Category Archives: Self-referential

Mathematics, music, philosophy, and Alain Badiou

IRCAM - 1

Panel at IRCAM, June 7, 2019.  Left to right:  François Nicolas, Yves André, Fernando Zalamea.  Alain Badiou is seated in the audience on the left.

To celebrate the publication of the third and final volume of Alain Badiou’s Being and Event trilogy, the organizers of the Paris MAMUPHI seminar — MAthématiques, MUsique, PHIlosophie — devoted a two-day conference at IRCAM (Institut de Recherche et Coordination Acoustique/Musique), under the title L’hypothèse du contemporain.

For the 20th anniversary of the Mamuphi seminar (mathematics-music-philosophy), these encounters are dedicated to L’Immanence des vérités, the latest work by the philosopher Alain Badiou, and, more particularly, to his theory of “works-in-truth”. How are works distinguished from “waste” and, incidentally, “archives”? The final part of the work by Badiou formalizes a limitless alternative to the oppression of finality. These days in June gather together mathematicians, musicians, philosophers, and the author to formulate their own hypothesis in the shadow of their reading of the contemporary in the 21st century.

Yves André invited me as one of the mathematicians, and because of my deep respect for André’s writings about mathematics — and of course for his mathematical work — I was pleased to accept the invitation.

Badiou’s three-volume system is heavily based on set theory and much of the third volume is devoted to the theory of large cardinals, with chapters on ultrafilters, theorems of Scott, Jensen, and Kunen, 0#, and much more.  I have no idea what the upcoming Columbia graduate workshop will make of all this.  My own presentation had nothing to do with set theory; my aim was to explain why Badiou was wrong to hint in his book, in passing, that the mathematics of Andrew Wiles belonged with the “waste,” or at best the “archives.”  You can watch my talk or you can read it (preceded by a couple of pages explaining my misgivings about the theme of the conference).

I have to confess a less highbrow motivation, though.  Here is an excerpt from a review by Stanley Chang of Mathematics without Apologies that appeared in Society, dated June 25, 2018.

Other reviewers, both academics and nonacademics, have quite forcefully deprecated his use of ideas without context, the irrelevancy of various sections, an unreadably poor organization, and a purposely opaque stream-of-consciousness that prohibits understand [sic] rather than encourages it. One of my own friends, an anthropologist in academia, laughingly said that his treatment of Badiou is something that you would expect from a bad first-year philosophy essay from a bad student at a bad university.

Although Chang gave excellent reasons for his evident dislike of the book, he went out of his way to give it a fair reading, and I have no problem with his review.  But why did he make up this part about Badiou?  MWA contains no “treatment of Badiou.”  According to the index, Badiou’s name appears three times, and only in endnotes.  Two of the references are direct quotations, without anything that can be construed as a “treatment,” and the third quotes Juliet Flower MacCannell’s comments on a quotation by Badiou regarding Lacan’s theory of love, along with Vladimir Tasić’s gloss on the quotation and the comments.

In the Q&A following my talk in Paris I got a laugh from Badiou by suggesting that the mere mention of his name would provoke the laughter of many American philosophers, not to mention anthropologists.  But I don’t think that explains Chang’s sentence.  Maybe he was confusing Badiou with Bourdieu?  Or maybe the treatment in question was on this blog, for example here?

I would fault the editors of Society for allowing the publication of that last sentence, or any sentence, on any subject whatsoever, that quotes an anonymous anthropologist — laughing no less — for the sole purpose of taking a cheap shot.  But in fact I have no idea what the sentence is about.

Short proofs

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

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

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

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

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

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

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

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

My guess — but it’s no better than anyone else’s — is that he decided that the priorities of Silicon Valley are just not compatible with those of the mathematical community, as he sees it.

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

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

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

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

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

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.

 

Jazz

Alexander-Jazz-of copy

Nearly three months have passed since I had the privilege of sharing the stage with Stephon Alexander at Book Culture, near Columbia.  MWA had been out for over a year, but I had put off reporting on the (very moderately attended!) event until Alexander’s book was available.  Alexander is both an accomplished theoretical physicist (“specializing,” as the event blurb indicates, “in the interface between cosmology, particle physics and quantum gravity”) and a respected jazz saxophonist.  “Respected” meaning:  when he walks into a downtown jazz club, the owner comes out to greet him.  

The Jazz of Physics is a fascinating read, as I’ll let you discover for yourselves.  Or perhaps you have already discovered the book; as of this writing , it is listed on amazon.com as #1 best-seller in quantum physics AND #2 best-seller in jazz, which must be a first.  Of course Alexander had to overcome the first obstacle that faces the author of any popular science book, namely:  when communicating ideas that only a few specialists really understand (and even then imperfectly and provisionally), how to draw the line between making them accessible and making them trivial?   Alexander uses jazz, and music more generally, as the basis for a series of increasingly complex and precise analogies with physics, especially his own work on the quantum mechanics of the early universe.  It works — readers and reviewers seem to be happy with the results — but I want to suggest that jazz is not merely used as a metaphor in this book.  If I understand the conclusion correctly, by the end Alexander is suggesting, plausibly, that the structure of the universe is itself improvisational, so that jazz turns out to be a surprisingly effective (even “unreasonably effective”) route to understanding cosmology.

I’ll leave the speculation at that.  When I was putting together material on the attitudes of musicians to mathematics, I did not search systematically but rather collected enough examples to establish what seemed to me general patterns, to wit:  classical musicians and rockers for the most part refused to acknowledge an affinity with mathematics, but African-American popular musicians — especially in rap and techno — seemed to hold mathematics in high regard.  (I met Alexander when I was putting this together and he gave me a few precious tips.)  I was frustrated to have found no meaningful material on the relations of jazz musicians to mathematics, but not frustrated enough to explore the question in a scholarly manner.

Alexander’s book doesn’t settle the question, but he does establish that some of the biggest names in jazz were seriously interested in physics.  He mentions Ornette Coleman, John Coltrane, and Yusef Lateef:

About a decade ago, I sat alone in a dim café on the main drag of Amherst, Massachusetts, preparing for a physics faculty job presentation when an urge hit me. I found a pay phone with a local phone book and mustered up the courage to call Yusef Lateef, a legendary jazz musician, who had recently retired from the music department of the University of Massachusetts, Amherst. I had something I had to tell him.…

“Hello?” a male voice finally answered.
“Hi, is Professor Lateef available?” I asked.
“Professor Lateef is not here,” said the voice, flatly.
“Could I leave him a message about the diagram that John Coltrane gave him as a birthday gift in ’67? I think I figured out what it means.”

There was a long pause. “Professor Lateef is here.”

The diagram is pictured in the Introduction to The Jazz of Physics, with the helpful caption “any other reproduction is prohibited.”  So you will have to read the book if you want to see what Alexander and Lateef had to say to each other.

The theologico-teleological apology

Schreiber

Comments on David Roberts’s Google+ page, May 30, 2016

David Roberts’s announcement a few months ago of his then-forthcoming review in the Gazette of the Australian Mathematical Society sounded like a warning shot, especially since I occasionally had the impression that he was trying to bait me on this blog.   The review is now out, and as far as I’m concerned it’s perfectly fair; the reviewer was even thoughtful enough to include what trade jargon calls a pull quote in the last paragraph, and you can expect to see it soon enough on the reviews page.

The review also provides (yet another!) opportunity to clear up some misconceptions, notably about charisma, as used in chapter 2.  I chose the word deliberately as a provocation, but it provokes different readers in different directions, and that’s beyond the author’s control.  The ambiguity of the word is already in Weber, it seems to me:  the charismatic leader is separated from the masses by an aura, while those possessed of routinized charisma are part of the mass of functionaries that make the community… function.  I tried to make it clear that chapter 2 was the (fictionalized) story of my acquisition of routinized charisma, in other words, of being accepted as a legitimate functioning member of the community.  So when Roberts writes

The ‘relaxed field’ that Harris discusses … is perhaps not the same for us as for those with charisma.

he is making a distinction that is quite alien to the spirit of the book; indeed, Roberts is displaying a paradigmatic form of charisma by publishing a book review in the Gazette of his learned society, and more consistently in his contributions to MathOverflow and other social media.

By the way, saying that chapter 2 was fictionalized is not the same as saying that it was made up; what I meant was, first, that it was written in acknowledgment of the narrative conventions of (a certain kind of) fiction; and that it didn’t matter for my purposes whether or not the events recounted were strictly true, as long as they were ideal-typical.

Roberts reads MWA as calling charisma a form of prestige whose acquisition is one of the motivations for doing mathematics, but this was not my intention.  No doubt mathematicians find it gratifying when our work is recognized, and much of the mass of chapter 2 is devoted to prizes and other forms of recognition, large and small, institutionalized or informal; but only André Weil is represented as actually craving prestige, and the context makes him recognizably an outlier.  An obsession with ordered lists and rosters of Giants and Supergiants is attributed to the community, rather than to individual mathematicians who hunger for recognition.  This obsession is such a visible feature of contemporary mathematics that it deserves explanation, and chapter 2 suggests an explanation that is so counter-intuitive that it seems not to have been noticed by anyone (on pp. 18-19):

The bearer of mathematical charisma… contributes to producing the objectification—the reality—of the discipline, in the process producing or imposing the objectification of his or her own position within the discipline.…The symbolic infrastructure of mathematical charisma is… the “objectification” of mathematics:  the common object to which researchers refer… In other words, it’s not just a theory’s contents that are defined by a social understanding:  so are the value judgments that organize these contents.

This brings me to Urs Schreiber’s instructive misreading of MWA‘s intentions, quoted above.  Most likely it’s a misreading based on no reading at all of MWA, because he seems not to be aware that the words “meaning” and “reality” that he cites as the aims of a self-aware mathematician are examined repeatedly in MWA, especially in chapters 2, 3, and 7.

Chapter 3 refers to three main forms of “apologies” for mathematics, labelled in keeping with the western philosophical tradition as “good, true, and beautiful.”  The word “tradition” is fundamental.  The one thing I find unforgivable when mathematicians make general comments about the values and aims of mathematics is the suggestion that they are saying something original.  Talk of values and aims is necessarily embedded in a philosophical and literary and social tradition; a failure to acknowledge this is merely a sign of ignorance, not of intellectual independence.  THAT is why MWA has nearly 70 pages of endnotes and more than 20 pages of references:  in order to record the author’s efforts to purge himself of the notion that his ideas are his own — and, no doubt, to encourage others to take the same path.

MWA cites those three main forms of “apologies” because they are the ones actually on offer; writing about them is my way of grappling with “reality.”  I attended the meetings described in chapter 10 not out of masochism (the champagne receptions were not bad at all) but because they were really happening, they were organized and attended by real decision-makers (“Powerful Beings”) whose decisions have real consequences for the future of the discipline; and the representations of mathematics (and of scientific research more generally) presented at those meetings were the real attempt of the community to procure the external goods necessary for its survival in its present form.  (I procured no pleasure, not even Schadenfreude, when I read the documents listed in the bibliography under “European commission”; but they are terribly important for anyone who is concerned about the future of mathematics.)

Anyway, Schreiber’s speculations cited above are irrelevant to MWA, but they are instructive nevertheless, because they exemplify what might be considered a fourth kind of apology that might be called Theologico-teleological.  One doesn’t need to believe in a supreme being to be a seeker of “answers to deep questions” or “meaning” or “reality,” but one has to believe in something.  I don’t know how to attach consistent meanings to the terms in quotation marks in the last sentence, and I don’t think Schreiber does either.  But I do know one name that has been given to the process by which meanings accumulate around terms like that:  tradition-based practice, specifically in the writings of Alasdair MacIntyre.  Two separate texts, both cited in the bibliography, led me to MacIntyre:  David Corfield’s article Narrative and the Rationality of Mathematics Practice and Robert Bellah’s book Religion in Human Evolution, which I read at the suggestion of Yang Xiao.  Both texts propose ethical readings of important human social phenomena, and this is important to me, because I have found that most arguments about the nature of mathematics, including Schreiber’s comments, turn out to be ethical arguments in disguise.

(Like “beauty,” the “answers to deep questions” or “meaning” or “reality” that Schreiber appears to be seeking can also be interpreted as euphemisms for “pleasure,” but I will leave this for another occasion.)