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.
Do you think you know the answer? Do you think others have the same answer? Imagine you could go back in time and ask this question to a number of top combinatorialists of the past 50 years. What would they say? Would you agree with them at all?
Turns out, these answers are readily available. I collected them on this page of quotes. The early ones are uncertain, defensive, almost apologetic. The later ones are bolder, prouder of the field and its status. All are enlightening. Take your time, read them all in order.
During my recent MIT visit, Jacob Fox showed me this blog which I found to be rather derogatory in its treatment of combinatorics and notable combinatorialists. This brought me back to my undergraduate days in Moscow, reminded of the long forgotten but back then very common view of combinatorics as “second rate mathematics”. In the US, I always thought, this battle has been won before my time, back in the 1980s. The good guys worked hard and paved the road for all younger combinatorialists to walk on, and be proud of the field. But of course the internet has its own rules, and has room for every prejudice known to men.
While myself uninterested in engaging in conversation, I figured that there got to be some old “war-time” replies which I can show to the Owl blogger. As I see it, only the lack of knowledge can explain these nearsighted generalizations the blogger is showing. And in the age of Google Scholar, there really is no excuse for not knowing the history of the subject, and its traditional sensitivities.
But while compiling the list of quotes linked above, I learned a few things. I learned how tumultuous was the history of combinatorics, with petty fights and random turns into blind alleys. I learned how myopic were some of the people, and how clever and generous were others. I also discovered that there is a good answer to the question in the title (see below), but that answer is not a definition.
What do authorities say?
Not a lot, actually. The AMS MSC (which I love criticizing) lists Combinaorics as 05, on par with about 70 fields, such as Number Theory (11), Geometry (51), Probability (60) and Computer Science (68). It is also on the same level as Nonassociative rings (17), K-theory (19) and Integral equations (51), which are perfectly fine areas, just much smaller. It is one of the 32 categories on the arXiv, but who knows how these came about.
At the level of NSF, it is one of the 11 Disciplinary Research Programs, no longer lumped with “Algebra and Number Theory” (which remain joined at the hip according to NSF). Around the country, Combinatorics is fairly well represented at the top 100 universities, even if breaking “top 10” barrier remains difficult. Some are firmly in the “algebraic” camp, some in “probabilistic/extremal”, some have a lot of Graph Theory experts, but quite a few have a genuine mix.
This all reminded me of a story how I found out “What is mathematics“. It started with me getting a Master of Arts degree from Harvard. It arrived by mail, and made me unhappy. I thought they made a mistake, that I should have been given Master of Sciences. So I went to the registrar office, asked to see the director and explained the problem. The director was an old lady, who listened carefully, and replied “let me check”. She opened some kind of book, flipped a few pages and declared: “Yes, I see. No mistake made. Mathematics is an Art.” Seeing my disappointed face, she decided to console me “Oh, don’t worry, dear, it’s always been that way at Harvard…”
What the experts are saying.
About the title question, I mean. Let’s review the quotes page. Turns out, a lot of things, often contradicting each other and sometimes themselves. Some are cunning and ingenuous, while others are misleading or plain false. As the management maxim says, “where you stand depends on where you sit”. Naturally, combinatorialists in different areas have a very different view on the question.
Few themes emerge. First, that combinatorics is some kind of discrete universe which deals with discrete “configurations”, their existence and counting. Where to begin? This is “sort of” correct, but largely useless. Should we count logic, rectifiable knots and finite fields in, and things like Borsuk conjecture and algebraic combinatorics out? This is sort of like defining an elephant as a “large animal with a big trunk and big ears”. This “descriptive” definition may work for Webster’s dictionary, but if you have never seen an elephant, you really don’t know how big should be the ears, and have a completely wrong idea about what is a trunk. And if you have seen an elephant, this definition asks you to reject a baby elephant whose trunk and ears are smaller. Not good.
Second theme: combinatorics is defined by its tools and methods, or lack of thereof. This is more of a wishful thinking than a working definition. It is true that practitioners in different parts of combinatorics place a great value on developing new extensions and variations of the available tools, as well as ingenuous ad hoc arguments. But a general attitude, it seems, is basically “when it comes to problem solving, one can use whatever works”. For example, our recent paper proves unimodality results for the classical Gaussian coefficients and their generalizations via technical results for Kronecker coefficients, a tool never been used for that before. Does that make our paper “less combinatorial” somehow? In fact, some experts openly advocate that the more advanced the tools are, the better, while others think that “term ‘combinatorial methods’, has an oxymoronic character”.
Third theme: combinatorics is “special” and cannot be defined. Ugh… This reminds me of an old (1866), but sill politically potent Russian verse (multiple English translations) by Tyutchev. I can certainly understand the unwillingness to define combinatorics, but saying it is not possible is just not true.
Finally, a piecemeal approach. Either going over a long list of topics, or giving detailed and technical rules why something is and something isn’t combinatorics. But this bound to raise controversy, like who decides? For example, take PCM’s “few constraints” rule. Really? Somebody thinks block designs, distance-regular graphs or coding theory have too few constraints? I don’t see it that way. In general, this is an encyclopedia style approach. It can work on Wikipedia which is constantly updated and the controversies are avoided by constant search for a compromise (see also my old post), but it’s not a definition.
My answer, after Gian-Carlo Rota.
After some reading and thinking, I concluded that Gian-Carlo Rota’s 44 y.o. explanation in “Discrete thoughts” is exactly right. Let me illustrate it with my own (lame) metaphor.
Imagine you need to define Russia (not Tyutchev-style). You can say it’s the largest country by land mass, but that’s a description, not a definition. The worst thing you can do is try to define it as a “country in the North” or via its lengthy borders. You see, Russia is huge, spead out and disconnected. It lies to the North of China but has a disconnected common border, it has a 4253 mile border with Kazakhstan (longer than the US-Canada border excluding Alaska), surrounding the country from three sides, it lies North-West of Japan, East of Latvia, South-West of Lithuania (look it up!), etc. It even borders North Korea, not that this tiny border is much in use. Basically, Russian borders are complicated and are a result of numerous wars and population shifts; they have changed many times and might change again.
Now, Rota argues that Combinatorics is similarly formed by the battles, and can only be defined as such. It is a large interconnected field concentrated (but not coinciding!) around basic discrete tools and problems, but with tentacles reaching deep into “foreign territory”. Its current shape is a result of numerous “wars” – the borderline problems are tested on which tools are more successful, and whoever “wins”, gets to absorb a new subfield. For example, in its “war” with topology, combinatorics “won” graph theory and “lost” knot theory (despite a strong combinatorial influence). In other areas, such as computer science and discrete probability, Rota argues there a lot of cooperation, a mutually beneficial “joint governance” (all lame metaphors are mine). But as a consequence, if one is to define Combinatorics (or Russia), the historical-cultural approach would go best. Not all that different from Sheldon’s approach to define Physics “from the beginning”.
In conclusion, let’s acknowledge that Combinatorics can indeed be defined in the same (lengthy historical) manner as a large diverse country, but such definition would be neither short nor enlightening, more like a short survey. As Danny Kleitman writes, in practice this lack of a clear and meaningful definition of the subject “never bothered him”, and we agree. I think it’s time to stop worrying about that. But if someone makes blank general statements painting all of combinatorics in a certain way, this is just indefensible.
UPDATE (May 29, 2013)
I thought I would add a link to this article by Gian-Carlo Rota, titled “What ‘is’ mathematics?” This was originally distributed by email on October 7, 1998. For those too young to remember the Fall of 1998, Bill Clinton’s testimony was released on September 21, 1998, with its infamous “It depends on what the meaning of the word ‘is’ is” quote. Rota’s email never mentions this quote, but is clearly influenced by it.
UPDATE (August 5, 2015)
Since this article was written, Russian borders changed again, and not in a way I could have imagined (or supported). I am guessing, Combinatorics boundaries also changed. See e.g. our latest blog post titled The power of negative thinking on the use of Computability in Enumerative Combinatorics.