Genetic determinism once more obnubilates French readers

After a French friend informed a few of his American colleagues that the center-right weekly magazine Le Point had printed a translation of Ted Hill’s article in Quillette, in which he alleges that an article of his had been censored by both the Mathematical Intelligencer and the New York Journal of Mathematics on political grounds, I decided I had no alternative to wasting half an hour familiarizing myself with a few of the details.  Having done so, I am just going to reproduce the message I sent to my French friend.

But first:  the French verb obnubiler is usually translated “to obsess,” which has nothing in common with the English cognate obnubilate, which means literally to cloud.  But in fact, French dictionaries interpret obnubiler quite differently:  someone is described as obnubilé whose judgment is clouded or impeded by an obsession.  The obsessive and repeated attempts to explain differences in power and status by genetic factors is a good example of obnubilation in this sense.

Now for my message:

I really don’t want to be wasting my time on this, but I’m afraid I’m going to have to.  Here is a description of Quillette:

and here is an article by Gowers analyzing the claims in Hill’s alleged study.
There is a second post, in which Gowers goes to extreme lengths to give Hill’s theses the benefit of the doubt, while remaining unconvinced.
I’m not going to comment on the editorial process at the Intelligencer or the New York Journal of Mathematics, which is a matter of very little interest.  What I see is just one more strained effort to disguise as scientific inquiry a thoroughly artificial and simplistic framing of a complex interaction of phenomena for which one has nothing resembling a coherent model, motivated solely by the demonstrate that the present distribution of power and resources has a natural basis.  All of this has dramatic political implications and the “libertarians” with whom Quillette identifies may belong to all kinds of tendencies — Dawkins used to be some kind of leftist, Pinker is a [censored!] liberal, Charles Murray is definitely right-wing — but the organized forces overlap significantly with the alt-right.
The problem with this sort of online debate is that it’s presented as intellectual censorship, while in fact it’s something else entirely.  Most of the liberals who are confused by this framing would never defend the right of creationists or climate change deniers — or holocaust deniers — to equal time, in the name of freedom of expression.  But there is a surprising openness to polemics disguised as scientific analysis when the aim is to prove that women are inferior at one thing or another.
To my mind, the best response to claims about hereditary differences in intelligence is still Gould’s The Mismeasure of Man, which illustrates the lengths to which defenders of inequality will go in attempting to prove their theses.  That was written nearly 40 years ago.  (You may remember that he mentioned that at one point IQ tests had been used as scientific proof that Jews were intellectually inferior to northern Europeans.)  Gould wrote a shorter but no less devastating review of The Bell Curve in 1994.
Unfortunately this particular vampire has not yet been nailed to its tomb once and for all.  Here is what I wrote about this in the middle of an article about the responsibility of mathematicians, for the celebration of Reuben Hersh’s 90th birthday.

I want to discuss an older story, one in which the mathematical sciences play at most a supporting role, but that I think illustrates well how philosophical confusion about the nature of mathematics can interfere with informed judgment. Here is a sentence that, syntactically at least, looks like a legitimate question to which scientific investigation can be applied:

Does mathematical talent have a genetic basis?

On the one hand the answer is obviously yes: bonobos and dolphins are undoubtedly clever but they are unable to use the binomial theorem. The question becomes problematic only when the attempt is made to measure genetic differences in mathematical talent. Then one is forced to recognize that it is not just one question innocently chosen from among all the questions that might be examined by available scientific means. It has to be seen against the background of persistent prejudices regarding the place of women and racially-defined groups in mathematics. I sympathize as much as anyone with the hope that study of the cognitive and neurological basis of mathematical activities can shed light on the meaning of mathematics — and in particular can reinforce our understanding of mathematics as a human practice — but given how little we know about the relation between mathematics and the brain, why is it urgent to establish differences between the mathematical behavior of male and female brains? The gap is so vast between whatever such studies measure and anything resembling an appreciation of the difficulties of coming to grips with the conceptual content of mathematics that what really needs to be explained is why any attention, whatsoever, is paid to these studies. Ingrained prejudice is the explanation that Occam’s razor would select. But I’ve heard it argued often enough, by people whose public behavior gives no reason to suspect them of prejudice, that it would be unscientific to refuse to examine the possibility that the highlighted question has an answer that might be politically awkward. It’s the numerical form of the data, I contend, and the statistical expertise brought to bear on its analysis, that provide the objectivity effect, the illusion that one’s experiment is actually measuring something objective (and that also conveniently forestalls what ought to be one’s first reaction: why has Science devoted such extensive resources to just this kind of question?) The superficially mathematical format of the output of the experiment is a poor substitute for thought. Maybe something is being measured, but we have only the faintest idea of what it might be.

More concisely:  if the question is not scientific, then the answer won’t be scientific either.  Or even more concisely:  garbage in, garbage out.
I added some emphasis that was not, I think, in the original article.  I just want to conclude with a particularly helpful paragraph from Gould’s review of The Bell Curve.
Like so many conservative ideologues who rail against the largely bogus ogre of suffocating political correctness, Herrnstein and Murray claim that they only want a hearing for unpopular views so that truth will out. And here, for once, I agree entirely. As a card–carrying First Amendment (near) absolutist, I applaud the publication of unpopular views that some people consider dangerous. I am delighted that The Bell Curve was written–so that its errors could be exposed, for Herrnstein and Murray are right to point out the difference between public and private agendas on race, and we must struggle to make an impact on the private agendas as well. But The Bell Curve is scarcely an academic treatise in social theory and population genetics. It is a manifesto of conservative ideology; the book’s inadequate and biased treatment of data display its primary purpose—advocacy.
I think, though, that Gould would not have been so delighted to see the publication of the theses of The Bell Curve in a journal that seeks to maintain editorial standards.

Justice, finally, for Maurice Audin

Place Audin

Photo taken in Paris by Ammine May 26, 2004.


It was announced today that French President Emmanuel Macron

would acknowledge that Audin “died under torture stemming from the system instigated while Algeria was part of France.”

More details can be found in the article published today in the Guardian , in a 3-minute video on and a long article by the inevitable Cédric Villani.  The role of Laurent Schwartz in the story was recalled on this blog in 2015.

There is also a Place Maurice Audin in Algiers:

PlaceAudinAlger From Cédric Villani’s blog,

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.

Short proofs

August 2018 was this blog’s busiest month in two years.  Practically all the visits came in the first two weeks, with much of the traffic arriving from Germany (1788 of 5574 views).  The explanation, apparently, is that Peter Scholze’s Fields Medal was announced the first day of the month, and the Hausdorff Institute of Mathematics in Bonn chose my blog post  as one of three “interesting and popular articles” on his work, along with the article Erica Klarreich published in Quanta two years ago, and my chapter in the book What is a Mathematical Concept? edited by de Freitas, Sinclair, and Coles.  Quanta‘s articles on mathematics are notoriously interesting and popular; my chapter on the “perfectoid concept” may or may not be interesting, but I can’t imagine why anyone would consider it “popular”; and the blog post — which, as you may remember, is a text that did not qualify for publication in The New Scientist, is somewhere in between.

Anyway, my WordPress dashboard informs me that the Hausdorff Institute’s recommendations were picked up by (Frankfurter Allgemeine Zeitung) as well as  These two sites, together with the Hausdorff Institute, my indefatigable colleague Peter Woit’s blog, and the inevitable Google and Facebook, accounted for most of August’s referrals.

This year’s Fields Medals were widely covered by the international press, with Scholze’s story featured most consistently, along with the unexpected drama of the theft of Birkar’s medal.  Apart from Ulf von Rauchhaupt’s rather insightful article (visibly influenced by my blog post, not always with full attribution), coverage was mainly as approximate as one might expect, and was more informative about the current state of science reporting than about the priorities of contemporary mathematics.  Most entertaining for me was the article on the French website, which included this surprising bit of news:

Scholze a donc incontestablement la bosse des maths, mais il ne s’agit pas de son seul talent. En effet, il faisait partie d’un groupe de rock à 17 ans, puis a été professeur d’histoire allemande à 24 ans.

Rough translation:  “Scholze unquestionably has the math bump [a French expression that derives from phrenological notions popular in the 19th century — apparently there really is such a cranial bump, though its connection to mathematics is dubious] but it’s not his only talent: he played in a rock band at age 17, then at age 24 became professor of German history [sic!]”  Instead of a byline the article refers to three sources:  DW, El País, and Quanta.  I strongly suspect the sources were consulted and consolidated by a robot reporter which offered its own intrinsically logical interpretation of the sentence that opens the El Pais article:

Con 17 años tocaba el bajo en un grupo de rock, con 24 se convirtió en el catedrático más joven de la historia de Alemania.

The news coverage also revealed something of the network of journalists’ local contacts.  Thus the New York Times consulted Jordan Ellenberg, while El País quoted José Ignacio Burgos; Le Monde went to the trouble of finding four different mathematicians to contribute sentences about each of the four medalists:  Laurent Fargues (for Scholze), Philippe Michel (for Venkatesh), Jean-Pierre Démailly (for Birkar), and, inevitably, Cédric Villani (for Figalli).

Practically every article alluded to Scholze’s refusal of the New Horizons Prize, already discussed on this blog in 2015.   This came as no surprise to me; in fact, I had already anticipated the hypothetical reader’s fascination with this telling detail in the article I had prepared for The New Scientist, with the following sentence about his motivations:

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.

This means something very specific to me, and it may mean something to mathematicians reading this post, but to the hypothetical New Scientist reader it means exactly that Scholze refused the prize because he refused the prize, a vacuous observation embellished with the enigmatic expressions “mathematical community” and priorities.”  As we already know, this sentence never made it into the pages of The New Scientist; but, much to my surprise, it was translated into Spanish, at least twice, and at least once into German.  In each case my sentence was promoted to the status of a “speculation,” although the journalists had absolutely no reason to treat me as an authority on the matter, and besides which, as, I already explained, in the context of a newspaper article my sentence was totally devoid of content.  (Though one could always hope that a particularly attentive reader will find it surprising that not only is these such a thing as a “mathematical community” – though the word “community” disappeared from the German version — but that it even has “priorities”.   The reader may be sufficiently intrigued to wish to learn more about this, in which case:  good luck!)

Apparently Scholze’s refusal of the $100,000 prize cried out so desperately for explanation that the journalists grabbed at the only straw they found.  If they had been a little more patient, though they could have waited until August 6, when Scholze’s own answer to the question appeared in his interview with Helena Borges in O Globo:

O que posso dizer é que aquele era um prêmio e que este é outro. E é tudo que vou comentar sobre.

Rough translation, which curious readers are invited to ponder:  “What I can say is that that [the New Horizons Prize] was one prize, and that this [the Fields Medal] is a different one.  And that’s the only comment I’m going to make about that.”

The other item mentioned in practically all the press coverage recalled how Scholze distinguished himself already at age 22 when (quoting O Globo again) he “transformou uma teoria de 266 páginas em um texto sucinto de 37 folhas” — “transformed a 266-page theory into a succinct text of 37 sheets.”  Most of the other sources, starting with Erica Klarreich’s article in Quanta in 2016, identified the overstuffed “266-page theory” as none other than my book with Richard Taylor.  There is an interesting lesson hidden in that story about radical abbreviation, but that’s a silly (as well as misleading) way of putting it.  I was hoping to explain why that’s the case before I present an overview of the proof of the local Langlands conjecture to the graduate reading group that meets at Columbia tomorrow afternoon, but unfortunately I have run out of time, and I’ll have to return to the question later.

Univalent foundations and mathematical practice

Pierre Deligne’s talk next week at the Voevodsky Memorial Conference at the Institute for Advanced Study is entitled What do we mean by “equal”.  His abstract suggests that he is broadly sympathetic to univalent foundations:

In the univalent foundation formalism, equality makes sense only between objects of the same type, and is itself a type. We will explain that this is closer to mathematical practice than the Zermelo-Fraenkel notion of equality is.

This could represent a turning point in the (brief) history of univalent foundations, and not primarily because of Deligne’s immense prestige.  Alongside his monumental contributions to mathematics, one of the qualities that made Deligne one of the most admired mathematicians is his unfailing lucidity, as evidenced in his own work as well as in his reformulations of the work of others (here, notoriously, but also here and here, just to mention a few of the examples that directly influenced my own thinking).  If he put his mind to it, Deligne could undoubtedly find a reformulation of univalent foundations that is consistent with (and not merely “closer to”) contemporary norms of mathematical practice.

Deligne’s use of the expression “mathematical practice” is itself remarkable in the light of IAS mathematicians’ historical rejection of sociological tendencies in science studies.  Deligne’s lucidity has been in evidence in his infrequent but eminently quotable interventions that touch on questions of philosophy or sociology of mathematics (see pp. 3-4 of this book, for example).  Unfortunately I will not be able to attend his talk this coming Tuesday but I look forward to watching the video.

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.”

Mathematicians as beneficiaries, and their patrons

What follows are the uncorrected notes for a presentation by videolink at the first workshop on Ethics in Mathematics, held in Cambridge April 20-21, 2018.

It’s a humbling experience for me to be asked to speak at this meeting, alongside some authentically legendary figures. Maurice Chiodo and Piers Bursill-Hall have assembled a stellar lineup in a remarkably short time. This is certainly a tribute to their energy and initiative, but the fact that so many speakers have agreed to participate is also a sign that Maurice and Piers have identified a need whose urgency is increasingly recognized across the profession. I do hope this week’s meeting will be remembered as the start of a genuine international movement to place ethics at the center of our work as mathematicians.

It’s a special honor to be invited to participate in a conference on mathematics and ethics that is taking place in Cambridge, home of G. H. Hardy, a mathematician whose commitment to pacifism and social justice is well-known even beyond the profession. Since mathematicians are constantly being asked why our work is useful, it’s appropriate to recall that Hardy once wrote that

A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.

Hardy was thinking particularly of military applications of science, as well as of the mathematical economics of his time. Had he lived a few years longer he would have witnessed the growth of mathematical game theory, whose destructive consequences in both domains have been developed assiduously by the RAND Corporation, which figures prominently in the biography of John Nash, among other mathematical heroes.

I consider Hardy a precursor of current proposals for mathematicians working on various applications to adopt “Hippocratic Oaths,” the ethics of abstaining from doing harm. In an article published last year entitled Do Mathematicians Have Responsibilities?, I mention some of the more recent applications of mathematics that are “useful” in Hardy’s sense, but my focus is different.

While pure mathematicians in particular may have wondered whether much of their work would ever be socially useful, it was generally believed that at least it caused no harm. Events of recent years have called that belief into question.  The sophisticated and often opaque derivatives developed by financial mathematics magnified the effects of a downturn in sectors of the US housing market into a global financial crisis whose consequences are still with us. Edward Snowden’s revelations in 2013 served as a reminder that contemporary cryptographic techniques based on number theory can also be used to facilitate general surveillance by governments. The rapid growth of Big Data has made it possible for commercial as well as public actors to track individual behavior with increasing precision, with grave implications for privacy.

In each of these applications of mathematics one finds the same three features: an approach to human activity that is purely instrumental; a disdain for democratic decision-making; and the empowerment of experts on the basis of their mathematical training. And in each case, a few mathematical scientists have pointed out that the power of mathematical technology imposes social responsibility on those who develop it, beyond putting trust in experts.

In this brief presentation I want to stress the second and third features, because they make it clear that the call to “do no harm,” important though it is, does not fully discharge our social responsibilities as mathematicians. The fact is that our very expertise, as academics and researchers, contributes to the reproduction of the social order that makes the abuses not only possible but often inevitable. We perceive the universities and research institutes in which we work as protected spaces and spaces to be protected, and this is true as far as it goes. But the primary function of the university is to reproduce existing relations of power and influence. In this sense, Hardy’s refuge in pure mathematics is itself part of the problem. Indeed, A Mathematician’s Apology fairly reeks of the elitism that, even in its current attenuated form, is an essential aspect of the image, or the brand, that distinguishes universities like Cambridge and Oxford and Harvard and Columbia and endows their professors with the expert status that so often serves to undermine the democratic process.

Let me add right away that I am fully aware of the dangers of this kind of talk in the face of climate denial and right-wing populism more generally. Nevertheless, I remain convinced that the primary role of the expert in public policy is to be mobilized in support of dominant interests, in the spirit of Margaret Thatcher’s There is no alternative. The article I just quoted has a good illustration of this in connection with the current massive growth of artificial intelligence, and the feverish promotion of the Internet of Things as a technological inevitability and a promising investment opportunity. The ethical implications of these developments seem to have been entrusted, in particular by the EU, to the AI industry itself:

In connection with [the risks of AI], it was announced that Facebook, IBM, Amazon, Google, and Microsoft had just formed the “Partnership on AI” for the purpose of “conducting research and promoting best practices.”

Since then Apple has joined (the big five + IBM) and there are now representatives of civil society (ACLU, EFF, and Center for Democracy and Technology, among others). Of course the relative weight of the corporate and civil partners in defining “best practices” remains to be seen.   My point, however, is that the vision of democratic decision-making still places the expert at the center.

By the way, I have not come to you today with an alternative and more democratic model. The problem is a profound democratic deficit in the society at large. That’s not a problem for this gathering to solve; but in my opinion it is inseparable from any serious reflection on the ethical obligations of mathematicians or any of our fellows in the elite sphere we inhabit.

My aim was rather to make a few remarks about research funding, and I will quote from my article in the Times Higher Education Supplement to indicate how difficult it is to avoid tainted sources.

[Tom Leinster’s] question hasn’t gone away: should we cooperate with GCHQ? The problem is that research funds have to come from somewhere; the survival of number theory depends on it. One veteran colleague likens mathematical research to a kidney; no matter where it gets its funding, the output is always pure and sweet, and any impurities are buried in the paperwork. Our cultural institutions have long since grown accustomed to this excretory function, and that includes our great universities. Henry VIII was a morally ambiguous character, to say the least, and a pioneer in eavesdropping as well as cryptography; but neither Hardy nor his friend Bertrand Russell refused his fellowship at Trinity on that account.  

It would be nice if the State could provide its own kidneys and impose an impermeable barrier between the budgets for research that is socially progressive, or at least innocuous, and the military and surveillance functions about which the less we know, the better. But States don’t work that way, and for the most part they never have. The only alternative to public funding, from whatever the source, is private philanthropy. America’s great private universities are monuments to the past and present generosity of some of our wealthiest citizens. That is not, however, what is most appealing about them. I find it demeaning to have to express gratitude for my research funding to practices of which I otherwise heartily disapprove — like hedge fund management, for example, or data mining — but that have given a few people the status of Ultra-High Net Worth Individuals … and thus in the position of being able to function publicly as philanthropists. Or to despots like the Emir of Kuwait, whose Foundation used to sponsor a generous lecture series at Cambridge.

It seems that anywhere you turn, you’re going to be someone’s kidney. But feeling demeaned is beside the point. As …Cathy O’Neil… put it in January 2014, “We lose something when we consistently take money from rich people, which has nothing to with any specific rich person who might have great ideas and great intentions.…” One of the things we lose: control of how decisions are made: “…the entire system depends on the generosity of someone who could change his mind at any moment.”

The more basic problem is that the very existence of UHNWI entails the concentration of power beyond the control of democratic oversight. Among billionaire patrons, Jim Simons stands out for his commitment to the values of working mathematicians — which is natural, given that he was a distinguished geometer before his management of the wildly successful hedge fund Renaissance Technologies made him an UHNWI. But the same high-frequency trading algorithms that fueled Simons’s philanthropy gave us Breitbart, courtesy of Robert Mercer, Simons’s former colleague at Renaissance. Mercer was much in the news earlier this year after it was revealed that, through his connection to Cambridge Analytica, he used psychologically targeted advertising on social media to intervene in the Brexit and Trump elections, possibly tipping the balance in both cases. Mercer has come to personify the sinister side of the UHNWI phenomenon, but even outspoken liberal billionaires like Facebook’s Mark Zuckerberg and Google’s Sergei Brin, who have been subsidizing pure mathematics indirectly through their cosponsorship of the extravagant Breakthrough Prizes, have built their fortunes on mathematical techniques that are no less threatening to privacy than GCHQ surveillance.

I could continue for quite a long time expressing my regret that the need to sustain our research places us in the uncomfortable position of dependence on ethically dubious sources of funding. In the interest of full disclosure, and to highlight the paradoxes of my own position, I ought to mention that this afternoon I will be heading to a conference in the Bavarian Alps, sponsored by the Simons Foundation! The first part of today’s presentation, however, was meant as a reminder that as researchers and academics our very salaries are being paid by institutions whose primary function is the preservation of the status quo. Insofar as the possibility of the most visible aberrations (Cambridge Analytica, NSA undermining of encryption standards, credit default swaps, drone guidance systems and so on) are built into the normal functioning of the status quo, and are justified by an ideology of expertise that is maintained by our universities and research institutes, our very existence as experts guarantees that our profession provides no refuge of ethical purity.

Interjection: How, by the way, did Trinity get to be so rich? I don’t know the answer; instead, I offer this bit of information as an ironic metaphor for our defense of ethics from our perches within the power structure:

At what is today Columbia University, there was a medal issued at graduation every year by the Manumission Society — many of whose members were slaveowners — for the best essay each year that opposed the slave trade (from a report by Eric Foner on Columbia’s website, as quoted in The Trinity Tripod of Trinity College, Connecticut, dated February 11, 2014)

(Of course, Columbia was hardly alone; Harvard, Penn, Dartmouth, William and Mary, and other leading universities of the time had interests in the slave trade.)

As I wrote in the THES piece:

[T]he immense privilege of devoting our lives to the research projects we have chosen freely imposes on us the obligation to speak out when our work is used for destructive ends, or when the sources of our funding do not share our values.

By “speaking out” I don’t mean simply reacting to abuses. I mean actively anticipating possible uses of our work, including our teaching of students, for purposes of which we do not approve. Here I would add that we are no less obligated to acknowledge the role of our institutions, and of our expert status within and through these institutions, in preserving existing power relations that are incompatible with democratic ideals.

The privilege of devoting our lives to our freely chosen profession makes us beneficiaries in the sense described in a recent book by my Columbia colleague Bruce Robbins. A great many people need to perform less rewarding work, or are rewarded less well for what they do, in order to provide us the means to pursue our professional goals.

Nevertheless, I want to conclude by stressing the importance of defending these benefits. I’m sure that each of you has been asked at one time or another some version of “how is what you do useful?” And if you are a pure mathematician you might have resorted not to Hardy’s definition of “useful” but rather to Hardy’s argument that mathematics is an art form, and therefore deserves to be pursued for its own sake. I suspect such an answer provides little defense against accusations of self-indulgence, irresponsibility, and a lack of due regard for the taxpayer’s money. Faced with such accusations — usually by individuals whose own position within the power structure leaves them open to challenge — I like to reverse the terms of the question: if mathematics is not to be pursued for its own sake, then for the sake of what? For profits, or Facebook “likes,” or to give Britain a leg up in the international marketplace? This should immediately pose the question of democracy, which in the present context includes the right to adhere to values that are not determined by the market and its ideologues and functionaries. All work should ideally be for its own sake. But this is an idea I am struggling to articulate, and I hope to have made some progress if and when we meet again.