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Don Draper, the impossibly cool protagonist of “Mad Men,” is in some ways what every 1960s advertising executive wished to be. His movie-star looks might have made Marilyn Monroe melt. His jet black hair is never out of place. When he yanks a Lucky Strike cigarette from the carton and draws it to his mouth, he could be the advertisement. His pouring an Old-Fashioned makes everyone around him want to find some rye whiskey, too. He prefers silence. He has money but doesn’t flaunt it and a gorgeous wife he only shows off on his terms. He is quiet but not aloof, mysterious but not indifferent, demanding but not uncompassionate. He also looks the part: crisp white shirt, custom-fitted gray suit, skinny black tie, barely peeking pocket square. Even his name—Donald F. Draper—fills the door of his corner office. It sounds dignified and refined. From the very first episode of the television series, this is the way Don is portrayed to pretty much everyone in his world. In the fifth episode of the show’s first season, Don wins an advertising award and appears wearing a resplendent tuxedo and bowtie in the next week’s Advertising Age. That black-and-white photo is Don Draper ideally depicted.

And yet the complexity of Don Draper as a character is that he is not, in fact, Don Draper. Before that episode, Don was Don, and that was enough. But he’s not Don, of course, and it’s the photo in Advertising Age that shatters his essential facade. Still basking in the media attention, Don receives a visit in the Madison Avenue headquarters of Sterling Cooper from someone named Adam Whitman. He claims to be Don’s brother. Don doesn’t have a brother! But the unscheduled appointment is jarring not simply because we realize we haven’t heard anything about Don’s family but also because this unflappable figure seems too rattled for the stranger to be anything but a stranger.

Adam is the brother of Dick Whitman, the person that became Don Draper. He is the link to the past for this Dick-Don hybrid—the first and most important reminder that Don Draper is not Superman on Madison Avenue but a product sold with Don’s greatest advertising campaign of all. It’s the process of peeling past the pocket square and bottle of rye to learn who Don Draper really is that’s the driving force behind “Mad Men.” There is no character in the show who’s more appealing than Don Draper, and there is no character more intriguing than Dick Whitman. It’s this attempt at reconciliation between the two that defines him.

A child between the ages of six months and eighteen months has an “instrumental intelligence” that is no more advanced than a chimpanzee’s, but when he looks at himself in the mirror, the child can “already recognize as such his image,” Jacques Lacan said 13 years before his 1949 lecture at the International Congress of Psychoanalysis in Zurich. That’s where he used his previous scholarship—something known as the “mirror stage”—as a baseline for examining “the formation of the I as we experience it in psychoanalysis.” The child then creates what Lacan calls the Ideal-I, which “situates the agency of the ego, before its social determination, in a fictional direction, which will always remain irreducible for the individual alone.” In plain English, the child will never not try to strive toward this fictionalized construct of himself.

The problem is that it’s an impossible standard. From there comes the dissonance. The child will try to overcome this separation between himself and what he sees in the mirror, but it can’t be overcome, because one is real and the other exists only in the imagination. Lacan argues that this is a self-defeating cripple. He calls it “paranoiac alienation.”

This 20th century French philosopher is more relevant than ever these days because so much of his work included visual elements. A flashback in a television show or movie, for example, is incredibly Lacanian (with a sprinkle of Freud for good measure). And what about a photograph? After the first 10 episodes of “Mad Men,” no one can say for sure whether Don Draper is really Don Draper. What changes all this in “Nixon vs. Kennedy” is a box of photographs.

Don is a self-made man who actually became someone else to achieve that idealized version of himself. The accounts man Pete Campbell used his last name more than the agency’s. Their backgrounds couldn’t be more different. In the first season of the show, Pete mistakenly comes into a collection of snapshots of Don his youth sent to Sterling Cooper by Adam Whitman after he identified his long-lost brother in the newspaper photo. Pete sifts through the photos and learns Don’s secret: He’s a fraud. When Pete confronts Don—sometime around the time Richard Nixon concedes the presidential election to John F. Kennedy—he denies the charge. Then he flees to mistress and begs her to run away with him. “We’ll go somewhere else—we’ll start over,” he says as he whisks a cigarette to his lips. She calls him on his bluff by referring to his father. Uh-oh! This brings about a flashback to Don’s youth. Finally we begin to understand how Dick became Don.

Dick was a poor boy born out of wedlock to a prostitute and an alcoholic father. He ran away from home, enlisted in the army and was sent to Korea. Dick was not the quarterback of his high-school football team, as his wife believed, nor was he a war hero deserving of the purple heart he stored in his desk at work. Instead he stole the tags of Donald F. Draper, his commander who died in a ditch where Dick was injured. He assumed this identity and went back to the United States. There he was no longer Dick Whitman but Don Draper. His fear of being discovered is a prevailing sense of a paranoia. Lacan couldn’t have written a better character if he tried: Don is completely alienated from the world around him. The more he believes himself to be Don Draper, the more he’s reminded that he’s not. Don knows better than anyone that it’s all a ruse.

A quick disclosure: It’s almost too easy to take Lacan’s lecture as gospel or use it to evaluate just about anybody in the world. But I think it can’t be avoided when it comes to Don Draper. He is an isolationist who trusts nobody and tells his mistress, a woman he hardly sees outside the bedroom, that she knows more about him than anyone. The fact that Don is never happy—or that no one really knows what happiness is to Don—is more than a product of his era. There is something deeply at fault within Don that can’t be solved as long as he is Don. So let’s just say it: When he urges his mistress to run away with him, he’s not running from his family or Sterling Cooper but from himself. He’s running to be Dick Whitman again. How does his mistress respond? “I feel sick,” she says.

Only after that does Don challenge Pete to tell Bert Cooper, he of the name on the door, about Dick Whitman. And when Pete shares his theory with Bert—“He is a deserter, at the very least, and who knows what else?”—the accusation lingers in the silent office. Don lights up a cigarette. Bert rises from his chair and walks past Don.

“Mr. Campbell,” he says. “Who cares? Even if this were true, who cares? This country was built and run by worse stories than whatever you imagined to hear.”

Pete calls Don a fraud and a liar and who-knows-what-else. Bert understands that he’s right but won’t hear any more. “A man is whatever room he is in,” he says. “And right now, Donald Draper is in this room, I assure you.”

I couldn’t help but think of a flashback later in the series when Don thinks back to the last time he saw his family. He was sitting on a train as it choo-chooed into the station. A woman sees him and thinks Don Draper has to tell Dick Whitman’s family of his death. “You’ve got your whole life ahead of you,” she says. “Forget that boy in the box.”

The very nature of Don’s job as creative director of Sterling Cooper is branding products to make customers think the way he wants them to. In a memorable scene at the end of the first season of “Mad Men,” Don describes the Kodak wheel this way: “It goes backwards, and forwards, and it takes us to a place where we ache to go again. It’s not called the wheel. It’s called the carousel. It lets us travel the way a child travels—around and around, and back home again, to a place where we know we are loved.”

In the sixth episode of the second season, “Maidenform,” Don invents a new campaign for the brassiere company Maidenform to compete with its rival Playtex’s flashier advertisements. The creative team settles on an advertisement with two women in their undergarments: one blonde bombshell (Marilyn Monroe) and the other a dark-haired vixen (Jackie Kennedy). One character says every woman is either a Jackie or a Marilyn. At the very least, every woman aspires to be one or the other when she looks in the mirror. The advertisement’s twist—its Draperian genius, if you will, and please do—is that the Jackie and the Marilyn are the same model in a different bra. “It’s a very flattering mirror,” Don tells the Maidenform executives.

At the same time Don is struggling with an identity crisis of his own for the first time since the near-panic attack with his mistress. During a Memorial Day party the Drapers attend, the servicemen in the room are asked to stand, and Don fulfills his obligation. Not long afterward, he finds an excuse to leave the party, run off to his new mistress and make love in a New York hotel room. When he sneaks up behind her, we can only see Don and Bobbie Barrett in the hotel room’s mirror.

“I want the full Don Draper treatment,” she says. “I wanted it, and I got it, and it’s better than they said.” Don backs away from this pillow talk and asks her to explain what she means. “You have lots of fans,” she says. “You have a reputation. Enjoy it.” Instead he ties her to the bed and leaves her there half-naked.

In the episode’s last scene, Don walks to his bathroom for his morning shave. His daughter, Sally, rushes in to watch him. He lathers his cheeks in shaving cream and stares at himself in his multi-paneled shaving mirror. A look of terror very clearly overcomes his face. He orders Sally to leave the room before he wipes the shaving cream and collapses onto the toilet.

It’s tempting in an essay like this to link such obvious imagery to Lacan. Why not! A mirror must represent the mirror stage, and thus Don’s inner conflict is positively Lacanian. And yet it seems obvious even in this case that “Mad Men” writers have brushed up on their Lacan. Twice in an episode about image and self-image Don looks into a mirror. Both times he finds his reflection to be repulsive. In fact, this utter separation between what he sees and what he wants to see induces an anxiety attack. The first time, when he’s with his mistress, Don hears her talk of his so-called reputation and that his sexual prowess has fans all over town. In a way, it’s representative of exactly what Don aspires to be. These women know nothing of Dick Whitman. To them he is fully Don Draper. He might never be closer to the idealized version of himself than he is at that very moment. Still he storms out of the hotel room because he knows that the “Don Draper treatment” isn’t even a real thing. This deeply guarded secret resurfaces and stares Don in the face at his home with his daughter in the room. He is not the hero that he pretends to be at the Memorial Day reception, nor is he even the person his precious daughter imagines him to be.

Season 3 of “Mad Men” starts and ends with flashbacks to Don’s youth. In the first scene, Don watches his own birth. The second occurs in the last episode, as Don relives the night of his father’s death and the unraveling of the modest life he knew. But it’s a dream-like state, not a flashback, in the middle of Season 3 that I’ll use in this Lacanian reading of our protagonist.

Loaded on two pills of Phenobarbital, fed to him by hitchhiking con artists, Don hallucinates that his father is sipping on moonshine in the corner of his cheap motel room. He tells him a corny joke about hillbillies—something Dick Whitman, if not Don Draper, would find funny—and Don cackles. Then his tone turns accusatory. “Look at you! You’re a bum, you know that?” he says. “Your hands, they’re as soft as a woman’s. What do you do? What do you make? You grow bullshit.” Don tries to protest but finally gives in. To be fair, he’s about to be knocked out by Phenobarbital. But also he can’t really respond to his own self-doubt. On the other side of the room, the two kids had taken him for a Cadillac-driving, slick-suited advertising man. His hallucinated father sees him as Dick Whitman. Some may believe the bullshit that Don grows. But he never will, and neither will his father.

Don wakes up in the morning with his face down on the carpet. He runs his hand over his face and feels a gash from slamming the floor after he had been knocked out. Then he staggers over to the motel room’s mirror and quickly stares at himself. (I mean, come on!) For a brief second, there are two people in the room. There is the man looking into the mirror and the man reflected in the mirror: Dick Whitman and Don Draper. And before he leaves, the camera sticks on the mirror in the room for an extra three seconds. Here come the usual caveats about mirrors and Lacan and all that. And yet again the mirror is more than the mirror. It represents the everlasting divide between Dick Whitman and Don Draper.

For one last time in this particular episode—also about the 1963 solar eclipse—Don confronts Dick once more with Bert Cooper, who had ignored the question of his identity in the first season. Bert is demanding that Don sign a contract. When he demurs, Bert coyly presents his trump card. “After all, when it comes down to it,” he says, “who’s really signing this contract anyway?” With a white bandage blanketing his nose, and with his hair just slightly disheveled, he picks up the pen and scribbles a signature in black ink: “DONALD F. DRAPER.”