How to Publish Counterexamples in 1 2 3 Easy Steps

I found some time to look over recent issues of Notices of AMS and I noticed a letter from Theodore P. Hill, a mathematician from Georgia Tech. The letter led me to read his article “How to Publish Counterexamples in 1 2 3 Easy Steps”. It’s a story that starts when he found errors in a mathematical paper that he got for review and traced those errors to a previously published article in Notices and ends almost two years later in pretty much a complete failure to cause any public admission of the problem and correction from the authors.

One of the interesting aspects of this story is the difference in attitude between the authors of the Notices article. The first author (called Au1) is a professor of political science at a leading U.S. university and the second (Au2) is an associate professor of mathematics. While Au2 was kind of receptive to criticism, Au1 idea was blame everything on Hill’s “misconceptions and misinterpretations”.
To me it looks like an example of clash of cultures between mathematics and humanities. While truth in mathematics is more objective, in humanities it is more of a matter of position in hierarchy. Au1 probably considers days when someone might effectively tell him he is wrong as long gone. Such clashes will happen more often as softer sciences become more mathematicised.

This is a Formalized Mathematics blog so I have to mention that of course such things can not happen when proofs are written in a formal proof language. I don’t want to defend Au1 but you can’t really say that a theorem is wrong if it is not clear what it says. When formalized mathematics becomes more popular similar clashes between the standards of truth will occur. Formalizers will be pointing errors in romantic math proofs and soft mathematicians won’t understand the problems.

There is a part missing in Hill’s story. Everybody makes mistakes in romantic math and one may be curious what Professor Hill does when someone finds an errror in his paper. I am in position to add that missing part. When I was a graduate student at The University of Tennessee I found such mistake in an article published in Proceedings of AMS by T. Hill and M. Spruill. When I notified them they sent an erratum to the publisher with all possible acknowlegments and thanks. So they did what they should. But in fact they did much more than that. Ted Hill invited me to Georgia Tech to give a talk on related subjects. As I was facing the prospect of a job search in near future it was an important opportunity to advertise my research. After the talk we went to an all-you-can-eat sushi bar and that was great. It was there when my fiance (now wife) started to accept that certain kinds of arthropods are edible and even tasty. So, my advice to every math graduate student: read Ted Hill’s papers very carefully and try to find a mistake. If you manage, you will have an opportunity to meet a remarkable person.

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