I heard this received wisdom first from M. Artin when I was a semester at MIT as a Sloan Fellow. My response – not agreeing with Artin, with whom I often conversed – was that the seriousness of errors depends on the article's ambitions and use of sophisticated mathematics. Also, if you don't fine-tune this opinion for mathematics, where would you? Finding referees for new, not-yet-assimilated tools is one tough problem. It requires interdisciplinary refereeing expertise and editors who follow the analysis. Or else, it is rife for abuse, applying stingy analysis to one author not applied to another.

An example will help convey what I mean by a serious error. A recent submission to Math. Annalen aimed to produce infinitely many primes supporting a conjecture of Ihara from the early '80s. Amidst technical details the author used the Galois closure of curve covers of prime-power degree as if the result would be a p-group. (Consider prime-power degree symmetric groups to see how wrong this is.) Only the prime 2 of his argument could be salvaged. Though then, the paper was inappropriate for the announced expectations of this journal. Yet, the initial editor accepted it and the author thanked the editor for catching the error.

Refereeing and proofs in practice: Two examples from my early experience informed my response to M. Artin. They also help me address the full gamut of Aczel's concerns. My first paper solved Schur's 1923 Conjecture [S]. There was a 500+ paper bibliography sent to me of earlier papers considering this easy-to-state conjecture. Most on particular polynomials not violating it. My short paper characterized properties an example would have. Then, using Riemann surfaces and group techniques, uncommon to the area, to show only the list of Schur could so contribute. The proof was everything.

My first Annals paper [AK] solved the main general problem posed by Ax and Kochen coming from their proof of a famous E. Artin conjecture. Prior to my result, someone well-connected to the area proved there could be no such theorem as mine. My paper had five referees. Four called my office at Stony Brook, with the same technical questions on my generalization of Chebotarev's theorem. A 5th revealed himself years later.

I got a fair hearing from a rare fair editor – Armand Borel – to whom I lobbied for my paper during a two year post-doctoral at IAS. My experience: Editors dominate in this process over referees.

More Recieved Wisdom: Here are other points of received wisdom complicated by the various parties and vested interests in the production of journals.

- Good work will get recognized [in the end].
- It's the judgment of your peers that counts.
- Most editors (resp. referees) are fair/work hard [see Bryan Cain's August 2007 Notices lament on the failure of this].
- It's the author's job to communicate well [immediately, prior to and independent of any referee feedback].
- Anyone can referee. What matters is results.

Checking the rewards for refereeing: Yet, this was half the target of Should Journals compensate Referees?. To pay/reward talented referees (responses to my article included further suggestions) would develop a responsive referee cadre. Still, how about those editors who take lack of referee response as if it is the

Correspondent #5 in my article asked, Why do mathematicians referee without compensation? He wondered, among the possibilities if it was a fair trade for having their papers refereed?

The experience for my last paper [L] sent out for refereeing says no, you might not even get that. [AK] and [S] from the beginning of my career were (respectively) key references for two papers – [A] (single-authored) and [B] (3-authored) – an IMRN editor asked me to referee.

Expansion from [AK] governed the first half of [A]. Vovodsky's relative Chow Motives – initially a black box to me – governed its second half. I opened direct communication with the paper's author for further clarification, but the editor said I was holding up the process and accepted the paper. A year later the author substantially revised and resubmitted his paper because of an error in the area of my questions.

The topic-opening result of [D] published in 2005 was the main result of [B] on which two of [B]'s authors had seen me lecture. I accepted the paper for publication with considerable comment separating what they did from previous results, but also asked they at least change their abstract. This consisted of just a special case for curves of [D]'s topic-opener. The authors took many months to gave a complicated response, refusing to acknowledge [D]. The IMRN editor accepted their paper without further revision, saying despite my immediate referee report, that the process had gone on long enough.

After 10 months here is the (same) editor's response to my submission [L]. "We have not been able to find a referee for your paper IMRN/82303 entitled 'Alternating Groups and Moduli Space Lifting Invariants.' In fact we get no answers whatsoever. We've decided not to continue but to simply return the paper without comment for submission elsewhere."

What is at stake?: There are many players in this process, including publishers who add little to our TeX-ed papers, and editors who edit precious little. The more players, the easier for any one to get off the hook. My opinion piece said: Refereeing is a hard task, and too few do it well. With more incentive, more mathematicians might develop the high skills that go with quick, quality refereeing.

I add here: Likely, too, editors would help improve the refereeing process by informing researchers more precisely of their journals' goals and analytical standards. Claiming, as many journals do, high standards, is not much guide for whether a particular paper belongs in a given journal.

I think it significant that another article in the May 2007 Notices was on the issue of journals. This was the announced resignation of Topology Journal editorial board. The disparate issues, however, raised by this and my opinion piece don't reconcile easily. The former appears as a revolt of elite mathematicians against venal publishers, while the latter insists on rectifying internal (to mathematics) difficulties between referees and editors. Further, as stated on p. 906 of [AMS07], 78% of the AMS operating budget is for publishing, especially aimed at keeping publication costs down.

[IM] comments on those issues by reconsidering some aspects of the Received Wisdom List.

Finally, my opinion piece started with the plight of a talented (I claim) mathematician with difficult dealings with the researchers in his area. So, it is hard to find referees for his ambitious projects. He's an example of another unanalyzed topic: Which mathematicians have the most troubles with getting timely refereeing? I contend some with the most difficulties are ambitious, technically and imaginatively, and the more modest and less reaching might find they get in print more easily. [IM] includes comments from some who wrote directly to me after my opinion piece. This amounts to a thesis, likely testable in several ways. Yes, qualifiers relate to the authors' having support within well-recognized schools.

[IM] makes two points on the ethics of mathematics: more on the received wisdom list above, and noting that the essence of mathematics – it's ability to define precisely, and then prove things from these definitions – fails to promulgate proof to the rest of science. My suggested reason – over specialization in the mathematics community itself – further motivates encouraging quality referees to rise above their specializations.

References

[A] J. Nicaise, Relative Motives
and the Theory of Pseudo-finite fields, IMRN to appear.

[Ac] J. Aczel, Response
to "Should Journals pay Referees," October
2007 Notices AMS.

[AMS07] From the AMS Secretary, Notices of the
AMS Aug. 07 **54, #7, 904–915.**

[B]
R. M. Guralnick, T. J. Tucker and M. E. Zieve, Exceptional Covers and Bijections on
Rational Points, IMRN to appear.

[AK] with G. Sacerdote, Solving
diophantine problems over all residue class fields of a number field
..., Annals Math. 104
(1976), 203–233. Picked up
(with Ken Ribet's help) from JStor
http://links.jstor.org/sici?sici=0003-486X%28197609%292%3A104%3A2%3C203%3ASDPOAR%3E2.0.CO%3B2-P.
Introduces the Galois Stratification procedure in its Original,
geometric form. Material corresponds roughly to Chap. 25 of the Fried Jarden book
(1985 edition).

[D] The place of exceptional covers among all
diophantine relations, J. Finite Fields 11 (2005) 367–433.

[IM] Who Reads Your Papers?: Achievement vs Success, Part I?
Andy Magid is departing and Steven Krantz is entering editor of the
Notices. Magid's article covered territory from the large AMS Notices
changes starting in 1995. I use the events that precipitated those
changes, at AMS and in departments, to discuss parallel changes
in refereeing/editing. It suggests there was another referee paradigm
that is still relevant today, but little used. Part II, in preparation
12/03/09, suggests it won't work to plead with everyone to be a
better referee. Rather, it suggests incentives journals could
apply to encourage those with the appropriate skills and interest to be
the referees we really need.

[L] Alternating groups and moduli space
lifting Invariants, Israel J. Math. (2009), 1–68, Arxiv #0611591v4. Starts with a description and properties
of spaces of 3-cycle covers, Arxiv #0611591. Serre asked me a question
with an incorrect guess at the answer. I gave a method that hinted at
the right answer, proved it in special cases and suggested a result
with further application. Serre agreed, used part of the method, and
completely answered his version of the question [Se].
I waited 10 years to write out three applications – two at the end of
the paper justifying my version of the result, one to producing special
canonical theta nulls on Hurwitz spaces, the other to analyzing a
possibility for a presentation of G_{Q} (a la, the proven, but less
canonical presentation in [FrV2]).

[S] On a
conjecture of Schur, Mich. Math. J. 17 (1970), 41--45.

[Se] J.-P. Serre, Revêtements
a ramification impaire et thêta-caractèristiques,
C.R.Acad. Sci. Paris 311
(1990), 547--552.