By “we” I mean those of us who earn (or, if retired, earned) our keep through mathematics. Mostly I mean those who prove theorems, though much of what I say applies to those who teach calculus, tell insurance companies what rates to charge, construct models to make hedge fund managers rich—things like that. By “different” I mean … well, not the same as the non-mathematical run of humanity. “Of course,” you may think, “we know more mathematics.” But there is more to it than that. I will point out some of the differences and give what I think are reasons for at least some of them. I will save them for later, especially because I will repeat them several times, and also because they are based on intuition and speculation. Until social scientists get around to conducting a parallel study of 2,000 mathematicians and 2,000 non-mathematicians over twenty or thirty years, speculation is all we have. However, speculation can be useful. The Riemann Hypothesis is speculation.
We can get at the differences seeing what the general public thinks of us. In the movies, for one place, we do not come out looking too well. Insanity is one of our main characteristics. In “A Beautiful Mind,” a 2001 film about John Nash, the mathematician who won a Nobel Prize for economics, it could not be helped because its subject was for many years insane by any definition of the term. The hero of “Good Will Hunting” (1997) is not clinically crazy, but someone with immense mathematical talent who chooses to work as a janitor and isolate himself from life; he qualifies as being on the far reaches of eccentricity. He also has an amazing memory and is able to do mental arithmetic with great facility, common characteristics of mathematicians in popular culture. This is annoying if you are someone like me, whose checkbook never balances at the end of the month. It is also annoying that, in the film, the difficult unsolved problem that Will Hunting solves can be disposed of in a few lines. Proofs are seldom that short. Well, that's Hollywood.
“Proof,” the good 2005 movie based on the even better 2000 play by David Auburn has as its center a father and daughter, both mathematicians. The father was completely gaga and doubts were raised about the daughter's sanity.