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?
Best Answer
Since the question is specifically about damage:
I think that what really causes damage to a mathematical area is when an important result is claimed by someone prominent in the field, but the proof is never completely written. Younger researchers are then likely to spend a lot of time and energy "cleaning up the mess", for little credit.
Things are even worse when there is some freedom of interpretation of what might have been proven. A younger researcher might want to use the announced result for some other purpose, but they might use a version of the theorem that ends up not being the one that got proved.
When a proof is (widely) accepted to be wrong or non-rigorous, or when someone retracts the claim of having proved a given result, that's when things are getting better for a field.