In Order of the Phoenix, Dumbledore tells Harry
'I did,' said Dumbledore. 'On a cold, wet night sixteen years ago, in a room above the bar at the Hog's Head inn. I had gone there to see an applicant for the post of Divination teacher, though it was against my inclination to allow the subject of Divination to continue at all. The applicant, however, was the great-great-granddaughter of a very famous, very gifted Seer and I thought it common politeness to meet her.
Why did Dumbledore want to discontinue Divination? Is it because he didn't much like the branch of magic or didn't believe in it? It does not sound plausible that he would remove a subject from Hogwarts just because of his personal opinions. (Moreover, Divination examinations are taken during the O.W.L.s so I doubt whether Dumbledore would, with his individual decision only, be able to remove it completely from Hogwarts). From Order of the Phoenix we also know that there is whole room dedicated to storing prophecies in the Department of Mysteries, so prophecies and divination are a real thing in the Harry Potter universe, from the point of view of us readers as well as the characters themselves. Then why would Dumbledore want to remove Divination?
Another thing to consider is that Professor McGonagall says, in the Prisoner of Azkaban, that true Seers are rare, so it might be that Dumbledore might have found it difficult to get someone competent enough for the job. This does not make much sense either, because if in the Harry Potter universe true Seers are rare, having someone teach the theory behind divination would at least help to keep the field alive. Also, he had trouble finding people for Defense Against the Dark Arts as well, but that does not mean he just gave up on the subject, rather, he made controversial hiring such as Professor Lupin and Moody to keep the classes running smoothly.
What exactly was the reason for Dumbledore not wanting to continue Divination at Hogwarts (until he met Professor Trelawny)?