Category Archives: Soundtrack

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.


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 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.

Me and Pink Floyd

Wikipedia has once again listed Mathematics without Apologies as an autobiography.  I was trying to figure out where I should send my complaints when I saw, just a few lines down, that Richard Wright’s Black Boy was listed as the autobiography of a musician.  Curious to learn more, I clicked on the name and was sent to the Wikipedia page of the late Pink Floyd keyboardist, who was two years old when his purported autobiography was published.  This is so cute that I really shouldn’t say anything, and I have to admit it’s kind of thrilling to be on a list that includes Santa Teresa de Avila, both Spike and Sir Christopher Lee (who wrote autobiographies with the same title), Casanova (who published the same biography twice and who will reappear in a future post as a mathematician), and Hulk Hogan.

So please, if you know the person who put my name on the list (at some point between June 10 and August 2), don’t say anything.  But why does anyone think my book is an autobiography?

Things that are not wrong with MWA, I: reasoning by lipstick traces


Image credit: John Bartelstone, from

Greil Marcus was teaching the world how not to be wrong many years before Jordan Ellenberg published a book by that title.  That, at least, was the opinion expressed by John Lydon—better known as Johnny Rotten—when asked about Marcus’s book Lipstick Traces.  To quote from the book’s back cover (with my emphasis added):

John Lydon:  “It’s so mad, it’s so daft, it’s so off the wall—it’s thoroughly enjoyable…”

Interviewer:  “But you don’t think he’s completely wrong?”

John Lydon:  “No, he’s not wrong.”

A foolproof way not to be wrong is to avoid making assertions to which the tired opposition between right and wrong can be meaningfully applied.  Actually, that’s on the back cover as well:

Marcus offer[s] interpretations that are meant to excite the reader to further imagining and thought rather than mere agreement or disagreement… (Anthony DeCurtis, Rolling Stone)

The brief proposal I submitted to Princeton University Press, accompanied by early drafts of what were to become Chapters 5, 6, and 9, opened with a promise to write a book in the same vein.

The idea for this book can be traced back to my reading more than ten years ago of Lipstick Traces by Greil Marcus. Marcus’s book, which has nothing to do with mathematics, is hugely entertaining; it is also a profound work of cultural archeology, tracing the roots of a striking and apparently unprecedented cultural phenomenon — the emergence of punk rock in the 1970s — through echos of millenarian movements and the forgotten history of striving for transcendence. My immediate reaction was: mathematics has been around a lot longer than rock and roll, and is much more pervasive in our lives, in our popular culture, and in our very way of thinking. Why is there no book that presents mathematics in a historical and cultural context rich enough to reveal mathematics as a fully human experience, with all the pathos that entails?

For a long time it was my firm intention to recycle that paragraph in the preface of the book PUP quickly agreed to publish.  The preface was the last thing I wrote, though, and either because I was exhausted or because I was concerned my meaning would be misunderstood or because the main text was already overloaded with rock and roll, I decided to try to adopt a more conventional tone in the first pages a potential reader was likely to see upon opening the book.  After more than half a year of writing this blog and reading the occasional bad review of MWA — I’ll be turning my attention to one of those before I’m through with this post — I am now convinced this was a mistake.  Besides, one of the outside reviewers to whom PUP sent the proposal before agreeing to sign a publishing contract reacted to this very paragraph:

I strongly urge  PUP to acquire Michael Harris’s (untitled) book.  Let
me put it quite simply.  Harris proposes to write a book that will stand
in the relation to mathematics that Greil Marcus’s LIPSTICK TRACES does
to cultural theory and the academic study of popular music.  You have
sent the proposal to the right reviewer, because I am a huge admirer of
LIPSTICK TRACES and think that Marcus’s book is the exemplar of the role
an academic press (in this case Harvard) can play in getting important,
complicated ideas from the academy out to the public.

To appreciate what Marcus accomplished with Lipstick Traces it’s not necessary to share his feeling that “every good punk record made in London in 1976 or 1977” is “the greatest thing you’ve ever heard.” (Lipstick Traces, p. 80; all references to the 1990 Harvard University Press paperback edition).  It’s enough to be willing to respect  “…the pursuit of a non sequitur for the pleasures only a non sequitur can bring… (p. 20) and to agree not only that “Real mysteries cannot be solved, but they can be turned into better mysteries” (p. 24) but that this can be the motto of a legitimate form of intellectual exploration going by the name of cultural criticism.

The anonymous reviewer accepted my proposal’s claim that, by writing book reviews and occasional articles,

I began to learn how to dissimulate a tightly directed narrative beneath an apparent series of free associations;

the reviewer added that “I cannot say confidently what [Chapter 6] is ABOUT but it is clear that something real is happening.”  Last week, in the course of a wide-ranging discussion of his two most recent books at the French bookstore Albertine in New York (pictured above), Marcus alluded to the appearance of free association in his writing, and for several minutes he explained his methods and his motives with impeccably quotable clarity.   Unfortunately I was not taking notes and, although some of the events at the Cultural Services of the French Embassy are recorded and made available on line — check out this conversation with John Nash and Cédric Villani if you haven’t already done so — I fear that all that’s left of Marcus’s words are the scattered traces that my memory will not be able to reassemble.

It would have been useful to be able to channel Marcus in responding to Mark Hunacek’s review of MWA in the American Mathematical Monthly.  This particular negative review appeared nearly three months ago, but I have been putting off linking to it on this page because I really don’t know what to make of it.  The conclusion, though, is unambiguous:

Final verdict: a reader will have to make an individual determination as to whether the benefits of this book outweigh the author’s rhetorical excesses and heavy-handed writing style. Undoubtedly, some will say that they do, but I’m afraid that I can’t count myself in that group.

Hunacek dislikes three things about the book:  its “opaque writing style,” its “self-congratulatory tone,” and its

stream-of-consciousness feel…, with the author jumping from one idea to the next, following no particular narrative pattern that I could discern.

I thought I had explained what was behind the “self-congratulatory tone” the last time a reviewer missed the point, in point 2 of this post.  But Hunacek obviously decided his assignment was to read the book, the whole book, and nothing but the book; he found MWA “opaque” because it refers to authors he had not read.   (By the way, I don’t understand why it’s more acceptable to announce one’s ignorance of Max Weber in the pages of the American Mathematical Monthly than it would be to write that one doesn’t “really know[…] anything about” Darwin, say, in a hypothetical analogous journal of sociology — or rather, I do understand the reason, and it doesn’t reflect well on our profession.)

Lipstick Traces, like many of Marcus’s other writings, is relevant to Hunacek’s third complaint, the one about stream-of-consciousness — or, as Marcus put it, free association.  Now Marcus is a professional writer and I am not, and I am under no illusion that I can make use of his stylistic innovations in cultural criticism with anything like his skill or effectiveness, not to say brilliance.  But Hunacek’s third objection, as far as I can tell, is not to my lack of skill nor even to the style he calls “heavy-handed” but to the very notion that a non sequitur, even a merely apparent non sequitur, can bring pleasure; so that his lack of discernment is the author’s fault, not his.

Each chapter of MWA is in fact organized around a “tightly directed narrative” — with the possible exception of the last part of Chapter 8 — and I’m not going to spoil the reader’s pleasure, such as it is, in figuring out where the direction points.  But I do want to share what I understood while listening to Marcus at Albertine, about why I needed to write in that particular way.  I have long felt that writing about certain particularly sensitive topics in the practice of mathematics leads invariably to dead ends.  (A good example is the controversy over whether mathematics is invented or discovered; there are many others.)  I didn’t want to avoid sensitive topics; on the contrary, as far as possible I wanted to write only about sensitive topics.  Tackling them head-on — taking a right vs. wrong position one way or the other — would clearly lead me to the same dead ends.  So I chose to follow Marcus and to argue by the evidence of lipstick traces — by means of analogies, anecdotes, stories, digressions, and often incongruous or “daft … off the wall” juxtapositions “to excite the reader to further imagining and thought rather than mere agreement or disagreement.”  (Brendan Larvor referred to this, eloquently, as “disclosure through juxtaposition and paradox,” which he considers a “Parisian activity.”  Marcus is well-versed in critical theory of the Frankfurt as well as Parisian variety, though he belongs to no -ism.)  And accessorily, to incite professionally qualified readers — historians, for example — to investigate some of the more suggestive lipstick traces that may have been overlooked.

Marcus said it much better at Albertine, in connection with his own aims as a writer, but those words are no longer available to me.

I have to write about one more paragraph in Hunacek’s review, the most problematic as far as I’m concerned, before I’m ready to post quotations from the review prominently on this blog.  I’ll also consider the question of whether, on balance, the publication of this kind of negative review isn’t a blessing in disguise.  But that’s for another time.

P.S.  Lipstick Traces is also indirectly and implausibly responsible for the inclusion of Figure 6.4 in MWA, and more directly and plausibly responsible for the brief discussion of Isidore Isou in the same chapter.

Fryderyk Chopin, a brilliant mathematician?


That’s the title, in Polish, of what appears to be a 4th-6th grade lesson plan developed for the composer’s bicentennial in 2010.  Another plan invites the student, in the words of Google translate,  “In the footsteps of Frederic Chopin [to] improve math skills“.   Five years have gone by, the 17th International Fryderyk Chopin Piano Competition is taking place at this very minute in Warsaw (the Impromptu in A flat major Op. 29, to be precise), and it may be the moment to ask ourselves:   what did Chopin think about mathematics?

The answer is:  apparently, not very much one way or the other.  But there is this entry in the painter Delacroix’s journal for April 7, 1848 (my rough translation):

I asked [Chopin] what established logic in music.  He helped me to understand the nature of harmony and counterpoint; so that the fugue is like pure logic in music, and to be a scientist in the fugue is to know the element of all reason and all consequence in music.… I thought how happy I would be to learn all these matters that so depress vulgar musicians.  This feeling gave me an idea of the pleasure that those who deserve to be called scientists find in science.  The true science is not what one ordinarily understands by this word, that is to say a part of knowledge distinct from art; no!  Science viewed in this way, demonstrated by a man like Chopin, is art itself, and on the other hand art is then no longer what the vulgar believe, that is to say a sort of inspiration that comes from who knows where, that works by chance and only presents the picturesque exterior of things.  It’s reason itself, ornamented by genius, but following a necessary path and governed by higher laws.

The poet Alphonse de Lamartine was effectively dominating the newly proclaimed Second Republic at the time of this conversation, and it was perhaps the time for the musician and the painter to talk about big questions like the relation between art and science.  Here’s a more contemporary big question in the same vein, which is the real motivation for this post.  During a break a Chopin specialist, talking about interpretation, claimed that a musical score is only an indication and  determines a performance only in the most general way.  If the Chopin competition means anything at all it could hardly be otherwise; after all, hundreds of contestants are asked to perform basically the same pieces, but the computer on which you are reading this can play the notes without ever making a mistake.

Now the big question is whether it can also be said that a written proof is only an indication and determines the presentation (“performance”) of the proof only in the most general way.  If we were to admit this as a possibility, then, while the developers of automated theorem provers may not exactly be seeking a chimera, it could well be argued that formalized mathematics is meaningless in the absence of the — presumably human — performer.

(But maybe fifteen years from now the 20th International Fryderyk Chopin Piano Competition will be won by a descendent of Deep Blue equipped with fingers and feet — having these appendages must be somewhere in the ground rules — and what will the contemporary Delacroix have to say about that?)

"Chopin denkmal wwa". Licensed under CC BY-SA 3.0 via Wikimedia Commons -

Return from Cusco


The fellow to the right of the title of the Cusco summer school is a khipu kamayuq, the (pre-conquest) Inca equivalent of an (applied) mathematician.

Eventually my Cusco lectures will be posted on YouTube:  three on automorphic forms and one on Matemáticas sin disculpas.   I will be slowly be catching up on the past few week’s developments.  In the meantime, you can listen to Las matemáticas by the Columbian group Los Alteños de la Sierra.

El amor
es como las matemáticas
así le dije a mi novia

El amor
es como las matemáticas
así le dije a mi novia

Y ella nomás se reía
ha hay hui hui
más amor menos dolor
entre tú y yo
por toda la felicidad
mi amor

Lateral influence


My attention has been drawn to the video of Alex Gamburd’s recent talk at the IHP on his joint work with Jean Bourgain and Peter Sarnak.  If I had been in town for the workshop, I would probably have attended, in which case I would have been struck, as have many of our colleagues, by Alex’s couture.  Cédric Villani was not in attendance, but the inevitable comparison to the Lady Gaga of mathematics has been recorded.  Is Alex the Bryan Ferry of mathematics?  Still too many lead singers, not enough of a rhythm section.