Proofs of false statements

Mark Dominus brought up an interesting question last month: have there been major screw-ups in mathematics? He defines a “major screw-up” to be a flawed proof of an incorrect statement that was accepted for a significant period of time. He excludes the case of incorrect proofs of statements that were nevertheless true.

It’s remarkable that he can even ask the question. Can you imagine someone asking with a straight face whether there have ever been major screw-ups in, say, software development? And yet it takes some hard thought to come up with examples of really big blunders in math.

No doubt there are plenty of flawed proofs of false statements in areas too obscure for anyone to care about. But in mainstream areas of math, blunders are usually uncovered very quickly. And there are examples of theorems that were essentially correct but neglected some edge case. Proofs of statements that are just plain wrong are hard to think of. But Mark Dominus came up with a few.

Yesterday he gave an example of a statement by Kurt Gödel that was flat-out wrong but accepted for over 30 years. Warning: reader discretion advised. His post is not suitable for those who get queasy at the sight of symbolic logic.

5 thoughts on “Proofs of false statements

  1. In 1991, a decade before he won the Fields Medal, Vladimir Voevodsky published with Michael Kapranov a paper “∞-groupoids and homotopy types” exploring the relationship between category theory and homotopy theory and seeking to extend a result by Grothendieck. Seven years went by before Carlos Simpson provided a counterexample that demonstrated that one of the main results of that paper was incorrect. However, he didn’t identify a particular problem with K&V’s proof, and Voevodsky was sure that Simpson had made a mistake — until 2013, when he found the error in his own paper.

    Voevodsky tells the story in the article linked below, along with other examples of published papers that contained major flaws. One is a paper by Spencer Bloch, that “was soon after publication found by Andrei Suslin to contain a mistake in the proof of Lemma 1.1. The proof could not be fixed, and almost all of the claims of the paper were left unsubstantiated.” The second is another paper by Voevodsky himself, published in 1993; examining it more carefully several years later he found that “the proof of a key lemma in my paper contained a mistake and that the lemma, as stated, could not be salvaged.”

    It was incidents like this that got Voevodsky interested in automated proof verification, and motivated his work on Homotopy Type Theory/Univalent Foundations.

    “The Origins and Motivations of Univalent Foundations”
    https://www.ias.edu/ideas/2014/voevodsky-origins

  2. Good examples and worth keeping in mind. They are literally remarkabe — worth remarking about — whereas analogous observations about software errors would not be remarkable.

    Also, the errors are in fairly esoteric corners of math, far removed from intuition and not of interest to large numbers of mathematicians. It would be hard to imagine, for example, that Navier-Stokes equations don’t have solutions — contra physical intuition — and that an error in an existence proof would go unnoticed since many mathematicians work in CFD.

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