What was the first non-constructive proof in mathematics that was widely believed? I did a course as an undergraduate where the course notes stated that Hilbert's original proof of the Nullstellensatz was the first, and at the time caused a revolution (e.g. Gordan's famous quote "das ist nicht Mathematik, das ist Theologie".) However, some other reading suggestes that some non-constructive proofs have been around for a long time, and the move away from constructive mathematics began at the start of the nineteenth century, and was a long time coming.
So, which was it? Was there a revolution, started by Hilbert or somebody else, or a gradual movement away from constructivism?