What are Faber's reasons for giving Montag his address in Fahrenheit 451?
Faber is a sweet old man who just happens to catch Montag's attention in the park one day. They talk about the weather for awhile and then Faber admits to being a former English professor. Faber quotes poetry and reminisces for an hour "without either acknowledging the fact that Montag was a fireman" (75). Faber must have thought that since he was quoting poetry for an hour that he had incriminated himself. He may have thought that Montag would follow him home anyway in order to report him, so why not give him his address and save him the trouble? As he gives Montag his address, he trembles and then says, "For your file. . . in case you decide to be angry with me" (75).
Montag did have his address in a file labeled "Future Investigations." The fact that Montag didn't turn Faber in a year ago shows that Montag's heart was warming up to books that far back; but it also shows that Faber was being brave that far back, too. He calls himself a coward, but it was very courageous of him to give out his address like that after quoting poetry and seemingly hiding a book in his jacket. So Faber must have been trying to be more than a coward at that time in his life by giving Montag his address. He must have seen something in Montag, too, though. Anyone who would actually sit and listen to the old man is worth making friends with. The lonely Faber was probably hoping for a friend.
The only thing Faber says at the time that he gives Montag his address is that Montag might someday decide to be angry at him. Later, at Faber's house, he says he thought Montag might come some day either with fire or friendship.
I think that there are two possiblities.
First, maybe Faber "sees" something in Montag that makes him think that Montag can be on his side sometime. He might think that having Montag as a contact will be useful.
A second option is that maybe he is sick of hiding like a coward. Maybe he is just taking a rash chance -- he's sort of thinking it might work, but he's sort of just wanting to do SOMETHING to prove he's capable of it.