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We deserve better journals
By and large, math journals treat the authors like a pesky annoyance, sort of the way a local electric company treats its customers. As in — yes, serving you is our business, but if you don’t like our customer service where else are you going to go? Not all editors operate that way, absolutely not all referees, but so many it’s an accepted norm. We all know that and all play some role in the system. And we all can do better, because we deserve better.
In fact, many well meaning mathematicians do become journal editors, start new journals, and even join the AMS and other professional societies’ governing bodies which oversee the journals. This helps sometimes, but they quickly burn out or get disillusioned. At the end, this only makes second order improvements while the giant sclerotic system continues its descent from bad to worse.
Like everyone else, I took this as a given. I even made some excuses: evil publishers, the overwhelming growth of submissions, everyone stressed and overworked, papers becoming more technical and harder to referee, etc., etc. For decades I watched many math journals turn from friendly if not particularly warm communal endeavors, to zones of hostility.
Only most recently, it occurred to me that it doesn’t have to be this way. We should have better journals, and we deserve a better treatment (I was really off the mark in my first line of this post). Demanding better journals is neither a fantasy nor a manifesto. In fact, physicists have already figured it all out. This post is largely about how they do it, with some lessons and suggestions.
What we have
If you don’t know what I am talking about, walk to any mathematician you see at a conference. If you have a choice, choose the one who looks bored, staring intensely at their shoes. Ask them for their most frustrating journal publishing story. You may as well sit down — the answer might take awhile. Even if they don’t know you (or maybe especially if they don’t know you), they will just unload a litany of the most horrifying stories that would make you question the sanity of people staying in this profession.
Then ask them why do they persevere and keep submitting and resubmitting their papers given that the arXiv is a perfectly fine way to disseminate their work. You won’t hear a coherent answer, but rather the usual fruit salad of practical matters: something about jobs, CVs, graduate students, grants, Deans, promotions, etc. Nobody will ever mention that their goal is to increase their readership, verify the arguments, improve their presentation style, etc., ostensibly the purpose of mathematical journals.
While my personal experience is a relatively happy one, I do have some scars to show and some stories to tell (see this, that and a bit in that blog posts on publishing struggles). There is no need to rehash them. I also know numerous stories of many people because I have asked them these questions. In fact, every time I publish something like this blog post (about the journals’ hall of shame), I get a host of new horror stories by email, with an understanding that I am not allowed to share them.
The adversarial relationship and countless bad experiences make it is easy to lose sight of the big picture. In many ways we are privileged in mathematics to have relatively few bad and for-profit actors. Money and grant funding matters less. We don’t have extreme urgency to publish. We have some relatively objective ways to evaluate papers (by checking the proofs). One really can work on the Moon, as long as one has a laptop and unlimited internet (and breathable air, I suppose).
We have it good, or at least we did when we started sliding into abyss. Because the alarms are not ringing, the innovation in response has stuttered. We are all just chugging along. Indeed, other than a few new online journals, relatively little has changed in the past two decades.
This is in sharp contrast with physics, which had very few of the advantages that math has (depending on the area). Besieged on all sides, physics community was forced to adapt faster and arguably better in response to changes in the publishing landscape. In fact, the innovations they made are so natural to them, their eyes open wide in disbelief when they hear how we continue to publish math papers.
The following is a story of the Physical Review E (PRE), one of the journals of the American Physical Society (APS). I will start with what I learned about the PRE and APS inner working, their culture, successes and challenges, some of which ring very familiar. Only afterwards I will get back to math publishing, the AMS and how we squandered our advantages.
What’s special about PRE?
I chose to write about the PRE because I published my own paper there and enjoyed the experience. To learn more about the journal, I spoke to a number of people affiliated with PRE in different capacities, from the management to members of the Editorial Board, to frequent authors and reviewers. These interviews were rather extensive and the differences with the math publishing culture are much too vast to summarize in a single blog post. I will only highlight things I personally found remarkable, and a few smaller things that can be easily emulated by math journals.
PRE’s place in the physics journal universe
PRE is one of five similarly named “area journals”: PRA, PRB, etc. More generally, it is one of 18 journals of the APS. Other journals include Physical Review Letters (PRL is APS’s flagship journal which published only very short papers), Physical Review X (PRX is another APS’s leading journal, online only, gold open access, publishes longer articles, extremely selective), Reviews of Modern Physics (APS’s highest cited journal which publishes only survey articles), and a number of more specialized journals.
The APS is roughly similar to the the AMS in its prominence and reach in the US. APS’s main publishing competition include the Institute of Physics (IOP, a UK physics society with 85 titles, roughly similar to the LMS), Nature Portfolio (a division of Springer Nature with 156 titles only a few of them in physics), and to a lesser extent Science by AAAS, various Elsevier, SIAM journals, and some MDPI titles.
Journal structure
The PRE editorial structure is rather complicated. Most of the editorial work is done by an assortment of Associate Editors, some of whom are employed full time by the APS (all of them physics PhD’s), and some are faculty in physics or adjacent fields from around the world, typically full time employed at research universities. Such Associate Editors receive a 2 year renewable contract and sometimes work with the APS for many years. Both professional and part time editors do a lot of work handling papers, rejecting some papers outright, inviting referees, etc.
The leadership of PRE is currently in flux, but until recently included Managing Editor, a full time APS employee responsible for running the journal (such as overseeing the work of associate editors), and a university based Lead Editor overseeing the research direction. The APS is currently reviewing applications for a newly created position of Chief Editor who will presumably replace Managing Editor, and is supposed to oversee the work of the Lead Editor and the rest of the editorial team (see this ad).
There is also an “Editorial Board”, whose name might be confusing to math readers. This is really a board of appeals (more on this later), where people serve a 3 year term without pay, giving occasional advice to associate editors and lending their credibility to the journal. Serving on the Editorial Board is both a service to the community and minor honor.
Submissions
The APS is aware of the role the arXiv plays in the community as the main dissemination venue, with journals as an afterthought. So it encourages submissions consisting of arXiv numbers and subject areas. Note that this makes it different from Nature and Science titles, which forbid arXiv or other online postings both for copyright reasons and so not to spoil future headline worthy press releases.
The submissions to all APS journals are required to be in a house two column style with a tiny font. Тhere are sharp word count limits for the “letters” (short communications) and the “articles”. These are rather annoying to calculate (how do you count formulas? tables?), and the journals’ online software is leaves much to be desired.
Desk rejections
At PRE, about 15-20% of all papers are rejected within days after the initial screening by managing or associate editors, who then assign the remaining papers according to research areas. Some associate editors are reluctant to do this at all, and favor at least one report supplemented by initial judgement. This percentage is a little lower than at the (more selective) PRL where it is reported to be 20-25%. Note that all APS journals pay special attention to the style, so it’s important to make an effort to avoid being rejected by a non-expert just because of that.
Curiously, before 2004, the percentage was even lower at PRL, but the APS did some rather interesting research on the issue. It concluded that such papers consume a lot of resources and rarely survive the review process (see this report). Of course, this percentage is relatively low by math standards — several math journals I know have about 30-50% desk rejections, with another 30-40% after a few quick opinions. On the other hand, at Science, over 83% papers get rejected without an external review.
Review process
Almost all the work is handled by associate editors closest to the area. The APS made a major overhaul of its classification of physics areas in 2016, to bring it to modern age (from the old one which resembles the AMS MSC). Note aside: I have been an advocate for an overhaul of MSC for a while, which I called a “historical anachronism” in this long MO answer (itself written about 14 years ago). At the very least the MSC should upgrade its tree structure (with weird horizontal “see also…” links) to a more appropriate poset structure.
Now, associate editors start with desk rejections. If the paper looks publishable, they send it to referees with the goal of obtaining two reports. The papers tend to be much shorter and more readable by the general scientific audience compared with the average math paper, and good style is emphasized as a goal. The reviewers are given only three weeks to write the report, but that time can be extended upon request (by a few more weeks, not months).
Typically, editors aim to finish the first round in three months, so the paper can be published in under six months. Only few papers lag beyond six months at which point, the editors told me, they get genuinely embarrassed. The reason is often an extreme difficulty in finding referees. Asking 4-8 potential referees is normal, but on rare occasions the numbers can be as high as 10-20.
Acceptance rate
In total, PRE receives about 3,500-4,000 submissions a year, of which about 55-60% get accepted, an astonishingly high percentage when compared to even second tier math journals. The number of submissions has been slowly decreasing in recent years, perhaps reflecting many new publications venues. Some editors/authors mentioned MDPI as new evil force (I called MDPI parasitic rather than predatory in this blog post).
For comparison, PRL is an even bigger operation which handles over twice as many papers. I estimate that PRL accepts roughly 20-25% of submissions, probably the lowest rate of all APS journals. In a more extreme behavior, Nature accepts about 8% submissions to publish about 800 papers, while Science accepts about 6% submissions to publish about 640 papers per year.
It is worth putting number published paper in perspective by comparing them with other journals. PRE and PRL publish about 1,800 and 2,100 papers per year, respectively. Other APS journals publish even more: PRD publishes about 4,000, and PRB close to 5,000 papers a year.
For math journals true acceptance ratios are hard to find and these numbers tend to be meaningless anyway due to self-selection and high cost of waiting for rejection. But numbers of published papers are easily available: Jour. AMS publishes about 25, Mathematika about 50, Proc. LMS about 60, Forum Math. Sigma in the range of 60-120, Bull. LMS in the range of 100-150, Trans. AMS about 250, Adv. Math. about 350, IMRN in the range of 300-500, and Proc. AMS about 450 papers per year. These are boutique numbers compared to the APS editorial machine. In the opposite extreme, MDPI Mathematics recently achieved the output of about 5,000 papers a year (I am sure they are very proud).
Publication
When a paper is accepted at PRE, it is sent to production which APS outsources. There are two quick rounds of approval of LaTeX versions compiled in the house style and proofread by a professional. It then gets published online with a unique identifier, usually within 2-3 weeks from the date of acceptance. Old fashioned volumes and numbers do exist, but of no consequence as they are functions of the publication date. There is zero backlog.
Strictly speaking there is still a print version of the PRE. I was told it is delivered to about 30 libraries worldwide that apparently are unconcerned with deforestation and willing to pay the premium. In truth, nobody really wants to read these paper versions. The volumes are so thick and heavy, it is hard to even lift them up from a library shelf. Not to dwell on this too much, but some graduate students I know are unaware even which building houses our math library at UCLA. It’s hard to blame them, especially after COVID…
Appeals
When a paper is rejected, the authors have the right to appeal the decision. The paper is sent to a member of the Editorial Board closest to the area. The editor reads both the paper and the referee reports, then writes their own report, which they sign and send to the authors. More often than not the decision is confirmed, but reversals do happen.
Since what’s “important” is ultimately subjective, appeals serve an important check on Associate Editors and helps keep peace in the community. Numerically, only about 3-5% of rejected papers are sent for an appeal, about 2-3 papers per Editorial Board member each year.
Embarrassingly for the whole field, I cannot think of a single math journal with an appeals process (except, interestingly, for MDPI Mathematics, which famously has the selectivity of a waste bucket). Even Nature has an appeals process, and nobody ever thinks of them as too friendly.
Note: some math journals do allow resubmissions of previously rejected papers. These papers tend to be major revisions of previous versions and typically go the same editor, defeating the point of the appeal.
Editorial system
The APS has its own online editorial system which handles the submissions, and has an unprecedented level of transparency compared to that of math journals I am familiar with. The authors can see a complete log of dates of communications with (anonymized) referees, the actions of editors, etc. In math, the best you can get is “under review” which brings cold comfort.
The editors work as a team, jointly handling all incoming email and submission/resubmission traffic. Routine tasks like forwarding the revision to the first round referees are handled by first person available, but the editorial decisions (accept/reject, choices of referees), are made by the assigned Associate Editor. If an Associate Editor has a week long backlog or is expecting some inactivity, his queue is immediately redistributed between other editors.
Relations between APS journals
Many PRE papers first arrive to PRL where they are quickly rejected. The editorial system allows editors from one journal see all actions and reports in all other APS journals. If the rejected PRL paper fits the scope of PRE and there are reports suggesting PRE might be suitable, PRE editors try to invite such papers. This speeds up the process and simplifies life to everyone involved.
For longer papers, PRE editors also browse rejections from PRX, etc. From time to time, business oriented managers at the APS raise a possibility of creating a lower tier journal where they would publish many papers rejected from PRA–PRE (translation: “why shouldn’t APS get some of MDPI money?”), but the approach to maintain standards keep winning for now. From what I hear, this might change soon enough…
Note: In principle, several editorial systems by Elsevier and the like, do allow transferring papers between math journals. In practice, I haven’t seen this feature ever used (I could be wrong). Additionally, often there are firewalls which preclude editors in one journal from see reports in the other, making the feature useless.
Survey articles
The APS publishes Reviews of Modern Physics, which is fully dedicated to survey articles. Associate Editors are given a budget to solicit such articles and incentivize the authors by paying them about $1,500 for completion within a year, but only $750 is the project took longer. The articles vary in length and scope, from about 15 to about 70 pages (when converted from APS to the bulky AMS style, these pages numbers would more than double). There are also independent submissions which very rarely get accepted as the journal aims to maintain its reputation and relevance. Among all APS publications, this journal is best cited by a wide margin.
We note that there are very few math journals dedicated to surveys, despite a substantial need for expository work. Besides Proc. ICM and Séminaire Bourbaki series which are by invitation only, we single out the Bull. AMS, EMS Surveys and Russian Math Surveys (in Russian, but translated by IOP). Despite Rota’s claim “You are more likely to be remembered by your expository work“, publishing surveys remains difficult unless you opt for a special issue or a conference proceedings. In the last two years I wrote two rather long surveys — on combinatorial interpretations and on linear extensions. Word of advice: if you want to have an easy academic life I don’t recommend doing that — they just eat up your time.
At PRE, there are no surveys, but the editors occasionally solicit “perspectives”. These are forward looking articles suggesting important questions and directions (more like public NSF grant applications than surveys). They publish about five such articles a years, hoping to bring the number up to about ten in the future.
Profiled articles
In 2014, following the approach of popular magazines, PRE started making “Editors’ Suggestions”. These are a small number of articles the editors chose to highlight, both formally and on the website. They are viewed as minor research award that can be listed on CVs by the authors.
Outstanding referee award
The APS instituted this award in 2008, to encourage quick and thorough refereeing. This is a lifetime award and comes with a diploma size plaque which can be hang on the wall. More importantly, it can be submitted to your Department Chair and your friendly Dean as a community validation of your otherwise anonymous efforts.
Each year, there are a total of about 150 awardees selected across all APS journals (out of tens of thousands referees), of which about 10 are from PRE. This selection is taken very seriously. The nominations are done by Associate Editors and then discussed at the editorial meetings. For further details, see this 2009 article about the award by the former Editor-in-Chief of Physical Reviews, which ends with
We feel that the award program has been most successful, and we will be continuing it at APS. [Gene D. Sprouse, Recognizing referees at the American Physical Society]
Note that such distinguished referee awards are not limited to APS or even physics. It’s a simple idea which occurred to journals across “practical” disciplines: accounting, finance, economic geography, economics, public management, regional science, etc., but also e.g. in atmospheric chemistry and philosophy. Why wouldn’t a single math journal have such an award?? Count be flabbergasted.
Community relations
As we mentioned above, in much of physics, the arXiv is a preferred publication venue since the field tends to develop at rapid pace, so strictly speaking the journal publications are not necessary. In some areas, a publication in Nature or Science is key to success, especially for a junior researcher, so the authors are often willing to endure various associated indignities (including no arXiv postings) and if successful pay for the privilege. However, in many theoretical and non-headline worthy areas, these journals are not an option, which is where PRL, PRE and other APS journals come in.
In a way, PRE operates as a digital local newspaper which provides service to the community in the friendliest way possible. It validates the significance of papers needed for job related purposes, helps the authors to improve the style, does not bite newcomers, and does not second guess their experimental finding (there are other venues which do that). It provides a quick turn around and rarely rejects even moderately good papers.
When I asked both the editors and the authors how they feel about PRE, I heard a lot of warmth, the type of feeling I have not heard from anyone towards math journals. There is a feeling of community when the editors tell me that they often publish their own papers at PRE, when the authors want to become editors, etc. In contrast, I heard a lot of vitriol towards Nature and Science, and an outright disdain towards MDPI physics journals.
It could be that my sample size was too small and heavily biased. Indeed, when I polled the authors of MDPI Mathematics (a flagship MDPI journal), most authors expressed high level of satisfaction with the journal, that they would consider submitting there again. One of my heroes, Ravi P. Agarwal who I profiled in this blog post, published an astounding 37 papers in that journal, which clearly found its target audience (so much that it stopped spamming people, or maybe it’s just me).
Note aside: Personally, the only journal I actually cared about was the storied JCTA where my senior colleague Bruce Rothschild was the Editor in Chief for 25 years, and where I would publish my best combinatorics papers. In 2020, the editorial board resigned in mass and formed Combin. Theory. I am afraid, my feelings have not transferred to CT, nor have they stayed with JCTA which continues to publish. They just evaporated.
Money matters
Despite a small army of professional editors, the APS journals provide a healthy albeit slowly decreasing revenue stream (about $43 mil. in 2022, combined from all journals, see 2022 tax disclosures on ProPublica website). The journals are turning a profit for the APS (spent on managers and various APS activities) despite all the expenses. They are spending more and making more money than the AMS (compare with their 2022 tax disclosures on ProPublica). There is much more to say here, but this post is already super long and the fun part is only starting.
Back to math journals
In the 20th century world with its print publishing, having a local peer review print journals made sense. A university of a group of universities would join forces with a local publisher and starts the presses. That’s where local faculty would publish their own papers, that’s where they would publish conference proceeding, etc. How else do you explain Duke Mathematical Journal, Israel Journal of Mathematics, Moscow Mathematical Journal, Pacific Journal of Mathematics, and Siberian Journal of Mathematics? I made a lot of fun at the geographical titles in this blog post, and I maintain that they sound completely outdated (I published in all five of these, naturally).
Now, in the 21st century, do we really need math journals? This may sound like a ridiculous question, with two standard replies:
- We need peer review, i.e. some entity must provide a certificate that someone anonymous read the paper and takes responsibility for its validity (sound weak isn’t it?).
- We need formal validation, i.e. we need to have something to write on our CVs. Different journals have different levels of prestige associated with them leading to distinctions in research recognition (and thus jobs, promotions, grants, etc.)
Fair enough, but are you sure that the journals as we have them are the best vehicles for either of these goals? Does anyone really believes that random online journals do a serious peer review? Where is this idea coming from, that the journals with its obvious biases should be conferring importance of the paper?
How are we supposed to use journals to evaluate the candidates, if these journals have uncertain rankings and in fact the relative rankings of two journals can vary depending on the area? Shouldn’t we separate the peer review aspect which makes multiple submission costly and unethical, from the evaluation aspects which desperately needs competition between the journals?
Again, this all sounds ridiculous if you don’t step back and look objectively at our publishing mess where a math paper can languish in journals for over a year, after which it is returned without a single referee report just because someone decided that at the end the paper is not good enough to be refereed. This happened to me multiple times, and to so many other people I lost count (in one instance, this happened after 3 years of waiting!)
Publishing utopia
Now, I know a lot of people whose dream publishing universe is a lot of run-by-mathematicians not for profit small online publications. It’s great to rid of Elsevier and their ilk, but it would not solve the issues above. In fact, this would bring a lot of anarchy and further loss of standards.
From my perspective, in a perfect world, “the people” (or at least the AMS), would create one mega journal, where the arXiv papers could be forwarded by the authors if they wish. Hundreds of editors (some full time, some part time) divided into arXiv subject areas, would make the initial screening, and keep say 30-40% of them to be send for review. Based on my reading of the arXiv stats, that gives about 10-15K papers a year to be refereed, a number way below what APS handles. The mega journal would only check validity and “publish” only based on correctness.
Publication at the mega journal would already be a distinction albeit a minor one. To ensure some competition, we would probably need to break this mega journal into several (say, 3-5) independently run baby megas, so the authors have a choice where to submit. In the utopia I am imagining, the level of rigor would be the same across all baby megas. It would also be a way to handle MDPI journals which would be left with a reject pile.
This wouldn’t take anything away from the top journals (think Annals) who would not want to outsource their peer review. In fact, I heard of major Annals papers studied by six (!) independent teams of referees, that’s above and beyond. But I also heard of Annals papers which seem to had no technical check at all (like this one by this guy), so the quality is maybe inconsistent.
So what about distinctions? The remnants of the existing general journals would be free from peer review. They would place bids on the best papers attracting them “modulo publication in the mega journal” with some clear set deadlines. The authors would accept the best bid, like graduate admissions, and the paper will be linked to the journal website in the “arXiv overlay” style.
Alternatively, some specialized or non-exclusive journals will make their own selections for best papers in their areas, which could be viewed as awards. One paper could get multiple such awards, and “best journal where the paper could be accepted” optimization issue would disappear completely. This would make a better, more fair world. At the very least, such awards would remove the pressure to publish in the top journals if you have a strong result.
Even better, one can imagine a competitive conference system in the style of CS theory conferences (but also in some areas of Discrete Math) emerging in this scenario. The conference submission could require a prior arXiv posting and later keep track of “verified” papers (accepted to the mega journal). When disentangled from the peer review, these conference could lead to more progress on emerging tools and ideas, and to even the playing field for researchers from small and underfunded universities across the world.
Note that there are already some awards for math papers given by third parties, but only a handful. Notably, AIM has this unusual award. More recently, a new Frontiers of Science Award was introduced for “best recent papers” (nice cash prize for a paper already published in the Annals and the like). Of course, most CS theory conferences have been giving them for decades (the papers later get published by the journals).
Would it work? Wouldn’t the mega journal be just another utility company with terrible service? Well, I don’t know and we will probably never get to find out. That’s why I called it a utopia, not a serious proposal. But it can hardly get any worse. I think pure math and CS theory are unique in requiring true correctness. When correctness is disentangled from evaluating novelty and importance, the point of the mega journal would be to help the authors get their proofs right and the papers accepted. Until then, journal editors (and referees to a smaller degree) have a conflict of interest — helping the authors might mean hurting the journal and vice versa. Guess who usually gets hurt at the end?
Back to reality
Obviously, I have no hopes that the “mega journal” would ever come to life. But NOT because it’s technically impossible or financially unsound. In other fields, communities manage somehow. The APS is a workable approximation of that egalitarian idea. Recently, eLife made another major experiment in publishing — we’ll see how that works out.
But in a professional society such as the AMS where new leadership handpicks two candidates for future leadership in a stale election? With a declining membership? Which claims the Fellow of the AMS award as it biggest achievement? Oh, please! Really, the best we can hope for is for a large “lower tier” journals with a high acceptance ratio. Why would AMS want that? I am glad you asked:
Case for higher acceptance rates at AMS journals
One argument why so few papers get published in good (think top 100) math journals is that math papers can be much longer than typical physics papers, so they take more print space and take longer to referee. However, this argument does not translate well into the digital age. Nor does that apply to Bull. LMS or Proc. AMS, of course, which publish mostly short papers. We mention in passing that while greater length is unavoidable sometimes, mathematicians tend to forget that brevity is a feature, not a bug.
Of course, math editors’ main argument in favor of low acceptance ratios is that this allows one to maintain high quality of papers. While true on its face, when applied uniformly this approach has major negative implications to the community.
Think of college acceptance rates. It’s true that Harvard maintains its prestige by having a ridiculously low acceptance ratio, and being private it’s hard to blame it (not that I am fan of the choices they make either, but this post is about something else). But should major public universities like UCLA do the same? What about community colleges? You see what I mean.
There is an obvious public good in AMS maintaining a large, free, friendly but thorough publication venue for papers that don’t meet the Trans. AMS threshold. This might not be the “mega journal” utopia, but it would be a major step forward. If SIAM, EMS, LMS and other major math societies set up something similar, we would actually be in a good place as the middle tier small journals would start changing their publishing model in response.
Short list of minor suggestions
As you can probably tell by now, in my opinion most math publishers are behind the curve in innovation and community relations. Let me summarize some basic ideas based on the discussion above that seem more approachable:
- Stop wasting paper and fully move to electronic publishing.
- Do not limit numbers of papers or pages. Rather, aim for as many good papers as you can.
- Improve your electronic editorial system to make it more transparent.
- Help editors work as a team, and incentivize them financially. Pay for 20% employment to experts across the world to help you run the journal.
- Set up new math journals fully dedicated to survey articles, both solicited and contributed.
- Create an appeals procedure and add a new type of senior editors who would take the job seriously.
- Institute a number of awards: for best long, short and survey articles in your journal, and for best referees. Make an effort to be fair by taking input from all editors.
Journal studies
If you read up to this point, you are probably wondering why most of these simple ideas hadn’t been widely discussed. Clearly, somebody is asleep at the wheel. Or, perhaps, doesn’t want to rock the boat (I am mixing my metaphors here, sorry). In case of for profit publishers like Springer and Elsevier, I can see why — they know all this stuff from their journals in other areas, but are very busy counting the money.
But the AMS Council can sure use a “Chair of journal innovation” whose job would be to conduct journal studies (like the many APS studies I mentioned above), or at least read other publishers’ studies. An amateur like me shouldn’t be able to tell you anything new that you couldn’t learn by googling. Perhaps, start by subscribing to an excellent newsletter Journalology fully dedicated to these ideas.
Acknowledgements.
I am extremely grateful to editors Dirk Jan Bukman, Alexander Kusenko, Valerio Lucarini, Mason Porter and Uwe Täuber, for kindly agreeing to be interviewed on the subject and for being so generous with their time. I am also thankful to several frequent APS contributors who wished to remain anonymous. If I misstated or misunderstood anything, the fault is all mine, obviously.
P.S. Mark Wilson kindly invited me to write a column for the AMS Notices on the issue of publishing. This prompted me to spend many hours thinking about the subject and talking to many physicists. At the end, I submitted a very short and non-polemical version of this blog post. If it ever gets accepted and published I will link it here.
UPDATE (August 14, 2024): The note has been accepted to the AMS Notices, and is available here. It is likely to appear in the January 2025 issue.
How I chose Enumerative Combinatorics
Apologies for not writing anything for awhile. After Feb 24, the math part of the “life and math” slogan lost a bit of relevance, while the actual events were stupefying to the point when I had nothing to say about the life part. Now that the shock subsided, let me break the silence by telling an old personal story which is neither relevant to anything happening right now nor a lesson to anyone. Sometimes a story is just a story…
My field
As the readers of this blog know, I am a Combinatorialist. Not a “proud one”. Just “a combinatorialist”. To paraphrase a military slogan “there are many fields like this one, but this one is mine”. While I’ve been defending my field for years, writing about its struggles, and often defining it, it’s not because this field is more important than others. Rather, because it’s so frequently misunderstood.
In fact, I have worked in other (mostly adjacent) fields, but that was usually because I was curious. Curious what’s going on in other areas, curious if they had tools to help me with my problems. Curious if they had problems that could use my tools. I would go to seminars in other fields, read papers, travel to conferences, make friends. Occasionally this strategy paid off and I would publish something in another field. Much more often nothing ever came out of that. It was fun regardless.
Anyway, I wanted to work in combinatorics for as long as I can remember, since I was about 15 or so. There is something inherently discrete about the way I see the world, so much that having additional structure is just obstructing the view. Here is how Gian-Carlo Rota famously put it:
Combinatorics is an honest subject. […] You either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. [Los Alamos Science, 1985]
I agree. Also, I really like to count. When prompted, I always say “I work in Combinatorics” even if sometimes I really don’t. But in truth, the field is much too large and not unified, so when asked to be more specific (this rarely happens) I say “Enumerative Combinatorics“. What follows is a short story of how I made the choice.
Family vacation
When I was about 18, Andrey Zelevinsky (ז״ל) introduced me and Alex Postnikov to Israel Gelfand and asked what should we be reading if we want to do combinatorics. Unlike most leading mathematicians in Russia, Gelfand had a surprisingly positive view on the subject (see e.g. his quotes here). He suggested we both read Macdonald’s book, which was translated into Russian by Zelevinsky himself just a few years earlier. The book was extremely informative but dry as a fig and left little room for creativity. I read a large chunk of it and concluded that if this is what modern combinatorics looks like, I want to have nothing to do with it. Alex had a very different impression, I think.
Next year, my extended family decided to have a vacation on a Russian “river cruise”. I remember a small passenger boat which left Moscow river terminal, navigated a succession of small rivers until it reached Volga. From there, the boat had a smooth gliding all the way to the Caspian Sea. The vacation was about three weeks of a hot Summer torture with the only relief served by mouth-watering fresh watermelons.
What made it worse, there was absolutely nothing to see. Much of the way Volga is enormously wide, sometimes as wide as the English channel. Most of the time you couldn’t even see the river banks. The cities distinguished themselves only by an assortment of Lenin statues, but were unremarkable otherwise. Volgograd was an exception with its very impressive tallest statue in Europe, roughly as tall as the Statue of Liberty when combined with its pedestal. Impressive for sure, but not worth the trip. Long story short, the whole cruise vacation was dreadfully boring.
One good book can make a difference
While most of my relatives occupied themselves by reading crime novels or playing cards, I was reading a math book, the only book I brought with me. This was Stanley’s Enumerative Combinatorics (vol. 1) whose Russian translation came out just a few months earlier. So I read it cover-to-cover, but doing only the easiest exercises just to make sure I understand what’s going on. That book changed everything…
Midway through, when I was reading about linear extensions of posets in Ch. 3 with their obvious connections to standard Young tableaux and the hook-length formula (which I already knew by then), I had an idea. From Macdonald’s book, I remembered the q-analogue of #SYT via the “charge“, a statistics introduced by Lascoux and Schützenberger to give a combinatorial interpretation of Kostka polynomials, and which works even for skew Young diagram shapes. I figured that skew shapes are generic enough, and there should be a generalization of the charge to all posets. After several long days filled with some tedious calculations by hand, I came up with both the statement and the proof of the q-analogue of the number of linear extensions.
I wrote the proof neatly in my notebook with a clear intent to publish my “remarkable discovery”, and continued reading. In Ch. 4, all of a sudden, I read the “P-partition theory” that I just invented by myself. It came with various applications and connections to other problems, and was presented so well, much nicer than I would have!
After some extreme disappointment, I learned from the notes that the P-partition theory was a large portion of Stanley’s own Ph.D. thesis, which he wrote before I was born. For a few hours, I did nothing but meditate, staring at the vast water surrounding me and ignoring my relatives who couldn’t care less what I was doing anyway. I was trying to think if there is a lesson in this fiasco.
It pays to be positive and self-assure, I suppose, in a way that only a teenager can be. That day I concluded that I am clearly doing something right, definitely smarter than everyone else even if born a little too late. More importantly, I figured that Enumerative Combinatorics done “Stanley-style” is really the right area for me…
Epilogue
I stopped thinking that I am smarter than everyone else within weeks, as soon as I learned more math. I no longer believe I was born too late. I did end up doing a lot of Enumerative Combinatorics. Much later I became Richard Stanley’s postdoc for a short time and a colleague at MIT for a long time. Even now, I continue writing papers on the numbers of linear extensions and standard Young tableaux. Occasionally, I also discuss their q-analogues (like in my most recent paper). I still care and it’s still the right area for me…
Some years later I realized that historically, the “charge” and Stanley’s q-statistics were not independent in a sense that both are generalizations of the major index by Percy MacMahon. In his revision of vol. 1, Stanley moved the P-partition theory up to Ch. 3, where it belongs IMO. In 2001, he received the Steele’s Prize for Mathematical Exposition for the book that changed everything…
Are we united in anything?
Unity here, unity there, unity shmunity is everywhere. You just can’t avoid hearing about it. Every day, no matter the subject, somebody is going to call for it. Be it in Ukraine or Canada, Taiwan or Haiti, everyone is calling for unity. President Biden in his Inaugural Address called for it eight times by my count. So did former President Bush on every recent societal issue: here, there, everywhere. So did Obama and Reagan. I am sure just about every major US politician made the same call at some point. And why not? Like the “world peace“, the unity is assumed to be a universal good, or at least an inspirational if quickly forgettable goal.
Take the Beijing Olympic Games, which proudly claims that their motto “demonstrates unity and a collective effort” towards “the goal of pursuing world unity, peace and progress”. Come again? While The New York Times isn’t buying the whole “world unity” thing and calls the games “divisive” it still thinks that “Opening Ceremony [is] in Search of Unity.” Vox is also going there, claiming that the ceremony “emphasized peace, world unity, and the people around the world who have battled the pandemic.” So it sounds to me that despite all the politics, both Vox and the Times think that this mythical unity is something valuable, right? Well, ok, good to know…
Closer to home, you see the same themes said about the International Congress of Mathematicians to be held in St. Petersburg later this year. Here is Arkady Dvorkovich, co-chair of the Executive Organizing Committee and former Deputy Prime Minister of Russia: “It seems to us that Russia will be able to truly unite mathematicians from all over the world“. Huh? Are you sure? Unite in what exactly? Because even many Russian mathematicians are not on board with having the ICM in St. Petersburg. And among those from “all over the world”, quite a few are very openly boycotting the congress, so much that even the IMU started to worry. Doesn’t “unity” mean “for all”, as in ∀?
Unity of mathematics
Frequent readers of this blog can probably guess where I stand on the “unity”. Even in my own area of Combinatorics, I couldn’t find much of it at all. I openly mocked “the feeling of unity of mathematics” argument in favor of some conjectures. I tried but could never understand Noga Alon’s claim that “mathematics should be considered as one unit” other than a political statement by a former PC Chair of the 2006 ICM.
So, about this “unity of mathematics”. Like, really? All of mathematics? Quick, tell me what exactly do the Stochastic PDEs, Algebraic Number Theory, Enumerative Combinatorics and Biostatistics have in common? Anything comes to mind? Anything at all? Ugh. Let’s make another experiment. Say, I tell you that only two of these four areas have Fields medals. Can you guess which ones? Oh, you can? Really, it was that easy?? Doesn’t this cut against all of this alleged “unity”?
Anyway, let’s be serious. Mathematics is not a unit. It’s not even a “patterned tapestry” of connected threads. It’s a human endeavor. It’s an assorted collection of scientific pursuits unconstrained by physical experiments. Some of them are deep, some shallow, some are connected to others, and some are motivated by real world applications. You check the MSC 2020 classification, and there is everything under the sun, 224 pages in total. It’s preposterous to look for and expect to find some unity there. There is none to be found.
Let me put it differently. Take poetry. Like math, it’s a artistic endeavor. Poems are written by the people and for the people. To enjoy. To recall when in need or when in a mood. Like math papers. Now, can anyone keep a straight face and say “unity of poetry“? Of course not. If anything, it’s the opposite. In poetry, having a distinctive voice is celebrated. Diverse styles are lauded. New forms are created. Strong emotions are evoked. That’s the point. Why would math be any different then?
What exactly unites us?
Mathematicians, I mean. Not much, I suppose, to the contrary of math politicians’ claims:
I like to think that increasing breadth in research will help the mathematical sciences to recognize our essential unity. (Margaret Wright, SIAM President, 1996)
Huh? Isn’t this like saying that space exploration will help foster cross-cultural understanding? Sounds reasonable until you actually think about what is being said…
Even the style of doing research is completely different. Some prove theorems, some make heavy computer computations, some make physical experiments, etc. At the end, some write papers and put them on the arXiv, some write long books full of words (e.g. mathematical historians), some submit to competitive conferences (e.g. in theoretical computer science), some upload software packages and experimental evidence to some data depositary. It’s all different. Don’t be alarmed, this is normal.
In truth, very little unites us. Some mathematicians work at large state universities, others at small private liberal arts colleges with a completely different approach to teaching. Some have a great commitment to math education, some spend every waking hour doing research, while others enjoy frequent fishing trips thanks to tenure. Some go into university administration or math politics, while others become journal editors.
In truth, only two things unites us — giant math societies like the AMS and giant conferences like ICMs and joint AMS/MAA/SIAM meetings. Let’s treat them separately, but before we go there, let’s take a detour just to see what an honest unrestricted public discourse sounds like:
What to do about the Olympics
The answer depends on who you ask, obviously. And opinions are abound. I personally don’t care other than the unfortunate fact that 2028 Olympics will be hosted on my campus. But we in math should learn how to be critical, so here is a range of voices that I googled. Do with them as you please.
Some are sort of in favor:
I still believe the Olympics contribute a net benefit to humanity. (Beth Daley, The Conversation, Feb. 2018)
Some are positive if a little ambivalent:
The Games aren’t dead. Not by a longshot. But it’s worth noting that the reason they are alive has strikingly little to do with games, athletes or medals. (L. Jon Wertheim, Time, June 2021)
Some like The New York Times are highly critical, calling it “absurdity”. Some are blunt:
More and more, the international spectacle has become synonymous with overspending, corruption, and autocratic regimes. (Yasmeen Serhan, The Atlantic, Aug. 2021)
yet unwilling to make the leap and call it quits. Others are less shy:
You can’t reform the Olympics. The Olympics are showing us what they are, and what they’ve always been. (Gia Lappe and Jonny Coleman, Jacobin, July 2021)
and
Boil down all the sanctimonious drivel about how edifying the games are, and you’re left with the unavoidable truth: The Olympics wreck lives. (Natalie Shure, The New Republic, July 2021)
What is the ICM
Well, it’s a giant collective effort. A very old tradition. Medals are distributed. Lots of talks. Speakers are told that it’s an honor to be chosen. Universities issue press releases. Yes, like this one. Rich countries set up and give away travel grants. Poor countries scramble to pay for participants. The host country gets dubious PR benefits. A week after it’s over everyone forgets it ever happened. Life goes on.
I went to just one ICM, in Rio in 2018. It was an honor to be invited. But the experience was decidedly mixed. The speakers were terrific mathematicians, all of them. Many were good speakers. A few were dreadful in both content and style. Some figured they are giving talks in their research seminar rather than to a general audience, so I left a couple of such talks in middle. Many talks in parallel sections were not even recorded. What a shame!
The crowds were stupefying. I saw a lot of faces. Some were friendly, of people I hadn’t seen in years, sometimes 20 years. Some were people I knew only by name. It was nice to say hello, to shake their hand. But there were thousands more. Literally. An ocean of people. I was drowning. This was the worst place for an introvert.
While there, I asked a lot of people how did they like the ICM. Some were electrified by the experience and had a decent enough time. Some looked like fish out of the water — when asked they just stared at me incomprehensively silently saying “What are you, an idiot?” Some told me they just went to the opening ceremony and left for the beach for the rest of the ICM. Assaf Naor said that he loved everything. I was so amused by that, I asked if I could quote him. “Yes,” he said, “you can quote me: I loved absolutely every bit of the ICM”. Here we go — not everyone is an introvert.
Whatever happened at the ICM
Unlike the Olympics, math people tend to be shy in their ICM criticism. In his somewhat unfortunately titled but otherwise useful historical book “Mathematicians of the World, Unite!” the author, Guillermo Curbera, largely stays exuberant about the subject. He does mention some critical stories, like this one:
Charlotte Angas Scott reported bluntly on the presentation of papers in the congress, which in her opinion were “usually shockingly bad” since “instead of speaking to the audience, [the lecturer] reads his paper to himself in a monotone that is sometimes hurried, sometimes hesitating, and frequently bored . . . so that he is often tedious and incomprehensible.” (Paris 1900 Chapter, p. 24)
Curbera does mention in passing that the were some controversies: Grothendieck refused to attend ICM Moscow in 1966 for political reasons, Margulis and Novikov were not allowed by the Soviet Union to leave the country to receive their Fields medals. Well, nobody’s perfect, right?
Most reports I found on the web are highly positive. Read, for example, Gil Kalai’s blog posts on the ICM 2018. Everything was great, right? Even Doron Zeilberger, not known for holding his tongue, is mostly positive (about the ICM Beijing in 2002). He does suggest that the invited speakers “should go to a ‘training camp‘” for some sort of teacher training re-education, apparently not seeing the irony, or simply under impression of all those great things in Beijing.
The only (highly controversial) criticism that I found was from Ulf Persson who starts with:
The congresses are by now considered to be monstrous affairs very different from the original intimate gatherings where group pictures could be taken.
He then continues to talk about various personal inconveniences, his misimpressions about the ICM setting, the culture, the city, etc., all in a somewhat insensitive and rather disparaging manner. Apparently, this criticism and misimpressions earned a major smackdown from Marcelo Viana, the ICM 2018 Organizing Committee Chair, who wrote that this was a “piece of bigotry” by somebody who is “poorly informed”. Fair enough. I agree with that and with the EMS President Volker Mehrmann who wrote in the same EMS newsletter that the article was “very counterproductive”. Sure. But an oversized 4 page reaction to an opinion article in a math newsletter from another continent seem indicative that the big boss hates criticism. Because we need all that “unity”, right?
Anyway, don’t hold your breath to see anything critical about the ICM St. Petersburg later this year. Clearly, everything is going to be just fantastic, nothing controversial about it. Right…
What to do about the ICM
Stop having them in the current form. It’s the 21st century, and we are starting the third year of the pandemic. All talks can be moved online so that everyone can watch them either as they happen, or later on YouTube. Let me note that I’ve sat in the bleachers of these makeshift 1000+ people convention center auditoriums where the LaTeX formulas are barely visible. This is what the view is like:
Note that the ICM is not like a sports event — there is literally nothing at stake. Also, there are usually no questions afterwards anyway. You are all better off watching the talks later on your laptop, perhaps even on a x1.5 speed. To get the idea, imagine watching this talk in a huge room full of people…. Even better, we can also spread out these online lectures across the time zones so that people from different countries can participate. Something like this World Relay in Combinatorics.
Really, all that CO2 burned to get humans halfway across the world to seat in a crowded space is not doing anyone any good. If the Nobel Prizes can be awarded remotely, so can the Fields medals. Tourism value aside, the amount of meaningful person-to-person interaction is so minimal in a large crowd, I am struggling to find a single good reason at all to have these extravaganzas in-person.
What to do about the AMS
I am not a member of any math societies so it’s not my place to tell them what to do. As a frequent contributor to AMS journals and a former editor of one of them, I did call on the AMS to separate its society business form the publishing, but given that their business model hinges on the books and journals they sell, this is unlikely. Still, let me make some quick observations which might be helpful.
The AMS is clearly getting less and less popular. I couldn’t find the exact membership numbers, but their “dues and outreach” earnings have been flat for a while. Things are clearly not going in the right direction, so much that the current AMS President Ruth Charney sent out a survey earlier this week asking people like me why do we not want to join.
People seem to realize that they have many different views on all thing math related and are seeking associations which are a better fit. One notable example is the Just Mathematics Collective which has several notable boycott initiatives. Another is the Association for Mathematical Research formed following various controversies. Note that there is a great deal of disagreements between these two, see e.g. here, there and there.
I feel these are very good developments. It’s healthy to express disagreements on issues you consider important. And while I disagree with other things in the article below, I do agree with this basic premise:
Totalitarian countries have unity. Democratic republics have disagreement. (Kevin Williamson, Against Unity, National Review, Jan. 2021)
So everyone just chill. Enjoy diverse views and opinions. Disagree with the others. And think twice before you call for “unity” of anything, or praise the ephemeral “unity of mathematics”. There is none.
2021 Abel Prize
I am overjoyed with the news of the Abel prize awarded to László Lovász and Avi Wigderson. You can now see three (!) Abel laureates discussing Combinatorics — follow the links in this blog post from 2019. See also Gil Kalai’s blog post for further links to lectures.
What if they are all wrong?
Conjectures are a staple of mathematics. They are everywhere, permeating every area, subarea and subsubarea. They are diverse enough to avoid a single general adjective. They come in al shapes and sizes. Some of them are famous, classical, general, important, inspirational, far-reaching, audacious, exiting or popular, while others are speculative, narrow, technical, imprecise, far-fetched, misleading or recreational. That’s a lot of beliefs about unproven claims, yet we persist in dispensing them, inadvertently revealing our experience, intuition and biases.
The conjectures also vary in attitude. Like a finish line ribbon they all appear equally vulnerable to an outsider, but in fact differ widely from race to race. Some are eminently reachable, the only question being who will get there first (think 100 meter dash). Others are barely on the horizon, requiring both great effort, variety of tools, and an extended time commitment (think ironman triathlon). The most celebrated third type are like those Sci-Fi space expeditions in requiring hundreds of years multigenerational commitments, often losing contact with civilization it left behind. And we can’t forget the romantic fourth type — like the North Star, no one actually wants to reach them, as they are largely used for navigation, to find a direction in unchartered waters.
Now, conjectures famously provide a foundation of the scientific method, but that’s not at all how we actually think of them in mathematics. I argued back in this pointed blog post that citations are the most crucial for the day to day math development, so one should take utmost care in making references. While this claim is largely uncontroversial and serves as a raison d’être for most GoogleScholar profiles, conjectures provide a convenient idealistic way out. Thus, it’s much more noble and virtuous to say “I dedicated my life to the study of the XYZ Conjecture” (even if they never publish anything), than “I am working hard writing so many papers to gain respect of my peers, get a promotion, and provide for my family“. Right. Obviously…
But given this apparent (true or perceived) importance of conjectures, are you sure you are using them right? What if some/many of these conjectures are actually wrong, what then? Should you be flying that starship if there is no there there? An idealist would argue something like “it’s a journey, not a destination“, but I strongly disagree. Getting closer to the truth is actually kind of important, both as a public policy and on an individual level. It is thus pretty important to get it right where we are going.
What are conjectures in mathematics?
That’s a stupid question, right? Conjectures are mathematical claims whose validity we are trying to ascertain. Is that all? Well, yes, if you don’t care if anyone will actually work on the conjecture. In other words, something about the conjecture needs to interesting and inspiring.
What makes a conjecture interesting?
This is a hard question to answer because it is as much psychological as it is mathematical. A typical answer would be “oh, because it’s old/famous/beautiful/etc.” Uhm, ok, but let’s try to be a little more formal.
One typically argues “oh, that’s because this conjecture would imply [a list of interesting claims and known results]”. Well, ok, but this is self-referential. We already know all those “known results”, so no need to prove them again. And these “claims” are simply other conjectures, so this is really an argument of the type “this conjecture would imply that conjecture”, so not universally convincing. One can argue: “look, this conjecture has so many interesting consequences”. But this is both subjective and unintuitive. Shouldn’t having so many interesting conjectural consequences suggest that perhaps the conjecture is too strong and likely false? And if the conjecture is likely to be false, shouldn’t this make it uninteresting?
Also, wouldn’t it be interesting if you disprove a conjecture everyone believes to be true? In some sense, wouldn’t it be even more interesting if until now everyone one was simply wrong?
None of this are new ideas, of course. For example, faced with the need to justify the “great” BC conjecture, or rather 123 pages of survey on the subject (which is quite interesting and doesn’t really need to be justified), the authors suddenly turned reflective. Mindful of self-referential approach which they quickly discard, they chose a different tactic:
We believe that the interest of a conjecture lies in the feeling of unity of mathematics that it entails. [M.P. Gomez Aparicio, P. Julg and A. Valette, “The Baum-Connes conjecture“, 2019]
Huh? Shouldn’t math be about absolute truths, not feelings? Also, in my previous blog post, I mentioned Noga Alon‘s quote that Mathematics is already “one unit“. If it is, why does it need a new “feeling of unity“? Or is that like one of those new age ideas which stop being true if you don’t reinforce them at every occasion?
If you are confused at this point, welcome to the club! There is no objective way to argue what makes certain conjectures interesting. It’s all in our imagination. Nikolay Konstantinov once told me that “mathematics is a boring subject because every statement is equivalent to saying that some set is empty.” He meant to be provocative rather than uninspiring. But the problem he is underlying is quite serious.
What makes us believe a conjecture is true?
We already established that in order to argue that a conjecture is interesting we need to argue it’s also true, or at least we want to believe it to be true to have all those consequences. Note, however, that we argue that a conjecture is true in exactly the same way we argue it’s interesting: by showing that it holds is some special cases, and that it would imply other conjectures which are believed to be true because they are also checked in various special cases. So in essence, this gives “true = interesting” in most cases. Right?
This is where it gets complicated. Say, you are working on the “abc conjecture” which may or may not be open. You claim that it has many consequences, which makes it both likely true and interesting. One of them is the negative solution to the Erdős–Ulam problem about existence of a dense set in the plane with rational pairwise distances. But a positive solution to the E-U problem implies the Harborth’s conjecture (aka the “integral Fáry problem“) that every graph can be drawn in the plane with rational edge lengths. So, counterintuitively, if you follow the logic above shouldn’t you be working on a positive solution to Erdős–Ulam since it would both imply one conjecture and give a counterexample to another? For the record, I wouldn’t do that, just making a polemical point.
I am really hoping you see where I am going. Since there is no objective way to tell if a conjecture is true or not, and what exactly is so interesting about it, shouldn’t we discard our biases and also work towards disproving the conjecture just as hard as trying to prove it?
What do people say?
It’s worth starting with a general (if slightly poetic) modern description:
In mathematics, [..] great conjectures [are] sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for their solution guides a large part of mathematics. Eternal fame awaits those who conquer them first. Remarkably, mathematics has elevated the formulation of a conjecture into high art. [..] A well-chosen but unproven statement can make its author world-famous, sometimes even more so than the person providing the ultimate proof. [Robbert Dijkgraaf, The Subtle Art of the Mathematical Conjecture, 2019]
Karl Popper thought that conjectures are foundational to science, even if somewhat idealized the efforts to disprove them:
[Great scientists] are men of bold ideas, but highly critical of their own ideas: they try to find whether their ideas are right by trying first to find whether they are not perhaps wrong. They work with bold conjectures and severe attempts at refuting their own conjectures. [Karl Popper, Heroic Science, 1974]
Here is how he reconciled somewhat the apparent contradiction:
On the pre-scientific level we hate the very idea that we may be mistaken. So we cling dogmatically to our conjectures, as long as possible. On the scientific level, we systematically search for our mistakes. [Karl Popper, quoted by Bryan Magee, 1971]
Paul Erdős was, of course, a champion of conjectures and open problems. He joked that the purpose of life is “proof and conjecture” and this theme is repeatedly echoed when people write about him. It is hard to overestimate his output, which included hundreds of talks titled “My favorite problems“. He wrote over 180 papers with collections of conjectures and open problems (nicely assembled by Zbl. Math.)
Peter Sarnak has a somewhat opposite point of view, as he believes one should be extremely cautious about stating a conjecture so people don’t waste time working on it. He said once, only half-jokingly:
Since we reward people for making a right conjecture, maybe we should punish those who make a wrong conjecture. Say, cut off their fingers. [Peter Sarnak, UCLA, c. 2012]
This is not an exact quote — I am paraphrasing from memory. Needless to say, I disagree. I don’t know how many fingers he wished Erdős should lose, since some of his conjectures were definitely disproved: one, two, three, four, five, and six. This is not me gloating, the opposite in fact. When you are stating hundreds of conjectures in the span of almost 50 years, having only a handful to be disproved is an amazing batting average. It would, however, make me happy if Sarnak’s conjecture is disproved someday.
Finally, there is a bit of a controversy whether conjectures are worth as much as theorems. This is aptly summarized in this quote about yet another champion of conjectures:
Louis J. Mordell [in his book review] questioned Hardy‘s assessment that Ramanujan was a man whose native talent was equal to that of Euler or Jacobi. Mordell [..] claims that one should judge a mathematician by what he has actually done, by which Mordell seems to mean, the theorems he has proved. Mordell’s assessment seems quite wrong to me. I think that a felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem. [Atle Selberg, “Reflections Around the Ramanujan Centenary“, 1988]
So, what’s the problem?
Well, the way I see it, the efforts made towards proving vs. disproving conjectures is greatly out of balance. Despite all the high-minded Popper’s claims about “severe attempts at refuting their own conjectures“, I don’t think there is much truth to that in modern math sciences. This does not mean that disproofs of famous conjectures aren’t celebrated. Sometimes they are, see below. But it’s clear to me that the proofs are celebrated more frequently, and to a much greater degree. I have only anecdotal evidence to support my claim, but bear with me.
Take prizes. Famously, Clay Math Institute gives $1 million for a solution of any of these major open problems. But look closely at the rules. According to the item 5b, except for the P vs. NP problem and the Navier–Stokes Equation problem, it gives nothing ($0) for a disproof of these problems. Why, oh why?? Let’s look into CMI’s “primary objectives and purposes“:
To recognize extraordinary achievements and advances in mathematical research.
So it sounds like CMI does not think that disproving the Riemann Hypothesis needs to be rewarded because this wouldn’t “advance mathematical research”. Surely, you are joking? Whatever happened to “the opposite of a profound truth may well be another profound truth“? Why does the CMI wants to put its thumb on the scale and support only one side? Do they not want to find out the solution whatever it is? Shouldn’t they be eager to dispense with the “wrong conjecture” so as to save numerous researches from “advances to nowhere“?
I am sure you can see that my blood is boiling, but let’s proceed to the P vs. NP problem. What if it’s independent of ZFC? Clearly, CMI wouldn’t pay for proving that. Why not? It’s not like this kind of thing never happened before (see obligatory link to CH). Some people believe that (or at least they did in 2012), and some people like Scott Aaronson take this seriously enough. Wouldn’t this be a great result worthy of an award as much as the proof that P=NP, or at least a nonconstructive proof that P=NP?
If your head is not spinning hard enough, here is another amusing quote:
Of course, it’s possible that P vs. NP is unprovable, but that that fact itself will forever elude proof: indeed, maybe the question of the independence of P vs. NP is itself independent of set theory, and so on ad infinitum! But one can at least say that, if P vs. NP (or for that matter, the Riemann hypothesis, Goldbach’s conjecture, etc.) were proven independent of ZF, it would be an unprecedented development. [Scott Aaronson, P vs. NP, 2016].
Speaking of Goldbach’s Conjecture, the most talked about and the most intuitively correct statement in Number Theory that I know. In a publicity stunt, for two years there was a $1 million prize by a publishing house for the proof of the conjecture. Why just for the proof? I never heard of anyone not believing the conjecture. If I was the insurance underwriter for the prize (I bet they had one), I would allow them to use “for the proof or disproof” for a mere extra $100 in premium. For another $50 I would let them use “or independent of ZF” — it’s a free money, so why not? It’s such a pernicious idea of rewarding only one kind of research outcome!
Curiously, even for Goldbach’s Conjecture, there is a mild divergence of POVs on what the future holds. For example, Popper writes (twice in the same book!) that:
[On whether Goldbach’s Conjecture is ‘demonstrable’] We don’t know: perhaps we may never know, and perhaps we can never know. [Karl Popper, Conjectures and Refutations, 1963]
Ugh. Perhaps. I suppose anything can happen… For example, our civilizations can “perhaps” die out in the next 200 years. But is that likely? Shouldn’t the gloomy past be a warning, not a prediction of the future? The only thing more outrageously pessimistic is this theological gem of a quote:
Not even God knows the number of permutations of 1000 avoiding the 1324 pattern. [Doron Zeilberger, quoted here, 2005]
Thanks, Doron! What a way to encourage everyone! Since we know from numerical estimates that this number is ≈ 3.7 × 101017 (see this paper and this follow up), Zeilberger is suggesting that large pattern avoidance numbers are impossibly hard to compute precisely, already in the range of only about 1018 digits. I really hope he is proved wrong in his lifetime.
But I digress. What I mean to emphasize, is that there are many ways a problem can be resolved. Yet some outcomes are considered more valuable than others. Shouldn’t the research achievements be rewarded, not the desired outcome? Here is yet another colorful opinion on this:
Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That’s what Gödel did for the Continuum Hypothesis. [Saharon Shelah, Rutgers Univ. Colloqium, 2001]
Why do I care?
For one thing, disproving conjectures is part of what I do. Sometimes people are a little shy to unambiguously state them as formal conjectures, so they phrase them as questions or open problems, but then clarify that they believe the answer is positive. This is a distinction without a difference, or at least I don’t see any (maybe they are afraid of Sarnak’s wrath?) Regardless, proving their beliefs wrong is still what I do.
For example, here is my old bog post on my disproof of the Noonan-Zeiberger Conjecture (joint with Scott Garrabrant). And in this recent paper (joint with Danny Nguyen), we disprove in one big swoosh both Barvinok’s Problem, Kannan’s Problem, and Woods Conjecture. Just this year I disproved three conjectures:
- The Kirillov–Klyachko Conjecture (2004) that the reduced Kronecker coefficients satisfy the saturation property (this paper, joint with Greta Panova).
- The Brandolini et al. Conjecture (2019) that concrete lattice polytopes can multitile the space (this paper, joint with Alexey Garber).
- Kenyon’s Problem (c. 2005) that every integral curve in R3 is a boundary of a PL surface comprised of unit triangles (this paper, joint with Alexey Glazyrin).
On top of that, just two months ago in this paper (joint with Han Lyu), we showed that the remarkable independence heuristic by I. J. Good for the number of contingency tables, fails badly even for nearly all uniform marginals. This is not exactly disproof of a conjecture, but it’s close, since the heuristic was introduced back in 1950 and continues to work well in practice.
In addition, I am currently working on disproving two more old conjectures which will remain unnamed until the time we actually resolve them (which might never happen, of course). In summary, I am deeply vested in disproving conjectures. The reasons why are somewhat complicated (see some of them below). But whatever my reasons, I demand and naively fully expect that my disproofs be treated on par with proofs, regardless whether this expectation bears any relation to reality.
My favorite disproofs and counterexamples:
There are many. Here are just a few, some famous and some not-so-famous, in historical order:
- Fermat‘s conjecture (letter to Pascal, 1640) on primality of Fermat numbers, disproved by Euler (1747)
- Tait’s conjecture (1884) on hamiltonicity of graphs of simple 3-polytopes, disproved by W.T. Tutte (1946)
- General Burnside Problem (1902) on finiteness of periodic groups, resolved negatively by E.S. Golod (1964)
- Keller’s conjecture (1930) on tilings with unit hypercubes, disproved by Jeff Lagarias and Peter Shor (1992)
- Borsuk’s Conjecture (1932) on partitions of convex sets into parts of smaller diameter, disproved by Jeff Kahn and Gil Kalai (1993)
- Hirsch Conjecture (1957) on the diameter of graphs of convex polytopes, disproved by Paco Santos (2010)
- Woods’s conjecture (1972) on the covering radius of certain lattices, disproved by Oded Regev, Uri Shapira and Barak Weiss (2017)
- Connes embedding problem (1976), resolved negatively by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright and Henry Yuen (2020)
In all these cases, the disproofs and counterexamples didn’t stop the research. On the contrary, they gave a push to further (sometimes numerous) developments in the area.
Why should you disprove conjectures?
There are three reasons, of different nature and importance.
First, disproving conjectures is opportunistic. As mentioned above, people seem to try proving much harder than they try disproving. This creates niches of opportunity for an open-minded mathematician.
Second, disproving conjectures is beautiful. Let me explain. Conjectures tend to be rigid, as in “objects of the type pqr satisfy property abc.” People like me believe in the idea of “universality“. Some might call it “completeness” or even “Murphy’s law“, but the general principle is always the same. Namely: it is not sufficient that one wishes that all pqr satisfy abc to actually believe in the implication; rather, there has to be a strong reason why abc should hold. Barring that, pqr can possibly be almost anything, so in particular non-abc. While some would argue that non-abc objects are “ugly” or at least “not as nice” as abc, the idea of universality means that your objects can be of every color of the rainbow — nice color, ugly color, startling color, quiet color, etc. That kind of palette has its own sense of beauty, but it’s an acquired taste I suppose.
Third, disproving conjectures is constructive. It depends on the nature of the conjecture, of course, but one is often faced with necessity to construct a counterexample. Think of this as an engineering problem of building some pqr which at the same time is not abc. Such construction, if at all possible, might be difficult, time consuming and computer assisted. But so what? What would you rather do: build a mile-high skyscraper (none exist yet) or prove that this is impossible? Curiously, in CS Theory both algorithms and (many) complexity results are constructive (you need gadgets). Even the GCT is partially constructive, although explaining that would take us awhile.
What should the institutions do?
If you are an institution which awards prizes, stop with the legal nonsense: “We award […] only for a publication of a proof in a top journal”. You need to set up a scientific committee anyway, since otherwise it’s hard to tell sometimes if someone deserves a prize. With mathematicians you can expect anything anyway. Some would post two arXiv preprints, give a few lectures and then stop answering emails. Others would publish only in a journal where they are Editor-in-Chief. It’s stranger than fiction, really.
What you should do is say in the official rules: “We have [this much money] and an independent scientific committee which will award any progress on [this problem] partially or in full as they see fit.” Then a disproof or an independence result will receive just as much as the proof (what’s done is done, what else are you going to do with the money?) This would also allow some flexibility for partial solutions. Say, somebody proves Goldbach’s Conjecture for integers > exp(exp(10100000)), way way beyond computational powers for the remaining integers to be checked. I would give this person at least 50% of the prize money, leaving the rest for future developments of possibly many people improving on the bound. However, under the old prize rules such person gets bupkes for their breakthrough.
What should the journals do?
In short, become more open to results of computational and experimental nature. If this sounds familiar, that’s because it’s a summary of Zeilberger’s Opinions, viewed charitably. He is correct on this. This includes publishing results of the type “Based on computational evidence we believe in the following UVW conjecture” or “We develop a new algorithm which confirms the UVW conjecture for n<13″. These are still contributions to mathematics, and the journals should learn to recognize them as such.
To put in context of our theme, it is clear that a lot more effort has been placed on proofs than on finding counterexamples. However, in many areas of mathematics there are no small counterexamples, so a heavy computational effort is crucial for any hope of finding one. Such work is not be as glamorous as traditional papers. But really, when it comes to standards, if a journal is willing to publish the study of something like the “null graphs“, the ship has sailed for you…
Let me give you a concrete example where a computational effort is indispensable. The curious Lovász conjecture states that every finite connected vertex-transitive graph contains a Hamiltonian path. This conjecture got to be false. It hits every red flag — there is really no reason why pqr = “vertex transitive” should imply abc = “Hamiltonian”. The best lower bound for the length of the longest (self-avoiding) path is only about square root of the number of vertices. In fact, even the original wording by Lovász shows he didn’t believe the conjecture is true (also, I asked him and he confirmed).
Unfortunately, proving that some potential counterexample is not Hamiltonian is computationally difficult. I once had an idea of one (a nice cubic Cayley graph on “only” 3600 vertices), but Bill Cook quickly found a Hamiltonian cycle dashing my hopes (it was kind of him to look into this problem). Maybe someday, when the TSP solvers are fast enough on much larger graphs, it will be time to return to this problem and thoroughly test it on large Cayley graphs. But say, despite long odds, I succeed and find a counterexample. Would a top journal publish such a paper?
Editor’s dilemma
There are three real criteria for evaluation a solution of an open problem by the journal:
- Is this an old, famous, or well-studied problem?
- Are the tools interesting or innovative enough to be helpful in future studies?
- Are the implications of the solution to other problems important enough?
Now let’s make a hypothetical experiment. Let’s say a paper is submitted to a top math journal which solves a famous open problem in Combinatorics. Further, let’s say somebody already proved it is equivalent to a major problem in TCS. This checks criteria 1 and 3. Until not long ago it would be rejected regardless, so let’s assume this is happening relatively recently.
Now imagine two parallel worlds, where in the first world the conjecture is proved on 2 pages using beautiful but elementary linear algebra, and in the second world the conjecture is disproved on a 2 page long summary of a detailed computational search. So in neither world we have much to satisfy criterion 2. Now, a quiz: in which world the paper will be published?
If you recognized that the first world is a story of Hao Huang‘s elegant proof of the induced subgraphs of hypercubes conjecture, which implies the sensitivity conjecture. The Annals published it, I am happy to learn, in a welcome break with the past. But unless we are talking about some 200 year old famous conjecture, I can’t imagine the Annals accepting a short computational paper in the second world. Indeed, it took a bit of a scandal to accept even the 400 year old Kepler’s conjecture which was proved in a remarkable computational work.
Now think about this. Is any of that fair? Shouldn’t we do better as a community on this issue?
What do other people do?
Over the years I asked a number of people about the uncertainty created by the conjectures and what do they do about it. The answers surprised me. Here I am paraphrasing them:
Some were dumbfounded: “What do you mean this conjecture could be false? It has to be true, otherwise nothing I am doing make much sense.”
Others were simplistic: “It’s an important conjecture. Famous people said it’s true. It’s my job to prove it.”
Third were defensive: “Do you really think this conjecture could be wrong? Why don’t you try to disprove it then? We’ll see who is right.”
Fourth were biblical: “I tend to work 6 days a week towards the proof and one day towards the disproof.”
Fifth were practical: “I work on the proof until I hit a wall. I use the idea of this obstacle to try constructing potential counterexamples. When I find an approach to discard such counterexamples, I try to generalize the approach to continue working on the proof. Continue until either side wins.”
If the last two seem sensible to you to, that’s because they are. However, I bet fourth are just grandstanding — no way they actually do that. The fifth sound great when this is possible, but that’s exceedingly rare, in my opinion. We live in a technical age when proving new results often requires great deal of effort and technology. You likely have tools and intuition to work in only one direction. Why would you want to waste time working in another?
What should you do?
First, remember to make conjectures. Every time you write a paper, tell a story of what you proved. Then tell a story of what you wanted to prove but couldn’t. State it in the form of a conjecture. Don’t be afraid to be wrong, or be right but oversharing your ideas. It’s a downside, sure. But the upside is that your conjecture might prove very useful to others, especially young researchers. In might advance the area, or help you find a collaborator to resolve it.
Second, learn to check your conjectures computationally in many small cases. It’s important to give supporting evidence so that others take your conjectures seriously.
Third, learn to make experiments, explore the area computationally. That’s how you make new conjectures.
Fourth, understand yourself. Your skill, your tools. Your abilities like problem solving, absorbing information from the literature, or making bridges to other fields. Faced with a conjecture, use this knowledge to understand whether at least in principle you might be able to prove or disprove a conjecture.
Fifth, actively look for collaborators. Those who have skills, tools, or abilities you are missing. More importantly, they might have a different POV on the validity of the conjecture and how one might want to attack it. Argue with them and learn from them.
Sixth, be brave and optimistic! Whether you decide to prove, disprove a conjecture, or simply state a new conjecture, go for it! Ignore the judgements by the likes of Sarnak and Zeilberger. Trust me — they don’t really mean it.
Some good news
Two of my former Ph.D. students won major prizes recently — Matjaž Konvalinka and Danny Nguyen. Matjaž is an Associate Professor at University of Ljubljana, Danny is a Lewis Research Assistant Professor at University of Michigan, Ann Arbor. Congratulations to both of them!
(1) The 2019 Robbins Prize is awarded to Roger Behrend, Ilse Fischer and Matjaž Konvalinka for their paper “Diagonally and antidiagonally symmetric alternating sign matrices of odd order”. The Robbins Prize is given in Combinatorics and related areas of interest is named after the late David P. Robbins and is given once every 3 years by AMS and MAA.
In many ways, this paper completes the long project of enumerating alternating sign matrices (ASMs) initiated by William Mills, David Robbins, and Howard Rumsey in the early 1980s. The original #ASM(n)=#TSSCPP(n) conjecture follows from Andrews’s proof of the conjectured product formula for #TSSCPP(n), and Zeilberger’s 84 page computer assisted proof of the the same conjectured product formula for #ASM(n). This led to a long series of remarkable developments which include Kuperberg’s proof using the Izergin-Korepin determinant for the six vertex model, the Cantini–Sportiello proof of the Razumov-Stroganov conjecture, and a recent self-contained determinantal proof for the number of ASMs by Fischer. Bressoud’s book (and this talk, slides) is a good introduction. But the full story is yet to be written.
(2) The 2018 Sacks Prize is awarded to Danny Nguyen for his UCLA Ph.D. dissertation on the complexity of short formulas in Presburger Arithmetic (PA) and many related works (some joint with me, some with others). See also the UCLA announcement. The Sacks Prize is given by the international Association for Symbolic Logic for “the most outstanding doctoral dissertation in mathematical logic“. It is sometimes shared between two awardees, and sometimes not given at all. This year Danny is the sole winner of the prize.
Danny’s dissertation is a compilation of eight (!) papers Danny wrote during his graduate studies, all on the same or closely related subject. These papers advance and mostly finish off the long program of understanding the boundary of what’s feasible in PA. The most important of these is our joint FOCS paper which basically says that Integer Programming and Parametric Integer Programming is all that’s left in P, while all longer formulas are NP-hard. See Featured MathSciNet Review by Sasha Barvinok and an overlapping blog post by Gil Kalai discussing these results. See also Danny’s FOCS talk video and my MSRI talk video presenting this work.
ICM 2018 Speakers
UCLA recently outed me (with permission) as a speaker at the next ICM in Rio. I am incredibly honored to be chosen, alongside my fantastic colleagues Matthias Aschenbrenner, Andrea Bertozzi, Ciprian Manolescu and Sucharit Sarkar.
P.S. I have more to say on the subject of ICM, but that can wait perhaps.
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FINAL UPDATE:
A complete list of ICM speakers is available here.
The power of negative thinking, part I. Pattern avoidance
In my latest paper with Scott Garrabrant we disprove the Noonan-Zeilberger Conjecture. Let me informally explain what we did and why people should try to disprove conjectures more often. This post is the first in a series. Part II will appear shortly.
What did we do?
Let F ⊂ Sk be a finite set of permutations and let Cn(F) denote the number of permutations σ ∈ Sn avoiding the set of patterns F. The Noonan-Zeilbeger conjecture (1996), states that the sequence {Cn(F)} is always P-recursive. We disprove this conjecture. Roughly, we show that every Turing machine T can be simulated by a set of patterns F, so that the number an of paths of length n accepted by by T is equal to Cn(F) mod 2. I am oversimplifying things quite a bit, but that’s the gist.
What is left is to show how to construct a machine T such that {an} is not equal (mod 2) to any P-recursive sequence. We have done this in our previous paper, where give a negative answer to a question by Kontsevich. There, we constructed a set of 19 generators of GL(4,Z), such that the probability of return sequence is not P-recursive.
When all things are put together, we obtain a set F of about 30,000 permutations in S80 for which {Cn(F)} is non-P-recursive. Yes, the construction is huge, but so what? What’s a few thousand permutations between friends? In fact, perhaps a single pattern (1324) is already non-P-recursive. Let me explain the reasoning behind what we did and why our result is much stronger than it might seem.
Why we did what we did
First, a very brief history of the NZ-conjecture (see Kirtaev’s book for a comprehensive history of the subject and vast references). Traditionally, pattern avoidance dealt with exact and asymptotic counting of pattern avoiding permutations for small sets of patterns. The subject was initiated by MacMahon (1915) and Knuth (1968) who showed that we get Catalan numbers for patterns of length 3. The resulting combinatorics is often so beautiful or at least plentiful, it’s hard to imagine how can it not be, thus the NZ-conjecture. It was clearly very strong, but resisted all challenges until now. Wilf reports that Richard Stanley disbelieved it (Richard confirmed this to me recently as well), but hundreds of papers seemed to confirm its validity in numerous special cases.
Curiously, the case of the (1324) pattern proved difficult early on. It remains unresolved whether Cn(1324) is P-recursive or not. This pattern broke Doron Zeilberger’s belief in the conjecture, and he proclaimed that it’s probably non-P-recursive and thus NZ-conjecture is probably false. When I visited Doron last September he told me he no longer has strong belief in either direction and encouraged me to work on the problem. I took a train back to Manhattan looking over New Jersey’s famously scenic Amtrack route. Somewhere near Pulaski Skyway I called Scott to drop everything, that we should start working on this problem.
You see, when it comes to pattern avoidance, things move from best to good to bad to awful. When they are bad, they are so bad, it can be really hard to prove that they are bad. But why bother – we can try to figure out something awful. The set of patterns that we constructed in our paper is so awful, that proving it is awful ain’t so bad.
Why is our result much stronger than it seems?
That’s because the proof extends to other results. Essentially, we are saying that everything bad you can do with Turing machines, you can do with pattern avoidance (mod 2). For example, why is (1324) so hard to analyze? That’s because it’s even hard to compute both theoretically and experimentally – the existing algorithms are recursive and exponential in n. Until our work, the existing hope for disproving the NZ-conjecture hinged on finding an appropriately bad set of patterns such that computing {Cn(F)} is easy. Something like this sequence which has a nice recurrence, but is provably non-P-recursive. Maybe. But in our paper, we can do worse, a lot worse…
We can make a finite set of patterns F, such that computing {Cn(F) mod 2} is “provably” non-polynomial (Th 1.4). Well, we use quotes because of the complexity theory assumptions we must have. The conclusion is much stronger than non-P-recursiveness, since every P-recursive sequence has a trivial polynomial in n algorithm computing it. But wait, it gets worse!
We prove that for two sets of patterns F and G, the problem “Cn(F) = Cn(G) mod 2 for all n” is undecidable (Th 1.3). This is already a disaster, which takes time to sink in. But then it gets even worse! Take a look at our Corollary 8.1. It says that there are two sets of patterns F and G, such that you can never prove nor disprove that Cn(F) = Cn(G) mod 2. Now that’s what I call truly awful.
What gives?
Well, the original intuition behind the NZ-conjecture was clearly wrong. Many nice examples is not a good enough evidence. But the conjecture was so plausible! Where did the intuition fail? Well, I went to re-read Polya’s classic “Mathematics and Plausible Reasoning“, and it all seemed reasonable. That is both Polya’s arguments and the NZ-conjecture (if you don’t feel like reading the whole book, at least read Barry Mazur’s interesting and short followup).
Now think about Polya’s arguments from the point of view of complexity and computability theory. Again, it sounds very “plausible” that large enough sets of patterns behave badly. Why wouldn’t they? Well, it’s complicated. Consider this example. If someone asks you if every 3-connected planar cubic graph has a Hamiltonian cycle, this sounds plausible (this is Tait’s conjecture). All small examples confirm this. Planar cubic graphs do have very special structure. But if you think about the fact that even for planar graphs, Hamiltonicity is NP-complete, it doesn’t sound plausible anymore. The fact that Tutte found a counterexample is no longer surprising. In fact, the decision problem was recently proved to be NP-complete in this case. But then again, if you require 4-connectivity, then every planar graph has a Hamiltonian cycle. Confused enough?
Back to the patterns. Same story here. When you look at many small cases, everything is P-recursive (or yet to be determined). But compare this with Jacob Fox’s theorem that for a random single pattern π, the sequence {Cn(π)} grows much faster than originally expected (cf. Arratia’s Conjecture). This suggests that small examples are not representative of complexity of the problem. Time to think about disproving ALL conjectures based on that evidence.
If there is a moral in this story, it’s that what’s “plausible” is really hard to judge. The more you know, the better you get. Pay attention to small crumbs of evidence. And think negative!
What’s wrong with being negative?
Well, conjectures tend to be optimistic – they are wishful thinking by definition. Who would want to conjecture that for some large enough a,b,c and n, there exist a solution of an + bn = cn? However, being so positive has a drawback – sometimes you get things badly wrong. In fact, even polynomial Diophantine equations can be as complicated as one wishes. Unfortunately, there is a strong bias in Mathematics against counterexamples. For example, only two of the Clay Millennium Problems automatically pay $1 million for a counterexample. That’s a pity. I understand why they do this, just disagree with the reasoning. If anything, we should encourage thinking in the direction where there is not enough research, not in the direction where people are already super motivated to resolve the problem.
In general, it is always a good idea to keep an open mind. Forget all this “power of positive thinking“, it’s not for math. If you think a conjecture might be false, ignore everybody and just go for disproof. Even if it’s one of those famous unsolved conjectures in mathematics. If you don’t end up disproving the conjecture, you might have a bit of trouble publishing computational evidence. There are some journals who do that, but not that many. Hopefully, this will change soon…
Happy ending
When we were working on our paper, I wrote to Doron Zeilberger if he ever offered a reward for the NZ-conjecture, and for the disproof or proof only? He replied with an unusual award, for the proof and disproof in equal measure. When we finished the paper I emailed to Doron. And he paid. Nice… 🙂
What do math journals do? What will become of them in the future?
Recently, there has been plenty of discussions on math journals, their prices, behavior, technology and future. I have been rather reluctant to join the discussion in part due to my own connection to Elsevier, in part because things in Combinatorics are more complicated than in other areas of mathematics (see below), but also because I couldn’t reconcile several somewhat conflicting thoughts that I had. Should all existing editorial boards revolt and all journals be electronic? Or perhaps should we move to “pay-for-publishing” model? Or even “crowd source refereeing”? Well, now that the issue a bit cooled down, I think I figured out exactly what should happen to math journals. Be patient – a long explanation is coming below.
Quick test questions
I would like to argue that the debate over the second question is the general misunderstanding of the first question in the title. In fact, I am pretty sure most mathematicians are quite a bit confused on this, for a good reason. If you think this is easy, quick, answer the following three questions:
1) Published paper has a technical mistake invalidating the main result. Is this a fault of author, referee(s), handling editor, managing editor(s), a publisher, or all of the above? If the reader find such mistake, who she/he is to contact?
2) Published paper proves special case of a known result published 20 years earlier in an obscure paper. Same question. Would the answer change if the author lists the paper in the references?
3) Published paper is written in a really poor English. Sections are disorganized and the introduction is misleading. Same question.
Now that you gave your answers, ask a colleague. Don’t be surprised to hear a different point of view. Or at least don’t be surprised when you hear mine.
What do referees do?
In theory, a lot. In practice, that depends. There are few official journal guides to referees, but there are several well meaning guides (see also here, here, here, here §4.10, and a nice discussion by Don Knuth §15). However, as any editor can tell you, you never know what exactly did the referee do. Some reply within 5 min, some after 2 years. Some write one negative sentence, some 20 detailed pages, some give an advice in the style “yeah, not a bad paper, cites me twice, why not publish it”, while others a brushoff “not sure who this person is, and this problem is indeed strongly related to what I and my collaborators do, but of course our problems are much more interesting/important – rejection would be best”. The anonymity is so relaxing, doing a poor job is just too tempting. The whole system hinges on the shame, sense of responsibility, and personal relationship with the editor.
A slightly better questions is “What do good referees do?” The answer is – they don’t just help the editor make acceptance/rejection decision. They help the authors. They add some background the authors don’t know, look for missing references, improve on the proofs, critique the exposition and even notation. They do their best, kind of what ideal advisors do for their graduate students, who just wrote an early draft of their first ever math paper.
In summary, you can’t blame the referees for anything. They do what they can and as much work as they want. To make a lame comparison, the referees are like wind and the editors are a bit like sailors. While the wind is free, it often changes direction, sometimes completely disappears, and in general quite unreliable. But sometimes it can really take you very far. Of course, crowd sourcing refereeing is like democracy in the army – bad even in theory, and never tried in practice.
First interlude: refereeing war stories
I recall a curious story by Herb Wilf, on how Don Knuth submitted a paper under assumed name with an obscure college address, in order to get full refereeing treatment (the paper was accepted and eventually published under Knuth’s real name). I tried this once, to unexpected outcome (let me not name the journal and the stupendous effort I made to create a fake identity). The referee wrote that the paper was correct, rather interesting but “not quite good enough” for their allegedly excellent journal. The editor was very sympathetic if a bit condescending, asking me not to lose hope, work on my papers harder and submit them again. So I tried submitting to a competing but equal in statue journal, this time under my own name. The paper was accepted in a matter of weeks. You can judge for yourself the moral of this story.
A combinatorialist I know (who shall remain anonymous) had the following story with Duke J. Math. A year and a half after submission, the paper was rejected with three (!) reports mostly describing typos. The authors were dismayed and consulted a CS colleague. That colleague noticed that the three reports were in .pdf but made by cropping from longer files. Turns out, if the cropping is made straightforwardly, the cropped portions are still hidden in the files. Using some hacking software the top portions of the reports were uncovered. The authors discovered that they are extremely positive, giving great praise of the paper. Now the authors believe that the editor despised combinatorics (or their branch of combinatorics) and was fishing for a bad report. After three tries, he gave up and sent them cropped reports lest they think somebody else considers their paper worthy of publishing in the grand old Duke (cf. what Zeilberger wrote about Duke).
Another one of my stories is with the Journal of AMS. A year after submission, one of my papers was rejected with the following remarkable referee report which I quote here in full:
The results are probably well known. The authors should consult with experts.
Needless to say, the results were new, and the paper was quickly published elsewhere. As they say, “resistance is futile“.
What do associate/handling editors do?
Three little things, really. They choose referees, read their reports and make the decisions. But they are responsible for everything. And I mean for everything, both 1), 2) and 3). If the referee wrote a poorly researched report, they should recognize this and ignore it, request another one. They should ensure they have more than one opinion on the paper, all of them highly informed and from good people. If it seems the authors are not aware of the literature and referee(s) are not helping, they should ensure this is fixed. It the paper is not well written, the editors should ask the authors to rewrite it (or else). At Discrete Mathematics, we use this page by Doug West, as a default style to math grammar. And if the reader finds a mistake, he/she should first contact the editor. Contacting the author(s) is also a good idea, but sometimes the anonymity is helpful – the editor can be trusted to bring bad news and if possible, request a correction.
B.H. Neumann described here how he thinks the journal should operate. I wish his views held widely today. The book by Krantz, §5.5, is a good outline of the ideal editorial experience, and this paper outlines how to select referees. However, this discussion (esp. Rick Durrett’s “rambling”) is more revealing. Now, the reason most people are confused as to who is responsible for 1), 2) and 3), is the fact that while some journals have serious proactive editors, others do not, or their work is largely invisible.
What do managing editors and publishers do?
In theory, managing editors hire associate editors, provide logistical support, distribute paper load, etc. In practice they also serve as handling editors for a large number of papers. The publishers… You know what the publishers do. Most importantly, they either pay editors or they don’t. They either charge libraries a lot, or they don’t. Publishing is a business, after all…
Who wants free universal electronic publishing?
Good mathematicians. Great mathematicians. Mathematicians who write well and see no benefit in their papers being refereed. Mathematicians who have many students and wish the publishing process was speedier and less cumbersome, so their students can get good jobs. Mathematicians who do not value the editorial work and are annoyed when the paper they want to read is “by subscription only” and thus unavailable. In general, these are people who see having to publish as an obstacle, not as a benefit.
Who does not want free universal electronic publishing?
Publishers (of course), libraries, university administrators. These are people and establishments who see value in existing order and don’t want it destroyed. Also: mediocre mathematicians, bad mathematicians, mathematicians from poor countries, mathematicians who don’t have access to good libraries (perhaps, paradoxically). In general, people who need help with their papers. People who don’t want a quick brush-off “not good enough” or “probably well known”, but those who need advice on the references, on their English, on how the papers are structured and presented, and on what to do next.
So, who is right?
Everyone. For some mathematicians, having all journals to be electronic with virtually no cost is an overall benefit. But at the very least, “pro status quo” crowd have a case, in my view. I don’t mean that Elsevier pricing policy is reasonable, I am talking about the big picture here. In a long run, I think of journals as non-profit NGO‘s, some kind of nerdy versions of Nobel Peace Prize winning Médecins Sans Frontières. While I imagine that in the future many excellent top level journals will be electronic and free, I also think many mid-level journals in specific areas will be run by non-profit publishers, not free at all, and will employ a number of editorial and technical stuff to help the authors, both of papers they accept and reject. This is a public service we should strive to perform, both for the sake of those math papers, and for development of mathematics in other countries.
Right now, the number of mathematicians in the world is already rather large and growing. Free journals can do only so much. Without high quality editors paid by the publishers, with a large influx of papers from the developing world, there is a chance we might loose the traditional high standards for published second tier papers. And I really don’t want to think of a mathematics world once the peer review system is broken. That’s why I am not in the “free publishing camp” – in an effort to save money, we might loose something much more valuable – the system which gives foundation and justification of our work.
Second interlude: journals vis-à-vis combinatorics
I already wrote about the fate of combinatorics papers in the Annals, especially when comparison with Number Theory. My view was gloomy but mildly optimistic. In fact, since that post was written couple more combinatorics papers has been accepted. Good. But let me give you a quiz. Here are two comparable highly selective journals – Duke J. Math. and Composito Math. In the past 10 years Composito published exactly one (!) paper in Combinatorics (defined as primary MSC=05), of the 631 total. In the same period, Duke published 8 combinatorics papers of 681 total.
Q: Which of the two (Composito or Duke) treats combinatorics papers better?
A: Composito, of course.
The reasoning is simple. Forget the anecdotal evidence in the previous interlude. Just look at the “aim and scope” of the journals vs. these numbers. Here is what Compsito website says with a refreshing honesty:
By tradition, the journal published by the foundation focuses on papers in the main stream of pure mathematics. This includes the fields of algebra, number theory, topology, algebraic and analytic geometry and (geometric) analysis. Papers on other topics are welcome if they are of interest not only to specialists.
Translation: combinatorics papers are not welcome (as are papers in many other fields). I think this is totally fair. Nothing wrong with that. Clearly, there are journals which publish mostly in combinatorics, and where papers in none of these fields would be welcome. In fact there is a good historical reason for that. Compare this with what Duke says on its website:
Published by Duke University Press since its inception in 1935, the Duke Mathematical Journal is one of the world’s leading mathematical journals. Without specializing in a small number of subject areas, it emphasizes the most active and influential areas of current mathematics.
See the difference? They don’t name their favorite areas! How are the authors supposed to guess which are these? Clearly, Combinatorics with its puny 1% proportion of Duke papers is not a subject area that Duke “emphasizes”. Compare it with 104 papers in Number Theory (16%) and 124 papers in Algebraic Geometry (20%) over the same period. Should we conclude that in the past 10 years, Combinatorics was not “the most active and influential”, or perhaps not “mathematics” at all? (yes, some people think so) I have my own answer to this question, and I bet so do you…
Note also, that things used to be different at Duke. For example, exactly 40 years earlier, in the period 1963-1973, Duke published 47 papers in combinatorics out of 972 total, even though the area was only in its first stages of development. How come? The reason is simple: Leonard Carlitz was Managing Editor at the time, and he welcomed papers from a number of prominent combinatorialists active during that time, such as Andrews, Gould, Moon, Riordan, Stanley, Subbarao, etc., as well as a many of his own papers.
So, ideally, what will happen to math journals?
That’s actually easy. Here are my few recommendations and predictions.
1) We should stop with all these geography based journals. That’s enough. I understand the temptation for each country, or university, or geographical entity to have its own math journal, but nowadays this is counterproductive and a cause for humor. This parochial patriotism is perhaps useful in sports (or not), but is nonsense in mathematics. New journals should emphasize new/rapidly growing areas of mathematics underserved by current journals, not new locales where printing presses are available.
2) Existing for profit publishers should realize that with the growth of arXiv and free online competitors, their business model is unsustainable. Eventually all these journals will reorganize into a non-profit institutions or foundations. This does not mean that the journals will become electronic or free. While some probably will, others will remain expensive, have many paid employees (including editors), and will continue to provide services to the authors, all supported by library subscriptions. These extra services are their raison d’être, and will need to be broadly advertised. The authors would learn not to be surprised of a quick one line report from free journals, and expect a serious effort from “expensive journals”.
3) The journals will need to rethink their structure and scope, and try to develop their unique culture and identity. If you have two similar looking free electronic journals, which do not add anything to the papers other than their .sty file, the difference is only the editorial board and history of published papers. This is not enough. All journals, except for the very top few, will have to start limiting their scope to emphasize the areas of their strength, and be honest and clear in advertising these areas. Alternatively, other journals will need to reorganize and split their editorial board into clearly defined fields. Think Proc. LMS, Trans. AMS, or a brand new Sigma, which basically operate as dozens of independent journals, with one to three handling editors in each. While highly efficient, in a long run this strategy is also unsustainable as it leads to general confusion and divergence in the quality of these sub-journals.
4) Even among the top mathematicians, there is plenty of confusion on the quality of existing mathematics journals, some of which go back many decades. See e.g. a section of Tim Gowers’s post about his views of the quality of various Combinatorics journals, since then helpfully updated and corrected. But at least those of us who have been in the area for a while, have the memory of the fortune of previously submitted papers, whether our own, or our students, or colleagues. A circumstantial evidence is better than nothing. For the newcomers or outsiders, such distinctions between journals are a mystery. The occasional rankings (impact factor or this, whatever this is) are more confusing than helpful.
What needs to happen is a new system of awards recognizing achievements of individual journals and/or editors, in their efforts to improve the quality of the journals, attracting top papers in the field, arranging fast refereeing, etc. Think a mixture of Pulitzer Prize and J.D. Power and Associates awards – these would be a great help to understand the quality of the journals. For example, the editors of the Annals clearly hustled to referee within a month in this case (even if motivated by PR purposes). It’s an amazing speed for a technical 50+ page paper, and this effort deserves recognition.
Full disclosure: Of the journals I singled out, I have published once in both JAMS and Duke. Neither paper is in Combinatorics, but both are in Discrete Mathematics, when understood broadly.
What’s the Matter with Hertz Foundation?
Imagine you have plenty of money and dozens of volunteers. You decide to award one or two fellowships a year to the best of the best of the best in math sciences. Easy, right? Then how do you repeatedly fail at this, without anyone notice? Let me tell you how. It’s an interesting story, so bear with me.
A small warning. Although it may seem I am criticizing Hertz Foundation, my intention is to show its weakness so it can improve.
What is the Hertz Foundation?
Yesterday I wrote a recommendation letter to the Hertz Foundation. Although a Fellow myself, I never particularly cared for the foundation, mostly because it changed so little in my life (I received it only for two out of five years of eligibility). But I became rather curious as to what usually happens to Hertz Fellows. I compiled the data, and found the results quite disheartening. While perhaps excellent in other fields, I came to believe that Hertz does barely a mediocre job awarding fellowships in mathematics. And now that I think about it, this was all completely predictable.
First, a bit of history. John Hertz was the Yellow Cab founder and car rental entrepreneur (thus the namesake company), and he left a lot of money dedicated for education in “applied physical sciences”, now understood to include applied mathematics. What exactly is “applied mathematics” is rather contentious, so the foundation wisely decided that “it is up to each fellowship applicant to advocate to us his or her specific field of interest as an ‘applied physical science’.”
In practice, according to the website, about 600 applicants in all areas of science and engineering apply for a fellowship. Applications are allowed only either in the senior year of college or 1st year of grad school. The fellowships are generous and include both the stipend and the tuition; between 15 and 20 students are awarded every year. Only US citizen and permanent residents are eligible, and the fellowship can be used only in one of the 47 “tenable schools” (more on this below). The Foundation sorts the applications, and volunteers interview some of them in the first round. In the second round, pretty much only one person interviews all that advanced, and the decision is made. Historically, only one or two fellowships in mathematical sciences are awarded each year (this includes pure math, applied math, and occasionally theoretical CS or statistics).
Forty years of Math Hertz Fellowships in numbers
The Hertz Foundation website has a data on all past fellows. I compiled the data in Hertz-list which spanned 40 years (1971-2010), listed by the year the fellowship ended, which usually but not always coincided with graduation. There were 67 awardees in mathematics, which makes it about 1.7 fellowships a year. The Foundation states that it awarded “over 1000 fellowships” so I guess about 5-6% went into maths (perhaps, fewer in recent years). Here is who gets them.
1) Which schools are awarded? Well, only 44 US graduate programs are allowed to administer the fellowships. The reasons (other than logistical) are unclear to me. Of those programs that are “in”, you have University of Rochester (which nearly lost its graduate program), and UC Santa Cruz (where rumors say a similar move had been considered). Those which are “out” include graduate programs at Brown, UPenn, Rutgers, UNC Chapel Hill, etc. The real distribution is much more skewed, of course. Here is a complete list of awards per institution:
MIT – 14
Harvard, Princeton – 8
Caltech, NYU – 7
Berkeley, Stanford – 5
UCLA – 3
CMU, Cornell, U Chicago – 2
GA Tech, JHU, RPI, Rice – 1
In summary, only 15 universities had at least one award (34%), and just 7 universities were awarded 54 fellowships (i.e. 16% of universities received 81% of all fellowships). There is nothing wrong with this per se, just a variation on the 80-20 rule you might argue. But wait! Hertz Foundation is a non-profit institution and the fellowship itself comes with a “moral commitment“. Even if you need to interfere with “free marketplace” of acceptance decisions (see P.S. below), wouldn’t it be in the spirit of John Hertz’s original goal, to make a special effort to distribute the awards more widely? For example, Simons Foundation is not shy about awarding fellowship to institutions many of which are not even on Hertz’s list.
2) Where are they now? After two hours of googling, I located almost all former fellows and determined their current affiliations (see the Hertz-list). I found that of the 67 fellows:
University mathematicians – 27 (40%)
Of these, work at Hertz eligible universities – 14, or about 21% of the total (excluding 3 overseas)
At least 10 who did not receive a Ph.D. – 15%
At least 13 are in non-academic research – 19% (probably more)
At least 8 in Software Development and Finance – 12% (probably more)
Now, there is certainly nothing wrong with directing corporate research, writing software, selling derivatives, designing museum exhibits, and even playing symphony orchestra or heading real estate company, as some of the awardees do now. Many of these are highly desirable vocations. But really, was this what Hertz had in mind when dedicating the money? In the foundation’s language, “benefit us all” they don’t.
I should mention that the list of Hertz Fellows in Mathematics does include a number of great academic success stories, but that’s not actually surprising. Every US cohort has dozens of excellent mathematicians. But the 60% drop out rate from academia is very unfortunate, only 21% working in “tenable universities” is dismaying, and the 15% drop out rate from graduate programs is simply miserable. Couldn’t they have done better?
A bit of analysis
Every year, US universities award over 1,600 Ph.D.’s in mathematical sciences, of which over a half go to US citizen (more if you include permanent residents, but stats is not easily available). So they are choosing 1.7 out of over 800 eligible students. Ok, because of their “tenable schools” restriction this is probably more like 300-400. Therefore, less than half of one percent of potential applicants are awarded! For comparison, Harvard college acceptance rate is 10 times that.
Let me repeat: in mathematics, Hertz fellows drop out from their Ph.D. programs at a rate of 15%. If you look into the raw 2006 NRC data for graduation rates, you will see that many of the top universities have over 90% graduation rate in the math programs (say, Harvard has over 91%). Does that mean that Harvard on average does a better job selecting 10-15 grad students every year, while Hertz can’t choose one?
Yes, I think it does. And the gap is further considering that Hertz has virtually no competition (NSF Fellowships are less generous in every respect). You see, people at Harvard (or Princeton, MIT, UCLA, etc.) who read graduate applications, know what they are doing. They are professionals who are looking for the most talented mathematicians from a large pool of applicants. They know which letters need to be taken seriously, and which with a grain of salt. They know which undergraduate research experience is solid and which is worthless. They just know how things are done.
Now, a vast majority of Hertz interviewers are themselves former fellows, and thus about 95% of them have no idea about the mathematics research (they just assume it’s no different from the research they are accustomed to). Nor does the one final interviewer, who is an applied physicist. As a result, they are to some extend, flipping coins and rolling dies, in hope things will work out. You can’t really blame them – they simply don’t know how to choose. I still remember my own two interviews. Both interviewers were nice, professional, highly experienced and well intentioned, but looking back I can see that neither had much experience with mathematical research.
You can also see this lack of understanding of mathematics culture is creeping up in other activities of the foundation, such as the thesis prize award (where are mathematicians?), etc. Of course a private foundation can award anyone it pleases, but it seems to me it would do much more good if only some special care is applied.
A modest proposal
There is of course, a radical way to change the review of mathematics applicants – subcontract it to the AMS (or IMA, MSRI, IPAM – all have the required infrastructure). For a modest fee, the AMS will organize a panel of mathematicians who will review and rank the applicants without interviewing them. The panel will be taking into consideration only students’ research potential, not the university prestige, etc. The Hertz people can then interview the top ranked and make a decision at the last stage, but the first round will be by far superior to current methods. Even the NSA trusts AMS, so shouldn’t you?
Hertz might even save some money it currently spends on travel and lodging reimbursements. The 13% operating budget is about average, but there is some room for improvement. Subcontracting will probably lead to an increase in applications, as AMS really knows how to advertise to its members (I bet you currently receive only about 40 mathematics applications, out of a potential 400+ pool). To summarize: really, Hertz Foundation, think about doing that!
P.S. It is not surprising that the 7 top universities get a large number of the fellowships. One might be tempted to assume that clueless interviewers are perhaps somewhat biased towards famous school names in the hope that these schools already made a good decision accepting these applicants, but this is not the whole story. The described bias can only work for the 1st year grad applicants, but for undergraduate applicants a different process seems to hold. Once a graduate school learns that an applicant received Hertz Fellowship (or NSF for that matter), it has every incentive to accept the student, as the tuition and the stipend are paid by the outside sources now.
P.P.S. Of course, mathematicians’ review can also fail. Even the super prestigious AIM Fellowship has at least one recipient who left academia for bigger and better things.
UPDATE (April 15, 2019). Over the years since this blog post, I have been contacted by people from the Hertz Foundation board. I have also followed up on the story and the recent fellowship recipients. I am happy to say that the foundation implemented various important changes vis-à-vis math interviews, to the visible effect. At the moment, the numbers are too small to report statistics and the changes I know are not a public information. I concluded that my criticism no longer applies, a happy ending to the story. I encourage now everyone to support the foundation financially as well as recommend your best students to apply.
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