Seriously, I get sick only once every 12d4 months or so. I've gone well over three years without illness before (once when I lived in chambana, and nearly the past three years here). If you searched back though my LiveJournal archives, you'd find nary another entry in which my mood is genuinely "sick". That'll teach me to go gallivanting off on
Today's whimsical title is a result of a weekend's convalescent rumination over the difference between a mathematician and a normal scientist or engineer. Now, we who are engineers do not like to admit to any inherent intellectual inferiority relative to science, and we who are scientists do not always like to suppose that there is anything magical in the so-called universal language that elevates itself above our mere mortal tongues. But let me just come right out and say it plainly: if you talk to a pure mathematician, the probability is high that, somewhere in the course of your conversation, you will recognize that he or she takes it for granted that there is something loftier about mathematics, the realm of abstract structures and patterns of change.
Maria Zamfir-Bleyberg, a professor emeritus of my department who was on the committee that recruited me, once called computer science a "selfish discipline", one that brooks no rivals of attention or preference. By this she meant that personal time, particularly that devoted to other creative and nurturing activities (yes, even parenthood!) sometimes had to be balanced against the demands of the discipline. What makes CS different from any other field, you ask? Well, first of all, Maria didn't say "balanced against", she said "given up to", and of course many of our graduate students - single or married, with children or childfree - might understandably balk at that characterization. But I digress. I came here today to talk abut math, and what is it that makes mathematics not only a selfish discipline, but a lonely and antisocial one. Herewith, my theories.1
Arrogation. Mathematicians are better. No, they don't just think they are: they have to be. Olympians walk alone. Which is as much to say: when a tradition - be it a religious sect, a secular order, a profession, or even an ethnicity - gets so caught up in its own legend, its peculiar perfectionism - that legend becomes as much a burden and a crushing responsibilty as it is an incentive. Carl Friedrich Gauss is said to have encouraged his sons to study law rather than mathematics, rather than taint the name of Gauss with mediocrity. How do you like them apples, zengeneral? Lawyers. Lawyers. (Fermat was a lawyer too, was he not?)
The terror of being second. Sometimes math appears to brook no seconds, either in ability or in temporal order. Gauss famously tantalized and taunted Jacobi, who would go to him with months of work at new theorems he had proven, new properties he had discovered - only to have Gauss reach into his cabinets and pull out unpublished notes from fifteen years before, independently tracing the same steps. In reading about the brilliant and promising career of Theodore Kaczynski, the convicted Unabomber, one gets the distinct impression that this kind of environment at least exacerbated his anti-technological singularity-phobia. I wonder if the same pressures that some withstand and cause others to snap is not a catalyst in some people's retreat from society. I don't think I am exaggerating when I say it is sometimes more than we can reasonably expect from human beings.
The urge to ineffability. Mathematics delights in the ineffable, the inscrutable. A true mathematician will grin (if inwardly) when you scratch your head and look duly intimidated by the massive sprawl of symbols he or she has neatly arranged on the board. An intelligent systems or programming language researcher is merely pleased that you find the derivation original, meaningful, and edifying; the grin is at your respect for the impressive display of sequent rules, denotational semantics, or algebra. A pure mathematician is glad you don't get it. This peculiar pathology is so common among mathematicians that my father used to regale me humorously with stories of his old chemical engineering classmates, who proposed to take handguns to their Ph.D. defenses. To shoot their committees if they failed? No - to shoot themselves if the committee understood the work, rendering it so tawdry and commonplace as to be readily comprehensible, applicable, et cetera.
That is no country for old men. Like chess, musical composition, gymnastics, and many other prodigious acts of youth, math demands too much, too soon - the early display of precocity, the sustained output during a "period of trial". When Evariste Galois lay mortally wounded from a pistol duel, he reputedly said to his brother, "do not cry; I need all my courage to die at twenty". Any aspiring mathematician I know, myself included, will tell you that it takes something of a greater level of courage to survive. Math, the Discipline, does not mind if we burn out: how few of the shining lights of the Elder Days did not? There is a romantic ideal of mathematics, to die prematurely as Abel, Galois, Ada Lovelace, or Ramanujan did, if you accomplished anything of worth, is to not have to worry about that messy rest of your life. To live to a ripe old age as Euler, Gauss, Newton, and Erdős did is to look over your shoulder every day, wondering if your moment of glory is past - and often knowing it has passed. Note that the four worthies I named were anomalously prolific throughout their long lives. You still wonder why some aspiring mathematicians plan on dying at 30? lao3 er2 bu1 si3 wei3 shi3 zei2, goes the Chinese aphorism: "the old who do not die are scoundrels" (that is, they rob the young of their birthright).
From mountaintop to mountaintop. Finally, mathematicians delight not in low obfuscation, but in astoundingly virtuous leaps. Like Asimov's "Eureka effect" of science, wherein scientists who happen upon a discovery by plodding methodicality feel obligated to "make up a story" about how "something wondrous happened", mathematicians also delight in jumping bodily from mountaintop to mountaintop without climbing down to earth. Abel admired Gauss with the remark, "He is like the fox, who effaces his tracks in the sand with his tail." Make no mistake - that is a compliment. I remember Uday Reddy, a type theorist whose group I worked in as a first-year graduate student, remarking of Gaisi Takeuti: "He is Sherlock Holmes! Everything is elementary to him." In 1994, at 75 years of age, Professor Takeuti was hailed as the Grandfather of Modern Proof Theory. I took his linear logic course, and was baffled week after week by the effortlessness with which he would produce proofs that took him seconds to our hours. He had an uncanny instinct: an automated deductive theorem prover would take days to come up with the same assertions, if it did at all. And he would do this with a good-natured but backhanded humility, calling every proof "kindergarten stuff". My point, though, is that Dr. Takeuti, like many who survived their twenties, thirties, and forties, knew what he was about. He knew exactly the width of the chasm he could leap, and by knowing it, could comfortably spend his fifties, sixties, and seventies leaping safely. It was a lot more than we could do, yet perhaps a young prodigy might do better. Dr. Takeuti acknowledged this: when faced with a very difficult proof, he would pause and say, "maybe this is first grade". I can only imagine what kind of near-impossible proofs would elevate a theorem to second grade status.
Method: eschew or embrace? Many computer scientists I know, some of them instructors of courses taken by students of higher mathematics, lament that the mathematician mindset (to be distinguished from the mathematical mindset) seems to shun any received methodology. Whether it be nomenclature, proof procedure, structure of communication (lemmas and intermediate steps), or just simply "using all the tools given", we really do seem to like to do things Our WayTM. In light of the above points, this is not so hard to explain: virtuosity is the one true virtue; methodicality is praised and starves, to paraphrase Juvenal. However, my point (and I do have one) is that method is a foundation of CS, of science, and yes, of engineering. And guess what, sports fans? It isn't a strict hierarchy, however much the numerati would like us to believe.
I wrote this for two reasons: First, because I have been sometimes thrilled, sometimes dismayed at the attitudes prevalent in my "home court", the CS-Math double major, and I wanted to sort out why things might be the way they are. Second, because I wanted to convey to the nonmathematician a little of the mindset: there really should be a field called Psychology of Mathematics (psych applied to math, not the other way around). Third, because I hope the above will shed a tiny bit of light for the bears in the mist: know yourself, and know it doesn't have to be any way but how you restrict or free yourself.
Oh, was that three reasons? Heh.
1 At some points in my discourse I will use "they" to refer to mathematicians, and at others I will use "I". This apparent schizophrenia is explained thus: sometimes I feel like a mathematician; sometimes I don't. Almond Joy's got nuts; Mounds don't.