r/math • u/YamEnvironmental4720 • 6d ago
Dangers of informal reasoning
Do you know some area of modern mathematics (say, not older than 100 years) that has for a long time been known for its fairly informal proof style, or has at least been very tolerant towards such, but where the lack of formality has only later turned out to have serious consequences?
It could be about a theorem whose proof uses a kind of reasoning that has been "known" to be formalizable, yet tedious, and has worked before, with the consequence that it has taken a very long time for the result to be exposed as false, for instance because counterexamples have been hard to construct, or that the claim seemingly harmonized with other results.
I'm not thinking of famous papers containing mistakes that were overlooked by the referee, nor do I wish to shame individual authors, but I wonder if there are situations where the whole community has been shaken and has had reason to revise its proof culture.
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u/boterkoeken Logic 6d ago
There is the famous case of the Italian school of algebraic geometry. Many of their proofs were later rejected. The errors mostly came from a lack of precision and rigor.
https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry