TITLE: Negative impact of wrong or non-rigorous proofs
QUESTION [38 upvotes]: The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well as the famous essay by Jaffe and Quinn (http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00413-0/) and responses to it (http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/), raises for me the following question:
What are some explicit examples of wrong or non-riogour proofs that did damage to mathematics or some significant part of it? Famous examples of non-rigorous proofs include Newton's development of calculus and the latter stages of the Italian school in algebraic geometry. Although these caused a lot of dismay and consternation, my impression is that they also inspired a lot of new work. Is it wrong for me to view it this way?
In particular, I'm told that people proved false theorems using Newton's approach to calculus. What are some examples of this and what damage did they do?

REPLY [14 votes]: A proof being wrong can mean many things:

By 1932, when the Hungarian-American mathematician John von Neumann claimed to have proven that the probabilistic wave equation in quantum mechanics could have no “hidden variables” (that is, missing components, such as de Broglie’s particle with its well-defined trajectory), pilot-wave theory was so poorly regarded that most physicists believed von Neumann’s proof without even reading a translation.
More than 30 years would pass before von Neumann’s proof was shown to be false, but by then the damage was done. The physicist David Bohm resurrected pilot-wave theory in a modified form in 1952, with Einstein’s encouragement, and made clear that it did work, but it never caught on. (The theory is also known as de Broglie-Bohm theory, or Bohmian mechanics.)

The problems is the interpretation of what has actually been proved. I don't like the No Free Lunch Theorems for Optimization, because their assumptions are unrealistic and useless in practice, but the theorem itself certainly feels true (but in a less trivial way than what is actually proved). And the conclusion is deeply flawed. It claims that there is no difference between a buggy implementation of a flawed heuristic and a correct implementation of a reasonable solution strategy. The conclusion should rather be that we should explicitly specify what our solution strategy is supposed to achieve, not just claim that it is a good black box search strategy.