Recently a friend and I had a chat about music, and he asked me if I do any composition of my own. Unfortunately I do not. I suppose I could blame it on a lack of improvisation skill, which in turn originates from a desire to play pieces “as they were originally intended”, i.e. sticking to the sheet music, though I guess the true answer also has something to do with a fear of failure of some sort.
In general, classical music sports a rather bold distinction between “performer” and “composer”. The composer is the person who creates the music, the performer is the person(s) who executes the music, and they need not be the same gal (indeed, the composer may write for an instrument she does not even know how to play! or write for an entire orchestra / ensemble / etc). The fact that classical music is “classic” also contributes greatly to the distinction: most of the great composers of yore are dead; the best we mortals can do is echo their masterpieces.
But being a classical performer is no shame. Indeed, some performers have risen to a demigod stature among the population (ok, among a very particular slice of the population, but it is a demigod stature nonetheless). These men and women have brought the art of execution of art to a grandmaster level. They are experts in their field; they tune every staccato and accent to picometer precision. They know each intricacy of each phrase by heart, mind, and finger.
Why am I telling you this? Because while in music both performers and composers are abundant, and both are respectable careers to aspire to, it seems to me that in high level mathematics, it is mostly the “composition” that is lauded. By “mathematical composers”, I mean research mathematicians, who explore the boundaries of the science, try to invent new mathematical structures and understand existing ones, and in general, prove a bunch of theorems, lemmas, corollaries, claims, propositions and remarks.
By “mathematical performers”, I mean those who take the work of the composers, and give the audience such a breathtaking show, that they’ll get a three-time standing ovation, eventually being forced to return to the stage to give an encore in the form of a “Using volume to prove Sperner’s Lemma” proof.
Yeah, I know, there aren’t much of the latter, and I think that we are all the poorer for it. What I envision is a mathematical lecturer virtuoso. Someone who can, through all the jumbled, messed up and interwoven six-part counterpoint of a proof, bring out a clear and lucid melody that will ring and resonate loud truth in the ears of the audience. Someone who can aptly tame the fierce and complex mathematical topics that generation upon generation of graduate students have failed to grasp, and finally bestow knowledge upon the ignorant. Someone who has studied the ancient texts and knows by heart, by mind, and by finger each intricacy of each phrase. Who can tune every theorem and lemma to picometer precision. An orator of great rhetoric, brilliant diction, and perfect handwriting. A lecture-hall veteran, who practices six hours a day and in the rest of her time finds out the best way to build a lecture series on a wide, demanding topic. In short, a full-time, professional, high level mathematics teacher.
Of course, the profession “full time teacher” is not unheard of. Yet, as far as I know, most teachers – i.e. most of those whose profession is to teach, and indeed do hone their presentation technique – are aimed at educating elementary and high schools. The number of such teachers at the academia level is small, if not infinitesimal. They do exist, for sure – at the Technion, as far as I know, at least two mathematics lecturers hold a full time position: Aviv Censor and Aliza Malek. They constantly receive much praise and awards, and their lecture halls are crammed so tightly, people stand in the hallways and peek through open doors just to hear them talk (though alas, I never chanced to study under them; this is in part because most of the courses they teach are aimed at non-mathematicians, and in any case are intended for undergraduates). But such men and women are a rarity.
Why is this? It’s quite understandable that many people would prefer to go into research rather than performance, but even then I would expect to see more performers than we have so far. Two other immediate reasons are: 1) lack of paying customers, lack of demand. 2) low social status when compared to research mathematicians (“Oh, you don’t invent anything of your own?”).
But this isn’t so with music, and *should not* be so with mathematics. I can therefore only hope that I live to see the day, when Carnegie hall is filled to burst with excited concert-goers; and when the lights turn on after an hour and a half of a dazzling performance of “The Nash Embedding Theorem”, there will not be a man or woman left unmoved, their hearts pounding with reborn youth, the math as music to their ears.