Category Archives: Alternative readings

Reuben Hersh, 1927-2020 (with text)

Here is the introduction to my article Do Mathematicians Have Responsibilities, published in Humanizing mathematics and its philosophy.Essays celebrating the 90th birthday of Reuben Hersh.Edited by Bharath Sriraman. Birkhäuser/Springer, Cham (2017) 115-123.

I have been an admirer of Reuben Hersh ever since I received a copy of The Mathematical Experience, then brand new, as a birthday present.  At that stage, of course, I was admiring the tandem Reuben formed then, and on other occasions, with his co-author Philip J. Davis.  It was only almost 20 years later, after I started reading What is Mathematics, Really? that I could focus my admiration on Reuben — and not only on the mathematician, the author, the thinker about mathematics, but on the person Reuben Hersh — the unmistakable and unforgettable voice that accompanies the reader from the beginning to the end of the book.  So unforgettable was the voice, in fact, that when Reuben, wrote to me out of the blue three years ago to ask me what I thought about a certain French philosopher, I so clearly heard the voice of the narrator of What is Mathematics, Really? (and no doubt of many of the passages of his books with Davis) that I could honestly write back that I felt like I had known him for decades, though we have never met and until that time we had never exchanged a single word.

The voice in question is the voice of an author who is struggling to put words on an intense and intensely felt experience, who has intimate knowledge of how it feels to be a mathematician and also a knowledge no less intimate of the inadequacy of the language of our philosophical tradition to do justice to that experience, so that all attempts to do so inevitably end in failure; but this knowledge is compensated by the conviction that the stakes are so important that we can’t choose not to try.   What makes Reuben’s authorial voice compelling is that it sounds just as we expect the voice of a person in the middle of that struggle must sound.[1]   It’s the strength of this conviction that comes across in Reuben’s writing, so that reading his books and essays is remembered (by me, at least) as a conversation, a very lively conversation, filled with the passionate sense that we are talking about something that matters.  Also filled with disagreements — because I don’t always agree with everything I read in Reuben’s books and essays; beyond questions of detail the difference might come down to my sense that Reuben is trying to get to the bottom of the mathematical experience, whereas I apprehend the experience as bottomless; or I might say that it’s the effort to get to its bottom that is at the bottom of the experience.  But the differences are of little moment; what stays with me after reading a few pages of Reuben’s writing is the wholeness of the human being reflected in his words, a human being who cares so deeply about his mathematical calling that he is ready to add his own heroic failure to the long list of admirable failures by the most eminent philosophers of the western tradition to account for mathematics; and without these inevitable failures we would not begin to understand why it does matter to us.

[1] As I wrote that sentence I remembered that I have still not met Reuben, nor have I ever spoken to him; but I checked one of the videos online in which he appears and, sure enough, his literal voice is very much as I expected.

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.”

Time to move on

wainua               Figure 6.1 (Clairaut's diagram)

Snail image:  Creative Commons licence courtesy of Te Papa; Clairaut’s love formula from Chapter 6 of MWA

My tireless editor Vickie Kearn at Princeton University Press has brought me the welcome news that Mathematics without Apologies will be coming out in a paperback edition next spring.   I started this blog for two reasons, and one of them — to clarify my intentions in writing the book — will vanish when I add two or three pages to the preface of the new edition.   The new pages — I have already written them — will devote one paragraph or so to each of four topics, provisionally under the headings charismamemoirsutility, and ethics; each paragraph will address some of the points raised by comments on this blog as well as in some of the more negative reviews.

My other reason  for starting this blog was to find some outlet for the wealth of material that I was not able to incorporate in the book.  Most of this material has remained untapped while I composed comments on current events or new findings, and I was idly wondering when I would get around to sifting through the 7 GB  or so that is gathering nanodust on my computer’s hard drive.  My Eureka! moment came when I realized that I had already devoted a considerable amount of my free time to writing the book during the better part of three years.  Perhaps I didn’t really want to return to the old material?  With the new preface, I can finally declare the book finished and move on to something else.

Will it be another book, maybe one that will win me the mythical seven figure advance?  Or will there be another blog, or the same one under another name?   That’s for the future to decide.  Meanwhile, this one will remain visible, but with no new entries.

My thanks to the regular readers and occasional visitors who helped keep the blog from slipping into solipsism.  And my special thanks to authors of comments who, by disagreeing, often sharply, with opinions expressed here, demonstrated that the meaning of mathematics is still a matter of controversy.

This was supposed to be the last entry, but I’m now thinking I should include part of the new preface material — or all of it, if PUP allows it.  Meanwhile, in order not to let anything go to waste, here is the post on which I was working when I realized that this blog had reached the end of its natural life…

I Cunfirenti

This was originally going to be an appendix to the playlist near the end of Chapter 8:  an exploration of the attitude to mathematics in the genre of organized crime ballads.  The deeper meaning of Rick Ross’s 2009 single Mafia Music was exposed even before it was released,  but I was unable to find an interpretation of the unexpected appearance of mathematics in the middle of this rap à clef:

I thought about my future and the loops I could pin.
Walked out on a gig and I turned to da streets,
Kept my name low key, I ain’t heard from in weeks.
I came up with a strategy to come up mathematically,
I did it for da city but now everybody mad at me.

Apart from Rick Ross, Gödel is the only person Google finds who can “come up mathematically.”  My guess is that Ross’s strategy (unlike Gödel’s) involves money.  But Ross is not really a gangster, and Mafia Music is not really a mafia song at all; in fact, by naming names the song breaks what I’m told is the most fundamental of all the rules of the Italian Malavita, namely the rule of omertà, the iron law of silence.

Now it struck me when I saw this that the mathematical profession has its own version of omertà, probably not very different from other forms of academic rules of silence, having to do with forms of behavior that straddle the line that divides the unpleasant from the unethical.  The behavior protected by mathematical omertà differs from other varieties in that it tends to inspire less literary commentary.  Instead it consists in scandalous rumors whispered in corridors when they are not being shouted across barroom tables, but that must under no circumstances be mentioned in public.  (There was a scurrilous exception in a well known literary magazine a few years ago, but I will not dignify it with a link.)

I am particularly sensitive to this rule just now, because in the past few weeks I was shocked to learn of abuse of power by several colleagues I would not have believed capable of such behavior (and by a few others I can easily believe capable of anything).  Whether being the repository of such confidences is one of the perks of my charisma, or whether it’s the abusers who feel newly entitled as a result of their own charisma, the mildest punishment I could expect if I chose to betray the dark secrets of the mathematical profession is not to be privy to such secrets in the future.  Breach of Mafia omertà is treated more harshly than that.  Many of the songs on the delightful album La Musica della Mafia are devoted to the many kinds of punishment the gangster ethic  —

Laws that don’t forgive those/Who break their silence

reserves for traitors — cunfirenti, in Calabrian dialect.  For example, the song entitled I cunfirenti promises that they will find “their final resting place in concrete walls” (‘Mpastati ccu cimentu e poi murati).

The album’s title is imprecise; it’s not a collection of songs of the Sicilian mafia but rather the ballads of their Calabrian declension, the ‘Ndrangheta, who deserve to be better known, and not only for their songs:

Its success at drug smuggling catapulted the ‘Ndrangheta past its more storied Sicilian rival, the Cosa Nostra, in both wealth and power. Italian authorities now consider the ‘Ndrangheta to be Europe’s single biggest importer of cocaine.

What I find most charming about this collection is the contrast between the lively rhythms of many of the songs and the uniformly grim, often bloody, content of the lyrics.  For example:

Malavita, malavita
Appartegnu all’Onorata
Puru si c’impizzu a vita
Eu nun fazzu na sgarrata

Which means

Malavita, malavita!
I am one of the honorable society.
And even if it costs me my life,
I will never surrender.

If you’re looking for mathematical content you have to skip to the last verse:

Ed eo chi tingu sangu ´nta li vini
Su prontu d’affruntari mille infami
A chista genti ci rispunnimu
Pidi sunu pronti centu lami

Which means

And I who have blood flowing through my veins
Am ready to face 1000 traitors
As they know all too well
That 100 sharpened knives are ready for them.

Is it common knowledge that anyone is fit to be US President?


A few weeks ago, Terry Tao used Donald Trump’s perceived lack of qualification for the presidency to illustrate the difference between mutual knowledge and common knowledge, in a blog post with the normative title It ought to be common knowledge that Donald Trump is not fit for the presidency of the United States of America.  It’s common knowledge that Terry Tao, in addition to being one of the Mozarts of mathematics, is a very sensible person, and like every sensible person he is appalled by the prospect of Trump’s election as president.  As an attempt to account for this unwelcome prospect, Tao suggested that the correctness of Proposition 1 above is a matter of mutual knowledge  —

information that everyone (or almost everyone) knows

but not (or not yet) common knowledge

something that (almost) everyone knows that everyone else knows (and that everyone knows that everyone else knows that everyone else knows, and so forth).

It seems to me, though, that Tao’s formulation of the question — whether Trump is “fit for the presidency” or, in the words of Proposition 1, is “even remotely qualified” — is ambiguous.  The only axiomatic answer is the one provided by Article II, Section 1 of the U.S. Constitution, which implies unequivocally that Trump, like me but (unfortunately) unlike Tao, is indeed “eligible to the office of President” — though I admit I haven’t seen his birth certificate — and eligible is here the only word that is unambiguous and legally binding.

Now I realize that, even if you are a mathematician and therefore legally or at least professionally bound to respect the axiomatic method, you will object (at least I hope you will) that Tao did not mean to suggest that Trump’s bare eligibility was in question, but rather that Trump did not meet the more stringent criteria of fitness or even remote qualification.  By analogy, no one would deny that  ø (the empty set) is eligible to be a set, according to the usual axioms of set theory, but rather that

  1. ø is hardly anyone’s favorite set;
  2. ø is in no sense a paradigmatic set; and
  3. ø is not the kind of set for which set theory was designed.

Thus, even if it were mutual or even common knowledge that Trump is, so to speak, the empty set of American politics, that would hardly count as a consensus on his fitness or even remote qualification.  I’m naturally sympathetic to this kind of argument, but Tao made it clear that only comments that

directly address the validity or epistemological status of Proposition 1

were eligible for consideration on his blog.  While I’m hardly a strict constructionist, I don’t see how to avoid interpreting the word epistemological in terms of the maximal epistemological framework I share with Tao, which in this case can only be Article II, Section 1 (together with the Zermelo-Fraenkel axioms, but I doubt they are of much help here).

I was already leaning to a different explanation of the Trump phenomenon before offered this helpful but depressing roster of the worst (and best) presidents in the history of the United States, according to (unspecified) “scholars.”  Running down the list, one sees that, although Barack Obama is undoubtedly one of the most fit of all the presidents, intellectually as well as academically speaking, he only shows up near the middle of the ranking.  Presumably this is because he has been less effective as a politician than the presidents at the top of the list.  Judging by his words, I would like to say that Obama is one of the most morally fit of the presidents on the list; judging by his deeds, on the other hand — these, for example, or these — the record is much less appealing.  Jimmy Carter has proved to be both intellectually and morally admirable since leaving the presidency, but he made two of the biggest foreign policy blunders in recent history while in office (he ranks quite poorly on the list, probably for different reasons).

It is clearly mutual knowledge that the notion of fitness to lead a modern democracy, in particular fitness for the presidency of the USA ,correlates strongly with a shocking disdain for the notion that elections are designed to reflect the popular will.   My sense is that Trump’s supporters, and their counterparts across Europe, would like this to be common knowledge.  Fortunately, they are not the only ones.

This will be the next-to-last post for the summer; the next post will explain why it may be time to put this blog to rest permanently.


Intelligent design — via higher categories?

Following a tip from [REDACTED] I discovered that my Paris colleague Pierre Schapira has translated and published an introductory article on sets, categories,  and higher categories, on an anonymously edited website called Inference and subtitled International Review of Science.  The caricature of Schapira that accompanies the article is terrible but the article is well-written — not surprisingly — and a logical next step for anyone (a literary theorist, for instance) whose first exposure to categories came in Jim Holt’s review of MWA.  The hypothetical literary theorist, if lazy, will not find it easy reading; Schapira prefaces the article with a dedication that is also a warning:

This essay is written for people like Gilles [Châtelet] open to new languages and new ideas—needless to say, not so many people, and not especially mathematicians.

This blog will undoubtedly get around to writing about Gilles Châtelet, the philosopher, gay activist, and professor of mathematics.  For the moment I will just mention that this “singular personality,” as Schapira calls him, was widely admired by people who otherwise would probably not agree on much of anything, especially for his book Les enjeux du mobile (available — barely — in translation as Figuring Space), which is quoted several times in MWA.  Schapira’s essay is much more demanding than the few paragraphs MWA devotes to categories but the determined literary theorist or social scientist will certain find it rewarding to read, and even well-informed computer scientists may find some passages enlightening.

The last section of Schapira’s essay bears the title Towards the Revolution, and the “major conceptual revolution” he has in mind is Voevodsky’s univalent foundations, much discussed on this blog.

In this new setting, the ∞-categories would become the usual categories, but in a totally new set-theoretical context. This would be a major conceptual revolution. But this idea of homotopy can also be found at the origin of another, much more tangible, revolution that will eventually allow for the validity of proofs to be checked using computers.

I find that last sentence surprising — in the context of Schapira’s essay, that is, not in reference to Voevodsky — and the next time I see Schapira I’ll have to ask him what he imagines will happen to pre-revolutionary mathematics if and when this comes to pass.  Not to mention counter-revolutionary mathematicians.  Because while Inference is reasonably eclectic — among its mathematical entries are a highly readable introduction by Gregory Chaitin to his thoughts on set theory, and a reprint of one of Pierre Cartier’s excellent articles on Grothendieck, it attracted some attention early on with a 3-part series by counter-evolutionary Michael Denton, called, appropriately enough, Evolution: A Theory in Crisis Revisited.  This series has been welcomed by the Intelligent Design crowd, not so much, apparently, by anyone else.  And this, in turn, has fueled speculation about the mind behind Inference.  The leading theory (or maybe the only theory, because it’s not clear how many people are paying attention) seems to be that the new online journal is the brainchild of the prolific David Berlinskiwho, among his other accomplishments, managed the difficult feat of making Richard Dawkins look sympathetic.

Whoever Inference‘s intelligent designer may be, he or she (or possibly an object of some alternative category) also appears to be an enthusiast for hidden variables theories in quantum mechanics.  The first issue reprinted an epistolary exchange on this topic between Sheldon Goldstein and Steven Weinberg, that dates back to 1996, when the Science Wars raged most furiously.   Bohmian mechanics may be one of the hidden variables behind the Inference editorial board, as it was behind the Science Wars themselves; Gross and Levitt, the authors of Higher Superstition, referred favorably (in one of their later publications, if I’m not mistaken) to an article by Dürr, Goldstein, and Zanghi (same Goldstein, and I think it was this article), and Jean Bricmont, author (with Alan Sokal) of Fashionable Nonsense, was also apparently sympathetic to hidden variables theories at the time, though not, I take it, sufficiently sympathetic to commit himself to one version or another.

I don’t know enough about quantum mechanics to have any opinion whatsoever about hidden variables, but I have observed that the main motivation of supporters of hidden variables theories seems to be the desire for objectivity — Goldstein writes that a physical theory

…should not involve any subjective notions in its very formulation, nor should it involve axioms concerned with measurement, since the notion of measurement is much too vague…

To my mind, a striking aspect of the “major conceptual revolution” that was category theory is that it radically calls into question the centrality of objectivity in philosophy of mathematics; that, at least, is the point of view of Chapter 7 of MWA.  It may be that Schapira sees the univalent axiom as a way to restore objectivity, and that would be a major conceptual revolution; but this is mere speculation.  Inference is eclectic enough not to require consistency with regard to such matters from one article to the next.


“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].


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.

Automated letter writing

to whom

Now back in New York after a quick visit to Harvard, having spent the way up writing letters of recommendation and nomination and the way back down writing more letters of recommendation and nomination, when I wasn’t wondering whether or not a train that rattled and shook so much could possibly have been designed by an artificial intelligence, it occurred to me to wonder, in addition, whether or not an artificial intelligence could possibly be programmed to write all these letters of recommendation and nomination.  This is also a response to this comment on an earlier post.  The comment claimed that a musician “has already written the answer” to one of my questions about artificial music performance.  I’m sure that it was not written by a musician because it has two authors, and I’m not sure it’s the answer, but it’s certainly an interesting and thought-provoking answer, and it inspired me, as I said, to wonder whether another person (composed of two or more people) would already have been inclined to write the answer to this question about artificial letter writing.  Or whether a machine would write a letter — to me, for example — answering my question.

I put a fair amount of time into writing letters of recommendation (and nomination) because I generally like the people I’m recommending (and nominating), which is very different from “liking” them.  A machine could easily find their social media footprints and “like” them but can it like them?  And if not, can its letters really substitute for letters written by a human?  After all, the department or institution that is considering hiring (or promoting, or rewarding) the candidate should be indifferent to the candidate’s non-objectifiable personal qualities and should only pay attention to facts — like citation statistics, for example.  And it turns out that machines are pretty good at reporting facts and even turning them into stories that are enjoyable, or at least “like”-able.  A machine willing to look beyond the mere citation indices and to explore the citations themselves will be able to write “the candidate’s work has been called interesting and thought-provoking” and every word of that will be objectively true.

Of course, having stepped this far out on a limb, one has to entertain the possibility that the entity doing the citing is (itself? themself?) a machine, and if you haven’t actually met the candidate but are only familiar with the candidate’s work — which is more likely than not if you are a letter-writing machine — then you have to be willing to imagine that the candidate may well also be a machine.   Whether I would have been able to follow this line of thinking to its logical conclusion if the Acela train had not made me seasick, or whether the line of thinking is itself a symptom of my seasickness, is a matter only the reader can judge, and then only if the reader knows what seasickness is like, and whether or not a machine has been invented that can experience seasickness, or whether it would even be ethically permissible to invent such a machine, are not questions I can begin to answer.

Red-blooded programmers with deadlines and bosses

The title “Category theory Lulz” grabbed my attention, especially in view of the (possibly obsolete) NY Times definition of “lulz”:

the joy of disrupting another’s emotional equilibrium…

Can that really be done with category theory, and how does the joy it procures compare with the pleasure of a non sequitur?

The answer is in Ken Scambler’s two-hour course on YouTube.  It’s probably a sign of the times that this and similar courses were posted at roughly the same time as the long exchange on HoTT and Univalent Foundations was taking place on this blog, and that Bartosz Milewski decided just over a year ago to write a book, apparently one online chapter at a time, entitled Category Theory for Programmers, even though functional programming has been using category theory for decades.  But it’s no longer only relevant to theoretical computer scientists.  Over the last few years, especially, “red-blooded programmers with deadlines and bosses,” as Scambler puts it, have been reading articles on whether “category theory [will] make you a better programmer” and tuning in to online courses like Scambler’s.

It can’t be a coincidence that so many computer scientists are attending year-long or semester-long programs on Univalent Foundations and/or HoTT while a growing number of programmers are trying to learn category theory.  Milewski himself mentions Voevodsky briefly in his fourteenth chapter:

For all intents and purposes isomorphic sets are identical. Before I summon the wrath of foundational mathematicians, let me explain that the distinction between equality and isomorphism is of fundamental importance. In fact it is one of the main concerns of the latest branch of mathematics, the Homotopy Type Theory (HoTT). I’m mentioning HoTT because it’s a pure mathematical theory that takes inspiration from computation, and one of its main proponents, Vladimir Voevodsky, had a major epiphany while studying the Coq theorem prover. The interaction between mathematics and programming goes both ways.

I won’t speculate on the reasons for this synchronicity because I don’t know anything about programming and I discovered this literature less than an hour ago.  But I’m ready to believe that the interaction goes both ways and I suspect I’ll learn more about programming by watching Scambler’s course and reading Milewski’s book than I would by following the traditional route.

P.S.  Confusion goes both ways too.  The “red-blooded programmers” seem to be having trouble with the notation g°f for composition at around 13’40” of Scambler’s course:  why is the g on the left when the arrow representing g is on the right?  Why indeed?