I was reminded of this artifact while reading David Sax’s “Save the Deli,” a new book recounting the rise and decline of Jewish delicatessens. Once as common in American inner cities as burrito joints are today, Jewish delis have largely faded as the communities that frequented them moved to the suburbs and changed their tastes. Traveling the world in search of a Platonic ideal of corned beef on rye, Mr. Sax finds that today’s successful Jewish delis are anomalies sustained by any one of four business models. Some are run by a charismatic owner-operator who devotes heart and soul to the place and usually dies with it, like the much-loved Wolfie Cohen Rascal House in Sunny Isles Beach, Fla. Some, such as Manhattan’s Carnegie Deli, resemble museums, preserving exaggerated, ancient menus and becoming tourist attractions. Others diversify, as did Zingerman’s in Ann Arbor, Mich., becoming a gourmet shop and supermarket brand instead of a restaurant. Or they find new a clientele—the new owner of Lou’s, an old-style inner-city Jewish deli in Detroit, discovered that African-Americans also love corned beef.

Of course, these models are unrelated to what Jewish delis once were: casual, unself-conscious sandwich counters serving a local Jewish clientele. That type of restaurant, it would seem, is going the way of herring cacciatore. One could fret, as Mr. Sax does, about what the apparent twilight of pastrami means for American Jewish culture. But it might be more revealing to consider what this fretting itself means, and why the calls to “save the deli” might themselves be a reason to rejoice.

The history of American food is really a history of immigration, and the nostalgia that comes with a cuisine’s decline is an indicator of an ethnic group’s confidence in its American identity. When a group first attains critical mass in America, its restaurants are mostly for its own members. Later, as these groups gain confidence, they begin selling their more accessible foods to a general public craving cheap exotic eats. These dishes then mutate into American form, a la chow mein, and the group’s American-born children typically spurn these foods as they try to assimilate. Around the third or fourth generation, the descendants of immigrants are secure enough in their American credentials to explore their “roots.” Shortly thereafter, food nostalgia sets in, and the quest for the “authentic” begins.