Category Archives: Words

Stéréotypes genrés problématiques


Long-time readers of this blog will remember how I solved a sticky gender stereotype problem, with the help of my friends, by replacing the “Actress” character in the dialogues in the chapters entitled “How to Explain Number Theory at a Dinner Party” with a “Performing Artist.”  The problem has now resurfaced during the preparation of the French version of the book, to be published by Éditions Cassini, whose director is a retired former colleague of mine at Paris 7.

The French translator has done an outstanding job.  I started writing in English upon moving to France in 1994, and I saw this as an emergency measure to protect my ability to express thoughts that I believed could not be captured in French.  I was still convinced that my English prose was untranslatable when I wrote MWA, and it is partly to guarantee that this would be the case that the syntax is often so convoluted.  But the translator — you’ll discover her name when the book is in print – managed to convey my intentions brilliantly.

Gender neutrality, however, is a challenge in French.  The translator initially used the word Actrice for “Performing Artist,” but I explained the issue and the dialogue is now between an Artiste and a Théoricien des Nombres.  This doesn’t completely solve the problem, however, and I’m not sure the problem, evident in the above excerpt, can be solved.

Non-binary gender grammar does exist in French, though you will not be surprised to learn that “L’utilisation de ces néologismes et de toute autre forme de langage inclusif est rejetée par l’Académie française32.”  Most of the stereotypes you have heard about the fustiness of the Académie française are true, but I don’t know how a thorough reform of the French dictionary would solve the problems indicated above.  A French number theorist is either a (male) théoricien or a (female) théoricienne.  Reactionaries (including some of my colleagues, I suspect) are still arguing that théoricien is adequate for all genders.  Progressives have for some years been addressing their exhortations to a gender-diverse community of théoricien(ne)s, or sometimes thé  But an individual is one or the other — or a self-identified non-binary individual might be a théoricæn (top choice, followed by théoriciem and theorician), if I am extrapolating correctly from the results of a survey published on the blog lavieenqueer.  But French doesn’t offer a genuinely gender-neutral translation of Number Theorist; non-binary is another box, alongside male and female.   The same goes for the adjective that specifies the Performing Artist’s gender; French Artistes can be patiente or patient or (again copying from the survey) patientæ, patienx, or patiens but they have to be one of those.

I found exactly two tell-tale adjectives in feminine form, applied to the Artiste in the dialogues — and many more in agreement with the number theorist’s designation as théoricien and not théoricienne nor théoricæn. I don’t know how to fix this issue in French, but I don’t even know how to begin to address it in Greek or Chinese, which are the other languages in which you can read the book — if you can read those languages, which I can’t.


This is the cover of the translation by Βερονα Πετρου, published by Ροπή, in which the Performing Artist is called a ΕΡΜΗΝΕΥΤΡΙΑ ΗΘΟΠΟΙΟΣ.  Although this seems to be a literal translation, and Google translate is of no help in determining whether or not this expression is gendered, when I type ΕΡΜΗΝΕΥΤΡΙΑ on Google, practically all the images that come up are of women.  Moreover, the text is unequivocal.  Here is the Greek version of the French text reproduced above, with the feminine ending circled.


Greeks must have come up with non-binary rules for adjectives, but I will leave it to Greek readers to help us figure them out.  Meanwhile, there seems to be no way to root out “problematic gendered stereotypes” worldwide, unless we want to imagine the dialogue taking place in the “theater of androids” which — as is recalled at the end of Chapter δ of MWA — was Maurice Maeterlinck’s emergency measure for preserving “the symbol,” “the dream,” and “art.”

The soul of a space

The paper Česnavičius and Scholze just posted on arXiv answers several longstanding open questions in fundamental algebraic geometry.  It also introduces a new definition with energetic new terminology:


I don’t pretend to know which of the authors had the idea of returning to Latin roots in order to find the appropriate word to designate the objects that Lurie had chosen to call “spaces,” as well as their cognates in other settings.  Scholze’s terminological innovations have been more than commonly successful up to now, but I predict that “animated sets” will be especially popular.  A whole thesis in philosophy of mathematics — and a second thesis in theology of mathematics? — could be devoted to the last sentence above.

It turns out that the expression “soul of a space” has been popular for some time among interior designers and architects.


A room designed by architect Vipin Bakliwala

Bakliwala’s reply also deserves our attention:

As architects, it is our duty to induce emotions into a space and create an ambience that brings forth our hidden calm, positive and spiritual side. We strive to expand the brief given by the client and create a space that elevates and improves his life. We struggle to provide an environment which is an enhanced reflection of his thoughts. We call such places soul shelters.  It is that space where the soul remains in its innate nature.

Do emotions inhere more spontaneously in “worldly” point-set “physical” topological spaces or in their animated “calm, positive, and spiritual” ghostly doubles?  Descartes and Spinoza might help us sort this out.


UPDATE:  T.G. pointed out that, if I had read past the introduction to the acknowledgments, I would have realized that

The terminology is due to Clausen, inspired by Beilinson; see the acknowledgements, and the first paragraph of section 5.1.

Here is a passage from Beilinson’s article Topological E-factors that sheds some light on his perception of the need for appropriate terminology, and about the desolation of ordinary category theory.


A. A. Beilinson, Topological E-factors, Pure and Applied Mathematics Quarterly 3, 357-391, (2007)

Beilinson, or my imagined recollection of him, expresses a rather different opinion of spaces on p. 202 of MWA.

Big Dada: A statistical found object at Cabaret Voltaire

Voltaire - 1

Here’s one even Jordan Ellenberg may not know.  There really is a Cabaret Voltaire in Zürich, right on the same corner as the original one, and while this year Zürich is celebrating the 100th anniversary of Dada, mostly at the Cabaret Voltaire, a few years ago there really was an exhibition with the title Dada x Statistik, and you can read about it on the city’s website.   The exhibition was held, naturally enough, at Cabaret Voltaire, where you can buy the exhibition catalogue (which features a lot of statistical diagrams and not so much Dada, as far as I could tell upon perusing it quickly while waiting for the show to start); but maybe it was at the city statistical agency as well, because the pretext for this exhibition was apparently that Dada and Zürich statistics are roughly the same age, and the two institutions are “a stone’s throw” from one other, according to the website:

Dass Statistik Stadt Zürich und Cabaret Voltaire gemeinsam eine Ausstellung realisiert haben, mag ungewöhnlich sein, überraschend ist es nicht. Beide prägt eine ähnlich lange Geschichte. Statistik Stadt Zürich ist das älteste städtische Statistikamt der Schweiz. Eine fast ebenso lange Geschichte hat der 1916 im Cabaret Voltaire begründete Dadaismus. Die beiden Institutionen liegen räumlich nur einen Steinwurf voneinander entfernt. Was lag also näher, als gemeinsam ein Projekt zu realisieren?

I don’t find this convincing enough to translate.  The catalogue had a slightly more plausible explanation:  the statistical agency and the dadaists have in common a propensity to collect a lot of stuff.   The Cabaret Voltaire’s website is not much more convincing (what does it mean to “bring the ideas of Dada into a connection with the morality and ethics of statistics and to create an instructive and entertaining third term”?).  But it does explain that 2013 was the “International Statistical Year”:

Dada x Statistik ist eine Zeitreise über die letzten 100 Jahre der Stadt Zürich. Mit situativen Interventionen und Exponaten wird das Arbeiten, Wohnen und Leben der Dadaisten um 1916 erfahrbar. Mit statistischen Daten und Prognosen wird zudem bis ins Jahr 2023 geblickt und so indirekt auch Dada in die Zukunft projiziert.

Dada x Statistik bringt die Ideen von Dada mit Moral und Ethik der Statistik in Verbindung und schafft ein lehrreiches wie unterhaltsames Drittes.  Dazu werden alle Räumlichkeiten sogar die Toiletten des Cabaret Voltaire bespielt.

Kuratoren von Dada x Statistik sind Adrian Notz vom Cabaret Voltaire und Statistik Stadt Zürich. Die Ausstellung ist ein partnerschaftliches Projekt, das im Rahmen des Internationalen Jahres der Statistik 2013 durchgeführt wird.

The big surprise, for me, was that the catalogue made no reference whatsoever to Big Data, although the pun is obvious; is it because the expression Big Dada is protected by a copyright since 1997?

The performance had nothing to do with statistics — unless that “instructive and entertaining third term” is now permanently installed at Cabaret Voltaire.   One of the performers, playing the part of André Breton, spoke this sentence twice:

And the wind gave way to mathematical illusions.

If this is a direct quotation, it’s unknown to Google, in English, French, or German, and I would be (moderately) grateful if someone could provide a source.  It turns out that both dada and surrealism were pursued by individuals who had studied mathematics, including  Tristan Tzara and Hans Bellmer; the Larousse Encyclopedia’s entry for dada alludes to the intention of Duchamp and Man Ray to “introduce a cold mathematical humor into life.”  And one recalls that, in the mid 30s, Max Ernst sketched and Man Ray took pictures of the objets mathématiques that they discovered during a visit to what the Centre Pompidou calls the Institut de Raymond Poincaré.

Given the widespread belief that nothing of world-historical importance, other than banking, has happened in Zürich since the First World War, and that the original Cabaret Voltaire was the center of the Zürich avant-garde at the time, I was expecting the present-day Cabaret to be a crass attempt to turn the subversion and anarchy of those pre-revolutionary days into a reliable revenue stream, an exercise in sanitized nostalgia for the TripAdvisor generation.  And perhaps it was, but I was surprised to observe that the standing-room crowd was uniformly hip (with at most one exception).  Maybe it’s because so many of them were American; the performers were from New York and Los Angeles, none of them could pronounce either the French names or the German names correctly, and it didn’t matter.  This naturally raises a paradoxical question.  The Breton character quoted a question Breton posed in 1922:

…among the objects said to belong to modernity, is a top hat more or less modern than a locomotive?

I ask:  is today’s Cabaret Voltaire hip because it attracts a hip crowd, or is the crowd hip because they’re at the Cabaret Voltaire?

Problematic gendered stereotypes, or “Who was that actress?”


One of the (positive) reviewers asked me who was the actress in the “How to explain number theory” episodes, and even hazarded a (wrong) guess, but what difference does it make?  The whole incident could have been made up, like the dialogues.   Or, just as the image above is a composite of several (real and imaginary) faces, the incident could have been a composite of (real and imaginary) incidents when, as happens to so many mathematicians, someone I just met asked me — “out of politeness, or perhaps desperation” as Tim Gowers accurately reports — to explain just what it is we do.  In other words, out of my many experiences of being put on the spot by strangers of various genders from various walks of life I could have concocted a fictitious ideal-typical incident in order to enliven the narrative and as a pretext for meeting my personal challenge of making the Birch-Swinnerton-Dyer conjecture meaningful to someone who knows nothing about mathematics.

But there really was such a dinner party, and I really was seated next to an actress, and the conversation really did touch on the topics mentioned, and the actress really did, unexpectedly, ask “what it was you number theorists do.”  I can now reveal, however, that I did make up one detail and alter another, in exercise of my newly-acquired literary license.  First, I have no idea whether or not the actress asked her question about number theory when dessert was served, or at some other point in the meal.  I must confess I was at first horrified when my editor told me early on that readers expect an author to be affirmative, and don’t want to read that “maybe it was around the time dessert was served, I honestly no longer remember.”  My editor was extremely lenient, perhaps excessively so, but she did instruct me to spare the reader apologies for my faulty memory.  Either I should say the fateful question was asked at the time of dessert, or I should find a way to frame the question without referring to the meal.  It seemed to me (reading the fine print on my literary license) that the story sounded better with a sideways glance at the meal, and that number theory goes better with dessert than with soup or salad, so dessert it was to be.

More momentous was my authorial decision to have the actress address her question to me directly, when — this I remember perfectly well — she had posed the question not to me but to my host, who is an admirable mathematician but not a number theorist.  He answered something along the lines of, “you should ask Michael” (or maybe he said “the person sitting next to you”), “he’s the number theorist in this room.”  It vastly improves the story to streamline the exchange between actress and author, but it is dishonest and frankly reprehensible, because it implies that the actress asked the question not out of politeness to her host, who was sitting across the table, but because she was fascinated and intrigued by the charming yet enigmatic number theorist seated to her right, and was perhaps looking for an excuse to prolong the conversation beyond the meal’s final course and… so it turns out the dessert was not such an innocent detail after all.

The episode, which serves as a prelude to Chapter α, was rewritten several times, to respond to criticism, but before I explain how that came about, here are three hints that won’t help you guess the actress’s identity.  First, I wrote that

She talked about the trials of being an actress, hinting that not all her peers suffered quite so much as she did

I can now reveal that she specifically speculated that she might do better by moving to France, where (the much older) Kristin Scott Thomas had made a fine career for herself.  (The woman at the dinner party is also doing quite well now, on the stage rather than in film, having left Britain, though not for France.)

Next, it’s not at all difficult to find her name on the internet, and she is sufficiently prominent that readers are invited to provide answers to the following questions (her name has been replaced by P.A.):

How much is English stage actress P.A.‘s net worth? How much does she make per week for her theatre work? Has P.A. done any endorsements or commercials? Does she have any other income sources beyond her acting career? Has she done any modeling?

Finally, I actually managed to track her down a year or so after the dinner party:  I wrote to tell her that her question had inspired me to write a twelve-page answer to the question (a first draft of what became the four chapters of “How to explain…”).  She invited me to send it to her; I did …and that was the last I heard from her.

Three years later, I mentioned to an academic couple I had just met that I was writing a book about mathematics that I hoped could be read with interest by non-mathematicians.  Since they insisted that they had no interest in reading a book about mathematics, I tested their resolve by showing them the first draft of the description of the dinner party encounter, which was as follows:

During the spring of 2008 I was invited by the Columbia University mathematics department to deliver the Samuel Eilenberg lectures — a perfect illustration of the Matthew effect described in the previous chapter.  The appointment involved living away from my family for several months.   Working late in the department one Friday evening, I must have looked even more forlorn than usual, because a colleague passing my open door decided on the spot to invite me home to dinner.  Several other mathematicians had been invited, along with a neighbor from another department, and the neighbor’s visiting friend, a pale young blonde British woman of medium height who turned out to be an actress between jobs — a real professional actress, with an agent and a long string of film and TV credits as well as a steady and successful career on the stage.  She talked about the trials of being an actress, hinting that not all her peers suffered quite so much as she did.  The younger mathematicians alluded to their own career anxieties, while their tenured colleagues offered reassuring but noncommital replies.  The actress glowed enigmatically during this part of the conversation, but when it came time to serve dessert, she turned to me without warning and asked, “What is it you do in number theory, anyway?”
The other mathematicians looked at me in unison, holding their collective breath.  I had stumbled into the awkward moment every mathematician dreads, my predicament highlighted by the questioner’s quiet radiance.*
*Theater and film reviewers feel obliged to use the word “beautiful” in connection with certain female roles.  The actress sitting opposite me that evening specializes in such roles.

Do you see what’s wrong?

It turns out that these particular academics were curious about many things but really did have no interest in reading a book about mathematics.   But as an act of friendship, they did make an effort to get into the text.  They encouraged me to make changes, starting with the story of the square root of 2, that for various reasons were impossible.  But their first comment was critical:

you need to explain in the first page why math problems continue to seduce you and other mathematicians over and over again and not use problematic gendered stereotypes to do so.

What was problematic about the gendered stereotypes?  They explained that the scenario of a woman asking questions of a male authority figure is itself a stereotype.  Why did I have to mention that my interlocutor was a woman?  Couldn’t I have just said that I was sitting next to an “actor” — after all, the Guardian uses the same word for female as well as male thespians — and leave the gender (or absence thereof) to the reader’s imagination?  (An “actor” who specializes in roles like Nora in A Doll’s House, or Masha in Three Sisters…)  But, I objected, it really was a woman who asked that question at the dinner party!  Nonsense, they replied, you’re telling a story, you can make any changes you want [and here I turned red and stammered, because I had indeed made the two changes mentioned above].

To make a long story short, the very next day I went to the colleague who had introduced me to the couple and asked what to do about those gendered stereotypes.  She read the passage carefully and observed, astutely, that the passage contains a physical description of the actress, and only the actress.   This is indeed the most insidious of gendered stereotypes:  to presume that a woman can only be mentioned in a story if accompanied by a description of her appearance.   So I took the hint:

1.  I removed “pale young blonde” and “of medium height”;

2.  I also removed the pointless footnote (though it was a literally accurate report on the reviews I read of her theatrical performances);

3.  I replaced “The younger mathematicians” by “The younger men and women among the mathematicians,” which was not altogether inaccurate;

4.  Most importantly of all, to allow for any conceivable gender coupling in the dialogues (and thus to avoid pernicious and problematic gender stereotypes), the character named “ACTRESS” became “PERFORMING ARTIST,” or P.A., while the gender of N.T., the number theorist, is never specified.

In spite of this last change, several reviewers believed the dialogues were between an actress and a male mathematician.  They were not wrong, because the reviewers who read this into the dialogue were all men, and there is no question that, in the later dialogues as well as in the initial encounter, I used my newly-minted literary license, and some strategic word placement, to attempt to heighten the dramatic tension by the merest hint of erotic tension.  The reviewers’ reactions prove that this attempt was at least moderately successful; but one question remains.  I wrote explicitly in the preface that N.T. is “one of the author’s alter egos,” but I intentionally gave the best lines to P.A., the other alter ego. So why did the reviewers identify with N.T. rather than P.A.?


5000 pages


John Sidles writes on Scott Aaronson’s blog:

The ongoing acceleration of global mathematical culture has produced a surplus of Grothendiecks and a paucity of Dieudonnés; in consequence “accelerated” intelligences already are walking among us; it is natural for STEM workers (young and old alike) to be simultaneously exhilarated and frightened by this reality.

The Stacks project, “maintained” by my office neighbor Johan de Jong, is a collective and strikingly successful solution to this problem that tonight is celebrating its 5000’th page.

The eight IHES volumes of EGA, in contrast, come to a total of 1814 pages; printed, however, on high quality paper.

A wonderfully disorienting experience

I’m halfway through W. G. Sebald’s Austerlitz, in which you can read, for example, the following comment on Newtonian time:

If Newton thought, said Austerlitz and pointed through the window down at the in the last reflection of the day glimmering water curve, that surrounded the so-called Isle of Dogs, if Newton really thought that time was a stream like the Thames, where then is the source of time and into which sea does it finally flow?

(That’s my translation, which captures just a bit of the tortuousness of Sebald’s German syntax that English can’t accommodate.)  Also this question:

In what way do things that are immersed in time differ from those that are never touched by it?

The relevance of this question to mathematics should be obvious.  The following quotations from a New Yorker article by Mark O’Connell, published on the 10th anniversary of Sebald’s death in a road accident, are not obviously relevant to mathematics, but some readers may find them relevant to the MWA‘s peculiar style, which some readers seem to find irritating.  It’s no accident that Sebald’s The Emigrants was one of the books I was reading while I was writing MWA:  I find that I naturally tend to adopt the rhythm and tone of whatever I’m reading in my own writing, and I choose what I read  as a function of what I’m trying to write.  (And that’s one reason I’m now reading Austerlitz.)

…it was out of frustration with the strictures of academic publication that Sebald turned to creative writing (a vague and ungainly term that, by default, winds up being the most accurate generic description of his work). “He’d originally taught German literature,” says Bigsby, “and had published the kind of books that academics do. But he got increasingly frustrated, and began to write in what he called an ‘elliptical’ way, breaching the supposed boundaries between fact and fiction—not what you’re supposed to do as an academic.” Sebald himself sometimes described his work as “documentary fiction,” which goes some way toward capturing its integration of apparently irreconcilable elements.

And then
Reading him is a wonderfully disorienting experience, not least because of the odd, invigorating uncertainty as to what it is, precisely, we are reading. His books occupy an unsettled, disputed territory on the border of fiction and fact, and this generic ambivalence is mirrored in the protean movements of his prose.

The “him” in question is Sebald, of course, but I can’t deny that “wonderful disorientation” is what I’m after when I read pretty much anything.


“I never want to see another Fourier series as long as I live”


The quotation is taken from a postcard from David Foster Wallace to Don DeLillo, written in 2002, reproduced on p. 274 of Every Love Story Is a Ghost Story: A Life of David Foster Wallace, by D. T. Max, and referring to what he elsewhere called his “wretched math book,” namely Everything and More.   Today’s image is the cover of a special issue of Lettera matematica pristem devoted to mathematics in the work and life of DFW, which has just been published, in Italian and English.   No one has been more adventurous than Italian mathematicians in exploring mathematics as a cultural activity — the last in the series of conferences organized by Michele Emmer in Venice took place this year and the series of volumes (also edited by Michele Emmer) is still available (in some sense) from Springer — but it seems to me that this particular initiative is especially successful.

I’m grateful to Roberto Natalini for inviting me to include my review of Everything and More in this special issue.  Natalini is

Direttore dell’Istituto per le Applicazioni del Calcolo “M. Picone”,
Consiglio Nazionale delle Ricerche

in Rome; but he is also coordinator of the website MaddMaths!,  chair of the EMS committee for Raising Public Awareness;  more to the point, he admits to an “obsession for… many years” with Wallace’s writing.  He and I have something in common:  I proposed to use mathematical ideas (in Bonus Chapter 5) as a scheme for organizing interpretations of the novels of Thomas Pynchon, while Natalini did the same with DFW’s Infinite Jest.  But if you read his essay “David Foster Wallace and the Mathematics of Infinity” in Lettera Matematica Pristem you will agree with me that his analysis is much more substantial than mine.  I limited my attention to conic sections as structural devices in Pynchon’s main novels; Natalini finds cardioids, lemniscates, and Möbius transformations, as well as all the conic sections in Infinite Jest.  He suggests that the two main story arcs (with main characters Hal Incandenza and Don Gately, respectively) form the two branches of a hyperbola (Gately above, Incandenza below), ten years before Pynchon used a similar device (as I claim in Bonus Chapter 5) in Against the Day.  And his proposal to read the fates of the main Incandenza characters in terms of inversion on the Riemann sphere is nothing short of brilliant.

Natalini’s essay originally appeared in a book entitled A Companion to David Foster Wallace Studies, which is as authoritative as it sounds.  My review of Everything and More originally appeared in Notices of the AMS 51(6), June/July 2004:632–638.  Springer, the publishers of Lettera Matematica Pristem, is attempting a kind of inversion of their own; you can read my review for free at the AMS website, but if you want to read it online in the Italian journal you’ll have to pay Springer $39.95,  €34.95, or £29.95.   If you want to make a donation to Springer — and really, they do deserve credit for going to the trouble of publishing Hausdorff’s Gesammelte Werke — save your money for Emmanuele Rosso’s self-referential DFW cartoon.  Or for Stuart James Taylor’s interview with Erica Neely, DFW’s technical consultant for Everything and More which, together with the D.T. Max book cited above, provide insights on DFW’s struggles with the book that I wish I had seen when I was writing my review.

If, on the other hand, you want to see a physical copy of a document that includes my literary reunion with Jordan Ellenberg in Italian, you may have to make the trip to Italy; the Italian edition of Lettera Matematica Pristem doesn’t travel much.  Here is an excerpt from Andrea Piazzi’s translation of my review (I’ve already mentioned, what an honor it is to be translated by the Italian translator of Fantastic Four comics and the cartoons of Will Eisner):

…nel mercatino sotto casa si trovano già titoli divulgativi sull’infinito. Anzi, a quanto pare ce ne sono proprio un bel po’. Uno di questi (Infinity: The Quest To Think the Unthinkable di Brian Clegg) è uscito quasi in contemporanea con E&M e i due sono stati recensiti insieme sul Guardian, dall’autorevole Frank Kermode.
Nonostante la domanda apparentemente illimitata per titoli del genere, una buona parte del sommario sembra essere predeterminata, il che può essere di non poco aiuto a chi fosse interessato a scrivere un proprio libro sull’infinito, oltre forse a dimostrarne di per sé l’esistenza.

Here there is a footnote meant to illuminate the comment about how books about infinity prove the existence of infinity:

4. Il recensore ha consultato cinque titoli divulgativi sull’infinito, tra i quali E&M e il libro di Clegg. I numeri tra parentesi indicano quanti discutono o fanno riferimento all’argomento in questione: il termine greco to apeiron per “infinito” [3], Pitagora [5], l’irrazionalità di √2 [5] e il destino di Ippaso [5]; i paradossi di Zenone [5]; Aristotele e l’infinito in potenza [5]; Archimede e L’Arenario [3]; La Città di Dio di Sant’Agostino [3]; la Summa Theologica di San Tommaso d’Aquino [4]; Nicolò Cusano [4]; le Due Nuove Scienze di Galileo [5]; le coordinate cartesiane [5]; Newton e Leibniz [5]; l’attacco di Berkeley contro gli infinitesimi (“fantasmi di quantità che furono”) [3]; il rifiuto di Gauss di ammettere gli infiniti in atto [5]; i paradossi dell’Infinito di Bolzano [5] e il suo pacifismo [3]; la Sfera di Riemann (con il punto all’infinito) [3]; la fama di Weierstrass come bevitore e spadaccino [3]; la trascendenza di Pi Greco [5]; le Sezioni di Dedekind [4]; il rifiuto dell’infinito da parte di Kroenecker e la sua persecuzione nei confronti di Cantor [4]; la teoria di Cantor degli Ordinali [4], la sua dimostrazione della numerabilità di Q [5], il metodo della diagonalizzazione [5], “Je le vois mais je ne le crois pas” (che Cantor scrisse in francese in una lettera a Dedekind, a proposito della sua dimostrazione della commensurabilità tra la retta e il piano) [5] e l’Ipotesi del Continuo [5]; la definizione di Peano degli interi in termini insiemistici [5]; il Paradosso di Russell [5]; l’Hotel di Hilbert [4]; il Teorema di incompletezza di Gödel [5] e la morte per inedia [5]; la dimostrazione di Cohen dell’indipendenza dell’Ipotesi del Continuo [5].


中文摘要 (Chinese summary of Chapter 1)


第一章The Veil小摘要, more precisely.  I can’t read that, I’m afraid, even though I’m convinced that the most interesting problem currently facing philosophy of mathematics is to clarify how or whether Chinese and European mathematics differ and how or whether these differences reflect differences in the respective metaphysical traditions.  One of the peculiarities of my own education is that I was exposed to classical Chinese philosophy (in translation!) before I read any of the canonical texts of European philosophy, so perhaps when I did come to the latter my philosophical bearings were already slightly askew.

Anyway, I was pleased to see that a summary — a “little summary,” as my recent Ph.D. student Lin Jie tells me the more precise title should be translated — has appeared on the website of a Chinese bookseller.   Here it is.

史学家Jeremy Gray认为职业自治(professional autonomy)是数学线代主义的标志。前现代的数学家的想象受限于数学-哲学和数学-科学之间的关系。在Gray看来,没有职业自治性,数学的现代转向就不会发生。现代主义之成为主流,是因为它恰当表述了数学家所在的新处境:数学被吸纳到现代研究型大学的结构(从而形成职业数学家的国际群体),数学的目标和主题的新形式得以形成。
如果本书是关于什么的话,那么本书是关于过一种数学家的双重人生是何种感觉:一方面数学家在这种职业自治之中的生活,只需要向同事负责;另一个是更广阔的世界中的生活。要解释数学家到底做什么是一件很难的事情,正如David Mumford所说的:“作为一个职业数学家,我已经习惯于生活在一种真空之中,这种真空的周围是那些以对数学一无所知而自豪的人。”这种困难使得后面的问题不被问出来:什么是数学家的目标?为什么要做数学?
本书并不追求得出定论,而是如 Herbert Mehrtens所说的,以“如何做数学”来说明“数学是什么”

The site also cites this passage from the Preface to MWA, which specifically refers to the cross-cultural comparative study of mathematics.

One of the most exciting trends in history of mathematics  is  the  comparative  study across  cultures,  especially between European (and Near Eastern) mathematics and the  mathematics of East Asia. These studies, which are occasionally (too  rarely) accompanied by no  less exciting comparative philosophy, is  necessarily  cautious  and  painstaking,  because  its  authors  are  trying  to  establish  a reliable basis for future comparisons.

Three mathematicians, three novels, only one movie, part 3



Many followers of this blog have undoubtedly read Daniel Kehlmann’s Measuring the World [Die Vermessung der Welt].  In contrast to the books of Fonseca and Désérable, it has been a major international success, winning too many prestigious awards to list.  Yet it has also attracted the attention and admiration of numerous literary scholars, many of whom nevertheless feel compelled to characterize it as “best-selling.”  “It was on the bestseller lists for weeks on end,” writes literary scholar Nina Engelhardt (in a private e-mail), “even competing with Harry Potter and Dan Brown.  It has also received a lot of critical literary attention and is generally viewed as a successful and innovative example of combining literature and science.”  My German-speaking friends tend to describe Kehlmann as a celebrity, often to be seen on TV talk-shows; he lives in Berlin and Vienna but also holds a visiting professorship at NYU.

Measuring the World devotes alternate chapters to the historical figures of Alexander von Humboldt and our very own Carl Friedrich Gauss, familiar to every German in the widely-circulated portrait reproduced above.  Soon after Kehlmann’s novel was translated into English, Frans Oort published a review in the AMS Notices.  It’s an understatement to say that Oort was disappointed with Kehlmann’s depiction of the Prince of Mathematics.  The review begins with a report of a dream, no doubt fictional:

The young Gauss started to smile, knowing that I recognized him, and remembered this story. Then his face and and figure changed into the beautiful portrait of the young Gauss published in the Astronomische Nachrichte, 1828…. He looked at my desk, and he started to talk to me. “I see that you are reading that book! What can this man mean, slandering me in this way?”… “why does this man have so little appreciation for the deep thoughts engendered in the beautiful things that I encountered and enjoyed in my life? Do you know where I can find this Kehlmann, so that I can explain to him the beauty of my ideas, and the reasons why I set out to measure things?”

It’s one of the most elegantly written and informative reviews I’ve ever read in the Notices, but the book I had just finished left a very different and altogether more positive impression.  So I wrote to Engelhardt, whom I had already consulted in connection with the Pynchon chapter of MWA, in search of clarification.  In her lengthy reply, she agreed with Oort that readers looking for historical accuracy in Measuring the World are likely to be unsatisfied.  But, as she explained (and as already should be clear from the title), that’s precisely the point.  I quote one of the articles* she has recently published on the book:

the humorous tone of the novel, the indirect discourse continuously indicating that events and dialogues are mediated, and the characterization of the eternally grumpy Gauss and an obsessed and naïve Humboldt can leave little doubt that Measuring the World is a work of fiction.

Here Engelhardt inserts a footnote with a reference to Oort’s review, naming at least one reader who failed to detect the telltale signs that Kehlmann’s novel is a specimen of historiographic metafiction.  Her comparative study of Measuring the World and Pynchon’s Mason & Dixon actually suggests that both novels belong as well to the rather different genre of scientific metafiction:

historiographic metafiction contests the accessibility of the past, an epistemological concern that does not challenge the reality of the past, while scientific metafiction problematizes the literally “natural,” namely the nature of the physical world, and thus introduces an ontological dimension.

The telltale signs include the consistent use of indirect speech in the German original, and pointers to Kehlmann’s “epistemological concern” are pretty hard to miss, frankly.  The one on the very first pages could not be more self-referential:

Even a mind like his own, said Gauß, would have been incapable of achieving anything in early human history or on the banks of the Orinoco, whereas in another two hundred years each and every idiot [Dummkopf] would be able to make fun of him and invent the most complete nonsense about his character.

The liberties Kehlmann takes with the empirical historical record — measurable liberties, one might say — are too numerous to mention.  For example, the 11-year-old Gauß discovered the curved geometry of the earth while flying in a hot air balloon with Pilâtre de Rozier, Montgolfier’s associate.  A quick calculation shows that Pilâtre had died several years before Gauss turned 11.  Oort made the calculation, as did the literary scholar Karina von Tippelskirch; yet they draw diametrically opposite conclusions — another illustration of the indeterminacy of measurement.

Tippelskirch reads the chapter entitled The Garden as a simultaneous enactment of reversals of Kafka’s The Castle and the Grand Inquisitor scene from The Brothers Karamazov.  The Humboldt segments are, if anything, even more meta.  Engelhardt:

Humboldt forges his journal when, afraid and refusing to go back into the jungle to shoot a jaguar, he is embarrassed about his actual behavior: “He decided to describe events in his diary the way they should have happened” (90).… it is not even certain whether it is Humboldt or a hallucination who tells his travel companion Bonpland that they “had climbed the highest mountain in the world. That would remain a fact, whatever else happened in their lives.” (152) Humboldt communicates the “fact” to Europe in “two dozen letters” (153), but it is incorrect on two accounts—readers witness that Humboldt and Bonpland have to turn back before reaching the summit and that, with the discovery of the Himalayas, Chimborazo proves not to be the world’s highest mountain.

The meta-sensitive reader is not surprised that in South America Humboldt encounters magic realism — story-telling boatmen named Carlos, Gabriel, Mario, and Julio! — as well as jaguars and crocodiles.  I expect that professionally-trained readers will detect in his travels across Russia a deliberately framing in the idiom of 19th century Russian realism.  So much of literary consequence has been written about Kehlmann’s book, in fact, and so much more will be written, that I will now turn to the question of particular concern to readers of MWA, namely:  how do these three novels depict provers of theorems not as abstractions but as live flesh-and-blood beings; in other words, how do they resolve the mind-body problem that is the topic of Chapter 6 of MWA?  More urgently, how do they contribute, if at all, to the canons of mathematical nudity?
Not at all, as I remember, in Coronel Lágrimas; the Grothendieck/Quijote figure smokes and drinks (too much) but is otherwise barely material at all.  Évariste features a single mystifying nude scene, in which the author undresses the frequently apostrophized but never visible character known as mademoiselle and then has her dress up as Galois in preparation for a fictional but unfulfilling love scene with Stéphanie.
The unwary reader who treats Kehlmann’s book as reliable history, on the other hand, will remember Gauss as quite the ladies’ man.  He visits the whores in Göttingen (not forgetting to think of numbers all the while) but he truly loves his first wife Johanna.  Barely 10 pages after their wedding night she dies in childbirth, in one of the most moving scenes in the book.    It’s the earlier scene, however, that reviews invariably highlight, specifically the moment in which the lovemaking is interrupted by Gauss’s discovery of the least squares method:

er schämte sich daß ihm ausgerechnet in diesem Moment klar wurde, wie man Meßfehler der Planetenbahnen approximativ korrigieren konnte…   weil er fühlte, daß sie erschrak, wartete er einen Moment, dann schlang sie ihre Beine um seinen Körper, doch er bat eine Verzeihung, stand auf, stolperte zum Tisch, tauchte die Feder ein und schrieb, ohne Licht zu machen:  Summe d. Quadr. d. Differenz zw. beob. und berechn. -> Min.

The scene is unlikely, as Oort points out, as well as historically inaccurate; and I haven’t yet figured out the author’s cunning purpose in placing this particular discovery at this particular point of the narrative.  And apparently it wasn’t enough to redeem the movie — I did mention that there was a movie, didn’t I?  A 3-D movie in fact, directed by Detlev Buck, with a screenplay by Buck and Kehlmann, starring Florian David Fitz as Gauß and Albrecht Abraham Schuch as Humboldt, and universally panned by German critics, in spite of an estimated 10 million € budget.   I found that figure on IMDB, where the film rates a miserable 5.7.  There are no reviews at all on Rotten Tomatoes, and I don’t know whether the film was even released to English-speaking audiences.

The trailer is up on YouTube, however, and if you want to add an image of a Gaussian bunda to your private canon of mathematical nudity you will find one at 0:39 (and another bunda a few seconds later).



*‘Scientific Metafiction and Historiographic Metafiction: Measuring Nature and the Past’. Twentieth-Century Rhetorics: Metahistorical Narratives and Scientific Metafictions. Ed. Giuseppe Episcopo. Napoli: Cronopio: 2014. 145–72. In press.