Absinthe is a drink made with green anise, a flowering plant whose taste is similar to licorice; it also has a high alcoholic content. Because so many writers and artists drank it, the aura of illicitness and mystery surrounding absinthe "has played into modern literature."
In Hemingway's story, Jig and the American may be drinking absinthe for the same reason that those living the Bohemian life in Paris, such as Hemingway himself, drank it: Supposedly, it greatly numbs the senses and even produces hallucinations. In her argument with the American, Jig may simply wish for a numbness to her fears and worries, even if it is temporary, for she intuitively sense the repercussions of having an abortion while the man perceives it only as a mechanical process and, therefore, something "perfectly natural" after which she will feel fine.
"We'll be fine afterward. Just like we were before." [man]
"What makes you think so?"
"That's the only thing that bothers us. It's the only thing that's made us unhappy."
Yet, the man cannot think of what is not yet, and he has another absinthe when he goes around the station to retrieve their bags. For, earlier when Jig asks him if after the operation he will
"...be nice again if I say things are like white elephants, and you'll like it?
like Scarlett O'Hara of Gone With the Wind, the American avoids thinking on troubling ideas,
"I'll love it. I love it now but I just can't think about it. You know how I get when I worry."
Jig has been waiting to experience the American's real love. But, she receives only non-commital and evasiveness on the man's part, a white elephant that is useless for her emotional need of feeling truly loved.