### Reliability

#### Posted by David Corfield

Melvyn B. Nathanson has an article on the ArXiv today – Desperately seeking mathematical truth – which questions the reliability of the mathematical literature. He observes,

When I read a journal article, I often find mistakes,

and reasonably concludes that

The literature is unreliable.

For deep, difficult and long proofs, we can only “rely on the judgments of the bosses in the field”. And so,

…even in mathematics, truth can be political.

A first line of response might try to lessen the worry. A dangerous situation is one where a result which is used frequently in other results hangs itself from a single perilous thread. But fortunately there’s some useful feedback in the system. To the extent that the result is used frequently, people will look more closely at it to see if anything extra can be extracted, and in particular submit its proof to closer scrutiny. Alternative proofs are likely to follow. A result upon which little depends may well not receive this attention, but then its reliability matters less.

And to the extent that people detect power in a result, this often coincides with an implicit conceptual fruitfulness whose elaboration often leads to its being tied in a variety of ways to other parts of the system. Think of quadratic reciprocity. Of course this process may take a long time on the scale of an individual’s career. Nathanson raises the long proof of the classification of finite simple groups as one for which we must rely on the bosses. Perhaps this is of little consolation to him now, but it is surely feasible that a century hence there will be a much better proof – the field will have found its Darwin:

…the classification of finite simple groups is an exercise in taxonomy. This is obvious to the expert and to the uninitiated alike. To be sure, the exercise is of colossal length, but length is a concomitant of taxonomy. Those of us who have been engaged in this work are the intellectual confreres of Linnaeus. Not surprisingly, I wonder if a future Darwin will conceptualize and unify our hard won theorems. The great sticking point, though there are several, concerns the sporadic groups. I find it aesthetically repugnant to accept that these groups are mere anomalies… Possibly…

The Origin of Groupsremains to be written, along lines foreign to those of Linnean outlook. (John Thompson)

But perhaps this line of response is a complacent one. If a single mathematician “often finds mistakes”, there must be a huge number out there. Now that so many papers are posted on the ArXiv, should it not be expected of you that on finding an error you email the author? A more radical and interesting solution would be to have a site arxiv-comments.org where any mistakes noticed, or even potential mistakes, could be recorded, along with useful thoughts on connections with other work, etc. This could do some of the work of our public reviews.

## Re: Reliability

It seems that the arXiv’s trackback system is geared along these lines.

On the other hand, several times I got the impression that arXiv trackbacks seem to be turned off somehow especially in parts of the math archive. (?)