Am I a number theorist?

 

1985ean Parisko École Normale Superieureko matematikako zuzendari eta irakasle hautatu zuten. Urte berean, Londresko Errege Elkarteko kide izendatu zuten. Emmanuel Collegeko kide izateaz gainera, Cambridgeko Matematikako Sadleiriar katedra ere lortu zuen. 1991n Cambridgeko Matematika Huts eta Estatistika Matematikako saileko zuzendari izendatu zuten.

Wikipedia lists 395 number theorists, from Euclid and Kamāl al-Dīn al-Fārisī to Jacob Tsimerman, but I am not on the list.  Actually, one should consider not one list but all the lists of number theorists, in languages from العربية to 中文, but I am not on any of the lists.

Some of the lists are easy to remember; for example, the Kazakhs only recognize Diophantus, Hadamard, Gauss, and Fibonacci (in alphabetical order:  Д, Ж, К, Ф); the Icelandic page only lists Dirichlet.  I wonder whether John Coates knows that he is the only number theorist not born in the 18th or 19th century, and only one of five number theorists of any time or place, to have a Wikipedia page in Basque, excerpted above; I will surely ask him the next time I see him.

I like to point out Wikipedia’s frequent errors, omissions, and oddities; it reinforces my possibly naive hope that there is a future for professional scholarship.  When I start writing anything I inevitably consult Wikipedia for source material, and I sometimes use “Wikipedia” as a stand-in for wired public opinion;  but I never quote it as a reliable reference, because too often it is not.  On this occasion I was looking for a list of number theorists — it should be easier to get that right than a list of autobiographies — because I had just come across an exchange on the n-Category Café in which Harvey Friedman took part, and in which Peano arithmetic was mentioned and I was wondering how many number theorists on the list would be able to recite the Peano postulates, and what that said about the state of our subject.  Surely Eratosthenes and John Coates’s four companions on the Basque page are exempt, but are contemporary number theorists really entitled to their places on the list?  To be continued…

 

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.

Four scientific societies react to the resignation of French experts

I am told that the previous post on the resignation of the ANR evaluation committee for mathematics and computer science was widely shared on Facebook, notably by researchers in the social sciences.  Today the Société Mathématique de France published a joint statement signed by the presidents of four professional organizations, as well as the text of a motion in support of the resignation, voted by the SMF at their national meeting last week.

The joint statement is reproduced below (in French).

Déclaration des sociétés savantes françaises de mathématiques et d’informatique

Société Française de Statistique (SFdS),

Société de Mathématiques Appliquées et Industrielles (SMAI),

Société Mathématique de France (SMF),

Société Informatique de France (SIF).

Mise  en  garde  sur  l’inadéquation  du  modèle  de  sélection  de  l’ANR  pour  les mathématiques et l’informatique.

Les sociétés savantes de mathématiques, statistique et informatique (SFdS, SMAI, SMF, SIF) alertent  les  pouvoirs  publics,  l’Agence  Nationale de  la  Recherche  (ANR)  et  la  communauté scientifique  sur  la  démobilisation  massive  des  mathématiciens  et  informaticiens  constaté  ces dernières années dans les appels à projets de l’ANR.

Cette démobilisation  apparaît comme une conséquence  du choix de l’ANR de ne pas tenir  compte  des  spécificités  disciplinaires  et  de  ne  pas  impulser  une  dynamique  qui soit réellement au service du développement de la science et de l’innovation en France.

Les  mathématiques,  les  statistiques  et  l’informatique  sont  fortement  moteurs  et  vont l’être  de  plus  en  plus  de  façon  directe,  transversale  et  interdisciplinaire  dans  tous  les changements  en  cours  concernant  le  développement  technologique,  les  enjeux  du numérique  et  la  capacité  d’innovation  en  France  et  à  l’international.  Pourtant,  le conseil  de  prospective  de  l’ANR  n’intègre  aucun  mathématicien  ni  informaticien  en son sein.

Le  Comité  d’Evaluation  Scientifique  de  l’ANR  en  mathématiques  et  informatique (CES 40) a  constaté  une  forte  baisse  du  nombre  de  projets  soumis  en  2016, conséquence immédiate d’une perte de la motivation des  collègues face au très faible  taux  d’acceptation  des  années  précédentes.  Il  souligne    également  la difficulté  de  mobiliser  les  collègues  pour  expertiser  des  projets  trop  souvent rejetés.

Or le nombre de projets soutenus est calculé par l’ANR proportionnellement au nombre de projets soumis. Cette année, nos deux disciplines auront donc encore moins  de  projets  acceptés,  amorçant  un cercle  vicieux  qui  met  en  danger  la vitalité de nos communautés.

En outre, les modalités d’élaboration du taux d’acceptation de l’ANR ne sont pas discutées  de  façon  ouverte  ni  diffusées  à  la  communauté  scientifique  (toutes disciplines  confondues).  Ce  taux est  déterminé  par  l’ANR,  de  façon  opaque  et  sans   aucune   concertation   avec   les   comités   après   leur   travail   d’évaluation scientifique. Il est fixé pour chaque défi, sans aucune considération disciplinaire qui permettrait  de  dégager  une  vision  pour  le  développement  de  la  science  et leur impact économique et sociétal. Les comités doivent aujourd’hui travailler en «aveugle», sans aucune information sur la politique de répartition des moyens, et sans prise en compte des critères scientifiques pour le classement final.

Les quatre sociétés savantes signataires  demandent donc que les comités scientifiques soient pleinement associés aux modalités d’élaboration des taux d’acceptation, qu’une enveloppe budgétaire soit décidée en amont du travail des comités et que le conseil de prospective de l’ANR soit plus représentatif pour les mathématiques et l’informatique. Porteuses  des  attentes  de  leur  communauté,  elles  souhaitent  rencontrer  le  ministère dans les plus brefs délais.

GÉRARD    BIAU,    Président    de    la    SFdS,

FATIHA    ALABAU,    Présidente    de    la    SMAI,

MARC    PEIGNE,    Président    de    la    SMF,

JEAN-­MARC    PETIT,    Président    de    la    SIF.

Not about Fibonacci

quadrivium - 1

Arithmetic, geometry, and music in Giovanni Pisano’s pulpit (1301-1310), Duomo di Pisa

Pisa is the international symbol of improbable constructions and therefore a fitting location for this week’s workshop.  Pisa is also a fitting location for meditating on the eternal and impossible question:  do we engage in mathematics because we find it beautiful, or do we find mathematics beautiful because of our programming?  Are Pisa’s medieval arcades beautiful because we are used to them or do we admire Pisa for the beauty of its medieval architecture?

In addition to the Roman sarcophagi that littered Pisa’s underworld and were recycled in medieval times to house the remains of political and military citizens “di primario spicco,” Pisa’s Camposanto contains the gigantic (5.6 x 15 m) 14th century fresco Il trionfo della morte which might have served as a reminder of the urgency of completing the program of this week’s workshop, but which is undergoing restoration and is therefore not visible to the public.  It seems to me the workshop provides a striking illustration of the complex interplay between freedom and inevitability in the design of a mathematical theory, in this case the mod p Langlands program, whose ultimate goals are being defined, democratically as far as I can tell, through workshops and conferences like this one.

Pisa’s medieval walls are also decorated with a variety of political statements.  Someone found it worth his or her while to design a stencil to celebrate an American mathematical personality:

Kaczynsky - 1

Seen on a wall in central Pisa. The caption reads “strike where it hurts the most.”

 

French expert committee resigns in protest

The members of the French Scientific Evaluation Committee in mathematics and computer science (CES 40) resigned unanimously on June 1 to protest “the confiscation of scientific choices by a purely administrative [i.e., bureaucratic] management.”

The role of the CES 40, and of similar committees in other disciplines, is to evaluate research proposals submitted to the Agence Nationale de la Recherche (ANR), which then decides which projects to fund.  The ANR (not to be confused with absolute neighborhood retract) was created in 2005 in emulation of the NSF, in order to shift priorities from long-term funding of laboratories and research teams to short-term funding of specific projects, “in a context of budgetary constraints [i.e. austerity],” according to Wikipedia.  Former French President Nicolas Sarkozy (currently under investigation for illegal campaign funding) explained the motivations of the move with his characteristic disdain for the scientific community:

Je souhaite qu’à cette nouvelle génération soit inculqué non plus le réflexe du financement récurrent mais la culture du financement sur projet, la culture de l’excellence, la culture de l’évaluation.

The text of the protest letter is copied below, and can also be read here, with comments, as well as on the website of the Société Mathématique de France.

Le Comité d’Evaluation Scientifique en mathématiques et en informatique de l’Agence Nationale de la Recherche démissionne en bloc pour protester contre la confiscation des choix scientifiques par une gestion entièrement administrative

Le 1er juin, à l’issue de trois jours d’évaluation scientifique, le comité en mathématiques et en informatique (CES 40) a décidé unanimement de ne pas transmettre ses conclusions à l’ANR. Ses membres refusent de servir de caution scientifique et déclineront toute sollicitation ultérieure de l’ANR dans les conditions actuelles.

Le comité conteste l’opacité du processus de sélection. A ce jour, le nombre de projets financés est déterminé en proportion du nombre de projets soumis, sans que les comités aient la maîtrise du seuil d’acceptation, ou la connaissance de l’enveloppe budgétaire attribuée. Or, loin d’être uniquement des informations financières ou administratives, ce sont des éléments scientifiques essentiels sans lesquels les comités ne peuvent élaborer une proposition cohérente.

L’addition des contraintes budgétaire et administrative conduit mécaniquement à un taux d’acceptation trop faible pour être incitatif. Or, la constitution d’un dossier de qualité exige un temps important, que de moins en moins de collègues accepteront d’investir au vu du taux de succès qui a cours. Cela s’est traduit par une diminution de plus de 20% du nombre de projets soumis dans le CES 40 qui entraîne à son tour une baisse du nombre de projets financés. L’ANR manque donc l’occasion de soutenir un nombre important de projets à fort impact.

Le comité s’inquiète aussi de la perte annoncée de son indépendance, puisque son président sera désormais employé par l’ANR.

Les membres du comité demandent à la direction générale de l’ANR la mise en place un nouveau mode de fonctionnement. Ils souhaitent un meilleur contrôle du processus de sélection, de manière à mettre en œuvre une politique scientifique cohérente qui respecte les spécificités de chaque discipline, au service de la stratégie nationale de la recherche.

Les membres du CES 40, unanimes :
– Christophe BESSE, Président du CES 40, Professeur de Mathématiques, Université Toulouse 3
– Marie-Claude ARNAUD, Vice-Présidente du CES 40, Professeur de Mathématiques, Université d’Avignon
– Max DAUCHET, Vice-Président du CES 40, Professeur émérite d’Informatique, Université Lille 1
– Mourad BELLASSOUED,  Professeur de Mathématiques, Université de Tunis El Manar
– Oliver BOURNEZ, Professeur d’Informatique, Ecole Polytechnique
– Frédéric CHAZAL, Directeur de Recherche en Informatique, INRIA Saclay
– Johanne COHEN,  Chargée de Recherches en Informatique, CNRS, Université Paris Sud
– François DENIS, Professeur d’Informatique, Université Aix-Marseille
– Bruno DESPRES, Professeur de Mathématiques, Université Paris 6
– Arnaud DURAND, Professeur de Mathématiques, Université Paris Diderot
– Alessandra FRABETTI, Maître de Conférence en Mathématiques, Université Lyon 1
– Jin Kao HAO, Professeur d’Informatique, Université d’Angers
– Tony LELIEVRE, Professeur de Mathématiques, Ecole des Ponts ParisTech
– Mathieu LEWIN, Directeur de Recherche en Mathématiques, CNRS, Université Paris Dauphine
– Gaël MEIGNIEZ, Professeur de Mathématiques, Université Bretagne Sud
– Sophie MERCIER, Professeur de Mathématiques, Université de Pau et des Pays de l’Adour
– Johannes NICAISE, Professeur de Mathématiques, Imperial College Londres
– Lhouari NOURINE, Professeur d’Informatique, Université Blaise Pascal
– Jean-Michel ROQUEJOFFRE,  Professeur de Mathématiques, Université Toulouse 3
– Alessandra SARTI,  Professeur de Mathématiques, Université de Poitiers

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.)

Leçon inaugurale

Leçon_inaugurale

Few of the 420 seats of the magnificent Amphithéâtre Marguerite de Navarre were empty when Claire Voisin performed the ceremony marking her entry to the “most prestigious institution of the French university” system, the 486-year old Collège de France.  Jean-Pierre Serre, whose work was amply cited by Voisin in her hour-long account of the branches of complex geometry — analytic, Kähler, and algebraic — sat in the middle of the front row, together with past and present Professors of the Collège.  The talk was systematic, organized, and comprehensive, like Voisin’s introductory two-part book on Hodge Theory and Complex Algebraic Geometry:  not a dinner party explanation by any means, it was “a rather complete tour of the subject from the beginning to the present” in the words of the review in the Bulletin of the AMS by Herbert Clemens, who noted “the break-neck pace of Voisin’s clear, complete, but ‘take no prisoners’ exposition.” Serre, who turns 90 (!) this September, was alert as always; colleagues to my left and right in the tenth row were happy with the pace and the content but speculated that the distinguished medieval historians and classicists in Serre’s row were already dozing off by the time Voisin defined complex structures in her second slide (my apologies for the blurry photos)

nombres complexes

and in any case long before she concluded her lecture with an allusion to her work on the generalized Hodge conjecture.

lastslide - 1

Readers of MWA will not be surprised to learn that the lecture was followed by a sumptuous champagne reception.  The new Professor has so many friends and admirers that the petits fours ran out well ahead of schedule, but there was (just barely) enough champagne for the jubilant crowd.  For Paris mathematics, the inauguration was undoubtedly the social event of this (very rainy) season.

I was wondering who would replace Don Zagier when he vacated his Chair in Number Theory a few years ago, so — as always in these situations — I consulted the best-informed of my colleagues.  He told me that no one had yet been named, but that it had been decided to create a Chair in Algebraic Geometry.  It was obvious for whom such a Chair was intended, but the formal announcement took some time to appear, and I was surprised to see that the press took no notice of what in the English-speaking world would certainly be considered an event of historic magnitude:  the naming of the first woman professor of mathematics to the most prestigious position in the French academy.  Journalists had no doubt chosen to heed Voisin’s own preemptive and scathing critique of this approach to “diversity”:

L’idée qu’augmenter le nombre de femmes à l’Académie des sciences aurait un impact sur la désaffection des femmes pour certains domaines des sciences est tout simplement grotesque. D’abord parce que l’Académie des sciences n’intéresse personne et ensuite parce que le choix de faire une carrière scientifique ne repose que sur les aspirations intellectuelles et le talent, et non sur des considérations mondaines. Personnellement, je supporte de moins en moins d’être passée en quelques années du statut de mathématicienne à celui de femme-mathématicienne, et de subir l’oppression grandissante de l’obsession paritaire, transportée à grand bruit par les médias.

Je souhaite que mon statut de femme, qui me plaît beaucoup, reste du domaine privé, et que l’évaluation et la reconnaissance de mon travail ne se trouvent pas polluées par la prise en compte de ce statut (ce qui est insultant en général : être une femme n’est pas un handicap !).

Je souhaite aussi ne jamais devenir Madame Quota, et surtout que cela n’arrive pas à mes filles. A supposer qu’on ne puisse pas parler d’autre chose que de la fameuse parité, serait-il possible de mentionner que les quotas sont à différents égards (dont certains non mentionnés ci-dessus) une menace pour les femmes scientifiques ?

Voisin’s point of view is rarely expressed so forcefully, but it is widely shared in France.  When I arrived from the United States in the early 1990s, it was disorienting, to say the least, to hear male mathematicians routinely making comments about the physical appearance of their female colleagues, behind their backs; but I also heard female mathematicians commenting (appreciatively or not) about the looks of their male colleagues.  There’s much more to be said, but I learned quickly enough that if I opened my mouth on this (or any other) subject I would be accused of being an “anglo-saxon” — and it was futile to brandish my Beowulf and point out that this status would be denied me in any actual English-speaking country.

After an hour of catching up with long-lost acquaintances I managed to push my way through the crowd of well-wishers to congratulate the newly-named Professor.  I hope she will not hold it against me that I have briefly extracted her “status of woman” from the private sphere.  She told me that she has a copy of MWA but that she hadn’t yet found the time to read it.  I advised her to skip ahead to Part II, and maybe chapter 4, which is where she would find the best jokes, and by all means avoid reading the boring chapter 3!  That’s the advice I give everyone, but after taking a look yesterday at his comment on David Roberts’s Google+ blog I feel I ought to make an exception for Urs Schreiber, who is specifically advised to reread the discussion of tradition-based practices on pp. 74-77 and to decide whether that discussion didn’t anticipate his objections.  I will have more to say on this topic when I comment on Roberts’s review for the Australian Mathematical Society.