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 .
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.
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.
Even if the math itself may be fine, the exposition may leave something to be desired:
Indeed, the exposition throughout puts one in mind of a stream of consciousness monologue in which temporal and logical order are not issues. While one may admire the technique in Joyce or Faulkner, it is a difficult way to learn science.
As a result of correspondence with the author, the reviewer realises that his attempt to understand the paper was unsuccessful; the criticism based on that attempt should be withdrawn. The reviewer does not understand the paper at all now.
The author asserts that “this book is written by an amateur for other amateurs, but we amateurs won't mind having the professionals reading over our shoulders''. The reviewer, a professional, would like to benefit from whatever insights the author may have. However, he has not been able to see through the blizzard of unusual (and largely unexplained) notations and formulas that fill the 274 hand-written pages of this book. If the author has a message for professional mathematicians, he will have to try again, using a language that is not a personal secret.
And even if the math and exposition is unimpeachable the reviewer may doubt whether it was worth bothering to write the paper at all:
The authors give a history of the problem and prove two propositions that produce some pretty large Smith numbers…. But the reviewer is not convinced thereby that Smith numbers are not a rathole down which valuable mathematical effort is being poured.
The authors… take the depraved view that this is the model of an optimal seducing policy for a dynamic continuous lover who at time t will have been done in by rivals or scorned women with probability 1−x(t). The optimal policy u(t) is computed in the cases U″<0, the concave lover; U(u)=u, the linear lover; and U″>0, the convex lover. No evidence is presented for the success of these policies in practice so we must conclude that the authors have had none.
This elementary problem occupies the whole of the lengthy paper under review. It is pursued with an unexpected maladroitness and many devious and useless complications.
There are those who claim to have solved a major open problem, with Fermat's Last Theorem a popular choice. As we saw here at 3QD, Andrew Wiles famously proved the validity of Fermat's Last Theorem in the mid 1990s. Some folks remain undaunted:
Herein the author states “her genuine concern'' about Wiles's purported proof of Fermat's last theorem which, after all, appeared in an “in-house publication in the Annals of Mathematics at Princeton''…. Of course, she has no such worries about the validity of her own, Euclidean-algorithm-inspired, proof of Fermat's last theorem.
Of course not all exceptional reviews are negative. Some are playful:
Such an element… is called evil by the authors, and their main result is that such elements do not exist…. Aesthetically, since Geck and Michel have defined “good'' elements… and proved that they exist in every conjugacy class…, it seems only right that someone should define “evil'' elements and prove their non-existence. By making the right definitions, evil has been eradicated and good has prevailed.
The publisher may be proud to have such a book in his collection. However, a price of $174 is a horror…. Overall, the book under review is a great gift. It is wonderfully written and consistently radiates a winning friendliness. The book is an excellent navigator through a babel of currently appearing papers and is an impressive account of many years of experience of many mathematicians. It is warmly recommended.
The king of the mellifluous review is Godofredo Iommi Amunátegui at Pontificia Universidad Católica de Valparaíso, Chile. He writes reviews in my area of research, representation theory, and I keep hoping he'll review one of my papers. Here's a taste:
The symmetric group Sn is an old, vast and gorgeous mathematical domain. Therefore, the discovery of a flower under a stone is not unusual…
Nowadays there is a kind of intellectual principle of least effort which consists in consulting the Web to get a rapid contour of almost anything under the sun. Disregarding my own irony, at the beginning of this review I shall play the forbidden game: “Vittorio Fossombroni (1754–1844) was a personage of multifold interests: mathematician, hydraulic engineer, economist, intellectual and political figure….
[Fossombroni] closes his “Discorso'' with a quotation—in English—of Addison: “If I can any way contribute to the diversion or improvement of the country in which I live, I shall leave it, when I am summoned out of it, with the secret satisfaction of thinking that I have not lived in vain.'' Let us conclude this review with the echo of these words.
The problem of inverse perspective may be stated as follows: given a painting, where is the painter's eye? The idea is to find the optimum position so that “when the eye is placed there, the picture is seen in its perfection''. There is no single solution…. A similar difficulty occurs when the present text is treated with care: where must the reviewer's eye be placed?
No discussion of reviews in mathematics can pass without mentioning Gian-Carlo Rota. While he was chief editor of Advances in Mathematics no one was spared the sharp knives of his editorial reviews. Perhaps none was more damning than his pithy review of a modern philosophy text: “When pygmies cast such long shadows, it must be very late in the day.” This along with a sampling of his other reviews can be found here.
Not all research papers are entirely serious. Alpher and Gamow invited Bethe to be a co-author so that they could have their paper's authorship read Alpher-Bethe-Gamow . And of course there is the infamous paper by Cox-Zucker .
 To protect the innocent and the guilty, I'll leave my quotes unattributed.
 To mimic the first three letters of the Greek alphabet.
 I trust no explanation is needed. In this case the authorship is an unfortunate coincidence.