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When to start reviewing

I was recently asked to become a MathSciNet reviewer, which, as far as I understand, is something that they ask almost any mathematician at a certain point in their career.

Should I accept? What are your motivations in both directions? Is this a service to the community that I should help with, like serving as a referee for journal papers?

I'll write my own pros and cons as an answer.

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    $\begingroup$ Unless you have something against MathSciNet (in which case, this is probably not the site to discuss it), the only downside is that it will be some additional work. On the plus side, you might find it interesting, you might learn something and some people might spot your name while reading a review. $\endgroup$
    – Niemi
    Commented Jun 25, 2012 at 14:07
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    $\begingroup$ Besides, it is not a career-changing decision and it's not like you cannot stop writing the reviews once you have started to do so. $\endgroup$
    – Niemi
    Commented Jun 25, 2012 at 14:08
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    $\begingroup$ You should contribute to the general production of mathematical knowledge in some way, such as refereeing for journals, reading grant applications, writing MathSciNet reviews or organizing conferences. Which particular method you choose should depend on your strengths and tastes. $\endgroup$
    – David E Speyer
    Commented Jun 25, 2012 at 14:27
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    $\begingroup$ Realated: mathoverflow.net/questions/14607/when-to-start-reviewing $\endgroup$
    – Mark Grant
    Commented Jun 25, 2012 at 14:27
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    $\begingroup$ @quid: I respectfully disagree: the other question is basically "am I experienced enough to review?", not "why should I do it?". I see that some of the answers given there would be more pertinent to this question than to the original one, but in my view this is an issue regarding those answers, not this question. [Incidentally: no, I am not the one that voted for reopening --- I am the original author and thus I prefer to abstain] $\endgroup$
    – Federico Poloni
    Commented Jun 26, 2012 at 10:06

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The question is natural but the answer is highly individual. For what it's worth, I'll comment from my own perspective as a now-retired reviewer who produced almost 500 reviews over many decades. In its early years the print version of Mathematical Reviews was still relatively slim but covered the most widely circulated journals and had as reviewers most of the active mathematicians of the time. It had the great advantage of bringing together both a print database of current literature and (often) helpful commentaries by specialists. For many decades the authors were compensated only by receiving free papers (sometimes books).

As mathematics and its offshoots proliferated in the 1960s and later, it became impossible for most people to skim all reviews. But the evolving classification scheme helped, even though it could never meet all needs. Until the Internet (and arXiv) developed far enough, the reviews and database played a mostly constructive role in communication. But managing the flow of papers and editing the submitted reviews required a lot of expensive professionals, as it still does. Some reviews were of course eccentric, such as one which simply quoted verbatim half of a two page note from a widely circulated Springer journal.

For me personally it was a way to keep in touch with a wider range of interesting mathematics than I actually worked on at the time. But to do the reviewing task well is time-consuming, since I always felt the need to delve into the related literature. (At least once I discovered an earlier proof in a slightly offbeat journal of a theorem published anew in a mainstream journal by an author who hadn't been aware of the earlier proof.) Sometimes you get correspondence (even arguments) from an author whose work you have reviewed. All very interesting, but optional activity like refereeing.

By now MathSciNet functions mainly as an excellent database, still very expensive to maintain, and reviewers are given AMS credits for their use. Fewer papers get full reviews, which is usually the right decision but not always. People rely more for up-do-date stuff on other Internet sources, but the organized and flexibly searchable database is worth the cost for those institutions which can afford it. (Not all can.)

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    $\begingroup$ "the organized and flexibly searchable database is worth the cost" --- yes, I agree that the database is invaluable, but what part of this value lies in the reviews? $\endgroup$
    – Federico Poloni
    Commented Jul 12, 2012 at 9:15
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Cons:

  • it takes time. I should rather worry about producing good-quality research and disseminating it through slides and talks, especially now that I am relatively young. Publishing slides and expository material will "help the world" more than an AMS review.

  • it is not something that has a great value for the community. It might be just me, but I find the reviews of little use, considering that the papers already have abstracts. Mathscinet overall is great as a literature database and search tool, but the presence of reviews does not add much to its value.

  • AMS is selling access to these reviews. I am probably getting paid peanuts, or nothing at all (they did not mention payment at all on the invitation e-mail), to help building a corpus that is sold commercially. The prices are not cheap for a single user. I'd rather contribute to a public database.

Pros:

  • I personally do not use often the reviews, but they might be very helpful for other people. Maybe someone else relies on then more than I do (waiting for feedback here).

  • I am afraid to look mean and ungenerous to the eyes of the "math community" if I refuse it. For instance. refereeing for a journal is a time-consuming job, but we should not refrain from doing it because it helps the community; maybe other mathematicians see these reviews in a similar way.

  • it will look good in my CV as an additional "editorial service"; it is somehow a recognition that one is a competent member of the math community.

Overall, the pros do not seem too strong to me, and I realize that they are all about "how people will see me", which is probably a sign that they are flawed arguments. So, unless you guys on Mathoverflow manage to convince me of the contrary, I think I will turn down the offer, at least for now.

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    $\begingroup$ The "payment" is 8 AMS points per review. They can be spent on the AMS website to buy books for example, and one point equals one dollar. $\endgroup$
    – YangMills
    Commented Jun 25, 2012 at 14:29
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    $\begingroup$ On the other hand Zentralblatt MATH pays 2.56 euros per review, which can be transferred to your bank account. $\endgroup$
    – YangMills
    Commented Jun 25, 2012 at 14:31
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    $\begingroup$ To add to the argument, the policy for some time has been to award for each review AMS points worth perhaps $8 toward books or other permitted transactions. Also, a well-written review of a paper with or without an abstract can add considerable value by pointing out things like errors or omissions that the author of the paper doesn't notice. Just as refereeing of papers is uneven, reviews vary widely in quality, so of course it's hard to generalize. $\endgroup$
    – Jim Humphreys
    Commented Jun 25, 2012 at 14:33
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    $\begingroup$ Reviewing for AMS (well, my personal experience is with Zentralblatt, but it should be similar) is not actually very time consuming, especially compared to refereeing for journals. And here’s an additional pro: from time to time, you get a book for free (though reviewing a book does take more time than an article). $\endgroup$
    – Emil Jeřábek
    Commented Jun 25, 2012 at 14:37
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    $\begingroup$ Let me strongly disagree with the point "It is not something that has a great value for the community". I can only speak for myself, but I find Math Reviews extremely helpful. They can be very efficient way of getting a good idea of what is known about a problem (especially by following the "From References" and "From Reviews" links in a given review) or finding out whether a certain theorem applies in the case you want to use it. Of course, those things depend heavily on reveiwers writing informative reviews. $\endgroup$
    – user5117
    Commented Jun 25, 2012 at 14:54

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