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In the early 50’s the U of Chicago math dept was a cauldron of productivity, the finest such crucible in the world. Marshall Stone was chair, other people included Andre Weil, Saunders Mac Lane, Serge Lang, Paul Cohen, Paul Halmos, occasionally von Neumann, Carnap.
Logic students included Bill Howard, Ray Smullyan, and Anil Nerode. Stan’s contention was that this was not at all accidental, a serendipity, but was directly related to university philosophy and structure as molded by Hutchins, which not only provided the environment, the milieu of freedom and minimal bureaucracy, so fundamentally necessary to intellectual ferment, but so constituted, attracted the most gifted, established and student, to come there in the first place. Thriving in this bastion of open enquiry he himself, as an undergraduate, produced important and original results in the newly developing arena of mathematical logic.
His philosophical, educational, and social orientation in regard to mathematics (and parenthetically everything else) burnished by the U of Chicago pure flame experience, Stan ever after carried this with him like a sheik’s fold-‐‑away tent in the desert, to be reassembled and dwelt within wherever he next set up camp. As I mentioned Stan didn’t enter your world, you entered Stan’s world, as disorienting and confusing as that might be for some.
The essential lesson from the U of Chicago was that mathematics (and science) is a deeply human activity dependent on individuals, their personalities, cultural orientation, character, prejudices, interests, oddities, propensities, their social nature ….. progress, or at least advances, in math, science, technology is not a linear tidal wave unto itself as it is now fashionable to portray it, but is intimately the outcome of very human endeavor, subject to the frailties, foibles, lack of communication, jealousies, start/stops, personal friendships and animosities, indeed ultimately as subject to human interaction and nature as anything else.
That being the case Stan was as much interested in the ‘sociology’ (education an element thereof) of mathematics as in the mathematics itself. In keeping with his penetrating understanding of ‘how things worked’, he kept in personal contact with enormous numbers of mathematicians, knew precisely what they were working on, what they were thinking about, and what their current problems were, and promoted advance by cross-‐‑pollinating, communicating, relaying the problems of one to another, the two quite possibly completely unaware of each other, a one-‐‑man (international) switchboard. Mathematicians by and large a solitary, solipsistic bunch, Stan was the inverse, a restless moving line ceaselessly, incessantly connecting the dots. In my time in Stan’s house, I was the witness, his shadow, a parrot on the shoulder, of this continual activity, hundred’s of phone calls placed here, there, everywhere, first call, ‘I have this lemma but it only works if this integral is bounded within such and such, but I can’t see it’…… second call (to England), ‘Tony, you did some stuff on integrals over the complex plane, what about this one?’…. and so on.
My impression is that Stan considered his very conscious role as quintessential Malcolm Gladwell ‘connector’ of equal importance to his own theoretical output regarding the advance of mathematics, as part of his ‘place’ within mathematics, perhaps more so.
For him development, discovery in present day mathematics was nearly a communal, living breathing enterprise, in which, the problems being as difficult as they now were, were subject to cracking only through collective effort, each contributing but a piece to the ultimate puzzle. Indicative of how central this was to his thinking, he very seriously commented that Feynman did not ‘hear the music’, was unconscious of how essential hands joined together, dance in a circle collaboration was. (On the other hand, he deeply admired the ‘Feynman Lectures on Physics’, a stupendous achievement Stan was certain Feynman would be remembered for, over and above his theoretical achievements.)
Regarding his ‘place’ in mathematics, I was present when a student asked him how good a mathematician did he himself think he was, rather impertinent, but one to which Stan responded: ‘average’…………………………………………………………………… some ‘average’. Sure if your evaluating basket also includes Gauss, Hilbert, Euler, Poincare, Godel, Riemann, Cantor, and Galois, maybe then you get an ‘average’ but, in so describing himself, Stan was actually indicating the standards to which he held himself.
This was strikingly evident in the amount of research in mathematical logic that he published ….. which was almost nothing. Stan had a religious reverence for the written word, and considered a work’s value measured only by its degree of permanence……………………………………………………. think of Euclid’s ‘Elements’, Newton’s ‘Principia’, Gauss’ ‘Disquisitiones’. With his great pride and unwavering honesty, Stan, regardless of his substantial (by any measure) of ongoing, original, creative theoretical achievement, would not publish, preferring to verbally spread his results about to the relevant people, my feeling that his ‘reach exceeded his grasp’ and that he did not feel his results sufficient (by comparison) to commit to posterity. But I think there was more to it than this, that this was in part ‘cover story’. Stan had a neurotic aversion to publishing that had little to do with quality of content or indeed ability to write (Witness the 5000 pages of notes he left behind him.) By way of contraindication, his spoken English was impeccable, clear, precise, consummately communicative. Further his writing echoed his speech, always sensitive of phrase, delicate of nuance, pithy of content, models of scholarly clarity. Indeed his very handwriting was elegant, well-‐‑formed letters, a flowing script, not the misshapen pinched hen scratching of the average mathematician, impossible to decipher. No, the ‘cover story’ is not enough. More to the point I think that Stan had a great fear of being ‘measured’, of being pigeon-‐‑holed, categorized, of putting down anything that might lead to misinterpretation and wrongful assumption, something at variance with his perception, results he wished at all costs to avoid. (Though he wasn’t much of a candidate for their business, Stan had a high regard for excellent publishers, Springer Verlag whom he urged me to visit on a drive through Germany being one, he on personal terms with the director.)
Stan, habitué of the ‘true’ university, held colleagues, friends, acquaintances (particularly those from the U of Chicago) to the same standards to which he held himself, faithful only to the coda of a member of a ‘true’ university, allegiance only to the elucidation and dissemination of truth. He had extraordinary disdain for the ‘publish or perish’ phenomena of the 60’s and 70’s, deeming it part of the corruption permeating academia and responsible for little more than clogging up the system with fourth rate garbage. Vigilant to sophistry, he sat in judgment from Olympian vantage, never shying from acerbic, infuriated comment. When Paul Halmos brought him his pre-‐‑publication notes of ‘Naïve Set Theory’, he said simply ‘Paul, this is shit’. (I am not sure there was ever any love lost between the two, when walking with Halmos to one of his talks at the U of New Mexico many years later, I mentioned Stan at which point Halmos asked if he was ‘still hanging around’, he clearly as much as anyone completely baffled by Stan’s existence. Never one to take a slight without parry, when Paul asked what I was doing, I said in barbed response, ‘hanging around’….. which quite amused him, his returning ‘I like a guy who leads with his chin.’…. I never did know Stan’s disapproval of the book, it being the text from which I learned my set theory.) Stan was particularly contemptuous of any debased writing that was done for money. When John Myhill was at Buffalo, he and Stan, long-‐‑standing friends, contentiously fell out over Myhill’s productions at the instigation of an educational publisher.
The more I came to know Stan, the more I was aware of various neuroses, a gestalt explicable I suppose by one of plumbless psychological insight. Among others, not content in pejorative critique of (in his eyes) the failings of friends and colleagues, he had a compelling compulsion to ‘get the goods on’, to ferret out for each and every some weakness, some moral lapse, some damaging attribute or act, some glaring fault of character. No one escaped. Readily accepting at that age reasonable, benign ‘explanations’ for any behavior of my Lancelot, in retrospect this seems more than strict adherence to searingly honest appraisal but rather some kind of setting himself apart, self-‐‑ affirmation of his purity, a leveling of the playing field, a girding of intellectual loins for future conflict.
Myhill’s sin was accepting the book contract, Feynman’s weakness was that he could be brought to his knees with ‘Richard, you can’t solve this.’ Anil Nerode tells of a party hosted at the house for a number of friends, some coming from considerable distance, during which Stan systematically insulted each present one by one. A smaller example, when a physicist friend, MIT PhD, passed through on his way from Caltech to visit MIT, excruciatingly funny guy whose every third sentence was sexual innuendo, inexhaustible, but one who described physics as a huge field of interlocking cogs, wheels, nuts, bolts, Stan remarked on the way back from delivering him to the train station that he was one of the few who escaped the indoctrinal horror that was MIT, but upon my demurrer, that he had only ‘almost’ escaped, did fully agree.
Stan’s antennae were hyper-‐‑sensitive to phoniness, quackery, sophistry, pedantry, perversion and obfuscation of ‘truth’ within the walls of academia, particularly as it was located in the philosophy department.
Though everyone had within them a component of ‘bullshit’, many (most?) academics made a living through the manufacture of it ….. these were the ‘cockroaches’…. Stan was RAID, the eradicator.
Within the math dept, Ralph Raimi was a cockroach, Stan scathingly dismissive of his grand-‐‑standing dramatics regarding a supposed cheating scandal at the U of Rochester, Raimi seizing the self-‐‑aggrandizing high ground, to pathetically pontificate the decayed state of student affairs for which he had the elixir (rant on huckster) to restore undergraduate moral health.
The philosophy department on the other hand was a nest of ‘cockroaches’, at times requiring Stan’s fumigation services wholesale. When Norman Stein’s brother, a noted scholar of Greek philosophers, addressed that department, Stan’s introduction was a torrent of (deserved) abuse towards the gathered so vitriolic he could hardly, nearing apoplexy, get it coherently out. One well known exemplar published a book in which Stan identified a 100 or so substantive errors, delighting in accosting the man quite by accident in a grocery store, leaving the hapless roach wondering with his ‘I have read your book.’, nothing further. After an afternoon conversation over a backyard fence at the Institute For Advanced Study with a rather talkative visiting historian, Stan’s rather terse assessment as we gained the back door, ‘A phony’.
With any individual Stan’s stance was ‘What can you do? What are you made of? Who are you ….. really?’ Relying completely on his direct experience of you (he once commented that anyone secure in their own mind, trustful of their own judgment, did so), he eschewed as irrelevant all secondary indicators, degrees, medals, awards, grade-‐‑point (he pointed out that the brightest kids tend to have spotty academic records because they simply can’t be bothered except with what they find interesting), commendations, scores, positions held, standing, in his assessment.
Himself without degree other than Bachelor of Philosophy, relying only upon his ‘white plume’, his actual achievement and the concomitant understanding of this by his professorial cohorts for his navigation through academia, he was deeply distrustful of any necessary relationship between one’s capability, potentiality, value and the secondary indicators of such. His stories of counter-‐‑examples were legion. One such, questioning the actual meaning of a PhD, related in his number theory class that fall semester, concerned an outstanding theorem, yet impervious to attack by sterling minds over decades, guessed to be certainly true, special cases calculated in significant number, PhD’s awarded for various attempts at proof. Undaunted a brash graduate student calculates the next case, finds the theorem fails ….. voila, question answered, surely worth a PhD? No??
More perniciously, Stan was horrified, vitriolic in his condemnation, when pursuit of the secondary indicators (grades for example) replaced the sacrosanct pursuit of knowledge, understanding, endeavors toward truth. This (most common, perhaps the norm?) was a perversion of the idea of a university, of your very purpose in being there. Most unfortunately, though, this pursuit was aided and abetted, encouraged, promoted by most institutions to the degree they diverted from the conception of true universities.
Perceiving the development of mathematics to be a deeply human enterprise, Stan placed great store in the interplay of its social connections, he intentionally promoting these through his proactive role as honey bee pollinator. Beyond this he was acutely attuned to and interested in the influences that people in the game had on each other, in particular the depth of influence of mentor (professor) on neophyte (student). The importance of these connections (connections that Stan assiduously tracked, kept mental notes of) is codified today in the website The Mathematics Genealogy Project which details mathematicians, their advisors, and students, providing a tree transparently revealing the lineage not only of subject matter but also of taste, prejudices, orientation, and philosophical slant. It would greatly appeal to his sense of humor the irony that Stan, with a profound influence on large numbers of mathematicians and students, is not included in this atlas.
Stan of course decried institutionalized influence, the schools and universities, which in the main he considered to be machines for the systematic stunting and crippling of initially healthy minds and spirits. In his one man guerilla war he personally had huge impact on large numbers, particularly students (I am a case in point.) Not simply a magnetic personality, I think Stan got to, reached, those he came in contact with through his scorching honesty and piercing sensitivity, an ability to comprehensively grasp whom he was dealing with married with consummate command of technique to enlighten. One illustrative example that comes to mind is Mel Nathanson, on his way to becoming a doctor, whose life was completely altered when he took up Stan’s suggestion to get into mathematics ….. but there are hundreds of such examples.
Particularly at high levels, vaunted intelligences are acutely concerned with their abilities and potential, as evidenced in their abrasive competitiveness and endless comparisons with their peers. In this setting, a teacher, professor can have extreme emotional impact on one of his highly strung charges. This can easily take a right turn, the field littered with the corpses of budding mathematicians, demoralized as to their capabilities by a callous, often idolized, figure. (It need not be a mentor, something else, like competitive exams, can have equal effect …… Stan once mentioned that Newc Greenleaf, a very good mathematician, never really got over not winning the Putnam competition …. or perhaps it was not being offered a position at Harvard.
Acutely aware of, deeply sensitive to the emotional state of a student, and ever intent not to inflict damage, Stan’s delicate interactions with a grasshopper could be nearly surgical in their insightful concentration. (Stan once said that he could never be a doctor, that he was not man enough to hold another person’s life in his hands.)
One afternoon Stan came storming into the house, clearly distraught, followed by a handful of gloomy faculty members, John Dollard and Norman Alling among them. These had been the committee to administer PhD orals to one of the grad students. Shaky to the point of collapse, the student lapsed into near catatonia under rather benign questioning, unable even to indicate how he might approach the problem of determining the curve of minimum time, the trajectory of a ball rolled from A down to B. Stan, appalled that he could be involved in what to him amounted to psychic torture, spent the afternoon denouncing any system that could culminate in such a spectacle.
Mathematical achievement so utterly dependent on such, Stan had an overweening interest in what constituted intelligence, innate capability. In no way did he imagine that humans are provided equal measure nor did he think that capacity was unlimited. His only point was what provided for the maximal realization of what was handed out. He was not dismissive of IQ tests, but thought of them as rather limited instruments for predicting achievement, measures that should in no way limit or affect one’s ultimate strivings.
He thought it child’s play to determine IQ, but more broadly innate capability in general, through direct observation and interaction, noting in passing that all truly able people secure in their own judgment made their judgments directly without reference to outside data or measures. He himself had a Santa’s bag full of mathematical concepts, puzzles, theorems with which to accurately determine basic ability, as he did with Mel Nathanson when he ran through the proof that an orthogonal vector set could not exceed in number that of a spanning set in a finite vector space, gauging Mel’s aptitude for mathematics by his ability to follow the proof, happily pronouncing in this case Mel’s undoubted bright future should he so choose to enter the game. He readily acknowledged that for every individual there was a limit to abstract understanding, commenting that if the abstraction was sufficient, over one’s head, for the individual it simply didn’t exist, wasn’t there. In one instance he held up one hand, fingers widely stretched and said if you didn’t immediately understand that the number of spaces between the fingers was one less than the number of fingers, then you could yet make a contribution to mathematics but it wouldn’t be much. One moment that occurred, Stan remarking on his failure to grasp a paper that had just surfaced, ‘Perhaps I just don’t have enough brains.’
With his intense antipathy towards repression of the individual, Stan equally loathed institutional bias, prejudice regarding any group of people, firm in the belief that all groups possessed equivalent native talent, albeit degrees of which were distributed among the individuals of a group. In particular he had a sublime sympathy for the American negro, contending with a vile racism inclusive of supposition of intellectual insufficiency. Viewing black ‘language’ of the ghetto not as degraded English but rather as an immensely clever, creative response to their appalling plight, Stan often spoke of his personal experience of ghetto kids, emphatic that they possessed abundant intellectual ability on the par with the whites and that, in the right hands, could easily demonstrate such.
Profoundly attuned to the emotional currents of this suppressed people Stan once asked me if I could hear the put-‐‑down of the whites in black music. Indelible is the memory of Stan one evening at the Institute For Advanced Study sitting at the feet (literally) of one of the black custodians, intent in that conversation as he would be in talking to Godel.
Perhaps because of his personal trials, Stan had an extraordinary interest in the effects, particularly negative ones, of emotion on normal thought processes, certainly as they related to progress in mathematical endeavor. He thought general mood significant, ‘You need to get a girlfriend.’, ‘The weather is really not good here.’, a general disaffection with one’s circumstance, or nebulous depression accounting for diminished achievement. More psychologically, he had acute sensitivity for hysteria, regarding it as a markedly disturbing factor, present to some degree in everyone, but dialed up in many to the point of dysfunction. He was an adept at dealing with and alleviating such, through vocal inflection, calmness of demeanor, measured speech, insightful questioning and suggestion. Much of his office time with students had little to do with mathematics but more to do with ameliorating hysteria …… His nephew after a rather poor showing on his high school SAT tests was shipped over to the house from Cincinnati for Stan’s ‘treatment’. Cognizant that the kid actually had enough brains but was terrified of the exams, he succeeded in sufficiently adjusting his mental set that on second taking he raised the overall score a couple of hundred points or so.
Stan was a humanist first (‘Man is the measure …. ’), a philosopher second, primarily Greek rationality though amongst the recent crop Wittgenstein was a favorite, a mathematician third ….. he was not interested in mathematics for its own sake, was not interested in it as a polyglot potpourri of aberrant and irrelevant mental machination, but rather regarded it the highest expression of rational thought, the jewel in philosophy’s crown, the noblest of pursuits which, Diogenes lantern in hand, led inexorably to ultimate truth. To the degree that we discussed mathematics it was mainly how he regarded it and its place in the universe of human thought. Which is not to say that we didn’t get down to cases, his grasp of the whole of it and his fundamental power to reveal the essence of any particular topic or branch or theorem always a source of wonder …… when I inquired about the calculus of variations, he adroitly explained its basic ideas. While waiting for an elevator in the math building on our way home, chatting about Galois theory, Stan incisively commented that really Galois had merely noticed X …. but followed this with ‘but it took 2000 years for someone to notice X’. Looking over my shoulder at my notes on normal subgroups, he casually advised me of their true significance, something which had completely escaped me up to that moment. Prior to beginning his class on number theory in that fall of 1966, Stan spoke of the lofty splendor, the majesty of that most ancient of mathematical endeavor. Remarking on the fact that its theorems are characterized as being both easily understood and extraordinarily difficult to prove, he then enunciated the proof of Euclid’s theorem that the primes are infinite, the pleasure (and instruction) being that in Stan’s rendition, in his clarion wording and measured cadence, the proof (which after all is quite simple) became a retelling by an elder of an honorable, venerable, ancient tale of the tribe.
Bespeaking his conception of the grandeur of mathematics as the pinnacle of human thought, Stan loathed any diminishment of it, any besmirchment thereof, any reduction to the common, any cavalier disregard or indifferent treatment. He detested Halmos’ introduction of the symbols IFF for ‘if and only if’ and the tombstone symbol indicating the end of a proof, his point being in the case of the latter that if you’re paying any attention at all, if you’re understanding what’s going on, then you certainly know when you’ve come to the end of a proof. Which does though beg the question, doesn’t the Latin QED (quod erat demonstrandum) do the same, Stan’s answer No! No! No!, QED is not a signal of the theorem’s end but rather a triumphal, voce bravura, ‘I have done it !’
Albeit handled in a highly sophisticated and emotional mature way Stan was exceedingly competitive. It goes with the territory, mathematicians pitting themselves against kryptonite hard Goliath problems, their sling shot brains their only weapon, they must not only have a faith in themselves, but also a faith that they can pretty much take care of anyone else around.
With Stan this evidenced itself in a variety of ways. In his teaching there was always an element of challenge, as a student you had to bring to the table a certain strength, a certain chutzpah.
Stan knew, and made it his quest to know, large numbers of accomplished (often well known, often famous) people, not only the finest mathematicians, but indeed those in virtually every field of art and science, philosophers, historians, physicists, writers, actors, businessmen, financiers, lawyers. Though I think a complex of motivations drove this, natural curiosity certainly one factor, I always had the impression of a competitive aspect, particularly with his mathematical equals, a desire not only to learn of, but also to come to grips with, a mano a mano face-‐‑off, a testing to see if he could hold his own, a determination to see, if he so chose, if he could fit into other shoes. He often characterized social interactions in terms of contest, winners and losers.
We were just dropping in on a mathematician at Cornell, spur of the moment, a friend from Chicago. All very casual, a few people there, a big mattress spread on the living room floor, the kids messing around.
Always with a multiple agenda, Stan brought up a number theory problem involving complex analysis that someone was working on in California, stuck at a certain point. This out of the way, conversation turned to hash, Stan just becoming aware of it, very curious.
The prof, something of a THC aficionado, mentioned a psychology experiment being run in the psychology lab on rats, loading them up with mega doses and trying to get them to run up an incline. Stan imagining a horde of high school students on the sidelines cheering the rats on, dissolved into paroxysms of laughter, little able to keep from falling off the couch, commenting with great glee as we got into the car, ‘He beat me, he beat me. ‘
At an occasion in which Feynman delivered a talk, following one by Feynman’s friend Freeman Dyson, Stan later talked to Dyson, also a friend of Stan’s, to get his take on Stan’s comments at a previous seminar featuring Oppenheimer. Dyson was in a somewhat dour mood, not really open to Stan’s importuning, Stan’s later explanation that Dyson was depressed that, in the back to back talks, Feynman had beaten him. Although he had misgivings about the Putnam competition’s meaning and validity, cautioning me as I left for the exam not to put too much emotional stake in the outcome, he nevertheless took the exam surreptitiously at home, pleased with his getting the majority of the problems. One evening, me driving, he sitting in the back of the car, I was surprised to hear him singing ‘Who’s afraid of Norman Stein?’ to the tune of ‘Who’s afraid of the big bad wolf?’. Amusingly, he would make a burlesque, wholly exaggerated competition, when on the basketball court with students or playing croquet with Dollard and the younger mathematicians.
In Stan’s world time was not the fourth dimension. It simply did not exist. Of no moment (so to speak), Stan lived his virtually structureless existence, subject only to his overarching, grandiose when he was on a roll at 36000 feet, plans and agenda. Time not a constraint, he was even freer to plot the course of each day by whim or design. Lectures ran over (rarely having begun at the appointed hour), the conversations ensuing sometimes moving to the student union or other convenient spot. Conversations started in the office would move over to the house, continue through a bite to eat (hardly could it be called dinner), proceed out on the lawn, carry on unabated into the night, day into night into day into night, merely a blur, of no consequence, in no way modulating the activities, a neutral backdrop to the (very) important matters to hand. All was free form, creatively fashioned, the evening with a visiting mathematician in which Stan impetuously decided to fly to Boston (when flying was still an exotic experience), immediately on the phone to see who was around, a typical case in point, the norm, not the exception. The house frequent of visitors, these folded seamlessly into the family flow, Stan always fluidly gave them his full attention, the conversations extending from moment of arrival to moment of departure, sometimes days later. Basking in this elevated milieu, reveling in this allegiance only to what was ‘important’, I yet cringe to this day, the world turned upside down, when I witness one of great stature and accomplishment with worlds of interesting things within, perhaps a Nobel prize winner, kowtowing to the tyranny of the clock, stopping (with apologies) his seminar on time, ‘Oh, I’m running out of time’….. when the audience should be on their feet, exhorting in one voice, ‘Don’t stop! Don’t stop! ‘
Stan was robustly, actively alert to the humor in any situation. Neither slapstick, surreal, nor common, his sense of humor was based in the predicaments of the human condition, eccentricities, situational improbabilities, absurdities, ironies, coincidence, serendipities, all exaggerated, bolstered, magnified by his fulminating imagination. When a calculus student detailed a fulsome yet completely incomprehensible answer, unknown terminology, to an exam question, Stan paraded it up and down the corridor amusing himself with the notion that the student was a great undiscovered genius revealing for the first time deep insights and heretofore unknown mathematical connections. The wife of an eminent expert in probability, one of Stan’s friends, approached him for some help on homework for a beginning statistics class she was taking. Asking her why she hadn’t asked her husband for help, Stan broke up when she said that she wasn’t really sure he understood the material.