Marguerite Lehr's writings
In addition to mathematical research papers, Marguerite Lehr published many interesting articles which give considerable insight into the way she thought about mathematics, scholarship and other things. We present below extracts from four of these papers giving, before each one, a brief indication of the context in which it was written.
- Marguerite Lehr was awarded the M Carey Thomas European fellowship to fund her studies in Europe in 1921-22 but she chose to postpone this, spending the two years 1921-23 at Bryn Mawr College, the first as a 'fellow in mathematics' and the second as a 'scholar in mathematics'. The M Carey Thomas European fellowship allowed her to spend 1923-24 in Rome and she sailed from New York on the SS President Arthur on 16 June 1923. In Rome she studied with Guido Castelnuovo, Federigo Enriques, and Vito Volterra. We give below extracts from her own description of her year abroad, written after she had returned to Bryn Mawr College. The paper is: A Marguerite Lehr, "Ave Roma", Goucher Alumnae Quarterly (March 1925), 6-9.Ave Roma.
Now that it is done, and there is only the joy of telling it left, I wonder at our nonchalance. The blithe announcement-" We're going to Rome to study!" was inevitably received with a questioning "I suppose you speak Italian?'', to which we answered with calm assurance, "Oh no." Curiously, that assurance, unbroken by the oft-repeated "I suppose" and by six weeks in fairy-tale towns like Göttingen with their winding cobbled streets and sagging houses, was shattered by a single day in Paris, for - we reasoned - we knew a little French and understood nothing; we knew no Italian. We took an express to Rome.
How inadequate that statement is! It means glimpses of bluest sea, green palms against creamy white houses, tunnel after sooty tunnel, soldiers in jaunty feathered hats at the Customs, the stretch of sun-beaten sands at Nice, the strange, beautiful names of Italian towns, and at last the terrific heat that was Rome in August. The days that followed seem now a curious jumble of lazy drifting and frantic endeavour. The mornings were spent for a while at the Summer School arranged for by the Italo-American Society with the University of Rome. It gives one an uncanny feeling to understand something of the "drift" of a lecture when the language is so appallingly new that perhaps one word in twenty can be disentangled. For three months, in the cool hours of the morning we read aloud to each other, listened with strained attention to those lectures of which we sometimes understood the general subject, or worked hard on translation and conversation with a teacher to whom we owe a great part of any success of the "wonderful year."
What wonder, then, that early afternoons, when the sun blazed on the wide piazza and the glare was intolerable, we speculated lazily about what the University was like, whether we would understand the lectures, how one got to Tivoli, what one asked for if one wanted a notebook, why the street which led straight out from the piazza changed its name at every corner whereas the Via Nazionale meandered proudly where it would, - a pleasant interval of rest in a darkened room with cool tiled floor. By five o'clock the air was surprisingly fresh and labyrinths of narrow streets were always luring us from the business of the moment, which was usually the task of buying soap or some such homely necessity, simple enough if you have the courage to say aloud the word you read in the dictionary.
Those months were broken by glorious holidays spent outside the city; the day at Tivoli, in the gardens of the Villa d'Este, where the ripple and drop and boom of water from brook and fountain and cascade drove away all memory of the city heat; the day that we walked the Appian Way and ate our lunch sitting back to back on the grass in the shadow of a ruined tomb, singing songs that I smile to remember, answering the cheerful greeting of the old peasant who wished us "Buon appetito"; the day that we spent under the pines high on the Janiculum, with Rome lying below dazzling white in the noon sun.
Then the University opened, and we met the men whose names we had seen on the backs of books, and we understood the lectures, and we could take notes, and suddenly we were settled in a round of work. Of course the Italian University is organised on a plan totally unlike the American one. The student entering the University at approximately the age of the average American college student begins at once to specialise; most of those with whom we came in contact were doing Mathematics, Physics and Astronomy. In general they carried a heavy lecture schedule - I heard of several with twenty-four hours a week; but it is possible to obtain the degree without attending classes, by passing the examinations. Registration is required only if the degree is desired; for attendance at lectures, only, there are no fees. Foreign students are exempt from registration fees upon presentation of proofs of their citizenship, which, incidentally, must be stamped and visaed and stamped again by consuls, Ministers of Foreign Affairs, and an endless succession of other officials.
The lectures at the Instituto Fiscio, where all advanced Mathematics was given, usually filled the morning, for after the lecture we often had to collaborate over the interpretation of difficult points; Sometimes the Italian was not clear, many times the theory needed talking over. Afternoons were still given over to rambles, sometimes through new, uncharted territory, oftener over the old ground up the hill along the Via Sistina to Monte Pincio, to lean on the wall and look over the spread of the Spanish Stairs with the flower stalls half emptied, across the piazza with its crowds sauntering from tea; to watch the sun sink and the blue haze spread over the hills, with the dome of San Pietro still showing softly white through the dark arch of the trees; to drift slowly up the narrow winding path, across into the Borghese Gardens, where children played, and nurses chatted, where scarlet-gowned German divinity students walked with clasped hands and murmuring lips. Those were intervals so calm and peaceful that often we turned home silently; only when we reached the noisy Via Nazionale and looked up at the magnificent jet of water that gives life to the whole avenue did we take up again the talk of small but pleasant things that made our walks such fun.
There were breaks, of course, in the University work; the most startling was the first. One morning we walked up to the entrance to the winding walk that led past the laboratories through a small garden to the Institute buildings, when to our amazement we found the great iron gates closed, with a notice stating that the University would remain closed until the students returned to a due respect of law and order! We had the most agreeable guilty feeling but we did not know what it was all about. We discovered that the students objected (and with some reason) to certain changes in the laws as to state requirements for professional people, but that they had rather lost their heads and insulted the "Rettore" of the University. There were no classes for ten days and then everyone returned in peace and quiet.
The two main vacations offered our only chance of seeing Italy. At Christmas time we went down to Sorrento, making that the base for excursions to Pompeii, Amalfi and Capri, because Naples seemed impossible when compared with living on a cliff surrounded by orange groves, with the Bay of Naples apparently lapping the very foot of Vesuvius. At Easter time we had three weeks of joyous holiday that included Assisi, Perugia, and Florence, and then we skipped to Ravenna and Venice. Our phenomenal good luck held out to the end; we saw Venice, grey and beautiful for two days, and then on the day that the King came, the sun shone, - shone on the gorgeous velvets and gold of the old gondolas that went to meet His Majesty, on the damasks and tapestries that draped the balconies of the palaces along the Grand Canal, on the three tremendous flags that floated in front of San Marco, and on the cheering crowd that filled the piazza. So I remember with equal longing a Venice blue-grey and still, and a Venice flaming in brilliant sun, flowing over with admiration and enthusiasm for a long-absent King.
The year seemed to rush to an end after that vacation. I was alone and very busy all day on notes of a course that I had hoped, in vain, would be given that year. The hour just before dinner, when the sun was low and the cool breeze swept down from the hills, was spent always in the big window overlooking the piazza, where children played on the broad steps of Santa Maria Maggiore, and the guard from the palace passed, swinging behind the inevitable band; watching the twilight settle over the pines on the Janiculum, so that the statue of Garibaldi no longer stood out boldly against clear sky, till at last the Janiculum light began flashing the "Red, white, green" that tells those far off "Here is Rome."
It is impossible to tell of all the kindnesses we met with; everyone seemed friendly and willing to help, without counting time or trouble. Perhaps I can make my feeling clear by telling a dream I had not long ago. I had read the headlines of a cross-word puzzle contest offering one thousand dollars as a prize; I dreamed that I had won the thousand dollars and a reporter came out to ask me what I planned to do with it. "I'm going to Italy, of course," I said. And he replied, his tone an apology for his stupidity, "Of course!"
- The following is taken from the Phi Beta Kappa Address by Marguerite Lehr to students at Bryn Mawr College at a time when the United States was actively involved in fighting World War II. It was published as: Marguerite Lehr, Towers of Ivory, Goucher Alumnae Quarterly (May 1945), 3-7.Towers of Ivory.
You are here in the name of Phi Beta Kappa, to honour good work done by students at various stages of the college experience. The topic chosen should be fitted to Phi Beta Kappa purposes, of concern to you who are students, and suited to my own particular interests and training. Phi Beta Kappa has always stressed philosophy, in the broad sense of a preoccupation with the attempts men make to bring some order into their chaotic experience, - with the concepts they form and the values they adhere to, in their struggle not merely to live, but to live a "good life." Phi Beta Kappa was formed in years torn by war, formed by men who knew that in such preoccupations lay their present strength and their future hope.
You are students in a college, familiar therefore with such aims; and you too are living under the impact of war, so you feel many kinds of pressure and you face many demands for action. I remember such a time: Twenty-six years ago, I was a senior here; the United States had been at war; in November it had signed an armistice, over which we rejoiced, not knowing how precise that word would prove to be. Now we are again at war, and every effort of man is judged against that hard fact. You scrutinise your college commitments, and often ask yourselves, "Is this my immediate duty, or could it wait for quieter days?"
Knowing that this question is in your minds, and often on your lips, I wish to speak not as a mathematician but as a teacher, because during the last seven or eight years, those of us who make part of the college undertaking have been forced to continual reassessment of our activity. In the middle thirties, in the colleges and universities, we began to see in our libraries, in the meetings of our research groups, and sometimes in our class-rooms, men and women whose names we recognised with honour, dispossessed now of their right to teach and their place to study. I recall vividly a day when a man who had been quietly investigating shelf after shelf in our mathematics library took down a volume, with a small smothered sound of surprise, and then, with its title page spread open, looked at me and said, "That was I ... " At first we were concerned about them personally, and then we began to know that the great and hard-won human right to think was being attacked at its very root. From that time, I have tried again and again to describe for myself this enterprise called knowledge, because a just description is the first step toward evaluation. With war an actuality, these attempts had an even stronger motive. All through the colleges, trained men and women not directly mobilised felt that they must somehow get closer to battle stations. Over and over as this kind of talk went on, I said, "No - I want to stay here, teaching, and I know why." My friends countered with "You're lucky - you're doing mathematics; which is important." And to my reply - "You've missed the point. If I were doing Latin or English Lit, I'd still stay now; this is part of what the fight is about ... " they laughed, "Oh, you like your Ivory Tower!" Of course that made me angry, and I had to think some more to find out why I was angry.
So I shall talk to you about this enterprise called knowledge. As an enterprise, it constitutes at once one of our greatest legacies from the past and one of our strongest tools for shaping the future. Especially in time of war it must be properly evaluated, since misconceptions will cause waste today and weakness tomorrow. I shall try to isolate some general aspects of the concept knowledge, so as to see the sources of its power and the basis for misconceptions that thwart that power. My examples spring of necessity from the fact that my special training is in mathematics. Forget that fact if you can; if I am fortunate in finding words for what I mean, they will fit your own special field too.
Our heritage of knowledge has its sources far back in the first making of words, and in the first signs for recording those words. Men find themselves bombarded by chaotic experience. They make groupings in this experience, and words to fit the groupings. Many of these first attempts seem to us later most inadequate. For example, there is a primitive language which uses two different sets of number words - one for animate and one for inanimate objects yet now most children perform a greater feat of abstraction when they learn to use "two," whether it be two dogs or two blocks. From this point of view, our enterprise has a cyclic aspect. We make words to draw distinctions; we use those words to deepen our experience, and we make new words at this new level. As the domain of ordered knowledge widens, we must review old words to be sure they still fit the wider range, and we must watch the use of more subtle words to be sure that. sentences made from them actually make sense. If I say to a child, "Jump into the air without taking your feet off the floor!" he knows that is a nonsensical command. Unfortunately not all nonsensical sentences are so easy to expose.
We might describe one fundamental aspect of our undertaking as the continual attempt by many people to assemble and put into good accessible order sound noncontradictory sentences, and to expose faulty contradictory ones. The essential procedures are the same, whether the matter be historical, literary, linguistic or scientific. First, a statement must be consistent in itself, as an assemblage of words with complex implications. Then, its sense having been established, it must further be consistent with the body of relevant fact, and it must be open to revision as new facts are learned, or new relationships are exposed. This undertaking needs contributions of many kinds - the slow gathering of observations from which useful generalisations may be made; the deeper analysis of relationships between particular and general; an occasional spectacular synthesis bringing wide domains under one clear view. Those who work to this end must if possible observe and amass without predilection for some one outcome. They must examine without prejudice all conclusions from any hypothesis. Only in this way can the necessary unifying concepts emerge clear and undistorted.
The most prevalent misconceptions of the activities labelled "academic" arise from the methods men have found useful in accumulating the body of organised knowledge. Abstractions serving to classify experience start from attention paid to common elements, disregarding differences. A child quickly learns the word chair, though chairs are of wood or metal, and may have arms, or rockers. The layman in a special field, seeing differences ignored, conclude that they have judged non-existent. As a result, in common usage "theory" has long been opposed to "practice"; only too few fathers have ever striven with their sons as did John Stuart Mill's: "I recollect also his indignation at my using the common expression that something was true in theory but required correction in practice; and how, after making me vainly strive to define the word theory, he explained its meaning, and showed the fallacy of the vulgar form of speech which I had used, leaving me fully persuaded that in being unable to give a correct definition of theory - and in speaking of it as something which might be at variance with practice. I had shown unparalleled ignorance."
Clearly, a necessary basis for knowledge is a body of verifiable fact. This body can be added to safely only in the exacting and disciplined way described. The scholar, exercising his hard-won fight to what is called free inquiry or disinterested investigation, has steadily widened the field of application, putting aside traditional dictations of thought, claiming as material amenable to his method problems previously settled by authority, whether of state, church, or antiquity. The layman, seeing this encroachment, often draws a false conclusion. He assumes that the scholar thinks this is the only way to work, - this is the only way to answer questions, - this is the only way to make decisions, and he says "Impractical." Or else he disposes of the whole affair by a label, "Ivory tower," with the indulgent comment, "Life isn't like that." A true scholar recognises this separation since it is a distinction he has drawn deliberately for his own good purposes. He knows what is properly in his domain, and in it he submits to this exacting discipline because that is one way to increase man's mastery over his experience. But he also knows that the domain amenable to reason is a restricted one. However it may widen as traditional dictations of thought are conquered, it will always be subject to subsequent evaluation on grounds other than consistency and authenticity.
Misconceptions of this kind become a crucial error in time of war, when no resources can be wasted. One of our resources is the full body of knowledge. Its use is effectively sabotaged by the theory-practice misunderstanding, or by evaluating activities too closely in term of immediate end. As an example of the waste due to mistrust of theory, I quote from the prefaces to a beautiful little text on Aircraft Analytic Geometry, published by a supervisor in one of the big western aircraft companies. Its author says, "The methods outlined ... have initiated an exactitude in tool design and tool fabrication by mathematical analysis which has aided in the evolution from cut-to-fit and drill-to-match methods to those of mass production which permit the manufacture of accurately formed detailed parts that arrive at the final assembly line for installation without further rework." Yet he wistfully remarks, "Since the application of this method is new to the industry, it should be welcomed by all technical men for reasons of their interest in progressive developments." And his superintendent comments, "The authors, in presenting this difficult subject in a practical, simplified, and self-explanatory form, have made a major contribution to the entire industry and to the war effort." Actually, this subject has been familiar for three hundred years; it is taught to many high school students in essentially the form here required. But this author knows it will meet such prejudice that trial-and-error methods will be preferred.
An example of the price we pay for too close attention to immediate practical ends comes to me from my experience in teaching one of the government courses in photogrammetry - mapping from aerial photographs. A map used for a work chart must be constructed in such a way that the desired earth measure - distance, direction - can be found from measurements on the map itself. A map of any large region must record the longitude-latitude network, and that network is quite special near the poles. Mapping of the polar region had been dismissed as mathematically easy in one way, and as practically complicated and unimportant. As a result of failure to encourage the work just for its own theoretic interest, we have had to spend valuable time developing under pressure the needed map projections. For your encouragement, I should say that the photogrammetry course gave me proof that we are learning the value of good theory. The Alaska branch of the Geodetic Survey realised early that vast areas would require repeated detailed mapping, and a government training course was established to supply, under civil service, workers for routine detailed work on maps being assembled for printing. Because they believed that people will work at a routine task more carefully and steadily if they understand something of the end result, they allotted three morning a week for five weeks to mathematics, and I was given only one instruction: "Give them anything that you think will make them more at home with the notion map." I assure you that that time paid dividends, both for the work and for the human beings concerned.
Another of our resources is the long-learned skill acquired by our scholars in the process of free inquiry. We know that this skill must now be turned to questions set by the moment's need; that because time presses, we must often find an acceptable compromise between the long careful check and the quick guess based on keen experience. We cannot test every fuse destined for a time bomb; but we must test some properly determined sample, or we pay men's lives to use duds. So now today many of our scholars are being useful precisely because they have life-long experience in these slow, impractical, abstract, theoretical procedures. Now under pressure, they are able to make the quick guesses, the safe compromises. But they do not confuse the two activities. We must keep clear our conviction, born out of the history of man's own ideas, that free inquiry in any field is a strong position, so strong that it cannot be tolerated by any authoritarian set-up; that at the same time, it is practical, since a good theory is the most efficient of tools.
The analysis so far is restricted to a concept of knowledge which concerns only a few people actively; that is how so many misconceptions of it can arise. There is however a second aspect of the enterprise which concerns all thinking people. During the long struggle to bring order into widening stretches of their complex experience, men have also studied themselves in the act of learning. By evaluating their successes and failures, they learn slowly how men think. They make careful descriptions of the way men's minds do work, they codify sequences called logical and recognise adherences to standards between values. As a result, we have come to make clear distinctions between matters of verifiable fact or information, matters of deduction on some permitted logical basis, and matters of opinion where there is an element of true choice. We know that as knowledge progresses, material may shift from one of these classifications to another, for example from the domain of opinion into the domain of verifiable fact, but our experience, already overwhelming, will seem to exhibit only more confusion if we lose sight of the classifications themselves.
One of the gains we hope for in college work is the acknowledgment of these distinctions. We hope that you will attempt to recognise matters of ascertainable fact as such, never implying a factual basis where none exists; that you will face demonstrable conclusions, and refrain from hence and therefore when logical basis is lacking; that you will acknowledge opinion as such, and choice of values as choice, subject to re-examination whenever wider experience provides a wider range for check. A choice steadily checked and reaffirmed becomes a position of strength. Such distinctions are difficult to draw, even under normal conditions and in the restricted fields of ordered knowledge. The ability to draw them is acquired slowly, being not of the nature of content, and in fact accessible only after content has been mastered. How much more difficult, then, to these distinctions in the conditions under which we make our daily judgments! But we hope here for a gain analogous to the one we saw in the case of trained experts, where long disciplined ways of study furnish a basis for good quick guesses under pressure. We are well acquainted with this phase of the learning habit in physical acts. The careful coaching for a dive (don't drop your head too soon!) or for a golf stroke (don't check at the ball, go on through!) is not intended for conscious use when the pressure is on. When there is no pressure, you take time for painfully slow, careful analysis, knowing that the chances for good performance are better with that back of you. When you must, you take a deep breath, and call upon that patiently acquired co-ordination. We hope for something analogous to that from your slow and at times painful discipline in knowledge. ... In the strict domain we work this way deliberately, by choice and not because we lack other abilities. We know that through the slow study of sound work and the careful analysis exposing faulty work, there can be built into us some slight faculty for recognition of sound and faulty, so that when we are pressed and must make decisions quickly, we will at least not contribute confusion, and we may at best make sound judgments.
As thinking people, citizens of a powerful nation involved in huge military undertakings, and facing stupefying decisions when the harsh necessity of war routines no longer gives automatic shape to our days, we will be making judgments, or acting in default of judgments. In the complex days to come, we shall need all aids, all the safeguards against unnecessary confusion that can be instilled into us. From a recent preface I quote, "The popular mind has grown so confused that it is no longer able to receive any statement of fact except as an expression of personal feeling." In our judgments, we will introduce tragic confusion if we have not learned, by careful work in less pressed situations, at least to distinguish the domain of knowledge from the domain of values, seeing them nevertheless as inextricably bound together. Knowledge without understanding is pedantry, but without knowledge the will to understand will be futile. This which was our privilege has become our safeguard and our weapon.
Only recently did I begin to wonder about the phrase Ivory Tower, feeling sure that originally its implication had not been derogatory. Bartlett says flatly that it was first used by Hugo describing Vigny, but it seemed to me that a poet using such a phrase without elaboration was trusting to an already formed association. And even in Bartlett it is clear that the derogatory tone comes with the later Anglo-Saxon adoption of the figure. My literary friends are on all kinds of trails, beginning with the medieval adorations of the Virgin - "Star of the sea, rose of Sharon, tower of ivory." Last night in the snow storm, the French Relief calendar on my desk blew open to August, and there under a picture of an old flower woman, shawled and huddled into the protection of a pillar, was a quotation: "On empêche un peuple de frapper, mais qui l'empêche d'attendre?" de Vigny. And I said, "Thank you for showing me that to Vigny it was a choice for strength, not a withdrawal."
You see why I am no longer bothered by the judgment implied in the current use of that term as a reproach. I know now that my ivory tower holds the records of a vast human heritage which must be transmitted steadily and undiminished; it flies a flag, flaunting in the face of an enemy that men dare to think; inside its walls are being made ideas which will bring to orderly view wider stretches of man's chaotic experience, just as in Detroit workshops are being made the planes that, flying tens of thousands feet high, bring back for study orderly records of man's disorderly activities. It is treasure-house, pill-box, workshop. Am I to desert it because some one flings a phrase at me?
- Marguerite Lehr did pioneering work as one of the earliest presenters of mathematics on television in 1952-53. WFIL, a broadcast radio station from Philadelphia, Pennsylvania, had joined with twenty-five different institutions in presenting University of the Air over Philadelphia's Channel 6. Bryn Mawr asked Lehr if she would present a course on mathematics, one of the courses with fifteen lectures. She described her course, called Invitation to Mathematics, in Marguerite Lehr, An Experiment with Television, Amer. Math. Monthly 62 (1) (1955), 15-21. Here are some extracts from the Introduction.Introduction to Invitation to Mathematics.
This is a report on a semester's experience with a mathematics program on a commercial network, concerned not so much with t he program itself as with t he powerful means at our disposal, for our purposes. I say "our" not only in the wide sense of national concern with an educational dilemma, if not disaster, in mathematical training but also in the special sense of mathematicians, with all that implies of the specialist's jealous concern for his own field of knowledge. Fifteen weeks convinced me that the best we can manage - best in a mature professional sense even in so supposedly esoteric a field as pure mathematics - pays off in that unknown general public at the receiving screen of a television camera in precisely the ways we want and need.
The opportunity and the risk were both great. A mathematics course offered with or without credit on an educational channel can be planned at some definite level of technical competence. Without this sifting of audience, lecturers tend to stress immediate practicality or to settle, perhaps reluctantly, for the thin fare of puzzles and pretty designs. I was sure one could whittle out fifteen topics exhibiting non-trivial mathematical ideas and strategies of thought, and present them in such a way that you would hear more if you knew more. Proper ordering and careful verbal echoes of past talks could give unity even though each question must be posed, developed and signed off in twenty-seven minutes. Surest of all was my knowledge that mathematics per se, shown as pure and driving human activity, not as accumulated results, intrigues people of all kinds and training's: I've never seen it fail. (What, never? Well, hardly ever!) Nevertheless I hesitated, aside from the work and the risk.
It is an ironic fact in academic fields that from outside the cry is, "Specialists can't communicate," while on both sides but particularly within the profession those who can or try to are suspect. They must be sugar-coating or watering down; even more likely, they are really devoted amateurs satisfied by over-simplifications which miss the root of the matter and misrepresent the activity. This is a costly prejudice, in a time when a proper diffusion of knowledge is one of our few safe-guards. No test situation however restricted can be neglected. Here, the syllabus permitted each talk to be conceived as sparking thought rather than completing some tiny piece of work. Topics with aspects being treated in one's advanced lectures would have the requisite overtones of current live activity, and close attention to verbalisation could satisfy a more informed listener yet not lose too soon the less trained. But what conception of the public would justify such use of time by a member of the faculty in a small department which must carry mathematics curriculum through the Ph.D. ?
The public includes people who hear a child's sums called his mathematics, and industries which classify as mathematicians little girls who punch adding machines. The public includes highly trained and able specialists who are naive about the intention of mathematics to the point of deep mistrust. This is our business, not to be dismissed with a smile even if our concern is entirely and narrowly vigorous growth of the pure field. It is a counsel of defeat to say, "Who would be watching T.V. at that hour?" (You'd be surprised at the spread that showed up even in chance encounters.) The program could be attacked only by forgetting "the public," except as people with minds to be intrigued by what had intrigued men before them, by trusting to thinking in front of them instead of instructing them. For us, the main problem is forcing that high level of attention from which ideas can spring with conviction. For many of them, reaching such a level even for a few minutes could be an exciting experience; one knows this from one's freshmen. My plan was an experiment in "talking from saturation"; the Monday morning time was a deliberate choice, as insurance against a divided mind. My intention was never to fight shy of abstractions; rather, to highlight the abstracting process wherever possible, from the child's first fresh leaps to the articulate strategies of the working mathematician.
- Marguerite Lehr presented the talk Of dice and men to the Seven Colleges Program, Detroit, Michigan in October 1955. A version of the talk was printed as 'Of dice and men', Goucher Alumnae Quarterly (Fall, 1956), 10-13. We present a version below.Of dice and men,
When Bishop Butler said "But for us, probability is the guide of life", and later Samuel Butler characterised life as "the art of drawing sufficient conclusions from insufficient premises", the contexts were literary, philosophical; the problem was the ancient one - how man is to conduct himself, admitting the basic uncertainty of all his knowledge. When George Gallup, interviewed on polling methods, says: "The people don't understand the laws of probability .... Most Americans vaguely suspect the polls of being a fascinating hoax," the word probability now has a mathematical ring, but alas, as so often, mathematics of the black-magic, uncomfortably mistrusted sort, or of the artificial kind, too formal for any successful use in the complex world of every day. Yet opinion polls (of which election ones are the least useful, highly dramatic though they be) and tests of drugs or vaccines (publicised or not) are just one form of an activity permeating our daily life; just one kind of attempt to face the everyday problems of decision posed by size and complexity, or by the necessity to act even when there is unavoidable risk of error.
I have no intention of discussing that unfortunate phrase "laws of probability", nor would I pretend that one can understand any powerful idea without long and serious work. I certainly have no intention of doing for you some carefully picked little exercise on bridge hands or flipped coins, where we add and multiply numbers and wave our hands over the result, calling it probability of something happening! I want to face the nature of the quandaries without being overwhelmed by their magnitude; we can start with very familiar examples.
In 1950, each of you saw or was reported to a census enumerator. The Bureau of the Census has undertaken, every ten years, to enumerate person for person, name by name, "the population of the United States", in a clearly stated meaning for that phrase. This takes tremendous organisation, planning and staff, and it seems only good common sense to make maximum use of such a set-up to gain relevant information for governmental decisions on the state of the nation, unemployment, etc. Contemplate now, after the fact, another piece of good common sense. The 1950 enumeration showed, listed on schedules for what is called "Continental U.S.", 150,697,361 names. A simple division shows you that each minute spent per name takes enumeration staff and wages for two and a half million hours of work, not to speak of clerical staff to process what is recorded. The dilemma is obvious and you can see the only way out. Enumerate and check a minimum information schedule for everybody, and use a longer schedule for some sample of the population. The questions now put themselves: - how big a sample? how selected? Remember, conclusions must be drawn from this smaller group which will affect the whole. That's our dilemma because of sheer size, and it's unavoidable. It is quite silly to say: "I don't trust sampling." We'd be bankrupt if we tried to ask everybody, and we'd be dead before the results were processed. It's better to have some information on record than none; it's better to use sample than somebody's guess or dictation. But what sample? Not only how big, but how selected in the light of the uses to be made of it?
Or take another example: If you use a flash bulb for a snap shot and it is faulty, you don't like it. Neither does the manufacturer! If he hopes to hold his market, some kind of trustable uniform quality has to be aimed at in the factory. But there is a catch here; if you test a bulb, you destroy it; so you are forced to decide about your output on some kind of sample testing. What kind? By what scheme? With what hope of assuring what you want: reasonably trustable quality in the untested part sold?
These problems have mathematical overtones of number, pattern, admissible conclusions. But the main change in attack from the classical accumulation of logical certainties is the recognition that here, in the very nature of the situation, some of our judgment will be invalid, and the existence of that uncertainty must be built into our premise before the logic takes hold. We are used to this as an over-all philosophy, and it is in no sense the counsel of despair - it is rather a measure of maturity and wisdom, as our first quotations show. Now, however, we are suggesting it as a strategy of thought in a particular domain - pure mathematics, - with the intention of developing strict vocabulary and techniques to aid us in clear-cut dilemmas of decision. We set ourselves a new undertaking: granted that errors of specific type cannot be eliminated, how far can we keep them in control, in size and in kind? These questions are not yet ready for development; you must not expect answers to them. "Answers" can be made only to well-delimited precise questions, stated in well-devised and clear vocabulary, for soundly conceived and well defined situations. Not every interrogatory sentence asks a question; not every well posed question has meaning to an untrained reader. One business of a mathematician is to recognise questions not properly posed, and either expose them or retool them so he an attack with the means currently at his disposal. (Incidentally, that steady reformulation over centuries, as current means change, is one of the most illuminating sides of the history of ideas.)
Suppose, now, that a telephone exchange A is to serve 2000 subscribers in another exchange B. It would be expensive, and wasteful of equipment, to install 2000 trunk lines, one for each customer, though this is the only way, mark you, that the company could be absolutely sure no customer ever has to wait for service. You have already said to yourself: "It's highly unlikely that all 2000 subscribers want to call at the same moment, and besides, an occasional 'Busy' won't matter if it's not too often." Let's take these ideas one at a time. The company will have in mind some standard of "too often"-e.g. that overall, not more than one call in a hundred should find all lines busy. How many trunks are needed to maintain service of this level? This is no concocted problem. It is precisely what faced early telephone companies, and it had to be met by pure idea. There was no backlog of experience. What can serve as guide? That vague "It is unlikely that ... " must be bettered; how unlikely? We need some acceptable measure of likelihood in these conditions, measure in the literal sense of a number to feed in to this argument, because the question we are asked requires a number as answer. How many lines ... 60? 100? 1000?
I remind you that over the long history of thought, man has had a reasonable success in this kind of attempt. From a recognised just as many, more, less, the number notions emerge to answer "How many?" From the simple number beginning, a recognised just as long, longer, not so long, develops into measures of length, with a meaning for plus and times in length and area. The underlying concepts are subtle; they exercised the best of Greek thought and the brilliant seventeenth century thinkers. Echoes of their subtlety penetrate into poetry and literature (see e.g. Book XI part l of War and Peace!) yet they are routinised so that today they can be every man's heritage. In the same way in our examples, a direct reaction more or less likely in an actual situation leads very naturally to the question: Can we make a scale? Can we devise some answer to "How likely?" in the strict number sense? Of course, if the initial reaction is confused or vague in its reference, then the attempt is doomed to failure. Let me make the simplest example.
Suppose I toss a coin, and it falls Head, H, H, .. always Head. Even if you are a complete innocent in such matters, even if you have never tossed a coin or laid a bet, if Head continues to fall, there will come a point where you say to me: "May I see it?" - or where you react in some way; (e.g. in a more primitive time, you might decide it is a sacred coin and the god sees to it that his face appears). In any case, the sequence observed calls for remark; you tend to "explain" in some way. Suppose after, say, 12 heads you look at the coin and it does have two designs. Do you jump to the conclusion it is weighted? or that I spin it cleverly? After how long a sequence of heads would you feel justified in accusing me? Or suppose in another set of 12 throws, you count 6 heads, 6 tails and think "That's more like it", at which point I exclaim: "Look! That's most unusual! lH, IT, 2H, 2T, 3H, 3T ... " (Read aloud, it's hypnotic.) Usual in one sense, unusual in another? How unusual? Is either reaction trustable? Is your own head swimming? No wonder! But that's the way people use words, and I think you see this will get us no place. We need in each case some very clear background against which the judgment more or less usual is made. We may then succeed in pulling out some measure for how unusual, and we might in the doing see possible meanings for "add the measures" and "multiply the measures." It is not surprising that the first successes are due to very great men and come from very simple situations. Here is the earliest beautiful example I know.
Early in the 1600's a gambler discussed with Galileo an apparent paradox. In scores with three dice, the score 10 can be made up in certain ways; e.g. from 1, 3, 6, or 2, 2, 6. He exhibited the different sets for score 9 and for score 10, six sets in each case. Yet as an experienced gambler he knew that he should favour 10 over 9. Note how sharply the problem emerges. He has an idea of the situation - it is a pure idea on how numbers add up; he has kept careful track of the way dice fall. His analysis does not agree with his experience, and something must give.
Galileo made a fresh attack. He wrote out triples in a careful pattern: A first block 111, 112, up to 116; then 121, 122 up to 126, and finally 661, 662 up to 666. The system make sure none are missed; in the first row all blocks begin with 1, in the last row all begin with 6. Then they ticked off the total score for each triple. Twenty-seven give score 10, but only twenty-five give score 9. This was not a simplified attack, adopted to convince a tyro; Galileo kept the page among his papers. Nothing further came of it, since the conflict of idea vs. experience had been resolved. Analysis and observation both favoured ten over nine.
If any of you, remembering high school algebra, has condescended a little to my gambler, murmuring "Permutations, not combinations", I beg of you, NO! Forget the later jargon. We are after something at once more innocent and more penetrating. We can see here two safeguards important in every probability attack:
1) For Galileo, 2 + 3 + 5 = 10 is a valid start but not an adequate description of the event being judged. (With coloured dice, the eye distinguishes between red 2, white 3, green 5 and R3, W2, G5.) He does not argue; he tabulates systematically and counts.
2) Comparing simple counts, 25 vs. 27, tacitly says that these new triples are given equal weight in the argument. Formulae later developed shorten the counting, but they are useful only after a primary analysis into cases has been made. The nobleman's careful breakdown could also be made to serve as conceptual background, if his triples are given different weights. The full scheme shows 235 in six places, but 333 occurs just once.
A little later, another gambler brought a sharp clear question on a similar conflict to the mathematician and philosopher Blaise Pascal. At the court, even stakes were bet for-against at least one 6 in four throws of a die, or for-against at least one double 6 in twenty-four throws of two dice. The second game often had to be terminated before it was decided, and the Chevalier de Mere really asked how to fix an equitable division of stake. But he stated frankly that his experience showed it advantageous to bet for in the first game and against in the second, and that arithmetic was a deceptive lead here. (His was!) Pascal was no gambler, but he was a very penetrating mathematician. He saw that division of stake called for very clear ideas on the possible way the game might finish after a stated moment, for impeccable logic in weighting these, and for efficient shortcuts to computation. Galileo's direct accurate enumeration of triples becomes massive even for four throws; it's almost hopeless for sets of twenty-four. The long correspondence between Pascal and Pierre Fermat, Toulouse lawyer and mathematician by avocation, started the enterprise from which our modern probability measures developed.
Post facto, we see that the successful start is made with simple clear-cut situations where we can line up an exhaustive set of outcomes accepted as "equally likely", and judged Yes-No against some clear simple criterion. For example, twenty-five of Galileo's triples gave score 9; the rest did not. The proportion of total i.e. 25/ 216 is a natural suggestion as a measure for likelihood of that score, but a number is of limited use unless one can do arithmetic with it. The first test of its power as a measure comes when we ask: Do certain specific combinations of event correspond to adding the numbers? to multiplying the numbers? In these first studies, each man worked alone, analysing on his own principles, devising his own short-cuts, and comparing only when the final number asked for had been computed. It is in one of these climaxes to an intricate analysis that Pascal wrote his friend: "Ah! I see that the truth is the same in Toulouse as in Paris!" (And, by the way, the numbers they reached agreed with the gambler's experience.)
First attacks on subtle concepts are almost always lengthy and difficult to follow. Formal mathematics comes about when, as a result of the mind's grapple with ideas, a powerful thought process is routinised so the mind is freed to move ahead in invention or to exercise judgment in the proper use of the new acquisition. The Swiss Bernoulli's, great Ars Conjectandi formalised the games-of-chance start as repeated trials of a simple event with an assigned probability number; De Moivre, Frenchman living in England, made a brilliant advance in evasion of laborious computation; and with these two accomplishments, the mathematical machinery is ready to attack, e.g., the telephone exchange problem - provided one can state it in the language of repeated trials. We want to know a likelihood of simultaneous calls. Clearly this depends on the heaviness of the traffic, and we would be wise to take care of the worst period. We could fit the classical method to this situation if we could think of each call as an event with a fixed probability number, but where are we to get that number? Our start came from an idea, matched against what happened; here, we must use the situation to form an idea. That measure can come only from the particular conditions, from some information or estimate such as average length of call over a peak hour. Suppose that average were two minutes; then we could attach the probability number , and the classical method will show that 87 lines will supply the required level of service.
In summary: I stress the deliberate attack through simplicity, by men who had the insight to see that the questions were deeper than they seemed. After that, there are interludes where the thought runs free of check against immediate answers sought; there are stages where the mass situation itself supplies a new element. Both sides contribute; each is necessary; neither predominates. Pure idea, yes; abstract and systematised, yes; but rooted in careful analysis of familiar notions, and like the giant Antaeus, stronger every time it is knocked down to touch the earth.
As the theory develops, the simple vocabulary of the game is carried over; don't be misled by that. It is naiveté, if not presumption, to read this figurative language at face value and decide it sidesteps reality! On the other hand, don't think mathematics proves anything about the real world. It can show what must be taken into account; it can weigh evidence, but nothing can take the weight of judging off of you. (In 1948, after the famous election poll failure, Gallup disarmingly said, "The trouble was not with our basic method - it was just our own stupidity.") Far from being surprised at failures, I more often feel with Einstein: "The most incomprehensible thing about the world is that it is comprehensible." And how? By what endeavour of man?
Not long ago, I read in a review of Proust: "The artist is the repository of a truth which is two-fold, for it 'rhymes' both with the world and with the inner order to which the artist refers." Mathematics has this "two-fold rhyming" character, as its servers know. Most mathematicians are craftsmen. Like other workers, they ply their trade; they push the marks around by rule, watch definitions and whittle careful sentences. But once in so often they are faced by just such moments of wonder as Einstein's. These systems, apparently narrow and restricted, special and over-simplified but exigent and disciplined, do serve to strengthen and deepen our comprehension of the almost overwhelming world of experience, precisely because they have been made with freedom to go where the thought demands. That is what the mind is like at its best.
Last Updated December 2021