## What is the error rate for published mathematical proofs?

Mathematical proofs are sometimes cited as the gold standard against which software quality should be compared. At school we rarely get to hear about proofs that turn out to be wrong and are inculcated with the prevailing wisdom that all mathematical proofs are correct.

There are many technical and social issues involved in believing a published proof and well known established mathematicians have no trouble pointing out that “… it is impossible to write out a very long and complicated argument without error, …”

Examples of incorrect published proofs include Wiles’ first proof of Fermat’s Last Theorem and an serious error found in a proof of a message signing scheme.

A question on mathoverflow contains a list of rather interesting false proofs.

Then, of course, there are always those papers that appear in journals that get written about more frequently on Retraction Watch than others.

What is the error rate for published mathematical proofs? I have not been able to find any collection of mathematical proof error data.

Several authors have expressed the view that because there so many diverse mathematical topics being studied these days there are very few domain experts available to check proofs. A complicated proof of a not particularly interesting result is unlikely to attract the attention needed to check it thoroughly. It should come as no surprise that the number of known errors in such proofs is equal to the number of known errors in programs that have never been executed.

Proofs are different from programs in that one error can be enough to ‘kill-off’ a proof, while a program can contain many errors and still be useful. Do errors in programs get talked about more than errors in proofs? I rarely get to socialize with working mathematicians and so cannot make any judgment call on this question.

Every non-trivial program is likely to contain many errors; can the same be said for long mathematical proofs? Are many of these errors as trivial (in the sense that they are easily fixed) as errors in programs?

One commonly used error rate for programs is errors per line of code; how should the rate be expressed for proofs? Errors per page, per line, per definition?

Lots of questions and I’m hoping one of my well informed readers will be able to provide some answers or at least cite a reference that does.

Michael de Villiers wrote in “The role and function of proof in mathematics” (1990, PDF at http://dynamicmathematicslearning.com/homepage4.html): according to an unnamed past editor of Mathematical Reviews, “approximately one half of the proofs published in it were incomplete and/or contained errors, although the theorems they were purported to prove were essentially true”. The original source of this claim is a book “Rigorous proof in mathematics education” by Gila Hanna (1983), but unfortunately this is long out of print. I wonder how the widespread use of Maple, MATLAB, etc. have affected this issue since then …

@Howell

Thanks for the link to this very interesting article. I think the following quote from Polya (not from his famous book) provides some useful background:

“… having verified the theorem in several particular cases, we gathered strong inductive evidence for it. The inductive phase overcame our initial suspicion and gave us a strong confidence in the theorem. Without such confidence we would have scarcely found the courage to undertake the proof which did not look at all a routine job. When you have satisfied yourself the theorem is true, you start proving it.”

If mathematicians believe the proof is true they are less likely to carefully pick over a proof. Undergraduate maths textbooks are known to be full of errors (sloppy copy-editing, author deadlines etc) and there is the story of Hilbert finding errors in Euclid’s proofs.

Amazon lists the Hanna book, but with none being available.

I think researchers will operate at the limit of what is possible. So the availability of mathematics software will result in them tackling harder questions rather than helping to ensure what the level they are currently at is error free.

A 2005 meeting of the Royal Society resulted in some very interesting papers (The nature of mathematical proof).