mathematics. The paper does not contain a clear problem statement, and the virtues encountered in the historical case are only commented on en passant, but the paper does not provide a structured discussion or a collected conclusion. … The paper lacks methodological reflections. Why is the case in question a good case to explore the philosophical questions at hand and why are the chosen sources the right sources to explore the case? It should be made crystal clear why a dispute over priority of a conjecture is a good case to explore the nature of mathematical values.

A second referee was more encouraging:

This submission represents an excellent idea for an article and a solid initial effort at fulfilling that idea, not yet suitable for publication in Synthese.

but this indulgence can be dismissed because the referee is someone I know.  And, as this referee accurately perceived,

the virtue ethics aspect currently reads as something tacked on to a stimulating but not philosophy-journal-ready fireside/blackboard-side chat about a curious and gossip-ready extract of the history of the theory of elliptic curves.

The third referee’s objections came under three headings:  “The philosophical upshot of the paper is thin,” “The paper neglects other work on priority disputes,” and most tellingly, “The search for a functional significance of the dispute may be overstated”:

Merton observes that, in some instances, battles over priority have no functional significance (i.e., the authors fight over the priority of a finding even if the discoveries are independent and epistemically equivalent). This is due in part to the reward system of science. Moreover, along the same lines, there is a psychological explanation in terms of ego-protective biases. The paper should examine this possibility more closely.

This last objection is silly.  Referee number 3 may never have met anyone as ego-protective as several of the protagonists of the controversy, but the point of my essay was to ask whether the controversy might not shed some light on the value system driving contemporary mathematics — and “functional significance” is not the point either.

In nearly all other respects all three referees were absolutely right about the article.  It did not connect to current philosophical debates, it did not engage in methodological reflections, it neglects the literature on priority disputes, and it was not written in the style of a philosophical paper.  But this is not what I thought the editors had in mind when they asked a mathematician for a contribution.

I accepted the invitation to contribute to the special issue in good faith, on the assumption that the issue’s editors had good reason to believe that the thoughts of a professional mathematician on the roots of an unusually bitter controversy in the field would have a place in the journal, and could provide useful raw material for analysis by philosophers who are curious about the value systems that actually guide the practice of professional mathematicians, even though the mathematician in question has never claimed to be a philosopher.    Had the editors of this special issue made it clear to me that my submission would be judged on the basis of familiarity with “the current debates on value ethics in mathematics,” as these are pursued by philosophers who “have been taught mathematics at university level,” or by its author’s efforts to situate the submission in relation to “relevant secondary scholarship,” I would have replied that I have neither the time nor the inclination to undertake a project on that basis.  That the editors of Synthèse and the referees find it helpful to erect artificial barriers* to dialogue between professional scientists and those philosophers who claim to be interested in the values of such scientists is not my concern.   But I consider it unprofessional as well as irresponsible on the part of the issue’s editors to have failed at any time to explain to me the unfamiliar standards that would be applied to my article.

This was my response to the rejection letter from Synthèse, which concluded:

I would like to thank you very much for forwarding your manuscript to us for consideration and wish you every success in finding an alternative place of publication.

Finding a place of publication on arXiv was easy and is a perfectly satisfactory alternative, but philosophers may not think to look there for raw material.  I consulted with the philosopher David Corfield — who first convinced me of the relevance of Alasdair MacIntyre’s virtue ethics to mathematical practice in the article he published in Circles Disturbed — and he offered to help draw out the philosophical material through a dialogue on the n-Category Café.  His first comment on the arXiv publication is already online.

*In view of the events of the last few weeks, I am now ready to acknowledge that fears of disciplinary cross-contamination may be justified in some circumstances.

Guest post by Kevin Buzzard

Kevin Buzzard wrote to let me know that WordPress rejected his comment on an earlier post, presumably because it was too long.  I reproduce it verbatim below.  It deserves to be read closely, in its entirety.  I have some thoughts about it, and I will write about them at some point, but for now I just want to leave you with this question:  do you agree with the claim in the last line that mathematicians “will have to come to terms with” the distinction he identifies, and will the “terms” necessarily be those defined by computer scientists?

This comment will somehow sound ridiculous to mathematicians, but since learning about how to formalise mathematics in type theory my eyes have really been opened to how subtle the notion of equality is.

A few months ago I formalised the notion of a scheme in dependent type theory, and whilst this didn’t really teach me any algebraic geometry that I didn’t already know, it did teach me something about how sloppy mathematicians are. Mathematicians think of a presheaf on a topological space as a functor from the category of open sets of the space to somewhere else (sets, groups, whatever). I have a very clear model of this category in my head — the objects are open sets, and there’s at most one morphism between any two open sets, depending on whether or not one is a subset of the other (to fix ideas, let’s say there’s a morphism from U to V iff U is a subset of V, rather than the opposite category, so presheaves are contravariant functors). But actually when formalising this definition you find that mathematicians do not use this category, they use an equivalent category (whose definition I’ll explain in a second). When formalising maths on a computer, this is a big deal.

Of course mathematicians are very good *indeed* at identifying objects which are “the same to all intents and purposes, at least when it comes to what we are doing with them right now”, e.g. two groups which are canonically isomorphic or two categories which are equivalent, and conversely I would like to suggest that actually computer scientists are quite bad at doing this — they seem to me to be way behind in practice (I had terrific trouble applying a lemma about rings in an application where I “only” had rings which were canonically isomorphic to the rings in the lemma, because in the system I was using, Lean, the automation enabling me to do this sort of thing is not quite ready, although progress is being made quickly). My gut feeling is that this situation is because there are too many computer scientists and not enough mathematicians involved in the formalisation process, and that this will change. In fact one of the reasons for my current push to formalise the notion of a perfectoid space in dependent type theory (note: not homotopy type theory) is to get more mathematicians interested in this sort of thing.

But back to the equivalence. Here is the surprising thing I learnt. Let X be a topological space. The actual category mathematicians use when doing sheaf theory is this. An object is a string of symbols in whatever foundational system you’re using, which evaluates to an open set. For example, X is an open set, as is (X intersect X), as is the empty set, as is (X intersect (the empty set)). Mathematicians instantly regard things like X intersect X as equal to X, because….well…actually why are they equal? They’re equal because two sets are equal if and only if they have the same elements — this is an axiom of mathematics. But when formalising maths on a computer, keeping track of the axioms you’re using is exactly what you have to do (or more precisely, getting the computer to invoke the axioms automatically when you need them is what you have to do). So X equals X intersect X, because of a *theorem* (or in this case an axiom, which is a special case of a theorem if you like; most theorems use several axioms put together in clever ways, this is a bit of a degenerate case). Mathematicians are so used to the concept of sets behaving like the intuitive notion of “a collection of stuff” that it’s very easy to forget that X = X intersect X is *not true by definition in ZFC*, it is true by the very first axiom of ZFC, but this is still a theorem. The elements are the same by definition, but equality of the elements implying equality of the sets is a theorem.

So the computer scientist’s version of the category of open sets is something like this: objects are valid strings of characters which one can prove are equal to open subsets of X, and there’s a morphism between U and V if and only if there’s a proof that U is a subset of V. In particular, in the example above there’s a morphism from X to X intersect X, and also a morphism from X intersect X to X, because both inclusions are theorems of ZFC (let me stress again that whilst both theorems are trivial, neither one is “true by definition” — both theorems need axioms from the underlying theory, absurd though it may sound to stress it). This makes the objects isomorphic, but not equal. Equality is a subtle thing for them!

The conclusion of the above (which of course a mathematician would regard as a fuss about nothing) is that computer scientists don’t work with the mathematician’s “skeleton” category, they work with an equivalent category, and hence get a notion of a sheaf which is canonically isomorphic to, but not strictly speaking equal to (in this extremely anal sense), the mathematician’s notion.

And how did I notice this? Why do I even care? It was when trying to prove that the pushforward of a sheaf F via the identity map id : X -> X was isomorphic to the sheaf you started with. I needed to come up with an isomorphism to prove this, and my first attempt failed badly in the sense that it caused me a lot of work. In practice one needs a map from F(U) to F(id U), for U any open set, with id U the image of U under the identity map (which equals U, by a theorem, which uses an axiom, and hence which is not true by definition). My first attempt was this: “prove id U = U, deduce that F(id U) = F(U), and use the identity map”. I then had to prove that a bunch of diagrams commuted to prove that this was a morphism of sheaves, and it was a pain because I really wanted this to be a complete triviality (as it is to a mathematician). I ran this past Reid Barton and he instantly suggested that instead of using equality to map F(U) to F(id U), I use the restriction map instead, because id U is provably a subset of U so there’s a natural induced map. I was wrong to use equality! I had too quickly identified U and id U because I incorrectly thought they were equal by definition. They are actually equal because of a trivial theorem, but to a computer scientist they are equal, but not definitionally equal, subsets of X, and this makes all the difference. Switching to res, all the diagrams commuted immediately from basic properties of the restriction map and indeed the computer proved commutativity of the diagrams for me. I was stunned.

In dependent type theory, there is at most one map from U to V, depending on whether or not there is a proof that U is a subset of V — all proofs of this give the same map. In homotopy type theory, different proofs give different maps, and in this particular situation this is not what we want — we actually get the wrong category this way — so presumably the homotopy type theory people have to do something else. I am not yet convinced that homotopy type theory is the right way to do all of mathematics (it works great for some of it, for sure). I am now convinced that dependent type theory can do all “normal” mathematics (analysis, algebra, number theory, geometry, topology) so I’m sticking here, but what I have learnt in the last year is that computer scientists seem to have several (competing!) notions of equality, and it is a subtlety which mathematicians are conditioned to ignore from an early age and which they will have to come to terms with one day.

Is the tone appropriate? Is the mathematics at the right level?

In the middle of December I was approached by an editor at New Scientist to write an article about “the work of Peter Scholze and its connections to the Langlands program, quantum theory, and anything else it might reasonably be said to have connections to.”   Since the publication of my book, various people have been encouraging me to devote some time to writing popular accounts of the contents of mathematics, including contemporary (“cutting-edge”) work, and not just what my book calls “the mathematical life.”  Scholze’s work is certainly cutting-edge, and I had already published a semi-philosophical account of his “perfectoid concept,” but the material seemed rather remote from what I imagined to be the concerns of the typical reader of New Scientist.  The editor naturally mentioned the rumor that Scholze would be receiving a Fields Medal at next month’s International Congress of Mathematicians in Brazil, but for reasons that were not clear to me he seemed to feel that Scholze’s work would somehow have more resonance for his readers than that of the other potential laureates.  Nevertheless, I accepted the challenge, and on February 1 I sent the editor a draft containing about 2/3 of the requested 2400 words, asking “whether the tone is appropriate and whether the mathematics is at the right level.”

Over the next two months there ensued the kind of lively give-and-take with the editor that I have always imagined to be the privilege of those who eke out their livings writing for the more intellectually ambitious of the mass-circulation magazines (Google tells me that New Scientist’s circulation in 2016 was 124,623).  The editor wrote back the very same day to warn me that mathematics articles are typically a hard sell for a magazine like his, but that “with the right approach” they can be successful.  It would be important for me to convince readers — at the very least, those “who might know something about Fermat’s Last Theorem or the Riemann Hypothesis” — that they should care about the material.  

Taking these suggestions to heart, I sent the editor three more drafts, and by the middle of March I was ready to see how it would be transformed by the process, mysterious to me, known as “editing.”  The result, when it arrived on March 28, was deeply discouraging.  Very little of my own text had survived the cuts.  In its place was an admittedly smoothly flowing narrative composed largely of the kinds of hackneyed metaphors and extraneous historical anecdotes that did nothing to clarify the originality of Scholze’s insight.  After rapidly exchanging a few polite messages, the editor and I agreed that it was pointless to continue, and that it would be best if the New Scientist could salvage what it could from our correspondence and my previous draft; the editor promised to “run [these extracts] past [me] for approval before use.”

Six weeks passed, and since I had heard nothing from the editor I assumed the article had been “killed” (an expression I’ve already encountered in my interactions with journalists).  But I checked during a lull in the middle of a lecture in Paris and was surprised to find that the New Scientist had gone ahead without notifying me and had published an article — a cover story! — under the Oscar-worthy title “The Shape of Numbers” (or the title “‘Perfectoid geometry’ may be the secret that links numbers and shapes”; or even “Theorem of everything: The secret that links numbers and shapes,” depending how you find it on the internet).

I’m not particularly happy that the author failed to let me know just how I was being quoted, and I don’t expect I’ll have anything to do with New Scientist in the future.  And I don’t think it’s very helpful to have described Aristotle as an “ancient Greek philosopher and mathematician.”  Still, even though the article doesn’t make much headway in explaining Scholze’s “secret that links numbers and shapes,” it could certainly have been worse.

The author preserved enough words from my final draft to render the draft unpublishable in any form, but I do believe I have the right to reproduce it on this blog.  Please be indulgent when reading it and bear in mind that it is still just a draft, written for the eyes of the sympathetic and professional editor who still exists, if only in my imagination.

Number theory and geometry, the two most ancient branches of mathematics, could hardly be more different, at least on the surface.   The former deals with the properties of integers — 1, 2, 3, and so on — and is designed to understand discrete objects. The latter studies spatial relations and measurements, and is built on our intuition of continuity. Aristotle thought they were separate because they applied to such distinct domains: “we cannot… prove geometrical truths by arithmetic,” he wrote, and he meant “and vice versa” as well.

Yet mathematicians have long speculated that features shared by arithmetic and geometry have common origins. The French mathematician André Weil described this to his sister in particularly vivid terms:

around 1820, mathematicians … permitted themselves, with anxiety and delight, to be guided by the analogy [between an arithmetical and a geometric theory]. [Now] gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur…

The unusual erotic charge of this letter, written in 1940, was stimulated by Weil’s pleasure in his recent solution of a geometric analogue of what then, as now, was the outstanding problem in number theory: the Riemann hypothesis. Like many problems in number theory, this one focuses on prime numbers, like 2, 3, 5: a number is prime if it can’t be factored as the product of two smaller numbers (unlike, say 6 = 2 x 3). There are infinitely many prime numbers, scattered among the integers according to no determinate pattern, but their frequency can be measured. The Riemann hypothesis predicts that this frequency follows the most natural possible rule.

The geometric version proved by Weil is the corresponding prediction for the frequency of points on a certain kind of curve. Just as prime numbers can be ordered by size, these points can be ordered by degree. Weil’s proof, which marks the beginning of the science with the most unaristotelian name of arithmetic geometry, showed that the number of points up to a given degree fits the prediction of the geometric Riemann hypothesis.

In a modern version of the analogy Weil found so delicious, prime numbers are points on a highly implausible kind of curve called Spec(Z), all stuck together by a strange sticky point that represents the familiar arithmetic of fractions. Ever since Weil proved his theorem about curves, and with increasing insistency in the last two decades, number theorists have believed that if one could make Spec(Z) genuinely curvy then fantastic consequences would follow — possibly including the Riemann hypothesis. Peter Scholze, today’s 30-year-old crown prince of arithmetic geometry, has not gone quite that far, but the p-adic geometry he has developed over the past 7 years has provided tantalizing hints of how a geometry of Spec(Z) might be built. In the process he has been transforming number theory at a rate that has the rest of us struggling to keep up.

Scholze, born in the former East Germany, would undoubtedly win a contest for World’s Most Popular Mathematician if there were such a thing; he has already received a long list of more conventional prizes. The Fields Medal is the highest honor for mathematicians under 40; most mathematicians are convinced that Scholze will be one of the winners at next August’s International Congress of Mathematicians in Rio de Janeiro. Scholze chose “p-adic geometry,” naturally enough, for the title of his prestigious plenary lecture at the Rio meeting. The “p” in “p-adic” denotes a prime number. Each prime has its own system of p-adic numbers, in which numbers become closer as their difference grows more divisible by p. In the 5-adic numbers, for example, 50 is 25 times closer to 2,000,000 than it is to 51 or 52. The 2-adic numbers are like binary numbers, but written in the wrong direction: our 16 is represented as 10000 in binary but is more like .0001 in the 2-adics.

There is an intrinsic geometry to the p-adic numbers, but it has little in common with Euclidean geometry.   A p-adic circle would be composed of infinitely many smaller circles, in a fractal pattern, while all p-adic triangles would be isoceles. But you can’t actually draw p-adic circles or triangles — in fact, you can’t connect any two p-adic dots by anything resembling a straight line. P-adics were introduced by Kurt Hensel in 1897 as a way of understanding solutions to diophantine equations — polynomial equations with whole number coefficients. Perhaps the most famous diophantine equations are the Fermat equations

Xn + Yn = Zn

where the exponent n is a positive integer. When Sir Andrew Wiles proved in the early 1990s that the Fermat equation has no solutions when n >2 — this is the famous Fermat’s Last Theorem — practically every step in the proof involved p-adic numbers. Hensel’s version of p-adic geometry was barely relevant to Wiles’s work.

Scholze takes a different approach to p-adic geometry, taking his cue from the radical expansion of geometry in the 1960s under the leadership of Alexander Grothendieck. In contrast to the system inherited from Euclid, which dissected circles and triangles as singular objects, or the analytic geometry of Descartes, which studied parabolas and ellipses as if they were drawn on graph paper, each of Grothendieck’s geometric objects is at all times considered in relation to every other object in its category — the technical term for the principles contemporary mathematicians use to organize objects of a given type.   So where a point in the Euclidean or Cartesian plane is just a familiar dot on a flat surface, a Grothendieck point is more like a way of thinking about the plane — which includes the possibility of drawing a triangle or an ellipse, or even squashing the surface of the globe into a planar map.

Grothendieck is usually considered the most influential mathematician of his time; the solution of Fermat’s Last Theorem, like every other major development in number theory over the past half century, would have been impossible without his innovations. Nevertheless the old Cartesian intuition, corrected by habits from algebraic calculation, largely sufficed when Grothendieck’s ideas were applied, notably by the French mathematician Jean-Marc Fontaine, who invented a series of new algebraic systems to bridge the gap between p-adic arithmetic and Grothendieck geometry. Scholze’s spaces, which retain some properties from familiar geometry and sacrifice some others, severely strain this intuition.

P-adic geometry can be viewed as the study of the geometry — in Grothendieck’s relational sense — that you would see hanging off the sticky curve Spec(Z) if you examined it under a microscope near the prime p.  Scholze was only 24 when his dissertation introduced the theory of perfectoid spaces, which combined the best properties of the many kinds of Grothendieck-style p-adic geometries that had been studied over the previous half century with Fontaine’s p-adic number theory. In the intervening years Scholze and his collaborators have used perfectoid geometry to solve or clarify so many outstanding problems in number theory and in other branches of arithmetic geometry that last year’s annual Arizona Winter School on perfectoid spaces attracted a record 400 graduate students and postdocs — double the previous record.

Perfectoid geometry is very much a work in progress, and its details are dispersed among hundreds of pages of difficult mathematics, but one can begin to see the point with the help of Weil’s “slightly adulterous” analogy between algebra and geometry, as applied to differential calculus. As developed by Newton and Leibniz, calculus permits the application of the notions of geometry on an infinitesimal scale, predicting the motion of a particle under the influence of external forces. In the most familiar cases, this motion can be described as a function of time t by a Taylor series. This is an infinite version of a polynomial function of t:

f(t) = ∑ antn  

where the coefficients an are constant real numbers. A p-adic number has a similar expression:

  ∑ anpn

where the the coefficients an are now integers, but the variable t has been replaced by the prime number p. The two expressions have a completely different character, however: whereas t is a variable, and can therefore take on infinitely many values and trace a geometric figure as time varies, the number p is itself a constant and the p-adic expression belongs to pure algebra.

The aim of perfectoid geometry, in a single sentence, is to make the constant p behave like a variable, and thus to apply geometric methods to the arithmetic of p-adic numbers, and from there to the rest of number theory. This has a most disconcerting implication.   Just as there are functions in calculus that depend on many variables — the forces on a vibrating string, for example, depend on the position along the string as well as time — perfectoid geometry makes it possible to clone a prime number, so that there can be several perfectoid versions of 3, taking independent values. With his theory of diamonds, a subsequent development of perfectoid spaces, Scholze managed precisely this.

Weil used a similar principle to prove his geometric Riemann hypothesis, which also depends on a prime number p. One can think of the curves he studied as trajectories of a particle parametrized by a time variable t. With a second variable u one can trace a second copy of the curve — a second particle—and Weil’s analysis of the frequency of points is based on using both copies simultaneously and comparing the places where the two particles coincide — this is the equation t = u — and where they have a fixed degree — this is given by a second equation (for example t = up means the point has degree 1). Similarly, in Scholze’s p-adic diamonds — take the prime p = 3 for concreteness —the 3-adic numbers stretch out into a kind of curve, and the excitement happens when my 3 gets close to colliding with your 3.

Apart from providing an especially rich framework for p-adic geometry, the most immediate applications of Scholze’s perfectoid spaces may be to the vast program outlined 50 years ago by Robert P. Langlands to unify number theory with the geometry of Lie groups, the systems of symmetries that are also central to mathematical physics. Mathematicians are aware that Wiles proved Fermat’s Last Theorem by establishing one particular consequence of the Langlands program; the last step was completed in collaboration with Richard Taylor. Scholze recently joined forces with Taylor and eight other mathematicians to push the argument of Wiles and Taylor in a new direction, one that would have been inaccessible without perfectoid spaces.

The full Langlands program is no more likely than the original Riemann hypothesis to be settled in the near future. But it also has a purely p-adic chapter. Scholze’s first published papers, before he invented perfectoid geometry, introduced a new perspective on this local Langlands correspondence — a subject on which I worked with Taylor about 20 years ago. More recently, the French mathematician Laurent Fargues proposed a way to use the cloning property of Scholze’s diamonds to provide a full solution of the p-adic side of the Langlands program. There are persistent rumors that Fargues and Scholze are working intensively on this proposal in advance of the coming summer’s meeting in Rio.

Scholze was briefly in the news in 2015 when he refused a $100,000 New Horizons Prize — the junior version of the $3 million Breakthrough Prizes awarded every year in a Hollywood-style extravaganza in Silicon Valley. Since he did not intend his decision as a public statement, guesses about Scholze’s motivations continue on the internet. What I can say is that parallels with the actions of Grigory Perelman, who solved the most famous problem in (traditional) geometry but refused the Fields Medal as well as the $1 million Clay Millenium Prize, before withdrawing from mathematics entirely, are completely off base.   Perelman was portrayed in Masha Gessen’s Perfect Rigor as a hermit and a crank, with rigid ideas of what is and is not proper. Scholze is gregarious, thoughtful, generous with his ideas, actively supportive of junior colleagues (some of whom are slightly older than he is). He doesn’t seek publicity, and he is most likely to be spotted at a conference drinking beer with his friends, but he doesn’t mind talking to the press when necessary. In every way he has shown that he is ready to accept the responsibilities that the mathematical community generally expects of its most influential and respected individuals. My guess — but it’s no better than anyone else’s — is that he decided that the priorities of Silicon Valley are just not compatible with those of the mathematical community, as he sees it.

Whatever his reasons, mathematics needs more individuals like Peter Scholze.   While the secrets of his success are not likely to be transmitted even to those who work most closely with him — and there are no prospects of cloning him in the near future — he has provided some insight into his goals as a mathematician, in a recent message that he has allowed me to share.

“What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.

Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.”

Number theory, GCHQ, and kidneys

If you can get past the paywall you can read some of my thoughts on research funding in an article published on March 8 in the Times Higher Education Supplement .

If not, here is a “fair use” excerpt:

Mathematicians have been reluctant to recognise that if our work interests generous donors, it is often precisely because it is “useful” according to a definition that Hardy proposed near the beginning of the First World War: “its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life”.

We will have to overcome this reluctance and draw uncomfortable conclusions. Wherever you turn as a mathematician, you’re going to be someone’s kidney: practically every potential source of research funds is tainted in some way.

(I’m afraid you’ll have to find a way to read the article if you want to know what that kidney is doing in that last paragraph.)

Am I a number theorist?


1985ean Parisko École Normale Superieureko matematikako zuzendari eta irakasle hautatu zuten. Urte berean, Londresko Errege Elkarteko kide izendatu zuten. Emmanuel Collegeko kide izateaz gainera, Cambridgeko Matematikako Sadleiriar katedra ere lortu zuen. 1991n Cambridgeko Matematika Huts eta Estatistika Matematikako saileko zuzendari izendatu zuten.

Wikipedia lists 395 number theorists, from Euclid and Kamāl al-Dīn al-Fārisī to Jacob Tsimerman, but I am not on the list.  Actually, one should consider not one list but all the lists of number theorists, in languages from العربية to 中文, but I am not on any of the lists.

Some of the lists are easy to remember; for example, the Kazakhs only recognize Diophantus, Hadamard, Gauss, and Fibonacci (in alphabetical order:  Д, Ж, К, Ф); the Icelandic page only lists Dirichlet.  I wonder whether John Coates knows that he is the only number theorist not born in the 18th or 19th century, and only one of five number theorists of any time or place, to have a Wikipedia page in Basque, excerpted above; I will surely ask him the next time I see him.

I like to point out Wikipedia’s frequent errors, omissions, and oddities; it reinforces my possibly naive hope that there is a future for professional scholarship.  When I start writing anything I inevitably consult Wikipedia for source material, and I sometimes use “Wikipedia” as a stand-in for wired public opinion;  but I never quote it as a reliable reference, because too often it is not.  On this occasion I was looking for a list of number theorists — it should be easier to get that right than a list of autobiographies — because I had just come across an exchange on the n-Category Café in which Harvey Friedman took part, and in which Peano arithmetic was mentioned and I was wondering how many number theorists on the list would be able to recite the Peano postulates, and what that said about the state of our subject.  Surely Eratosthenes and John Coates’s four companions on the Basque page are exempt, but are contemporary number theorists really entitled to their places on the list?  To be continued…


Not about Fibonacci

quadrivium - 1

Arithmetic, geometry, and music in Giovanni Pisano’s pulpit (1301-1310), Duomo di Pisa

Pisa is the international symbol of improbable constructions and therefore a fitting location for this week’s workshop.  Pisa is also a fitting location for meditating on the eternal and impossible question:  do we engage in mathematics because we find it beautiful, or do we find mathematics beautiful because of our programming?  Are Pisa’s medieval arcades beautiful because we are used to them or do we admire Pisa for the beauty of its medieval architecture?

In addition to the Roman sarcophagi that littered Pisa’s underworld and were recycled in medieval times to house the remains of political and military citizens “di primario spicco,” Pisa’s Camposanto contains the gigantic (5.6 x 15 m) 14th century fresco Il trionfo della morte which might have served as a reminder of the urgency of completing the program of this week’s workshop, but which is undergoing restoration and is therefore not visible to the public.  It seems to me the workshop provides a striking illustration of the complex interplay between freedom and inevitability in the design of a mathematical theory, in this case the mod p Langlands program, whose ultimate goals are being defined, democratically as far as I can tell, through workshops and conferences like this one.

Pisa’s medieval walls are also decorated with a variety of political statements.  Someone found it worth his or her while to design a stencil to celebrate an American mathematical personality:

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Seen on a wall in central Pisa. The caption reads “strike where it hurts the most.”


“Je m’en f…,” he wrote

Pierre Colmez has pointed out a few passages published in the Serre-Tate Correspondence where Serre and Tate express their opinions about the correct way to identify that conjecture about which so much ink has been spilled.  The date is October 21, 1995, the papers of Wiles and Taylor-Wiles have been published, and Tate is confiding that

I am tired of the Sh-Ta-We question. But it doesn’t go away.

Joe Silverman had just written to him about the correct nomenclature for the Japanese translation of his book (presumably one of his books about elliptic curves):

Springer-Tokyo wondered if we still wanted to call it the  “Taniyama-Weil Conjecture,” since they say that everyone in Japan now calls it the “Shimura-Taniyama Conjecture.” I certainly agree that Shimura’s name should be added to conjecture (Serge’s file is quite convincing), but I don’t feel strongly about whether Weil’s name should be omitted. I hope you won’t mind that I told Ina that for the Japanese edition it would be all right to call it « Shimura-Taniyama », although I suggested that they add a phrase « now proven in large part by Andrew Wiles ».

Tate is finished for the moment with Sh-Ta-We but wonders whether the new theorem should be named after Wiles or Taylor-Wiles, at which point he wrote the censored passage above.

Before we move on to the substance of the question, let’s speculate as to the reason for Tate’s ellipsis.  We know that Tate was born in Minneapolis, and we heard a lot about Minnesota nice earlier in this election cycle; could it  just be the way of Minnesotans to talk in ellipses?  Or maybe the passage was censored by the Société Mathématique de France, publisher of the Serre-Tate Correspondence.

The significance of Tate’s expletival indifference is easier to ascertain.  If anyone occupies the pinnacle of charisma in contemporary number theory, it’s John Tate; but here he is choosing not to exercise his charisma to manipulate public opinion in favor of one historical label or another.  There’s no doubt in my mind that his survey article on elliptic curves was largely responsible for popularizing the conjecture and associating it with Weil’s name (“Weil [82] has the following precise conjecture…”) — but there’s no reason to think this was his intention; he had called it Weil’s-Shimura’s conjecture in a letter to Serre dated August 4, 1965 (on p. 262 of the first volume of the Serre-Tate Correspondence — it’s August and Tate writes “Bonnes Vacances” at the end of his letter).

Colmez adds a footnote to the 1995 letter.  I quote verbatim:

Il me semble qu’une bonne part de la dispute vient de ce que l’on s’obstine à considérer deux conjectures bien distinctes comme une et une seule conjecture. Si E est une courbe elliptique définie sur Q, considérons les deux énoncés suivants :

two statements

La théorie d’Eichler-Shimura [Ei 54, Sh 58], complétée par des travaux d’Igusa [Ig 59] et de Carayol [Cara 86], permet de prouver que le second énoncé implique le premier. Réciproquement, le second est une conséquence de la conjonction du premier, de la théorie d’Eichler-Shimura, et de la conjecture de Tate pour les courbes elliptiques sur Q. Comme la véracité de la conjecture de Tate n’est pas vraiment une trivialité, cela donne une indication de la différence entre les deux énoncés. Une différence encore plus nette apparaît quand on essaie de généraliser les deux énoncés à un motif. Le premier se généralise sans problème en : la fonction L d’un motif est une fonction L automorphe – une incarnation de la correspondance de Langlands globale. Généraliser le second énoncé est plus problématique, et il n’est pas clair que ce soit vraiment possible.

Colmez’s mathematical gloss is impeccable, but he provides no evidence that anyone actually confused the two statements.  Shimura certainly did not; when I was in Princeton the year after Faltings proved the Tate conjecture for abelian varieties, he (unwisely) asked me whether I thought the proof was correct, instead of consulting any of the numerous experts on hand.

Only a professional historian of mathematics can determine the accuracy of Colmez’s explanation for the disagreement.  Serre, in any case, was never convinced by Lang’s file.  In his message to Tate on October 22, 1995, he wrote

La contribution de Weil (rôle des constantes d’équations fonctionnelles + conducteur) me parait bien supérieure à celle de Shimura (qui se réduit à des conversations privées plus ou moins discutables).

Serre clarified his position in a letter to David Goss, dated March 30, 2000, and reprinted two years later in the Gazette of the Société Mathématique de France.  I alluded to this letter in the text quoted in the earlier post.  To my mind, the most interesting part of this letter is his explanation of what he sees as Weil’s contribution to the problem.

b) He suggests that, not only every elliptic curve over Q should be modular, but its “level” (in the modular sense) should coincide with its “conductor” (defined in terms of the local Néron models, say).

Part b) was a beautiful new idea ; it was not in Taniyama, nor in Shimura (as Shimura himself wrote to me after Weil’s paper had appeared). Its importance comes from the fact that it made the conjecture checkable numerically (while Taniyama’s statement was not). I remember vividly when Weil explained it to me, in the summer of 1966, in some Quartier Latin coffee house. Now things really began to make sense. Why no elliptic curve with conductor 1 (i.e. good reduction everywhere)? Because the modular curve X0(1) of level 1 has genus 0, that’s why!

One can agree or disagree with Serre’s criterion — who apart from Serre has proposed numerical verifiability (or falsifiability!) as the boundary between meaningful and questionable (“discutable”) conjectures? — but at least it has the merit of being stated clearly enough to serve as the starting point for a philosophical consideration of authorship of a conjecture.  It’s not merely rhetorical.

The Taniyama, Shimura, Weil controversy in Herts.


Hatfield Galleria,  by Cmglee (own work) CC BY-SA 3.0 via Wikimedia Commons


In honor of the Abel Prize Committee’s decision to award credit for the celebrated conjecture on modularity of elliptic curves to Shimura, Taniyama, and Weil (in that order) in the course of awarding the Abel Prize to Andrew Wiles, I am publishing here for the first time an excerpt from the text of my talk, entitled Mathematical Conjectures in the Light of Reincarnation, at the conference Two Streams in the Philosophy of Mathematics that took place in 2009.  (The remainder of the talk was reworked and expanded into Chapter 7 of MWA.)  The conference was organized by David Corfield and Brendan Larvor and was held on the campus of the University of Hertfordshire in Hatfield, England.  There are no photographic records of the conference, which left practically no internet trace whatsoever, apart from the program posted on the FOM website, so I have included a Wikimedia commons photo of the Hatfield Galleria Shopping Mall, which is where the conference dinner was held, across a traffic circle (roundabout) from the university campus.

For years I have wanted to write a comprehensive article about the controversy over the name of the conjecture for a philosophy of mathematics journal.  But I have never had the patience to organize the themes of the controversy, and instead, as a way of relieving my persistent irritation with the way the controversy has been addressed, I have been inserting cranky fragments of arguments into articles and presentations where they don’t necessarily belong.   Here is an example from an October 2009 draft that is actually called PHILOSOPHICAL IRRITATION.  The original reason for my irritation is explained in the second paragraph, and it goes back to Serge Lang’s earliest interventions on behalf of Shimura, before Wiles proved Fermat’s Last Theorem; the words “chagrin” and “avalanche” allude to an incident that took place some five years after Wiles’s announcement, about which, perhaps, more will be written later.

Even before the Science Wars erupted, I had observed with increasing distress a bitter debate over the appropriate nomenclature for a conjecture of fundamental importance for my own work in number theory. As a graduate student I had been taught to refer to the conjecture as the “Weil conjecture” but a few years later, after a series of consultations I have not attempted to reconstruct among senior colleagues, the name was changed to “Taniyama-Weil conjecture.”  By the beginning of my career, which fortuitously coincided with the official consecration, in the form of the four-week Corvallis summer school, of the Langlands program, this name designated this program’s iconic prediction and unattainable horizon, though even at the time it was understood to hold this position only as an effect of convergence after the fact, since Langlands had found his way to his program by another route, and the conjecture was primarily iconic for number theorists.[1] About ten years later the conjecture underwent another promotion when it was discovered[2] that it implied Fermat’s Last Theorem as a consequence. Serge Lang then began an energetic campaign (some details are recorded in the AMS Notices) to change the name on the grounds that, after an initial hesitant formulation by Taniyama, it had been proposed in a more precise form by Shimura, the first to suggest the idea to Weil who, after a period of skepticism, not only published the first paper on the conjecture but wrote both Taniyama and Shimura out of the story. At this point the name of the conjecture underwent several bifurcations: Shimura-Taniyama-Weil for those inclined to generosity (and alphabetical order), Taniyama-Shimura-Weil for those with a certain view of history, Taniyama-Shimura or Shimura-Taniyama for those in Lang’s camp (including Shimura himself, as I later learned to my chagrin) who saw Weil as a treacherous interloper, and occasionally Taniyama-Weil for those who took pleasure in baiting Shimura.  An often acrimonious exchange of opinions on the question, which enjoyed a brief revival after Wiles proved the conjecture in sufficient generality to imply Fermat’s Last Theorem and Taylor and his collaborators proved it in complete generality, has led to the current impasse where there is still no consensus on what the conjecture (now a theorem) should be called: French sources generally include Weil’s name, whereas many if not most American authors do not[3].

What I found and continue to find most disheartening is that none of the quarrel’s protagonists saw fit to provide any guidance to resolving similar conflicts in the future. Indeed, with the exception of a relatively late[4] contribution by Serre, to which I return below, no one acknowledged that it might be of interest to consider the dispute as other than sui generis, and the discussion remained largely in the forensic mode initiated by Lang. How and to whom to attribute ideas — a more or less isolated conjecture, a research program (such as the Langlands program), or a key step in a proof — is the question of most moment in the development of individual careers, the distribution of power and resources, or the evolution of the self-consciousness of a branch of mathematics.  It can be fruitfully analyzed by historical or sociological methods, specifically by science studies in one or another of its incarnations. It can also be given the status of a philosophical question. Indeed, the claim that such a question has philosophical content beyond what is accessible by history or sociology — that it can in some sense be analyzed in terms of principles whose nature remains to be determined — is itself a philosophical claim, and one that is likely to be contested. When such a question arises I would like to be able to answer it on the basis of principled arguments and not by joining a transitory alliance or actor network. It is not the sort of problem that typically appeals to philosophers.

The problems about mathematics that do appeal to philosophers, according to the Oxford Handbook[5], include (for example)

  1. What, if anything, is mathematics about?
  2. …how do we know mathematics [if we do]?
  3. To what extent are the principles of mathematics objective and independent of…?

and so on. I see no room on this list for an account of how to attribute authorship to a conjecture. The word “conjecture” does not even appear in the index of this 800-page handbook (“theorem” occurs, but sparsely; more popular index entries are “truth,” “proof,” “proposition,” and “sentence,” as well as topics like “arithmetic,” “geometry,” and “number”). No guidance is forthcoming from the handbook as to whether a conjecture is ontological rather than epistemological or methodological. A conjecture must be a matter of importance to mathematicians, though, if so many of them, and not only the rival claimants and their friends, are willing to sacrifice valuable working time for fruitless belligerence in order to arrive at an accurate attribution.

It may just be that the conjecture is something the working mathematician is forced to cite by name, and that one attempts to trace it back to its original source in order to avoid irritating short-tempered colleagues. I briefly thought I had found a way to evade responsibility by referring to it as “the conjecture associated with the names of” followed by three names, a formulation that is undoubtedly objectively true (and admitting of ostensive proof in this very text); and at least one colleague followed me[6] along this deceptively innocuous path until the day we walked into an avalanche. Don’t follow my lead in suggesting that the problem can be evacuated into a matter of typographical convenience:  do we know what kinds of “something” we can be “forced” to treat in this manner, and just what is this “force” that holds us in its grip? As of this writing many colleagues have given up on nominal attribution altogether and refer to the conjecture by what it says (the “Modularity Conjecture for elliptic curves,” for example). But how can a conjecture “say” anything?[7]

[1] Langlands’s own priorities were elsewhere, as he has frequently pointed out. His insights have been so influential in so many branches of mathematics that he can hardly be said to own the Langlands program any more. But he certainly has been clear and consistent about his own reasons for formulating the program that bears his name.

[2] By Frey, Serre, and Ribet. The word “discovered” is used here in the colloquial sense and expresses no philosophical commitment.

[3] I haven’t checked Japanese practice.

[4] Its publication came late in the story, but it’s clear from discussions with French colleagues that Serre had expressed his opinions on the matter quite early on.

[5] Stewart Shapiro, Chapter 1, p. 5; letters mine.

[6] Langlands used a similar formulation in a recent article in Pour la Science but I have no reason to think he didn’t come up with it on his own.

[7] This at least looks more like a Handbook question — but which one?

By the end of this text my irritation has expanded from the “short-tempered colleagues” who had promoted the controversy to include, uncharitably, the contributors to the Handbook who didn’t think to include material that would help to clarify what, if anything is really at stake (beyond personal antipathies, which should not be underestimated) in controversies of this kind.

Serre’s “relatively late” contribution to the debate is quite interesting, and has influenced a number of colleagues, but it looks like I never got around to revising the draft to keep the promise to explain what Serre added to the discussion.  Maybe I will on this blog, or maybe I’ll actually write that philosophical article.



How MWA promotes the (oppressive?) hierarchy


From Langlands, Is there beauty in mathematical theories?, Lectures at Notre Dame University, January 2010, published in The Many Faces of Beauty, Vittorio Hösle, ed.

Finally I can begin to fulfill the promise I made last August to point out a few of the things that are really wrong with Mathematics without Apologies.   The diagram reproduced above is taken from an article Langlands wrote when the editor invited him to contribute an article on mathematical beauty to the volume cited above, with the title The Many Faces of Beauty.  (In the summer of 2011 I was invited to contribute an article on mathematical beauty to an issue of the Portland-based literary journal Tin House with the title Beauty.  So Langlands and I have at least that in common.)   The diagram lists  “the names of some of the better-known creators of the concepts” that contributed to the solution by Andrew Wiles of Fermat’s Last Theorem, which Langlands chooses as the starting point of his account of the theory of algebraic equations and, ultimately, automorphic forms.

One will have noticed that all the names belong to men, practically all of them European.  This is a problematic feature of the hierarchical nature of mathematics, but it’s not my topic today.  The question instead is Who speaks for mathematics? which (it seems to me) is at least implicit in Piper Harron’s identification of the oppressiveness of mathematical hierarchy.  Much to my regret, MWA did not break with the convention of quoting the reflections of Giants and Supergiants and those most visible among our contemporary colleagues, the people whose names appear in lists and diagrams like the one copied above.   Thus the question Who speaks for mathematics? is answered by pointing to those who occupy the most prominent positions in the (oppressive?) hierarchy.

This is in part due to the unsystematic nature of my research and in part due to structural features of the hierarchy, which I emphasize on p. 39, in what I have already identified as the key passage in Chapter 2:

We’ll see throughout the book  quotations by Giants and Supergiants in which they conflate their own private  opinions and feelings with the norms and values of mathematical research,  seemingly unaware that the latter might benefit from more systematic  examination.  One of the premises of this chapter is that the generous licence  granted hieratic figures is of epistemological as well as ethical import.

My own experiments with the expression of what appear to be my private opinions resemble this model only superficially and only because they conform  to the prevailing model for writing about mathematics.  My friend’s point was  that even my modest level of charisma entitles me not only to say in public  whatever nonsense comes into my head…

In other words, it’s much easier to be quoted if you have published your thoughts in the first place, and it’s much easier to get your thoughts published if you are identified as a consequential mathematician.  I don’t know how to overcome this (possibly oppressive) characteristic of the mathematical hierarchy, and it’s one of the main reasons I am hoping sociologists will pay closer attention to mathematics.

Having said that, I should dispel any notion that Langlands took advantage of his mathematical eminence, in the article from which Diagram B is taken, to write “whatever nonsense” came into his head.  On the contrary, while the modesty of his intentions is evident throughout, there is no nonsense but rather a good deal of profound and unconventional thinking about the nature of our vocation.  So I will take the risk of promoting the (oppressive?) hierarchy once again and encourage you all to read the article, if you have not done so already; and I will quote a few of its more memorable passages.

On his own limitations and uncertainties:

I learned, as I became a mathematician, too many of the wrong things and too few of the right things. Only slowly and inadequately, over the years, have I understood, in any meaningful sense, what the penetrating insights of the past were. Even less frequently have I discovered anything serious on my own. Although I certainly have reflected often, and with all the resources at my disposal, on the possibilities for the future, I am still full of uncertainties.

On the effects of (what MWA calls) charisma:

Because of the often fortuitous composition of the faculty of the more popular graduate schools, some extremely technical aspects are familiar to many people, others known to almost none. This is inevitable.

On Proposition 78 of Book X of Euclid’s Elements:

This is a complicated statement that needs explanation. Even after its meaning is clear, one is at first astonished that any rational individual could find the statement of interest. This was also the response of the eminent sixteenth century Flemish mathematician Simon Stevin…

On the beauty of Galois theory:

What cannot be sufficiently emphasized in a conference on aesthetics and in a lecture on mathematics and beauty is that whatever beauty the symmetries expressed by these correspondences have, it is not visual. The examples described in the context of cyclotomy will have revealed this.

On cooperation — and charisma! — in mathematics:

Like the Church, but in contrast to the arts, mathematics is a joint effort. The joint effort may be, as with the influence of one mathematician on those who follow, realized over time and between different generations — and it is this that seems to me the more edifying — but it may also be simultaneous, a result, for better or worse, of competition or cooperation. Both are instinctive and not always pernicious but they are also given at present too much encouragement: cooperation by the nature of the current financial support; competition by prizes and other attempts of mathematicians to draw attention to themselves and to mathematics.


What today’s number theorists need to know


Of course I’m not going to answer the question in the title.  If you want to know the answer, you can try to get it past the ferocious gatekeepers at MathOverflow; here, as always, the point is to question questions, not to answer them.

The above excerpt from Kummer’s  De numeris complexis qui unitatis radicibus et numeris integris realibus constant treats the familiar construction of units in the cyclotomic field of 5th roots of unity.  It’s a matter with which all number theorists are or should be familiar, but it’s less clear that it’s necessary today to be able to read Latin (I can’t).   It’s also arguably  unnecessary for today’s number theorists to be able to construct norm tables like this one later in Kummer’s article:

Kummer            Contemporary number theory is highly dependent on different kinds of invariant theory; for example, Vincent Lafforgue’s construction of Langlands parameters for automorphic representations over function fields is based on R.W. Richardson’s study of orbits in powers of a reductive group under the diagonal action.   But explicit invariant theory in the style of Paul Gordan’s late 19th century work is only of interest to specialists:


German seems to have gone the way of Latin, as far as number theorists (other than native German-speakers) are concerned, although it’s not so long ago that it was considered necessary to be able to read the works of Hecke and Siegel in the original language.  Although Vincent Lafforgue has provided an English-language survey of his recent work, and has devoted some serious thought to automatic bilingual publication, number theorists still do need to be able to read French, whatever Jeremy Paxman thinks.

Gordan, who apart from his work in explicit invariant theory is remembered for being Emmy Noether’s thesis advisor, reputedly exclaimed

Das ist nicht Mathematik, das ist Theologie!

when Hilbert’s Basis Theorem put an end to Gordan’s business of proving finite generation of rings of invariants.  Colin McLarty’s unpackaging of this anecdote, which was published in (the underappreciated) Circles Disturbed, should be essential reading for anyone with a serious interest in philosophy of mathematics.

Essential reading as well for number theorists, whose business of compiling norm tables has been supplanted by considerations some might qualify as theological.  Apart from the bits of mathematics number theorists need to know that have unfortunately not yet been invented — for the proof of the Tate Conjecture, for example, or Beilinson’s conjectures — there are the branches of mathematics in the process of being invented that are so massive that they will displace more mundane concerns if number theorists choose to make room for them in our finite minds.  In search of potential applications to the Langlands program I find myself tempted to dip into the vast treatise of Gaitsgory and Rozenblyum, published by installments at the bottom of this page.  It is not in Latin:


In their Preface the authors characterize their project as “A stab in the back” (I hope they mean “stab in the dark”), offer advice for “practical-minded readers,” and provide this helpful account of the contents of their book that can serve as a classification of mathematical activities more generally:

The substance of mathematical thought in this book can be roughly split into three modes of cerebral activity: (a) making constructions; (b) overcoming difficulties of homotopy-theoretic nature; (c) dealing with issues of convergence.  Mode (a) is hard to categorize or describe in general terms. This is what one calls ‘the fun part’.

Mode (b) is something much better defined: there are certain constructions that are obvious or easy for ordinary categories (e.g., define categories or functors by an explicit procedure), but require some ingenuity in the setting of higher categories. For many readers that would be the least fun part: after all it is clear that the thing should work, the only question is how to make it work without spending another 100 pages.

Mode (c) can be characterized as follows. In low-tech terms it consists of showing that certain spectral sequences converge. In a language better adapted for our needs, it consists of proving that in some given situation we can swap a limit and a colimit (the very idea of IndCoh was born from this mode of thinking). One can say that mode (c) is a sort of analysis within algebra. Some people find it fun.

Bearing in mind that the purpose of these 1000 or so pages is to lay the groundwork for a categorical understanding of the geometric Langlands program — which is undoubtedly itself only the first of an infinite chain of increasingly categorical, not to say theological Langlands programs — the authors offer encouragement in the form of a poem by Osip Mandelshtam: