"If you could see your whole life from start to finish, would you change things?"
Louise's journey in the film is very much summed up in this one statement. While the film is progressing forward in the narrative we finally learn that Louise has gained the ability to see into the future, and even so--with all of the hardship and pain--she chooses not to change anything.
"If you learn it, when you really learn it, you begin to perceive time the way that they do. So you can see what's to come. But time, it isn't the same for them. It's non-linear."
Louise has figured out that the weapon the heptapods have come to give to humanity is their language, which when learned allows them to perceive the future. And with this knowledge, the opportunity to save the world and its people from destruction.
"You approach language like a mathematician."
Through working together, Ian and Louise have formed a mutual understanding and appreciation of each other's way of thinking. This quote reflects their combined perspectives, which they have learn to adapt in order to work together towards a common goal.
"We are so bound by time. By its order...Come back to me. Come back to me."
This quote, from the film's opening scenes, establish the central concern of the film: the human experience and how it is limited by our understanding of time as linear. Rather than seeking to "order" her experiences, Louise has come to understand time as fluid and non-linear due to her encounter with the heptapods. The repetition of the phrase "come back to me", which she repeats again at her daugher's death, reinforces this idea of circularity and non-linearity. It also develops the emotional dimension of the film; as Louise pleads with her deceased daughter to "come back" to her, the audience is struck with the full force of her grief.
