Noncommutative Analysis

Category: politics in math

Thirty one years later: a counterattack on Halmos’s critique of non-standard analysis

As if to celebrate in an original way the fifty year anniversary of Bernstein and Robinson’s solution to (a generalization of) the Smith-Halmos conjecture (briefly, that if T is an operator such that p(T) is compact for some polynomial p, then T has an invariant subspace), several notable mathematicians posted a interesting and very nonstandard (as they say) paper on the arxiv.

This paper briefly tells the story regarding the publication of this paper, in which Bernstein and Robinson use Robinson’s new theory of non-standard analysis (NSA) to prove the above mentioned conjecture in operator theory. This was one of the first major successes of NSA, and perhaps one would think that all of the operator theory community should have accepted the achievement with nothing but high praise. Instead, it was received somewhat coldly: Halmos went to work immediately to translate the NSA proof and published a paper proving the same result, with a proof in “standard” operator theoretic terms. (See the paper, I am leaving out the juicy parts). And then, since 1966 until 2000 (more or less), Halmos has been apparently at “war” with NSA (in the paper the word “battle” is used), and has also had criticism of logic; for example, it is implied in his book that he did not always consider logic to be a part of mathematics, worse, it seems that he has not always considered logicians to be mathematicians. (When I wrote about Halmos’s book a few months ago, I wrote that I do not agree with all the opinions expressed in the book, and I remember having the issue with logic and logicians in my mind when writing that).

In the paper that appeared on the arxiv today, the authors take revenge on Halmos. Besides a (convincing) rebuttal of Halmos’s criticisms, the seven authors hand Halmos at least seven blows, not all of them below the belt. The excellent and somewhat cruel title says it all: A non-standard analysis of a cultural icon: the case of Paul Halmos.

Besides some feeling of uneasiness in seeing a corpse being metaphorically stabbed (where have you been in the last thirty years?), the paper raises interesting issues (without wallowing too much on either one), and may serve as a lesson to all of us. There is nothing in this story special to operator theory versus model theory, or NSA, or logic. The real story here is the suspicion and snubbish-ness of mathematicians towards fields in which they do not work, and towards people working in these fields.

I see it all the time. Don’t kid me: you have also seen quite a lot of it. It is possible, I confess, that I have exercised myself a small measure of suspicion and contempt to things that I don’t understand. As the authors of the paper hint, these things are worse than wrong – they might actually hurt people.

Anyway, many times people who are ignorantly snobbish to other fields end up looking like idiots. Stop doing that, or thirty years from now a mob of experts will come and tear you to shreds.

P.S. – It seems that the question of who was the referee of the Bernstein-Robinson paper is not settled, though some suspect it was Halmos. Well, if someone could get their hands on the (anonymous!) referee report (maybe Bernstein or Robinson kept the letter?), I am quite sure that if it was Halmos, it would be clear. In other words, if Bernstein or Robinson suspected that it was him on account of the style, then I bet it was.


P.P.S. – regarding the theorem starting this discussion the quickest way to understand it is via Lomonosov’s theorem. The invariant subspace theorem proved by Bernstein and Robinson (polynomially compact operator has an invariant subspace) is now superseded by Lomonosov’s theorem (google it for a simple proof), which says that every bounded operator on a Banach space that commutes with a nonzero compact operator has a non-trivial invariant subspace.

Something sweet for the new year

Tim Gowers recently announced the start of a new journal, “Discrete Analysis”. The sweet thing about this journal is that it is an arxiv overlay journal, meaning that the journal will act like most other elctronic journals with the difference that all it does in the end (after standard peer review and editorial decisions) is put up a link on its website to a certain version of the preprint on the arxiv. The costs are so low, that neither readers nor authors are supposed to pay. In the beginning, Cambridge University will cover the costs of this particular journal, and there are hopes that funding will be found later (of course, arxiv has to be funded as well, but this does not seem to incur additional costs on arxiv). The journal uses a platform called Scholastica (which does charge something, but relatively low – like $10 per paper) so they did not have to set up their webpage and deal with that kind of stuff.

The idea has been around for several years and there are several other platforms (some of which do not charge anything since they are publicly funded) for carrying journals like this: Episciences, Open Journals. It seems like analysis, and operator theory in particular, are a little behind in these initiatives (correct me if I am wrong). But I am not worried, this is a matter of time.

The news of the baby journal made me especially happy since leaders like Gowers and Tao have been previously involved with the creation of the bad-idea-author-pay-journals Forum of Mathematics (Pi and Sigma), and it is great that their stature is also harnessed for a decent journal (which also happens to have a a nice and reasonable name).

Just a link to another blog I would like to put up

One of my favorite blogs written by a mathematician is Izabella Laba’s “The accidental mathematician“. The title of her blog in itself is enough to make it one of my favourites. Some of her posts on political-academic issues, especially gender, were eye openers for me. Her recent post is another powerful piece.

There is one particularly troubling paragraph. (Brought here out of context. You have to read her post for context).

Here is the first half of the paragraph, which is a statement worth considering in the context of academia, even without the context of discrimination:

We gerrymander research areas so as to keep in the people we choose and exclude those we would rather keep out. Even those gerrymandered borders can fluctuate, expanding when more names are needed on a funding application and then shrinking back when the benefits are shared. We define “interesting and exciting” as that which interests and excites those colleagues whose opinions we respect, and we respect them the most when they agree with us. We cite the “enthusiasm” of colleagues, or lack thereof, as though it were an objective and quantifiable measure of worth.

For completeness, here is the last part of the paragraph – a statement that can be made also outside the context of math, and is too worthy of consideration:

We care deeply about those women and minorities who are absent, hypothetical, or nonexistent, devising elaborate strategies to attract them and treat them fairly, but ignore those who are already there, standing right in front of us and asking for the same resources that their colleagues have been enjoying all along.

Interesting figure

I found an interesting figure in the March 2014 issue of the EMS newsletter, from the article by H. Mihaljevic´ -Brandt and O. Teschke, Journal Profiles and Beyond: What Makes a Mathematics Journal “General”?

See the right column on page 56 in this link. (God help me, I have no idea how to embed that figure in the post. Anyway, maybe it is illegal, so I don’t bother learning.) One can see the “subject bias” of Acta, Annals and Inventiones.

On the left column, there is a graph showing the percentage of papers devoted to different MSC subjects in what the authors call “generalist” math journals (note carefully that these journals are only a small subclass of all journals, chosen by a method that is loosely described in the article). On the right column there is the interesting figure, showing the subject bias. If I understand correctly, the Y-axis is the MSC number and the X-axis represents the corresponding deviation from the average percentage given in the left figure. So, for example, Operator Theory (MSC 47) is the subject of about 5 percent of the papers in a generalist journal, but in the Annals there is a deviation of minus 4 from the average, so if I understand this figure correctly, that means that about 1 percent of papers in the Annals are classified under MSC 47. Another example: Algebraic Geometry (MSC 14), takes up a significant portion of Inventiones papers, much more than it does in an average “generalist” journal.

(I am not making any claims, this could mean a lot of things and it could mean nothing. But it is definitely interesting to note.)

Another interesting point is that the authors say that of the above three super-journals, Acta “is closest to the average distribution, though it is sometimes considered as a journal with a focus on analysis”. That’s interesting in several ways.


Worse than Elsevier, worse than …

I recently received the following email from Cambridge University Press:

Submit your article online now

Dear Dr Shalit
We are delighted to announce that the online submission systems forForum of Mathematics, Pi and Forum of Mathematics, Sigma are now live. Forum of Mathematics offers fully open access publication combined with peer-review standards set by an international editorial board of the highestcalibre, and all backed by Cambridge University Press and our commitment to quality.
Don’t forget:
  • For the first three years Cambridge University Press will waive the publication charges
  • After this, a publication charge for authors will be set at £500/$750, this charge being based on real publishing costs and overheads
etc., …,
Authors benefit from:• Peer-review by experts

• Free, permanent, worldwide access to your article
• High editorial and production service
• The author will hold the copyright of published papers via a Creative Commons license
• State-of-the-art online hosting, including forward reference linking and extensive content alerts
• Free online colour
• Global dissemination of your paper
Kind regards,
Cambridge Journals

To which I replied:

Dear ************,

I will not submit a journal to FOM because I strongly object to author processing charges, and your journal endorses this practice.

Kind regards,
Orr Shalit

I blogged on this subject before. I just want to add here that in my opinion, the fact that FOM does no collect money from authors for the first three years does not make it better than other predatory journals, it makes it worse. Cambridge University Press uses its prestige to endorse the practice of author charges, and it uses its money to make it look sustainable, to get us used to the idea.

I really hope that mathematicians will not flock behind the leaders of this initiative. The overall impression I get is that my hopes are hopeless. So here is one last cry: you are going in the wrong direction! Even if dpearments change so much that everybody gets a fair budget for publishing (which I find hard to believe), do we really need another item in our academic lives where we have to fill in a form to get $750 for a simple academic activity? And how can one possibly consider submitting a paper to FOM when one can submit to some other journal for free and save one’s department $750?

 Here is an example of how to do it right.

Reflections on the New York Journal of Mathematics

As I have just announced in a previous post, Matt Kennedy and I have just published a paper in the New York Journal of Mathematics.

The New York Journal of Mathematics is a nonprofit electronic journal, which posts papers openly online so that anyone can read them without any subscription fee. And of course (funny that this has to be noted) it does not require that authors pay for having their papers published. It exists simply for the benefit of mathematical research and the mathematical community. This is how journals should be. There are others like it: there is the BJMA in which I have published in once. See also the list of free online math journals here.

The NYJM is more than a community project – it is a good general math journal. How do I know? The same way I know that other good journals are good: first, I take a look at the editorial board, and I see that there are distinguished mathematicians among the editors (and most importantly for me, I check that there is an editor who is close to my field so he/she will know what to do with my submission); second, I check to see if mathematicians whom I know and highly respect have published there; third, just to be on the safe side, I can browse the index and see if any famous mathematicians which I have heard of have published there too; fourth, I check to see if the journal is on MathSciNet’s Citation Database Reference List (it is); after that I may or may not decide to submit (and this of course also depends on what my coauthor thinks), and if I submit I also get an impression of how professional, smooth and fast the publishing process is. My impression from my recent experience is that the publishing process in NYJM is as professional, smooth and fast as I could hope for.

Unfortunately, some committees which make decisions regarding tenure and promotion also need to decide if the journals in which candidates publish are good journals. There are several “bibliometrical” tools which help committees and administrators figure out if journals are any good. Here at BGU the tool usually used is something called ISI Web of Knowledge. Now guess what ISIWoK says about NYJM. Seriously, guess: do you think that ISIWoK says that NYJM is a good journal or an OK journal or a bad journal?

HA! Trick question! According to ISIWoK, the New York Journal of Math doesn’t exist. There is no such journal. Now, the NYJM has been coming out since 1994, so somebody at ISIWoK hasn’t been doing a very good job. Or maybe they have?

Well: luckily my university has decided to treat NYJM as a real journal (I am sorry to admit that I probably would not have published there otherwise). Unfortunately, there is still a way to go: my university still uses ISIWoK to count citations, so for this paper of mine there will be no data. I hope that this will change before I am up for promotion.


UPDATE February 5, 2013: Mark Steinberger commented below that NYJM is now covered by Thomson-Reuters Web of Science, and that this is retroactive to Volume 16 (2010).

Worse than Elsevier

Recently, I received the following email:

Dear Dr. Shalit,

I am writing to inquire whether you have received our previous email inviting you to submit an article to the Special Issue on “Uncertain Dynamical Systems: Analysis and Applications,” which will be published in Abstract and Applied Analysis, and the deadline for submission is October 19th, 2012.

Looking forward to hearing from you.

Best regards,


To this, I replied:

Dear ************,
I am sorry, I did not realized that you were waiting for an answer from me.
The special issue sounds interesting, but I do not submit papers to journals that require processing charges from the authors.
Best regards,
Orr Shalit

This has been my opinion for a long time, and it didn’t change when Gowers and Tao joined the bad guys. Here’s what I think is bad about the publishing model where authors pay to have their papers published.

  1. There is an obvious conflict of interests here, which might corrupt science.
  2. These journals always seemed to me to be a nasty way to wring money out of mathematicians that either don’t know better, don’t believe in their own worth, or couldn’t (for some reason) publish their work in a normal journal.
  3. It will decrease mobility: it creates another obstacle for mathematicians with no grant money or from weaker institutions, making it harder for them to eventually get grants and move to perhaps stronger institutions.
  4. And even if I do have grant money, that’s not how I want to spend it.

And don’t tell me that in the eighteenth century or ancient Greece scientists payed to have their work published: because here people are not paying to have their work published – everybody’s work is published on the web if they wish it – here people are paying to have their work published inside a journal, meaning that they are buying their work’s credibility.

Only two good things about this model. First, it is open-access, which is great, but as I’ve said that doesn’t matter any more, since all papers are open access anyway (even if the official journal version isn’t). Second good thing, and this is really a good thing: in this model people have to think about what they are sending for publication, because publishing also has a price. So hopefully this can create eventually a situation where people publish a little less papers, but these papers are more complete and contain less repetition.

That last point is really is something to think about. I can think of at least one different means of attaining this goal: tenure.