Somewhere (?) in the writings of Gian-Carlo Rota, I recall a statement that old-fashioned Aristotelean syllogisms are not used in modern mathematics. I know of one gaudy counterexample, and wondered whether there are others.
The major premise is Matiyasevich's theorem, proved in 1970:
All recursively enumerable sets are Diophantine.
The minor premise is a discovery in the 1930s, I think by several people including maybe, Kleene, Turing, and Church:
Some recursively enumerable sets are non-recursive.
(Matiyasevich built on work of Julia Robinson, Hillary Putnam, and Martin Davis, done over a couple of decades.)
The conclusion crosses the 10th item off of Hilbert's famous list of problems:
Some Diophantine sets are non-recursive.