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

patiente

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 théoricien.ne.s.  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.

mathimatika-horis-apologies-9786185289188-200-1303183

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.

Greek

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?

mutualknowledge

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 fivethirtyeight.com 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”

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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].