The story of “Desiree’s Baby” takes place in the southern United States during the early 1800’s, in a time and setting in which one's family name was everything for the Southern aristocracy. The patriarch of the family had dominance not only over slaves, but over the women and children, as well. It was his role to control them in order to maintain his family honor, and to maintain the prestige of his name at all costs.
Therefore, Armand Aubigny’s decision to marry Desiree, a white woman who is beautiful and cultured but whose name is unknown, is out of the cultural norm. Having been abandoned as an infant, she was raised by Madame and Monsieur Valmonde. In spite of having been raised a lady, Desiree's doubtful origins would be considered a blight on her worthiness in the eyes of the Southern aristocracy. We are told that when Armand falls instantly in love and determines to marry Desiree, “he was reminded that she was nameless.” The fact that Chopin doesn’t tell us who reminds him gives us the sense that it is likely more than one person, representing the societal norm that Southern aristocracy must marry into an honorable family name.
Yet Armand, it appears, is somewhat impulsive and overly confident that his own impeccable family name will be enough to cover for her. It is ironic that he judges her acceptable based on the color of her skin. Likewise, when their child is born slightly dark-skinned, he again judges her. He completely disowns Desiree and her baby, telling her, “it means that you are not white.” Based on his cultural prejudices and fear for his family name, he falls out of love with her as instantly as he fell in love.
The final, ironic twist in the end is the letter Armand finds in which his mother reveals that she “‘belongs to the race that is cursed with the brand of slavery.’” He might not be “nameless” as Desiree is, but his contemporaries would deem him lower than she if they knew his true heritage.
The story gives clues enough to this truth, again involving skin color. Chopin tells us that Desiree sometimes fears but always loves “Armand's dark, handsome face,” and at his accusation that she is not white, she cries, “Look at my hand; whiter than yours.” Yet because he is the son of a wealthy aristocrat, it seems that no one has ever questioned his ethnicity. Chopin leaves the reader to wonder at Armand's reaction to the news of his heritage. Based on the knowledge that Desiree and the baby never return to L'Abri, coupled with Armand´s prejudiced cultural mindset and the fact that he has been obliterating all the artifacts of his marriage in a fire, we can assume that he will burn the evidence which would taint his name, yet live in self-hatred for the rest of his days.
This phrase comes at the beginning of the story when we are told of how Armand fell in love with Desiree "like a pistol shot." Although Valmonde, Desiree's guardian, does his best to remind Armand of Desiree's "obscure origin" (which refers to her probable illegitimate state), and the way that she does not have a "name" (referring to her uncertain parentage and lack of connections), Armand insists that he must marry her with no delay. Note his reasoning:
What did it matter about a name when he could give her one of the oldest and proudest in Louisiana?
Of course, there is intense irony in the phrase you have mentioned. Although he is cautioned by Valmonde as much as possible and her uncertain extraction, he still insists on marrying Desiree, only to find that the child they have together is obviously mixed race. Armand leaps to the conclusion that because of Desiree's background, she must be responsible for this, whilst all along it is actually his own hidden past that is responsible.