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Mathematicians at Alma College do not do research primarily to advance our collective mathematical knowledge. They do mathematical research because being engaged in research helps them convince their students that mathematics is a living enterprise and helps them communicate to their students how mathematics is done rather than just its dry facts.
Given their teaching and service responsibilities (and their level of interest in research), these mathematicians publish no more than one or two papers every several years. Given their low level of engagement in research, they are not well connected to any research community. As far as I can tell, their papers are not of significant interest.
Refereeing is the only way mathematicians at Alma get feedback on their mathematical research. Refereeing is also the only way that anyone can easily certify whether these mathematicians are doing genuine mathematics. One can argue about how well the journal system works for them, but it does provide them at least some feedback as well as certification.
In effect, our current system provides a small time subsidy for research by mathematicians at Alma, since they rarely, even far more rarely than they publish papers, have the time or expertise to referee a paper themselves. The journals organize this subsidy by telling individual prospective referees it is their turn to help out. Experts can and do refuse to referee, but there is some pressure generated by the system to accept at least once in a while.
Some of you are thinking about systems that are meant to cover only a small portion of our research and publication activities. I am not worried about your proposals. Some of you, however, are thinking about systems that at least have the potential of taking over all of mathematical publishing. The current journal system does not seem viable if the core of the research community abandons it. If your system succeeds, how will it serve the mathematicians at Alma? Will it simply throw them overboard? Does that mean students at Alma will then get no exposure to mathematical research? How will other faculty at Alma, for example those serving on the tenure and promotion committee, have any inkling whether their mathematics faculty actually do mathematics? How will graduate admissions committees at Central Michigan or Michigan Tech evaluate letters written by Alma professors recommending Alma students for their graduate programs?
Systems based on purely voluntary, crowd-sourced comments and ratings will not work for mathematicians at Alma and their papers because there is no crowd reading these papers. Arguments about whether refereeing provides good peer review are besides the point, because these papers won't get any other peer review. The current system at least ensures that every paper has at least one person, the referee, responsible for reading it in some detail. On occasion the referee shirks this responsibility, but most mathematicians take this responsibility seriously.
I urge everyone to think carefully about how systems under discussion will affect the mathematicians at Alma and the over one thousand schools like it that provide most undergraduate education in the United States. I believe mathematicians in smaller developing countries face similar issues. If your system will not serve them,
make sure it leaves room for other systems that do.
Alma College is a small college of about 1400 students in Michigan. It has three mathematicians with doctorates on its faculty. I do happen to personally know one mathematician who received his undergraduate degree from Alma, earned a Ph.D. in the last 10 years, and published several papers of some importance. There is no strong reason I picked Alma as an example. I just happen to know about it, and I picked it essentially randomly out of the two or three hundred such schools I have heard of.
For those who are not familiar with institutions such as Alma College, Alma is a private, not-for-profit small college offering only undergraduate degrees. It receives its money both from tuition and from philanthropic donations, including a small endowment. There are many such colleges in the United States, including some quite well-known ones attracting very talented students. (Alma is not one of those, but its students are significantly above average.) Students choose Alma because they believe that the dedication to teaching of its faculty and the personal attention possible with small classes at a small college outweigh any potential benefits one might have from taking classes taught by more active researchers. Personally, I believe that all but the most precocious geniuses get a better education at a school like Alma rather than at a research university. Almost all students at Alma live on campus in dormitories, and faculty are often interested in and to some extent involved in the students' non-academic lives as well as teaching classes. In particular, faculty often play a leading role in organizing the intellectual life of the school outside of classes by bringing in speakers and organizing discussions.
I'm at a college even smaller than Alma. A lot smaller. I'll echo your concerns. One approach would be something along the lines of this: In order to submit a new paper for review, one must first have reviewed a paper in his or her area of expertise. Thus, people will always have reviewed at least as many papers as they have written. Papers should only be reviewed for correctness and originality. Other comments can be made, but not as part of the official record. The fact that mathematicians are (usually) human and humans like to gossip and compete will ensure this.
Papers to read would be distributed randomly. If no unreviewed paper is available, then an already reviewed paper can be re-reviewed by whomever is waiting to submit. This sort of system accomplishes a number of things, including providing an inducement on the part of reviewers to get their work done quickly.
I know this doesn't answer your concern about folks at Alma being unable to competently review papers, but, I would say that, if you can't review a paper in your area of expertise, you probably shouldn't be writing one.
@John - I am of course being selfish here, but at my level of expertise I generally hope my papers are refereed (and hopefully therefore read) by people who are more expert (in the general area, not the specific problem) than myself. This isn't a question of competence but excellence.
Any good system requires experts to do more than their share, because an expert will do a better job at refereeing my paper than I will at refereeing theirs.
I don't disagree, Alexander. I'd much much rather have an expert evaluate my work. So would everyone else. What is likely to happen under my proposal, though, is that the top work will be evaluated by the top people - and maybe others, too. And it will be done more quickly. This is key, I think, and I should have emphasized this point in my post above. There are many, far too many journals for which the average submission to publication time is over a year. This is because we are relying too heavily on only a few people to do the work of review.
I'd also say that when I've refereed papers, I have gained greatly from it. It is probably some character flaw in me, but I read very carefully when I review, in a way in which I almost never read when I am just reading a paper. Doing this yields huge dividends, but is costly; like you I have very little time for research and I must balance the need to read broadly and deeply. I can't have both (or sometimes either :) ). Anyway, I think you make some good points, I just think we all owe the discipline the same work, even if the quality of that work varies.