The new issue of the Notices of the American Mathematical Society turned up in my mailbox today. It features an interesting, if slightly disturbing, editorial (PDF format) by CUNY mathematician Melvyn Nathanson. He wonders about how confident we can really be regarding the proofs that appear in our research journals:
But why the delay? Surely, any competent person can check a proof. It’s either right or wrong. Why wait two years?
The reason is that many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community.
But the community itself is tiny. In most fields of mathematics there are few experts. Indeed, there are very few active research mathematicians in the world, and many important problems, so the ratio of the number of mathematicians to the number of problems is small. In every field, there are “bosses” who proclaim the correctness or incorrectness of a new result, and its importance or unimportance. Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation.
He goes on to write:
Part of the problem is refereeing. Many (I think most) papers in most refereed journals are not refereed. There is a presumptive referee who looks at the paper, reads the introduction and the statements of the results, glances at the proofs, and, if everything seems okay, recommends publication. Some referees do check proofs line-by-line, but many do not. When I read a journal article, I often find mistakes. Whether I can fix them is irrelevant. The literature is unreliable.
How do we recognize mathematical truth? If a theorem has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it’s not a theorem. But even in mathematics, truth can be political.
Interesting stuff, and largely accurate, alas. I think Nathanson overdoes it a bit in likening the authorities in a field to political bosses or gang leaders, but otherwise I agree with every word of his editorial.
His remarks about peer review are especially on the money. Refereeing a research paper in mathematics is an awful, time-consuming task. The style among research mathematicians is to make brevity the highest of virtues, and anyone who writes three consecutive sentences of jargon-free exposition will immediately be accused of excessive wordiness. When reviewing a paper you can reasonably expect to be stopped dead several times, banging your head in frustration and asking yourself, “How the heck does this sentence follow from anything that came before?”
There is a practical reason for this, and also a sinister one. The practical reason is that paper is expensive. (On the other hand, I have not noticed that papers published in electronic journals are any clearer than their low-tech counterparts.) The sinister reason is that crystal clear writing would immediately make apparent the triviality of most research papers.
It is also true that there is a large element of faith in modern mathematics. As Nathanson writes, if the bosses say the result is valid, that is good enough for just about everyone. In my own field of algebraic graph theory, there is an article that I have cited in several of my research papers. It was authored by one of the true giants of the field, appeared in a prestigious journal, and its main result is certainly one that feels right given everything else I know about the subject. The fact remains that despite several attempts, both on my own and with my research collaborator, I can not make heads or tails out of the proof. It starts off sensibly enough, but then quickly fades to black as I go farther into it. But I cite the result anyway.
There are very few mathematicians who actually enjoy picking up a research paper and reading it. Even if the paper comes right from your own little corner of the mathematical world, it is still going to be difficult to read. What is fun, on the other hand, is going to a conference, meeting the author of the paper, and asking him to explain what, exactly, he did. It’s quite rare for the explanation to take more than a few minutes.
I often tell my students that math is easy, it’s math classes that are hard. Mathematics deals in clever ideas and moments of clarity. There is nothing quite so satisfying as looking into darkness and mystery, and, using nothing more formidable than your own brainpower, creating light. Math classes, alas, deal in tedious symbol manipulation, excessive abstraction, and in doing everything possible to make the subject appear to the world as a conspiracy against the laity. As Nathanson points out, it’s not just students that have that feeling. Even professionals find themselves at the mercy of those who know more.