How we cite our quotes: All quotations are from Memento.
Quote #1
LEONARD: My wife deserves vengeance. It doesn't make any difference if I know about it. Just because there are things I don't remember doesn't make my actions meaningless. The world doesn't just disappear when you close your eyes. Anyway, maybe I'll take a photograph to remind myself. Get another freaky tattoo.
Leonard knows that, despite his condition, the world goes on just like it used to. The actions he takes are real and so are their consequences. But just because they're real doesn't make them meaningful. We like his confidence, but how does he actually know the world is still there?
Quote #2
In the first scene (which happens in reverse), Leonard shoots Teddy and then takes his picture. He immediately flaps the picture to develop it, looking at it as he does so. The very first shot of the movie is of the Polaroid picture of a dead Teddy.
In lieu of memory, Leonard has created a system of picture to keep track of things. But when do the pictures stop being representations of the real and start becoming more meaningful than the events, objects, and people they depict? It's almost like Leonard has created his own type of world, a unique existence in the pictures of things. He kills Teddy but is more interested in looking at the picture of dead Teddy. The picture is the only thing that's important.
Quote #3
LEONARD: There are things you know for sure.
NATALIE: Such as?
LEONARD: I know what that's gonna sound like when I knock on it. I know what that's gonna feel like when I pick it up. See? Certainties. It's the kind of memory you take for granted. You know, I can remember so much. The feel of the world... and her—she's gone. And the present is trivia, which I scribble down as f***ing notes.
Leonard may not have short-term memory but that doesn't mean he's clueless. He feels the world just like the rest of us do. There are things you just know to be true even if you don't remember learning them. As a fun side note, Leonard's explanation of certainties is apparently similar to Nolan's explanation when he was pitching the movie.