The theory of Sobolev spaces has grown into a central part of the study of partial differential equations, to the point where many authors are unable to discuss a PDE except in this language. Sobolev spaces quantify the smoothness of functions — their “regularity”. Thus the discussion of a PDE in these terms requires one to attend to the smoothness of its solutions all the time, even when that isn’t the point of the investigation.

I see an analogy. The nouns in languages like French and Italian have genders, so it is impossible to mention a fork or a knife without specifying if it is feminine or masculine. Of course, the French know that the femininity of a fork and the masculinity of a knife are artifacts, but they have to get the artifacts right if they want to speak correctly. And sometimes grammatical gender can be helpful, when one is discussing people or animals that have a biological gender.

Is there a cost? Well, you decide. Memorizing la fourchette and le couteau feels like extra work to outsiders learning a language, introducing distinctions that may seem distracting or comical; but children growing up with the language barely notice. Many mathematicians are like that these days, native speakers of Sobolev spaces.

[25 May 2025]