Category Archives: Uncategorized

AMS letter on the immigration ban

From the AMS website.  See also the letter dated January 31, signed by 164 organizations and universities.

AMS Board of Trustees Opposes Executive Order on Immigration
Monday January 30th 2017

Providence, RI: The members of the Board of Trustees of the American Mathematical Society wish to express their opposition to the Executive Order signed by President Trump that temporarily suspends immigration benefits to citizens of seven nations.

For many years, mathematical sciences in the USA have profited enormously from unfettered contact with colleagues from all over the world. The United States has been a destination of choice for international students who wish to study mathematics; the US annually hosts hundreds of conferences attracting global participation. Our nation’s position of leadership in mathematics depends critically upon open scientific borders. By threatening these borders, the Executive Order will do irreparable damage to the mathematical enterprise of the United States.

We urge our colleagues to support efforts to maintain the international collegiality, openness, and exchange that strengthens the vitality of the mathematics community, to the benefit of everyone.

We have all signed the online petition of academics opposing the ban. We encourage our colleagues to consider joining us in signing it and in asking the Administration to rescind the Executive Order.

Robert Bryant, President of the AMS
Kenneth Ribet, President-Elect of the AMS
Ruth Charney
Ralph Cohen
Jane Hawkins
Bryna Kra
Robert Lazarsfeld
Zbigniew Nitecki
Joseph Silverman
Karen Vogtmann

UPDATE: The online petition is experiencing a delay in accepting emails and displaying new names. [1/31/17]

Contacts: Mike Breen and Annette Emerson
Public Awareness Officers
American Mathematical Society
201 Charles Street
Providence, RI 02904
Email the Public Awareness Office


Founded in 1888 to further mathematical research and scholarship, today the American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.


Did Simon Jenkins get it wrong?



That MWA currently (but temporarily) occupies the top two spots in the Amazon UK ranking in philosophy of mathematics [sic] can be entirely attributed, I believe, to its mention in yesterday’s Guardian column by Simon Jenkins.  The journalistic charisma of Jenkins is such that his readers were willing to sail behind him into MWA‘s uncharted waters even though the point of his allusion is by no means clear.  Judge for yourselves:

I agree with the great mathematician GH Hardy, who accepted that higher maths was without practical application. It was rather a matter of intellectual stimulus and beauty. A new book by Michael Harris, Mathematics Without Apologies, goes to the extremes of this stimulus, to the categorical ladder, incompleteness theory and the Black-Scholes equation, used to assess financial derivatives. It ends in the “inconsistency nightmare”, that nought might possibly equal one.

I’ve read the sentence with my name in it ten times, not out of vanity but because I still can’t figure out what it means, nor what it has to do with the point Jenkins wants to make: which is that what British pupils really need from school is “crowded out by a political obsession with maths.”

It seems that Jenkins cites MWA as proof of the “extreme” uselessness of mathematics.  There is absolutely nothing like that in the book.  Nor does the word “nightmare” occur even once (much less “incompleteness theory”) and I have no idea what the “It” is that “ends” in the nightmare equation 0 = 1.  The syntactical confusion of this paragraph — which didn’t dissuade a few dozen Amazon UK customers — can be blamed on an overzealous Guardian editor, or perhaps on a Jenkins assistant  who was assigned to read (or rather, to “read”) MWA (or perhaps any recent book that alludes to Hardy) and who got muddled in its disorganized snobbery.  The article’s motivation, on the other hand, is a recurring theme in Jenkins’s columns.  In 2008, he called his study of quadratic equations a “waste of time” and wrote

Maths and science self-justify as economically worthwhile in a way that law or economics or management studies do not dare.

In 2014, he returned to the charge with at least articles, including one entitled  For Britain’s pupils, maths is even more pointless than Latin.   There have been more diatribes against mathematics and science education over the years, so that Stephen Curry wrote memorably that

Arguing about science with Simon Jenkins is like trying to wrestle with a fart — you can’t miss the odious stink but there’s almost nothing to get hold of.

The malodorous simile didn’t discourage me from trying to cash in on the moment, in my small way, by renaming the link to Princeton University Press “PURCHASE THE BOOK.” Curiously enough, the link in the Jenkins article is to this blog — that’s how I detected the existence of his article — and not to the Guardian bookshop, which reports that my book is in stock and which may be the last place on earth where you can still see the rejected initial cover.

Jenkins’ “miserabilist” complaint puts me in mind of an analogy Ed Frenkel has used repeatedly, notably in the December 5, 2013 issue of The Economist, where Jenkins should have read it:

Imagine you had an art class in which they taught you how to paint a fence, but never showed you the great masters. Of course, you would say; ‘I hate art.’ You were bad at painting the fence but you wouldn’t know what else there is to art. Unfortunately, that is exactly what happens with mathematics.

On the evidence of his public disagreement with Andrew Hacker — this tweet seems to be the most recent instance — Frenkel would undoubtedly reject Jenkins’s arguments about the place of mathematics in the curriculum.  Frenkel’s fence-painting analogy helps to clarify just what Jenkins got wrong in his (or his assistant’s) reading of MWA.   Nowhere is it claimed in MWA that mathematics is useless.  Rather, what MWA has to say about the utility of mathematics has two parts.  The less important part is that utility, whatever that means, is not a primary motivation for pure mathematicians, nor even a secondary or tertiary motivation.   The much more important point is that justification of mathematics on the grounds of utility begs the question of what utility really means, and how the ideology of utility is used by decision-makers (or Powerful Beings) to foreclose reasoned discussion of social priorities, replacing them by invocations of purely technocratic criteria which leave no alternative.  The ideology of utility serves in practice to protect the interests of the powerful.  Jenkins has certainly not broken with this ideology; he just disagrees with the details of its elaboration.

Pétition contre l’ingérence du président du CNRS dans les élections de l’UPMC

Voici le texte de la pétition, que vous pouvez  signer, mise en ligne deux jours avant les élections à l’Université Pierre et Marie Curie (Paris 6).

Dans un courrier récent envoyé aux personnels CNRS de l’Université Pierre et Marie Curie, le président du CNRS, M. Fuchs, a apporté son soutien ès-qualités à une liste particulière de candidats aux élections des conseils centraux de l’UPMC.

Ce courrier serait déjà au-delà de l’admissible si l’expression publique de ce choix était faite en son nom, balayant démocratie et devoir de réserve. Cependant, M. Fuchs s’autorise à déclarer que “le CNRS soutient la liste X”. Or, le CA du CNRS n’a pas été consulté sur ce choix et des collègues CNRS figurent sur la liste opposée à celle à laquelle il apporte son soutien. Certes les statuts du CNRS donnent à son directeur le pouvoir de représenter l’établissement dans ses relations extérieures, mais on peut s’interroger sur la légitimité et la représentativité de la parole de cet organisme auprès de ses membres et de la société dès lors qu’il se trouve enrôlé, malgré lui, derrière l’initiative toute personnelle de son PDG.

Cette ingérence du président du CNRS dans la vie démocratique d’une université, pour venir en aide à l’un de ses homologues, serait anecdotique si elle ne venait ajouter l’insupportable à une longue suite de réformes et d’incidents conduisant à une dépossession de nos métiers et de nos libertés d’exercice.

Nous, universitaires et membres du CNRS, n’approuvons pas l’expression d’un pouvoir solitaire qui ne dit pas son nom et souhaitons par ce message le faire savoir publiquement.

Nous ne sommes pas des salariés du CNRS: nous sommes le CNRS.
Nous ne sommes pas des salariés de l’Université: nous sommes l’Université.

Nous attendons de vous, Madame Vallaud-Belkacem, Monsieur Mandon, un rappel des libertés académiques et du respect de la démocratie universitaire, rappel qui devra s’imposer au directeur du CNRS que vous avez reconduit dans ses fonctions.

Voir aussi la réaction de la CGT, la réaction de la FSU, ainsi qu’un communiqué de plusieurs organisations syndicales :

Les organisations syndicales : 








SUD Education


sont scandalisées.

Nos collègues du CNRS ont reçu vendredi dernier un mail de la déléguée régionale du CNRS leur transférant le message d’Alain Fuchs, Président du CNRS,  adressé à Jean Chambaz, président de notre Université. Dans ce message, que vous pouvez trouver ci-dessous dans son intégralité, Alain Fuchs écrit « Le CNRS soutient la liste « Réunis » que vous portez pour les élections prochaines …. » !

Nous nous  inquiétons de cette ingérence du CNRS dans des élections locales universitaires. De quel droit M. Fuchs se permet d’engager le CNRS et par là-même ses personnels sans les avoir consultés ? Beaucoup de nos collègues CNRS partagent notre stupéfaction  et nos inquiétudes.

L’engagement institutionnel  de M. Fuchs est une ingérence inacceptable dans le fonctionnement de l’Université. Nous espérons que l’équipe présidentielle  actuelle n’est pas complice de cette grave atteinte à la démocratie.

Nous appelons nos collègues à se rendre massivement dans les urnes demain 16 février où ils pourront s’exprimer librement sur ce scandale. 

Pour rappel, toutes les informations sur les élections (listes, professions de foi, bureaux) sont consultables  ici :

Blinkered devotion to useless knowledge

The latest entry in the register of futile utility is a letter to the New York Review of Books, signed

Bob Eisenberg
Bard Professor and Chairman Emeritus
Department of Molecular Biophysics and Physiology
Rush University Medical Center
Chicago, Illinois

objecting to a display of “profound ignorance of the role of mathematics in creating the world we live in” in Jim Holt’s “otherwise admirable review” of MWA.  After enumerating some useful applications of mathematics (design of buildings, electronic devices, imaging technology, an amplifier), Eisenberg affirms that

The necessity of mathematics is indeed obvious to everyone in computer science, to anyone who has programmed at all, and of course to all engineers and almost all scientists.

and formulates a wish:

Perhaps it would be helpful if the public, or at least the educated public, realized that our standard of living is directly the result of that tiny part of the world that mathematics describes accurately (with simple equations and unchanging parameters).

Holt replied by politely pointing out that Eisenberg is “absolutely right” about the “technological usefulness of mathematics” — even the Rolling Stones need calculus, Holt agrees — and that Eisenberg simply misread the review.  That’s certainly true, but since some of the most negative reviews of MWA have been motivated by a similar misreading, perhaps it would be helpful if we tried to figure out what accounts for such persistent reading in to my book, and now to Holt’s review, of something that is clearly not there.

My reactions to negative comments on this blog have been mild because I know their authors’ words, like Eisenberg’s, are not their own.  Stefan Collini’s article in the latest issue of the London Review of Books offers a clue as to how these words entered their authors’ minds.  Collini has for years been chronicling the attempts of successive governments to browbeat British academia into mindless utilitarianism, and I shamelessly mined his articles for material for MWA.    Collini points out that the expression “value for money”

occurs four times in the one-page foreword to the new Green Paper written by Jo Johnson, the minister for universities and science

The British way has the merit of crude clarity, and one senses the British touch in expressions like “entrepreneurial mindset” (as in “Insufficient entrepreneurial mindset among students,” an obstacle to EU Strategic and Priority Initiatives in a report dated 2009 — the term is American, however) sprinkled throughout MWA, particularly in Chapter 10.   This helps explain why Collini is able to write so clearly:

Much of our contemporary discourse about universities still draws on, or unwittingly presumes,  [a] pattern of assumptions [that remained stable for 30 years or so after 1945]: the idea that the university is a partly protected space in which the search for deeper and wider understanding takes precedence over all more immediate goals; the belief that, in addition to preparing the young for future employment, the aim of developing analytical and creative human capacities is a worthwhile social purpose; the conviction that the existence of centres of disinterested inquiry and the transmission of a cultural and intellectual inheritance are self-evident public goods; and so on.

If that boldface passage (my emphasis) looks familiar, it’s because the self-evidence in question was invoked on p. 70 of MWA and already quoted on this blog:

if the notion of “general [or public] interest” means anything at all, it should be a matter of general interest that work be a source of pleasure for as many people as possible.

That’s the sort of thing that nowadays gets you in trouble with revolting taxpayers, but MWA, like Collini, invites readers to imagine a time not so long ago when it really was self-evident.   Revolting taxpayers are officially the motors of the change of attitude, as Collini reports, but the real culprits are elsewhere:

If ‘prosperity’ is the overriding value in market democracies, then universities must be repurposed as ‘engines of growth’. The value of research has then to be understood in terms of its contribution to economic innovation, and the value of teaching in terms of preparing people for particular forms of employment. There are tensions and inconsistencies within this newer conception, just as there are in the larger framework of neoliberalism: neoliberal thinking promotes ‘free competition’ in international markets, while the rhetoric of national advantage in the ‘global struggle’ often echoes mercantilist assumptions. But, gradually, what we still call universities are coming to be reshaped as centres of applied expertise and vocational training that are subordinate to a society’s ‘economic strategy’.

That’s MWA‘s thesis on “usefulness” in a nutshell, and Collini puts it into words much better than I ever could.   This post’s title is taken from Collini’s very next paragraph, and you should probably just skip this post (and MWA) entirely and just read Collini’s columns on British universities, working back from the most recent one.  Collini has his fellow literary scholars in mind, no doubt, but it applies just as well to pure mathematics.  This is harder to perceive, of course, because the Golden Goose argument makes a superficially persuasive case for the unpredictable utility of blue skies research in theoretical science.  So even if Eisenberg’s underlying motivation is to fend off the inevitable accusation (by legislators in Illinois, North Carolina, or elsewhere) that academic institutions are spending revolting taxpayers’ money to support people (useless or otherwise) doing useless work, what he actually writes seems to make sense.

The utility of work in mathematics, however, is not self-evident — or at least it’s less self-evident that “the existence of centres of disinterested inquiry and the transmission of a cultural and intellectual inheritance” used to be.  Eisenberg hopes to settle the matter by pointing out that mathematical work is useful because it is responsible for the persistence of something whose utility is self evident, namely our standard of living.  Depending on how “our” is understood, however, the self-evidence of this criterion for utility can immediately be disproved.   Algebraic geometry turns out to be useful, according to an essay entitled “Mathematics: Accepting the Increasing Energy Demand Challenge”  for Shell’s Algebraic Oil project.  Consider

a collection of measurements of physical production quantities. This collection of quantities may be considered as causing the production; the associated indeterminates are called the causing indeterminates. The production itself is considered as the effect of the action of the causing variables. The production is – assumed to be – an element of the special ring under consideration. This is admittedly still a modeling assumption, albeit – relatively – weak: the production is assumed to be a polynomial, but its structure – coefficients and support – are not known upfront. Now there will of course be relations among the causing variables, that is polynomials in these variables that when evaluated over [the measurements], vanish. From the point of view of value – size – of the production, all production polynomials that differ by a relation in the causing variables are the ‘identical’ an in particular for Shell sensible point of view. So production polynomials are considered modulo relations among the causing variables. Mathematically speaking this means that the production is considered in another ring, in which the relations among the causing variables are declared to belong to the zero of that new ring.

The essay was the contribution of Dr. Jan van der Eijk, Group Chief Technology Officer of Shell Research, to the book Mathematik—Motor der Wirtschaft which was quoted at some length in chapter 10 of MWA.  The above paragraph is on pp. 95-96; there are also extensive discussions of applications of PDE and control theory.  All of this is useful because

The world’s growing population and the rapid development of new economies will result in a sharp rise of energy demand. Although sustainable energy sources will play an increasingly important role, fossil fuels will remain the backbone of the global energy supply for the foreseeable future.

Thus mathematics is directly responsible not only for the maintenance of our standard of living, as Eisenberg would have it, but the improvement of the standard of living in “new economies.”  Ivar Ekeland, in his Syndrome de la grenouille, has a different notion of utility:

…the problem of global warming is posed solely in economic terms, and human beings have been reduced to their utilitarian dimension.  We have made them into machines for optimizing their individual well-being.  The miracle of the market is that individual self-interests lead finally to a collective optimum, but in the case of global warming, and other public goods, the miracle did not take place, the invisible hand is not operating, and the mere search for individual economic advantage leads to a collective catastrophe.

Less catastrophically, MWA invokes the eminent cultural critic Tom Waits to ask how to balance the necessity of number theory for e-commerce against the utility of independent bookstores and record shops.

The point is that it’s meaningless to say that mathematics contributes to our standard of living without specifying when and for how long; that applications of mathematics are necessary or useful without specifying for whom?  That should be obvious, but if it were obvious, why do people like Eisenberg (and certain hostile reviewers) feel the need  to insist on the utility of mathematics in the abstract, as if the notion made sense?

General Knowledge

The error of characterizing MWA as an autobiography has now attained a degree of absurdity that can only be called sublime.  What appears to be a carbon copy of the Wikipedia list of autobiographies already mentioned here in October, and still not corrected (still including the silly Richard Wright listing, for example), has now been posted by something called — gk as in General Knowledge — on a page entitled “List of autobiographies by celebrities.”  So if you know me, or if you know someone else on the list (for instance:  Fidel Castro, Alyssa Milano, or Goro Shimura), you can tell all your friends that you know a celebrity.  Even better, you can write that you know a celebrity the next time you take

exams like UPSC,SSC,Clercical Exams,State PSC,IBPS,Railways,GATE etc.

since that’s where General Knowledge apparently comes in handy.

Problems of a problematic vocation

Readers who (like the author) persist in wondering what I was trying to say after they have finished the book may find it useful to take the book’s subtitle more literally.  If you believe mathematics is a problematic vocation, it doesn’t necessarily follow that you believe that the problems have solution, much less that the book’s author has found them.  Just identifying the problems may help to clear up misunderstandings (for example, that certain questions necessarily have answers).  Assigning problems to appropriate categories may be even more helpful.  With this in mind, here is a short but far from exhaustive list of some of the problems examined (but not solved) in  MWA, divided among four categories:  ethical problems (taking a stance on one of these problems entails an ethical commitment, and it is difficult to avoid taking a stance); historical problems (what appears to be a question about some intrinsic feature of mathematics is better addressed by investigating the questions’ history; best left to historians); linguistic problems (the imaginative resources we can apply to understanding the problem are limited by our language); and other.

Ethical problems

1.  Is mathematics elitist and/or hierarchical, and must it be? (Mainly addressed in Chapter 2)
2.  Who should pay for mathematics?  (Mainly addressed in Chapters 3, 4, and 10)
I wrote a three-part post about this last spring after realizing that what I wrote in the book lent itself to misinterpretation.  But the misinterpretations continue, in part as a result of some of the most recent reviews so I will write another post on this topic that I hope (but don’t expect) will settle it once and for all.
3.  Are mathematicians responsible for the uses to which our work is put?  What are the implications of “Faustian bargains” with funders? (Mainly addressed in Chapters 4 and 10)
4.  How to explain number theory (or topology, or dynamical systems) at a dinner party? (Mainly addressed in the obvious place)
5.  Must mathematicians have bodies?  (Addressed briefly in Chapter 6 and even more briefly in Chapter 7)

Historical problems

1.  Does mathematics belong to high or low culture? (Mainly addressed in Chapter 8)
2.  Should mathematics be seen as an Art or a Science, or both? (Mainly addressed in Chapters 3 and 10)
3.  How are Foundations of mathematics chosen? (Mainly addressed in Chapters 3 and 7)

Linguistic problems

1.  Does mathematics have a beginning and an end?  (Mainly addressed in the short first and last chapters, and in Chapter 7)
2.  Is mathematics created or discovered?  (Mainly addressed in Chapter 7; also in Chapter 3.  See also realism vs. nominalism, etc.)


1.  Is the image of mathematics in popular culture accurate, and if not, what can be done about it? (Mainly addressed in Chapters 6 and 8, as well as in the “How to Explain” dialogues)
2.  What does mathematics have (structurally and socially) in common with the arts, especially the visual arts?  (Mainly addressed in Chapter 3 and Chapter 10 and near the end of Chapter 8)
3.  What should mathematicians write or think about the motivations of literary authors, specifically (but not exclusively) authors of fiction, who make allusion to sophisticated mathematics in their writings?  (Mainly addressed in Bonus Chapter 5, but also in the Science Wars)

I was inspired to write this list by some of Mike Shulman’s recent comments (this one, for example), and more directly after reading a recent post on the always-engaging MathTango website.  The inimitable Shecky Riemann showed excellent taste in choosing Siobhan Roberts’s biography of John Conway as his “favorite popular math book” of 2015.  And I can’t fault his taste in picking MWA for the second slot.  Shecky writes:

Harris, more than any mathematician I’ve read, has a knack for saying things that sound interesting, but that are just vague or ambiguous enough to leave one uncertain of what his exact point is. That sounds like a criticism, but in some perverse way it makes his writing all the more thought-provoking and engaging…

Perhaps the exact point is that I’m uncertain (and maybe you should be too).