I'd known R for years: a friend from libertarian activist days. I was the one who gave him the referral card that admitted him into paradise: the 8709, a bathhouse two doors from my gay gym. The 8709 attracted all the hot numbers from West Hollywood: they all worked out with weights; and they ranged in appearance from well above average to absolute knockouts.
I'd occasionally run into R at the baths, or share a meal with him at a libertarian convention. Where do I meet R now? Like a black comedy screenplay: FLASH FORWARD TO: John and R taking I.V.'s together at the doctor's office. R said that our medical situation was "hanging by our fingernails from the cliff." "Well," I said, "just a few minutes ago we were dancing there."
It was a cliff I had been born to dance on.
My perception of being different began very early in life. I was brought up in the same One True Holy Catholic and Apostolic Church that Voltaire called the Infamy in his famous injunction, "Ecrasez l'Infame."
For seven years I attended a Catholic grade school. I call that saga, "Procrustes Unbound." The nuns and priests suppressed every difference they could detect, and still tried to ferret out more. Everyone had to wear uniforms. The girls had to wear a sort of tartan plaid skirt and a white blouse. The boys had to wear white shirts and bowties for the younger boys and neckties for the older. The discipline was so intense that when I attended public school, a classmate commented "Dentinger went to some queer military school where they made you stand up to answer."
The pettiest deviations received instant censure. We were forced to carry "conduct cards." When you crossed over the line -- never clearly delineated -- a nun would say imperiously, "Give me your conduct card." She would pull a paper punch from within the folds of her uniform, and punch a hole in for that period. If at the end of the period (a week, I think) there were no holes, a little star was licked and stuck on.
There was physical violence as well. I saw a nun grab a kid by the ear and bang his head into the blackboard. I saw nuns break wooden rulers over the hands of accused (ergo guilty) miscreants. I saw kids forced to walk to the front of the room and stand on their own fingers while the nun paddled them... hard. And in eighth grade, a priest -- a hot-tempered ruddy Irishman -- thought I'd sailed I paper plane. Since I hadn't, I denied it. Judge, jury, and instant executioner, he slapped me hard enough in the face to knock my glasses all across the room -- and, I think, damage the frame. The next day, having cooled down, the bastard exhibited the shred of grace necessary to apologize in front of the class. I refused to accept his apology, and that was the end of the matter.
The people running the place were crushingly ignorant. For three years none of these educational experts grokked the reason I presented a conduct problem in class -- to wit, I was undchallenged beyond description.
In the second grade I was expelled for rolling a pencil down my desk during Miss Dolan's French class. She said, "If you do that one more time, I'll take you down to the office and have you expelled." How could I resist? You can only know the contour of the darkened cliff -- and the exhiliration of dancing on the edge -- by risking your footing.
She told me to go to the office. I refused, thinking that if I stayed in class, she might cool off. She grabbed me, and in a panic, I held onto the desk. She dragged me -- and the desk -- up the aisle and out into the hallway. Then she trooped down to the head hatchetnun and had me expelled.
For what it's worth, my parents stood behind me, baked extra cakes for the nuns, wrote them letters, wheedled and cajoled them to let me back in. The nuns relented. The next year, in third grade, I got into similar trouble for the usual reason:boredom. So a new head nun, ordered me demoted to the first grade, which she taught.
She assigned me some simple-minded exercise which ideally required a ruler that I didn't have. I was supposed to do drawings in seven neat panels on the paper. I took the paper in portrait orientation, and folded it into three rows. Then, very carefully, I folded the top row and the bottom row in half -- and the middle row into thirds. This display of ingenuity was apparently when it penetrated -- after nearly three years -- that I was ahead of the other kids, not behind them.
Meanwhile, my father had me perform a trick for the principal. Other kids my age were barely ready for fun with Dick and Jane; I read a newspaper story out of the Chicago Tribune to her. I was given special tests.
These were I.Q. tests, and I was decreed "gifted," I think. This word was okay, even within the culture of anti-deviance, for it had an air that the Church had bestowed the "gift" (though it had actually been smothering it). The word "genius" was taboo; it had the sense of being self-willed and deviant.
So it was arranged that I would study at home during the summer and skip the fourth grade. I was young for my grade and short for my age; so I was a runt when I was put into the fifth grade with my sister, whom this embarrassed.
One of my stigmata in grade school was left-handedness. While I was in the first or second grade, we were instructed on the precise manner in which to pick up a report card.
An inflexible instructions was that you must take the card with your right hand. I piped up, "I can't do that. I'm left-handed." So the nun (part of the nervous system emanating from the God-brain to coordinate all us little centipede feet), forced me to walk up to her desk three times and practice accepting the paper with my right hand. I was pleased to discover some use for that hand, but I didn't care for the manifest contempt for the pettiest, almost unnoticeable deviations from The Norm.
Little did I know at the time, but the same institution was doing far worse to us sinister deviants. It was forcing students to write right-handed. This causes significant, sometimes severe, neurological problems in left-handers.
In 1987, my brother and I went back to the suburb of our youth. We went by the Catholic school and a priest happened by. We asked about the school. There were no more nuns -- no one was going into the convent any more. I was pleased that the Church was further losing its iron grip on society's throat.
About ten percent of the population is naturally left-handed. (Though some authorities believe that if all barriers to left-handedness were removed, one in three people would be left-handed.) By coincidence, about ten percent of the population are gay; no doubt more would be, without the stigma.
Sometimes lefties seem part of the same sort of fraternity as gays. Often I will have to sign something in front of someone who will comment, "Oh, another leftie." I'll wink and say, "All the best people are."
I arrived at Caltech to study mathematics. I expected this field to be a bastion of rationality, the ramparts of reason invulnerable, the drawbridge to madness drawn up. It proved chimerical. There was no escape, even in mathematics, from conflict and irrationality.
One time, like other graduate students, I gave a seminar on a topic of my selection. Charles de Prima, did not question my proof. Instead, he exposed me to an unexpected and far more fundamental attack: "Why is this important?" Then he let loose one of his famous raucous laughs. We called him Chuckling Charlie.
He offended me with the impertinence of raising the stakes, of bumping the argument up from the mathematical to the meta- mathematical. But I shouldn't have gotten mad; he was merely inoculating me against an ever-present danger: the trap-door in the mathematician's self-esteem. Mathematics is a game, and the player is always exposed to the risk that people will differently value his contribution -- that the other players will pack up their marbles and go home, leaving him as alone as he felt as a little child genius.
A fellow student finally asked a senior professor: What does it mean for something to be an important problem? Conceivably this could mean a real-world application, like secure data encryption. But no. The professor explained it: an important problem is one that a lot of people have tried to solve, without success. There are fads in math, some set off by the selection of the Fields Medal, which is sort of a Nobel Prize in mathematics. Thus is the great cosmic meta-mathematical question -- What is Important? -- answered: in the same manner sheep decide which way to stampede next.
Sometimes the hierarchy of importance is established not with a proof, but with a sneer. One time I talked with my thesis advisor, W.A.J. Luxemburg, about non-measurable sets, which are an ugly but necessary consequence of the seemingly reasonable Axiom of Choice. "Non-measurable sets are an invention of the devil," he jibed. He dealt with them by ignoring them, the same way most of us dealt with turbulence and other then-intractable problems in physics. (These were later dealt with brilliantly by chaos theory; all this fascinating stuff was under my nose during the seventies; but all I saw was the dirt under the rug.) I found the rationality of mathematics was as substantial as a cloud: solid at a distance; fog close up.
This discovery took time. At first, math was still the one area in which you proved a theorem, and all doubt was banished; carping objections were utterly crushed, like ants underfoot. In this I saw power, and safety -- a mirage of youth. As I came closer to the mirage, it proved elusive. There were more disquieting incidents.
In my third (and last) year as an undergraduate, I was taking a graduate course in complex variables. Our distinguished professor, Morris Marden, had a credibility-packed diploma from Harvard, and we were using as a textbook Harvard mathematician's Lars V. Ahlfor's Complex Variables -- the third edition, I believe. In the first few chapters, an occasional exercise discussed metric spaces, of which the complex plane is a very special case.
Professor Marden was only a couple of years from retirement, and had a charming sort of befuddlement about him at the blackboard. He would finger his suit and face, and smudge them all over with chalkdust. One Friday he assigned us several homework problems from Ahlfors, among them to prove this theorem: given two closed subsets C[1] and C[2] of a complete metric space, at least one of the sets compact, then there exists a point in one and a point in the other minimizing the distance between the two. I bent my brain on this exercise for an entire weekend, but no matter how I tried, there was a gap in my proof that I could not paper over.
Finally --duh! -- the light-bulb went on and I thought: maybe it's false. So in a matter of minutes I constructed a counter example.
Later I showed the example to my real variables instructor, the cigar-chomping D.W. Solomon. He puffed away, gloated with me, and said, "I love it. This is an insultingly simple counter example. You know, I had to prove this theorem when I was an undergraduate, and I always thought there was something fishy about it." The reason I had found the error was not that I was some sort of genius, but due to the oddity that I had taken a second-year graduate course in Hilbert spaces the previous summer. Normally students take the courses in the opposite order, and by the time they know enough about Hilbert spaces to construct the counter example, they've forgotten the fishy theorem altogether.
So on Monday, when all of the grad students in the class turned in their "proofs" of this theorem, I turned in this counter example. Marden was a bit flustered, and chalkdusted his face more than usual. But a little embarrassment would wash off. What didn't so readily wash off for me was the sense of disquiet of a little of the solid floor sagging and creaking. How could this error have been propagated for so long in this bastion of certainty?
Well, I could write it off -- it was just an exercise outside the main scope of the book, probably most professors didn't really assign it. If anyone had found the counter example, they had never bothered to communicate it to Ahlfors. I myself did not do so for another eight months, when I sent it, rudely written over a photocopy of the Harvard Math Department's form letter rejecting my application to graduate school there. Ahlfors' reply was a great deal more polite than my tart little note. And in the next edition, the error was corrected by positing that both C[1] and C[2] be compact (which makes the proof trivial). So I was able to put it out of my mind for the time being.
But as Freud tells us, the repressed always returns.
It returned the next year, my first year of graduate school at Caltech. I arrived on campus with tremendous insecurity. My first acid trip the summer before had been a bad one; I feared I had destroyed my mind and my future. I felt the stigma so deeply that I only confided my tortured fears to two people. One, mercifully, was Richard Dittman, my favorite physics professor. He reassured me that there was no evidence of anything like that happening. But I was still horribly depressed and fearful.
Added to this was a disappointment. When I arrived, age 19, I expected at least to have the ego-balm of being the youngest graduate student in the math department. But Arthur Rubin (who was to become my office mate) was sixteen. He had gone directly from high school to grad school, enrolling simultaneous- ly as an undergraduate.
He selected Tech because it was the only school that would allow him to do this. It registered him as an undergrad so he could continue entering the annual Putnam Competition. This was a tough annual mathematics exam, relying more on ingenuity than detailed study. In the previous year, I had barely made it into the top 200. Arthur scored number one in the country for four years in a row. I was still pretty impressed with myself, but all delusions of being "a young Newton" were wiped away.
As were additional illusions. Every year, an assortment of top-notch outsiders serve as visiting professors at Tech. There was a good-natured professor from Ireland, Sean T., from whom I took a course in ring theory. He had a delightful Irish accent, the sort Americans are charmed by. Sean invited Arthur Rubin and me over to his place for dinner. So we got to know each other a little better.
One day I was walking through the math department and saw his door open, and I came in and asked him about his research. Although he was teaching ring theory, his own research was in a different area of algebra, namely finite groups. The particular problem he was working on had developed into a race to see whether computer proofs or an analytic approach would provide an answer first.
A couple of weeks later, I dropped by to see how Sean was progressing on the problem. Well, he said, since I'd last spoken with him, someone "published a paper purporting to prove it." However, as he stated, the guy was notorious for sloppy work. I got a clear sense that it wouldn't kill Sean if mathematical knowledge had to wait, as long as the credit wasn't whisked out from under him.
This is a perfectly understandable human reaction; I've felt that way about getting scooped on articles: I'd done all the work, but gotten no credit because I wasn't fast enough. Yet up to this point I'd maintained a bit of the idealistic notion that science was a great cooperative effort to roll back the chaos of ignorance. Here I was seeing through this. I even got the sense that it wouldn't exactly be the end of the world for Sean if mathematical knowledge has to wait forever for the result, if the alternative were for someone else to get the credit for proving it.
That's the thing about ivory towers -- they're so hard to keep clean.
Another aspect of the irrationality of mathematics is the problem personalities it attracts. People who are incapable of dealing with others as human beings are attracted to this intellectual battlefield; it offers clear objective rules by which you can intellectually obliterate people; you can shut them up with just the raw power of symbols.
One of the spookiest examples of this was a young hotshot professor, nicknamed Johnny Mac. He was blond and taciturn and had slightly rugged angular features, like a clothing model or an Ayn Rand hero.
In my second year at U.W.M., I took Mathematical Analysis. It was the first mathematical field I studied while lacking a pre-existing mesh of understanding, like a net-shaped skin graft on a serious burn; and however large the gaps these traceries left, many fish were caught in the net. But I had never developed good studying skills; this left me out of my depth.
Johnny Mac wasn't out of his. In fact, he let on that he resented being forced to teach an undergraduate class (of only eight students, if memory serves), even a senior level class like this one, and even if it were his only class. But dutifully, he lectured on this tough subject, without reference to notes. He performed this same feat for months, every Monday, Wednesday and Friday. Then one day in the middle of a particularly difficult proof, he pulled a piece of paper from the inside pocket of his sports jacket, referred briefly to notes in his small, pointed handwriting, and secreted the folded paper back inside his jack- et.
This happened only one other time. And those two times, rather than exposing feet of clay, served to italicize his daunt- ing intelligence.
Further daunting was his clear aversion to other humans. Every day at the end of class, he would have his galoshes and raincoat on before any of us could even pack our notes. And presto: he'd be gone. His office hours ranged from brief to non-existent.
One day I walked by his house near campus, and saw what I assumed were his wife and a couple of kids -- handsome little blond clones of their father. Well, I thought, he must have some interpersonal skills.
I returned to U.W.M. in August, 1989 to visit the physics and mathematics departments. When I went to the math department, my old professor Robert Moore happened to be around. I asked about Johnnie Mac, and he was only able to enlighten me a little. The aging wunderkind, it seemed, had been trying to crack an extremely difficult problem, when it happened.
My curiosity had already been piqued by the secretary of the department chairman. I didn't find Mac's name among the mailboxes, and so I inquired after him. The secretary was cagey. "He's not around," she said. I explained that I'd been a student of his, was in town for the first time in ten years, and hoped to see him. Did she know where he was? At last she said, "He's in the county mental institution."
I had the queer, giddy sensation of having survived a near miss that hit another person instead. MacMillan was now hanging by his fingernails from a cliff where I had danced.
What a mathematician proves when he cracks a difficult problem, is not a theorem. What he proves is that he is a better mathematician than anyone else. MacMillan, I conjectured, had all of his ego invested in proving this; and when he couldn't crack the problem, the problem cracked him.