by Jonathan Kujawa
While I was in graduate school the film “Trekkies” was released. You can see the trailer here and the full film here. What could easily be mocking is in fact a heartfelt look at a group of people who choose to devote their lives to something they love. After seeing the film my friends and I semi-seriously suggested that mathematicians would make a great subject for a documentary. We have more than our share of interesting folks. And, like Trekkies, there is an entire subculture.
One corner of that subculture is Mathematical Reviews. An arm of the American Mathematical Association, Math Reviews is a compendium of everything published in mathematics. It was founded in 1940 and contains over three million publications, with the earliest published in 1810. What makes Math Reviews invaluable is the reviews. Each research paper, monograph, book, etc., is assigned to a volunteer mathematician who has the expertise to write a review of the work. Short of personal attacks, slander, and the like, the reviewer is pretty much free to write what they choose. The usual thing is to give a summary of the work along with commentary. As a reviewer you might discuss how the results fit in the broader field or highlight aspects of the work which might be of particular interest. Oftentimes it's hard to tell from the title and abstract if a paper, say, contains needed results. Well written reviews can save the reader countless hours in the library.
Since reviewers have a free hand there are plenty of exceptional reviews amongst Math Reviews's vast collection. Ten years ago my colleague, Kimball Martin, began a compilation of truly great reviews. If you have access to a library with a subscription to Math Reviews, you can read his entire collection for yourself. Some are rave reviews, but there are some real zingers in there as well (see the title of this essay) which I thought the readers of 3QD would appreciate [1].
With decades worth of publications, some truly terrible papers have appeared. Reviewers aren't ones to let rubbish slide through. Sometimes it is the mathematics itself which is questionable:
It is hard to imagine in a single paper such an accumulation of garbled English, unfinished sentences, undefined notions and notations, and mathematical nonsense. The author has apparently read a large number of books and papers on the subject, if one looks at his bibliography; but it is doubtful that he has understood any of them…. What is amazing to the reviewer is that such a thing was ever printed.
Or:
Not every text containing mathematical formulae or terminology may be considered as a scientific work. Sometimes it is a mere imitation. My impression is that this is exactly the case of the paper under review. The paper deals with some relations between Riemann theta functions, but I have a feeling that the authors have only a rather vague notion about this subject. I doubt that they have read items 1,2,3,6 of their own references. All of the authors' statements are either tautological or false.