The last line of The Great Gatsby. The preceding paragraph is:

Gatsby believed in the green light, the orgastic future that year by year recedes before us. It eluded us then, but that's no matter -- tomorrow we will run faster, stretch out our arms farther. ... And one fine morning ---- So we beat on, boats against the current, borne back ceaselessly into the past.

Nick feels that Gatsby's entire life was to live in the past. To live the dream he had as a boy, when he was "full of hope" and had "infinite possibilities." Gatsby achieved wealth, fame, and the lifestyle he worked so hard for, but he never had Daisy. He got as close to her as he could, but would not shatter the perpetual dream he had of her. Somehow, five years later, Daisy doesn't live up to this dream. She chooses Tom over him, and doesn't lend herself to the dream as fully as Gatsby did.
Gatsby's dream and hope of her and himself is what drove him, with the hope that she would come to his parties or that he would visit East Egg. And when his dream was so close to being reality, it shattered.
I have a different interpretation of this passage.

As a culture, Americans (and perhaps humans in general) are constantly seeking to distance themselves from the past. It's everywhere I look. Think about the great masses of teenagers whose sole aim in life is to "get out of this stupid town." We're always trying to escape the past, to lead better lives than our parents, to find something new and exciting. Think about the line "tomorrow we will run faster, stretch out our arms farther." We live our lives constantly striving to better our situation tomorrow, or next month, or next year. We live for a dream, seeking a future that we imagine in our minds.

The character of Gatsby is the manifestation of this desire. He is a dreamer. In his futile pursuit of Daisy, he seeks an unattainable, shapeless future that reflects all of our hopeless dreams. Daisy could be that coveted promotion, the corner office, maybe an artistic achievement, like the Great American Novel or rock stardom.

But it never works. The final sentence says as much. We are rowing our boats toward the future, but we will never get there, because the current flows toward the past. No matter how hard we row, it will all be futile, because we'll always end up in the past.

Yet Fitzgerald stops short of condemning this pursuit; instead, he acknowledges that it is a part of human nature. Notice how he says, "So we beat on..." like it's inevitable that we row our boats this way, despite the current flowing in the opposite direction. This is what makes The Great Gatsby a tragedy. It's like Madame Bovary by Gustave Flaubert, which recognises the tragedy of the pursuit of the romantic life. It is a part of who we are. It is human.

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