"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.
LOUISE: If all I ever gave you was a hammer...
COLONEL WEBER: Everything's a nail...
Louise and the Colonel discuss language and the reality of what is being said between the humans and the heptapods. She is explaining how easily words can become misinterpreted and cause catastrophe if not fully worked out to be understood.
"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.
"(Heptapods) can't seem to follow our algebra. But complex behavior? That clicks."
This quote reinforces the association of the heptapods with the lack of order. Even though they couldn't comprehend the systematic way of thinking, they could utilize more complex concepts. Moreover, this is also reflected in their language, which conveys the meaning of words, rather than a specific spelling.
