Category Archives: Self-referential

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.)

The seven figure advance

It’s not a myth.  Two people I know reported this spring that they were offered, and accepted, advances for forthcoming books in excess of $1,000,000.  The authors are serious and knowledgeable people and their books will be informative and will probably sell pretty well.  After their agents take their cuts, they will be able to live comfortably in New York City for several years while planning their next projects.

But will the publishers recover their investment?  [The contributor of this comment will want to skip the rest of this paragraph.]  They will undoubtedly try.  The authors both warned me that accepting a sizable advance entails surrendering control to the publisher of the final product, as well as much of one’s free time.  There will be unwelcome editorial changes as well as a grueling schedule of book tours, often with wine and cheese.   But a quick calculation (not taking into account eventual sales of merchandise, video games, TV contracts, ringtones, and secret revenue streams that the profession has yet to reveal to me) suggests that the publisher will have to count on six figure sales to recover a seven figure advance.  How realistic is that?

Not very, according to Lynn Neary, reporting last September on Weekend Edition:

So what is a good sales figure for any book?

“A sensational sale would be about 25,000 copies,” says literary agent Jane Dystel. “Even 15,000 would be a strong enough sale to get the publisher’s attention for the author for a second book.”

But if that second book doesn’t sell, says Dystel, odds are you won’t get another chance.

By this measure, Eugenia Cheng’s How to Bake π: An Edible Exploration of the Mathematics of Mathematics has been a sensational success, according to a richly illustrated feature article that appeared in the New York Times two weeks ago.   The Times reports that Cheng’s book has sold just about 25000 copies in the US alone, not counting translations.   As for other aspiring writers, Neary’s news is not good.

Just over 1,400 full- and part-time writers took part in the [2015 Authors Guild survey], the Guild’s first since 2009. There has been a 30 percent decline in author income since then and more than half of the respondents earned less than $11,670 (the 2014 federal poverty level) from their writing related income.

“No one likes to see the word ‘poverty level’ on a survey that has anything to do with people you know,” says Roxana Robinson, president of the Authors Guild. “You used to be able to make an absolutely living wage as a writer. You wrote essays and you published them in journals. You wrote magazine pieces and you got paid very well for those. And you wrote books and you got good advances. So being a writer, it didn’t usually mean you would be rich, but it had meant in the past that you could support yourself.”

What happened?

The Authors Guild blames the decline in writers’ income on a combination of factors: online piracy of digital material, consolidation within the publishing industry which has led to more focus on the bottom line, the dominance of Amazon and the rise of self-publishing which has cut into the market for traditional publishers.

The Times portrait of Cheng was charming but mathematically slight.  Steve Strogatz had a cameo, which is not so surprising, since he’s well-known to Times readers for his Joy of X columns, and maybe also because he has managed the exceedingly rare feat of writing successful popular books about mathematics with real content.  More surprisingly, there is also a quote from John Baez that gives the reader only the slightest hint of his legendary exuberance:

“Eugenia has gone all the way in,” he said. “She’s trying to explain math to everybody, with or without pre-existing expertise, and I think she’s doing wonderfully.”

The Times author, it’s safe to say, did not go “all the way in.”  There is Cheng’s helpful illustration of associativity with a custard recipe.

You must first combine the sugar and egg yolks and whisk them into a froth before you pour in the cream.

Blend the ingredients in a different order, [Cheng] said, “and you end up with a runny mess.”

To illustrate mathematics, on the other hand, the author chose a problem from elementary algebra; hardly the Baez-Dolan Cobordism Hypothesis.

Now for the question on all your lips:  how is MWA doing, on literary agent Jane Dystel’s scale?  Well, it has a long way to go before anyone will call its sales “sensational,” and it may never get there.  But the book is in its second printing, Princeton University Press has more than recovered its (modest!) advance, and discussions of motion picture rights — as it says in my contract

(b) For the licensing of dramatization, public reading, radio, video, sound recording, and motion picture rights, you will receive 75% of the net amounts received by the Press. —

have yet to begin.

Another thing wrong with MWA : an occasional series

Originally (in the summer of 2015, that is) I was planning to make a single long list of flaws in the book.  But this obviously didn’t happen, so I will just continue to list flaws when I think of them, as I already did a few weeks ago.

Item:  The author’s literature search left huge gaps.  For example, Ian Stewart published an unusually clear and concise explanation of the Black-Scholes equation and its role in the 2008 financial crisis in the February 11, 2012 issue of the Guardian.  Two months later, when I began writing Chapter 4, Tim Harford posted an article quoting Stewart on the BBC website.   This article was cited in note 36 to Chapter 4, and you can check that Stewart’s name appears, without explanation.  Did the author go to the trouble of consulting the original source?  No, he didn’t; if he had he would certainly have included the excellent concluding paragraph of Stewart’s article:

Despite its supposed expertise, the financial sector performs no better than random guesswork. The stock market has spent 20 years going nowhere. The system is too complex to be run on error-strewn hunches and gut feelings, but current mathematical models don’t represent reality adequately. The entire system is poorly understood and dangerously unstable. The world economy desperately needs a radical overhaul and that requires more mathematics, not less. It may be rocket science, but magic it’s not.

How MWA promotes the (oppressive?) hierarchy

LanglandsNotreDame

From Langlands, Is there beauty in mathematical theories?, Lectures at Notre Dame University, January 2010, published in The Many Faces of Beauty, Vittorio Hösle, ed.

Finally I can begin to fulfill the promise I made last August to point out a few of the things that are really wrong with Mathematics without Apologies.   The diagram reproduced above is taken from an article Langlands wrote when the editor invited him to contribute an article on mathematical beauty to the volume cited above, with the title The Many Faces of Beauty.  (In the summer of 2011 I was invited to contribute an article on mathematical beauty to an issue of the Portland-based literary journal Tin House with the title Beauty.  So Langlands and I have at least that in common.)   The diagram lists  “the names of some of the better-known creators of the concepts” that contributed to the solution by Andrew Wiles of Fermat’s Last Theorem, which Langlands chooses as the starting point of his account of the theory of algebraic equations and, ultimately, automorphic forms.

One will have noticed that all the names belong to men, practically all of them European.  This is a problematic feature of the hierarchical nature of mathematics, but it’s not my topic today.  The question instead is Who speaks for mathematics? which (it seems to me) is at least implicit in Piper Harron’s identification of the oppressiveness of mathematical hierarchy.  Much to my regret, MWA did not break with the convention of quoting the reflections of Giants and Supergiants and those most visible among our contemporary colleagues, the people whose names appear in lists and diagrams like the one copied above.   Thus the question Who speaks for mathematics? is answered by pointing to those who occupy the most prominent positions in the (oppressive?) hierarchy.

This is in part due to the unsystematic nature of my research and in part due to structural features of the hierarchy, which I emphasize on p. 39, in what I have already identified as the key passage in Chapter 2:

We’ll see throughout the book  quotations by Giants and Supergiants in which they conflate their own private  opinions and feelings with the norms and values of mathematical research,  seemingly unaware that the latter might benefit from more systematic  examination.  One of the premises of this chapter is that the generous licence  granted hieratic figures is of epistemological as well as ethical import.

My own experiments with the expression of what appear to be my private opinions resemble this model only superficially and only because they conform  to the prevailing model for writing about mathematics.  My friend’s point was  that even my modest level of charisma entitles me not only to say in public  whatever nonsense comes into my head…

In other words, it’s much easier to be quoted if you have published your thoughts in the first place, and it’s much easier to get your thoughts published if you are identified as a consequential mathematician.  I don’t know how to overcome this (possibly oppressive) characteristic of the mathematical hierarchy, and it’s one of the main reasons I am hoping sociologists will pay closer attention to mathematics.

Having said that, I should dispel any notion that Langlands took advantage of his mathematical eminence, in the article from which Diagram B is taken, to write “whatever nonsense” came into his head.  On the contrary, while the modesty of his intentions is evident throughout, there is no nonsense but rather a good deal of profound and unconventional thinking about the nature of our vocation.  So I will take the risk of promoting the (oppressive?) hierarchy once again and encourage you all to read the article, if you have not done so already; and I will quote a few of its more memorable passages.

On his own limitations and uncertainties:

I learned, as I became a mathematician, too many of the wrong things and too few of the right things. Only slowly and inadequately, over the years, have I understood, in any meaningful sense, what the penetrating insights of the past were. Even less frequently have I discovered anything serious on my own. Although I certainly have reflected often, and with all the resources at my disposal, on the possibilities for the future, I am still full of uncertainties.

On the effects of (what MWA calls) charisma:

Because of the often fortuitous composition of the faculty of the more popular graduate schools, some extremely technical aspects are familiar to many people, others known to almost none. This is inevitable.

On Proposition 78 of Book X of Euclid’s Elements:

This is a complicated statement that needs explanation. Even after its meaning is clear, one is at first astonished that any rational individual could find the statement of interest. This was also the response of the eminent sixteenth century Flemish mathematician Simon Stevin…

On the beauty of Galois theory:

What cannot be sufficiently emphasized in a conference on aesthetics and in a lecture on mathematics and beauty is that whatever beauty the symmetries expressed by these correspondences have, it is not visual. The examples described in the context of cyclotomy will have revealed this.

On cooperation — and charisma! — in mathematics:

Like the Church, but in contrast to the arts, mathematics is a joint effort. The joint effort may be, as with the influence of one mathematician on those who follow, realized over time and between different generations — and it is this that seems to me the more edifying — but it may also be simultaneous, a result, for better or worse, of competition or cooperation. Both are instinctive and not always pernicious but they are also given at present too much encouragement: cooperation by the nature of the current financial support; competition by prizes and other attempts of mathematicians to draw attention to themselves and to mathematics.