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Ten Lessons I wish I had been Taught
Gian-Carlo Rota
MIT, April 20 , 1996 on the occasion of the Rotafest
Allow me to begin by allaying one of your worries. I will not spend
the next half hour thanking you for participating in this conference, or
for your taking time away from work to travel to Cambridge.
And to allay another of your probable worries, let me add that you are
not about to be subjected to a recollection of past events similar to the
ones I've been publishing for some years, with a straight face and an occasional
embellishment of reality.
Having discarded these two choices for this talk, I was left without
a title. Luckily I remembered an MIT colloquium that took place in the
late fifties; it was one of the first I attended at MIT. The speaker was
Eugenio Calabi. Sitting in the front row of the audience were Norbert Wiener,
asleep as usual until the time came to applaud, and Dirk Struik who had
been one of Calabi's teachers when Calabi was an undergraduate at MIT in
the forties. The subject of the lecture was beyond my competence. After
the first five minutes I was completely lost. At the end of the lecture,
an arcane dialogue took place between the speaker and some members of the
audience, Ambrose and Singer if I remember correctly. There followed a
period of tense silence. Professor Struik broke the ice. He raised his
hand and said:
"Give us something to take home!" Calabi obliged,
and in the next five minutes he explained in beautiful simple terms the
gist of his lecture. Everybody filed out with a feeling of satisfaction.
Dirk Struik was right: a speaker should try to give his audience something
they can take home. But what? I have been collecting some random bits of
advice that I keep repeating to myself, do's and don'ts of which I have
been and will always be guilty. Some of you have been exposed to one or
more of these tidbits. Collecting these items and presenting them in one
speech may be one of the less obnoxious among options of equal presumptuousness.
The advice we give others is the advice that we ourselves need. Since it
is too late for me to learn these lessons, I will discharge my unfulfilled
duty by dishing them out to you. They will be stated in order of increasing
controversiality.
-
Lecturing
-
Blackboard Technique
-
Publish the same results several times.
-
You are more likely to be remembered
by your expository work.
-
Every mathematician has only a few tricks.
-
Do not worry about your mistakes.
-
Use the Feynman method.
-
Give lavish acknowledgments.
-
Write informative introductions
-
Be prepared for old age.
1 Lecturing
top
The following four requirements of a good lecture do not seem to be altogether
obvious, judging from the mathematics lectures I have been listening to
for the past forty-six years.
a. Every lecture should make only one main point The German philosopher
G. W. F. Hegel wrote that any philosopher who uses the word "and" too often
cannot be a good philosopher. I think he was right, at least insofar as
lecturing goes. Every lecture should state one main point and repeat it
over and over, like a theme with variations. An audience is like a herd
of cows, moving slowly in the direction they are being driven towards.
If we make one point, we have a good chance that the audience will take
the right direction; if we make several points, then the cows will scatter
all over the field. The audience will lose interest and everyone will go
back to the thoughts they interrupted in order to come to our lecture.
b. Never run overtime Running overtime is the one unforgivable
error a lecturer can make. After fifty minutes (one microcentury as von
Neumann used to say) everybody's attention will turn elsewhere even if
we are trying to prove the Riemann hypothesis. One minute overtime can
destroy the best of lectures.
c. Relate to your audience As you enter the lecture hall, try
to spot someone in the audience with whose work you have some familiarity.
Quickly rearrange your presentation so as to manage to mention some of
that person's work. In this way, you will guarantee that at least one person
will follow with rapt attention, and you will make a friend to boot.
Everyone in the audience has come to listen to your lecture with the
secret hope of hearing their work mentioned.
d. Give them something to take home It is not easy to follow
Professor Struik's advice. It is easier to state what features of a lecture
the audience will always remember, and the answer is not pretty. I often
meet, in airports, in the street and occasionally in embarrassing situations,
MIT alumni who have taken one or more courses from me. Most of the time
they admit that they have forgotten the subject of the course, and all
the mathematics I thought I had taught them. However, they will gladly
recall some joke, some anecdote, some quirk, some side remark, or some
mistake I made.
2 Blackboard Technique
top
Two points.
a. Make sure the blackboard is spotless It is particularly important
to erase those distracting whirls that are left when we run the eraser
over the blackboard in a non uniform fashion.
By starting with a spotless blackboard, you will subtly convey the impression
that the lecture they are about to hear is equally spotless.
b. Start writing on the top left hand corner What we write on
the blackboard should correspond to what we want an attentive listener
to take down in his notebook. It is preferable to write slowly and in a
large handwriting, with no abbreviations. Those members of the audience
who are taking notes are doing us a favor, and it is up to us to help them
with their copying. When slides are used instead of the blackboard, the
speaker should spend some time explaining each slide, preferably by adding
sentences that are inessential, repetitive or superfluous, so as to allow
any member of the audience time to copy our slide. We all fall prey to
the illusion that a listener will find the time to read the copy of the
slides we hand them after the lecture. This is wishful thinking.
3 Publish the same result several times
top
After getting my degree, I worked for a few years in functional analysis.
I bought a copy of Frederick Riesz' Collected Papers as soon as the big
thick heavy oversize volume was published. However, as I began to leaf
through, I could not help but notice that the pages were extra thick, almost
like cardboard. Strangely, each of Riesz' publications had been reset in
exceptionally large type. I was fond of Riesz' papers, which were invariably
beautifully written and gave the reader a feeling of definitiveness.
As I looked through his Collected Papers however, another picture emerged.
The editors had gone out of their way to publish every little scrap Riesz
had ever published. It was clear that Riesz' publications were few. What
is more surprising is that the papers had been published several times.
Riesz would publish the first rough version of an idea in some obscure
Hungarian journal. A few years later, he would send a series of notes to
the French Academy's Comptes Rendus in which the same material was further
elaborated. A few more years would pass, and he would publish the definitive
paper, either in French or in English. Adam Koranyi, who took courses with
Frederick Riesz, told me that Riesz would lecture on the same subject year
after year, while meditating on the definitive version to be written. No
wonder the final version was perfect.
Riesz' example is worth following. The mathematical community is split
into small groups, each one with its own customs, notation and terminology.
It may soon be indispensable to present the same result in several versions,
each one accessible to a specific group; the price one might have to pay
otherwise is to have our work rediscovered by someone who uses a different
language and notation, and who will rightly claim it as his own.
4 You are more likely to be remembered by your expository work
top
Let us look at two examples, beginning with Hilbert. When we think of Hilbert,
we think of a few of his great theorems, like his basis theorem. But Hilbert's
name is more often remembered for his work in number theory, his Zahlbericht,
his book Foundations of Geometry and for his text on integral equations.
The term "Hilbertspace" was introduced by Stone and von Neumann in recognition
of Hilbert's textbook on integral equations, in which the word "spectrum"
was first defined at least twenty years before the discovery of quantum
mechanics. Hilbert's textbook on integral equations is in large part expository,
leaning on the work of Hellinger and several other mathematicians whose
names are now forgotten.
Similarly, Hilbert's Foundations of Geometry, the book that made Hilbert's
name a household word among mathematicians, contains little original work,
and reaps the harvest of the work of several geometers, such as Kohn, Schur
(not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri
and several other Italians.
Again, Hilbert's Zahlbericht, a fundamental contribution that revolutionized
the field of number theory, was originally a survey that Hilbert was commissioned
to write for publication in the Bulletin of the German Mathematical Society.
William Feller is another example. Feller is remembered as the author
of the most successful treatise on probability ever written. Few probabilists
of our day are able to cite more than a couple of Feller's research papers;
most mathematicians are not even aware that Feller had a previous life
in convex geometry.
Allow me to digress with a personal reminiscence. I sometimes publish
in a branch of philosophy called phenomenology. After publishing my first
paper in this subject, I felt deeply hurt when, at a meeting of the Society
for Phenomenology and Existential Philosophy, I was rudely told in no uncertain
terms that everything I wrote in my paper was well known. This scenario
occurred more than once, and I was eventually forced to reconsider my publishing
standards in phenomenology.
It so happens that the fundamental treatises of phenomenology are written
in thick, heavy philosophical German. Tradition demands that no examples
ever be given of what one is talking about. One day I decided, not without
serious misgivings, to publish a paper that was essentially an updating
of some paragraphs from a book by Edmund Husserl, with a few examples added.
While I was waiting for the worst at the next meeting of the Society for
Phenomenology and Existential Philosophy, a prominent phenomenalist rushed
towards me with a smile on his face. He was full of praise for my paper,
and he strongly encouraged me to further develop the novel and original
ideas presented in it.
5 Every mathematician has only a few tricks
top
A long time ago an older and well known number theorist made some disparaging
remarks about Paul Erdos' work. You admire contributions to mathematics
as much as I do, and I felt annoyed when the older mathematician flatly
and definitively stated that all of Erdos' work could be reduced to a few
tricks which Erdos repeatedly relied on in his proofs. What the number
theorist did not realize is that other mathematicians, even the very best,
also rely on a few tricks which they use over and over. Take Hilbert. The
second volume of Hilbert's collected papers contains Hilbert's papers in
invariant theory. I have made a point of reading some of these papers with
care. It is sad to note that some of Hilbert's beautiful results have been
completely forgotten. But on reading the proofs of Hilbert's striking and
deep theorems in invariant theory, it was surprising to verify that Hilbert's
proofs relied on the same few tricks. Even Hilbert had only a few tricks!
6 Do not worry about your mistakes top
Once more let me begin with Hilbert. When the Germans were planning to
publish Hilbert's collected papers and to present him with a set on the
occasion of one of his later birthdays, they realized that they could not
publish the papers in their original versions because they were full of
errors, some of them quite serious. Thereupon they hired a young unemployed
mathematician, Olga Taussky-Todd, to go over Hilbert's papers and correct
all mistakes. Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems.
There was one exception, a paper Hilbert wrote in his old age, which could
not be fixed; it was a purported proof of the continuum hypothesis, you
will find it in a volume of the Mathematische Annalen of the early thirties.
At last, on Hilbert's birthday, a freshly printed set of Hilbert's collected
papers was presented to the Geheimrat. Hilbert leafed through them carefully
and did not notice anything.
Now let us shift to the other end of the spectrum, and allow me to relate
another personal anecdote. In the summer of 1979, while attending a philosophy
meeting in Pittsburgh, I was struck with a case of detached retinas. Thanks
to Joni's prompt intervention, I managed to be operated on in the nick
of time and my eyesight was saved.
On the morning after the operation, while I was lying on a hospital
bed with my eyes bandaged, Joni dropped in to visit. Since I was to remain
in that Pittsburgh hospital for at least a week, we decided to write a
paper. Joni fished a manuscript out of my suitcase, and I mentioned to
her that the text had a few mistakes which she could help me fix.
There followed twenty minutes of silence while she went through the
draft. "Why, it is all wrong!" she finally remarked in her youthful
voice. She was right. Every statement in the manuscript had something wrong.
Nevertheless, after laboring for a while, she managed to correct every
mistake, and the paper was eventually published.
There are two kinds of mistakes. There are fatal mistakes that destroy
a theory; but there are also contingent ones, which are useful in testing
the stability of a theory.
7 Use the Feynman method top
Richard Feynman was fond of giving the following advice on how to be a
genius. You have to keep a dozen of your favorite problems constantly present
in your mind, although by and large they will lay in a dormant state. Every
time you hear or read a new trick or a new result, test it against each
of your twelve problems to see whether it helps. Every once in a while
there will be a hit, and people will say: "How did he do it? He must
be a genius!"
8 Give lavish acknowledgments top
I have always felt miffed after reading a paper in which I felt I was not
being given proper credit, and it is safe to conjecture that the same happens
to everyone else. One day, I tried an experiment. After writing a rather
long
paper, I began to draft a thorough bibliography. On the spur of the moment,
I decided to cite a few papers which had nothing whatsoever to do with
the content of my paper, to see what might happen.
Somewhat to my surprise, I received letters from two of the authors
whose papers I believed were irrelevant to my article. Both letters were
written in an emotionally charged tone. Each of the authors warmly congratulated
me for being the first to acknowledge their contribution to the field.
9 Write informative introductions top
Nowadays, reading a mathematics paper from top to bottom is a rare event.
If we wish our paper to be read, we had better provide our prospective
readers with strong motivation to do so. A lengthy introduction, summarizing
the history of the subject, giving everybody his due, and perhaps enticingly
outlining the content of the paper in a discursive manner, will go some
of the way towards getting us a couple of readers.
As the editor of the journal Advances in Mathematics, I have often sent
submitted papers back to the authors with the recommendation that they
lengthen their introduction. On occasion I received by return mail a message
from the author, stating that the same paper had been previously rejected
by Annals of Mathematics because the introduction was already too long.
10 Be prepared for old age top
My late friend Stan Ulam used to remark that his life was sharply divided
into two halves. In the first half, he was always the youngest person in
the group; in the second half, he was always the oldest. There was no transitional
period.
I now realize how right he was. The etiquette of old age does not seem
to have been written up, and we have to learn it the hard way. It depends
on a basic realization, which takes time to adjust to. You must realize
that, after reaching a certain age, you are no longer viewed as a person.
You become an institution, and you are treated the way institutions are
treated. You are expected to behave like a piece of period furniture, an
architectural landmark, or an incunabulum.
It matters little whether you keep publishing or not. If your papers
are no good, they will say, "What did you expect? He is a fixture!"
and if an occasional paper of yours is found to be interesting, they will
say, "What did you expect? He has been working at this all his life!"
The only sensible response is to enjoy playing your newly-found role as
an institution.
He interrupted her. Close at hand is a stable where two beautiful ponies are kept. They are snowy white, and are consecrated to the goddess Ku-wanon, the deity of mercy, who is the presiding genius of the temple. They are in the care of a young girl, and it is considered a pious duty to feed them. Pease and beans are for sale outside, and many devotees contribute a few cash for the benefit of the sacred animals. If the poor beasts should eat a quarter of what is offered to them, or, rather, of what is paid for, they would soon die of overfeeding. It is shrewdly suspected that the grain is sold many times over, in consequence of a collusion between the dealers and the keeper of the horses. At all events, the health of the animals is regarded, and it would never do to give them all that is presented. On their return from the garden they stopped at a place where eggs are hatched by artificial heat. They are placed over brick ovens or furnaces, where a gentle heat is kept up, and a man is constantly on watch to see that the fire neither burns too rapidly nor too slowly. A great heat would kill the vitality of the egg by baking it, while if the temperature falls below a certain point, the hatching process does not go on. When the little chicks appear, they are placed under the care of an artificial mother, which consists of a bed of soft down and feathers, with a cover three or four inches above it. This cover has strips of down hanging from it, and touching the bed below, and the chickens nestle there quite safe from outside cold. The Chinese have practised this artificial hatching and rearing for thousands of years, and relieved the hens of a great deal of the monotony of life. He would not have it in the scabbard, and when I laid it naked in his hand he kissed the hilt. Charlotte sent Gholson for Ned Ferry. Glancing from the window, I noticed that for some better convenience our scouts had left the grove, and the prisoners had been marched in and huddled close to the veranda-steps, under their heavy marching-guard of Louisianians. One of the blue-coats called up to me softly: "Dying--really?" He turned to his fellows--"Boys, Captain's dying." Assuming an air of having forgotten all about Dick¡¯s rhyme, he went to his place in the seat behind Jeff and the instant his safety belt was snapped Jeff signaled to a farmer who had come over to investigate and satisfy himself that the airplane had legitimate business there; the farmer kicked the stones used as chocks from under the landing tires and Jeff opened up the throttle. ¡°Yes,¡± Dick supplemented Larry¡¯s new point. ¡°Another thing, Sandy, that doesn¡¯t explain why he¡¯d take three boys and fly a ship he could never use on water¡ªwith an amphibian right here.¡± Should you leave me too, O my faithless ladie? And years of remorse and despair been your fate, That night was a purging. From thenceforward Reuben was to press on straight to his goal, with no more slackenings or diversions. "Is that you, Robin?" said a soft voice; and a female face was seen peeping half way down the stairs. HoMElãñÔóÂÜÀ³ó
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