Wednesday, November 01, 2006

Symptoms of withdrawal

When I first saw mention of a proof of the existence of a solution to the Navier-Stokes equations, I assumed that it had more in common with "proofs" of P=NP or the Riemann hypothesis. In fact, P. Smith had some keen original insights, even if the paper ultimately needed to be withdrawn. There is no shame in this, of course. As G. Kuperberg has illustrated with many examples posted next to his office door, quite capable mathematicians will from time to time push something out the door before it can stand on its own two feet; many of these errors are discovered when a colleague provides a counter-example to one of the central proofs of the article, which I would find considerably more embarrassing than discovering a "serious flaw" in antecedent peer-reviewed literature. Real shame lies in leaving papers up after their erroneousness has been plainly illustrated; as of this writing, this is true of those who wrote about the LP formulation of the TSP just a few weeks ago.

This became a sufficiently hot topic in the blogosphere community that Seed magazine ran a short write-up of the brief history of Smith's work and the comments made from as high up as P. Woit's site. Unfortunately, S. Ornes took the opportunity to sensationalize the matter, suggesting that these events "might ... give mathematicians of the future a strong incentive to be hyper-meticulous about their work", and "[p]erhaps [they] will have to reconsider posting their work on arXiv."

As if there isn't already sufficient incentive. The referee process is ponderous, and it is simply not in anyone's interest, least of all the researcher's, to submit rough work to peer-reviewed journals. Sloppy work will take even longer to make it to press, or possibly be rejected outright. In our "publish or perish" world, six extra months between publications can have a severe impact on a performance review; while on the job market, it can make the difference between being employed and not.

And no one I know will change their practice of submitting to (or defying) the arXiv. The Seed article suggests that many members of the mathematical community will fear the public scrutiny and response that developed in the wake of the Navier-Stokes paper and therefore reduce their use of the electronic pre-print service. The fact is that extremely few of us are doing such high profile work that it will result in attention beyond the regular readership of our subclassifications, and thus have nothing of the sort to fear. Events such as this, as much as they might make for exciting news stories, have a minimal impact on the day-to-day life of working mathematicians.

The mechanisms that I see at work are two-fold: first, the increasing speed of mathematical communication via the internet, and second, the volume of positive attention mathematics has garnered in recent years in the culture at large. We just happen to be looking at a point in history where these two phenomena have interacted more dramatically than they have before, causing an unprecedented event, the speculated impact of which has been tremendously overstated.

Fifty years ago, preprints were mailed to colleagues at remote institutions, journals played a critical role in the communication of new research, and conferences were one of the rare opportunities for intimate interaction between mathematicians from different regions. Now, journals are published both in print and online, almost every author either uses the arXiv or maintains a collection of their preprints on their own website, and email is routinely used to disseminate results or facilitate collaborations. Conferences, while still providing a unique opportunity to travel to exotic locales and spend some uninterrupted quality time with like-minded researchers, no longer play their critical role in discourse, as conference proceedings are made available electronically and air travel costs for one-on-one collaboration are at stupendous historical lows. The trend always has been to take advantage of new technologies whenever they might be useful; the arXiv will maintain its utility, and will therefore see no decline in its usage.

Thanks to movies, television, and the popular press, a larger segment of the population has taken an interest in mathematics than ever before. This means not only that laypeople are more likely to have a demand for news from the math world, but also that more specialists are willing to fulfill that demand (such as in this forum, solipsistic as it may be). The result is one that has been seen time and time again; once a critical mass of anonymous commentators has accrued, the signal will be overwhelmed by noise. It's very sad that in this case some of that noise took the form of personal criticism of Prof. Smith by non-mathematicians who have no experience with the process of mathematics.

We all make mistakes; those of us who live on the cutting edge make a lot of them. We catch most of them before sharing our work with the world, but a few slip through, are caught later, and rectified. It's far better to live by the pursuit of truth than the fear of error.

As a fortune I got after a chinese meal once told me:

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