A Scientist's Comfort Zone

Scientists usually have better things to talk about than how hard it is doing research; and popular accounts of science, even more, concentrate on the glamour of achievement and dismiss the months of self-doubt and confusion: "suddenly she made another stunning and serendipitous discovery." Occasionally, though, even really top-class scientists confess to spending some time on the same level as the rest of us; and I find this humility comforting and at the same time inspiring.

Please email me with more quotes!

Hermann von Helmholtz

"I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty."

Posted here by goodreads.com.

Alexander Grothendieck

Since then I've had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group who were more brilliant, much more 'gifted' than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

Translated from Récoltes et Semailles. I found this in a post at the blog lesswrong.com.

Jacques Hadamard

For instance, there exists a theory, the theory of groups, the importance of which, in our science, grew increasingly for more than one century, especially since the work of Sophus Lie at the end of the nineteenth century. Some mathematicians, especially contemporary ones, have improved it by most beautiful discoveries. Some others—I confess that I belong to the latter category—though being eventually able to use it for simple applications, feel insuperable difficulty in mastering more than a rather elementary and superficial knowledge of it. Psychological reasons for that difference, which seems to me incontestable, would be interesting to find.

In Jacques Hadamard, The Mathematician's Mind, Princeton, 1945 (but I was led to it by this interesting entry on Peter Cameron's blog).

Kenneth A. Ribet

Ribet: At the end of my graduate career, I was very surprised that people were actually offering me jobs. I thought, "Who cares? I'm doing this thing with abelian varieties, it's not likely to attract any attention."
Notices: So everybody goes through those doubts.
Ribet: Absolutely. Doing mathematics is just a permanent situation of being wracked with self-doubt. Where is my next theorem? Am I still any good? Is this proof really right? Will anyone care about it?
Notices: Those are comforting words for the young people entering the profession. They know they are not alone.
Ribet: You bet!

From "Interview with New AMS President Kenneth A. Ribet", Notices of the AMS, vol. 64, no. 3, March 2017; online here.

Michael Atiyah

In 2001, [Ed Witten] invited me to Caltech, where he was a visiting professor. I felt like a graduate student again. Every morning I would walk into the department, I'd go to see Witten, and we'd talk for an hour or so. He'd give me my homework. I'd go away and spend the next 23 hours trying to catch up. Meanwhile, he'd go off and do half a dozen other things. We had a very intense collaboration. It was an incredible experience because it was like working with a brilliant supervisor. I mean, he knew all the answers before I got them. If we ever argued, he was right and I was wrong. It was embarrassing!

In an interview with Siobhan Roberts for Quanta Magazine

Gian-Carlo Rota

There are numerous moments in Ten Lessons I wish I had been Taught to cheer you up and inspire you simultaneously!

John Wallis

Quamquam enim hanc spes non exigua visa est affulsisse, lubricus tamen quem prae manibus
habemus Proteus tam hic quam superius non raro elapsus, spem fefellit.
"Although no small hope seemed to shine, what we have in hand is slippery, like Proteus, who
in the same way, often escaped, and disappointed hope."

On squaring the circle. The translation is in the first of two lovely articles by Jacqueline A. Stedall here.

Piet Hein

Problems worthy of attack,
Prove their worth by fighting back!

I saw this on J.D. Phillips' research page at homepage.

Nathaniel Bowditch

I never came across one of Laplace's "Thus it plainly appears" without feeling sure that I have many hours of hard work before me.

This accompanies the MacTutor Archive entry for Bowditch

Cornelius J. Everett

Quoted from G. T. Q. Hoare "Stanisław Ulam 1909–1984", Mathematical Gazette, 83, no. 496, 1999, 10–24.

Stanisław Ulam

Ulam never concealed his dependence on Everett: "I had some general, sometimes only vague, ideas. Everett supplied the rigor, the ingenuities and the details of the proof, and final constructions."

From p. 343 of Norman Macrea's John von Neumann, American Mathematical Society, 1999.

From an H.E. Robbins' review of Ulam's autobiography.

Carol Robinson

When I look at where I've got to I think I'm just an ordinary person in the street. I've been very lucky, I've not given up, I'm quite determined and competitive, but I'm not a genius or anything! I don't think people should think that they can't do something because they haven't got an enormous IQ. It's how you apply yourself.

From an interview on a website called moretolifethanshoes.com, but this website has turned into a book and a facebook page and I'm not sure if or where the interview features. There's another good interview with Robinson, full of humility and wisdom, here.

Stephen Smale

At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. And that background consists of many threads, some strong, some weak. My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematians to understand an argument. The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understanding, not before.

I read this at math.berkeley.edu/~smale/biblio/chaos.ps (postscript file, page 5, there is a pdf copy here).

That I have been able to accomplish anything in mathematics is really due to the fact that I have always found it so difficult. When I read, or when I am told about something, it nearly always seems so difficult, and practically impossible to understand, and then I cannot help wondering if it might not be simpler. And on several occasions it has turned that it really was more simple!

Found in The Honours Class: Hilbert's Problems and their Solvers by Benjamin Yandell, AK Peters, Natick, MA, 2002.

On the life sciences: "je dirais que c'est un salon superbe tout resplendissant de lumière, dans lequel on ne peut parvenir qu'en passant par une lonque et affreuse cuisine"

From Introduction à l'étude de la médecine expérimentale. Source: Claude Bernard, " Introduction à l'étude ..." (click 'read online'; the above sentence is on page 12, para. 2)

I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.

Found on Donald Simanek's quotations page

We are in the ordinary position of scientists of having to be content with piecemeal improvements: we can make several things clearer, but we cannot make anything clear.

Found on Donald Simanek's quotations page

Every honest researcher I know admits he's just a professional amateur. He's doing whatever he's doing for the first time. That makes him an amateur. He has sense enough to know that he's going to have a lot of trouble, so that makes him a professional.

Found on Donald Simanek's quotations page

It was absolutely marvelous working for Pauli. You could ask him anything. There was no worry that he would think a particular question was stupid, since he thought all questions were stupid.

Found on Donald Simanek's quotations page

Physics is very muddled again at the moment; it is much too hard for me anyway, and I wish I were a movie comedian or something like that and had never heard anything about physics!

Quoted from a letter to R. Kronig, 25 May 1925, in Donald Simanek's quotations page

Irving Kaplansky

We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.

from Paul Halmos: Celebrating 50 Years of Mathematics, I found the quote in the Kaplansky entry at MacTutor.

Mathematics is not a deductive science — that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

From I Want to be a Mathematician, Washington: MAA Spectrum, 1985 (Carrie Rutherford told me about it.)

Eizabeth Scott in a tribute to Robinson, as quoted in the Association for Women in Mathematics article on Robinson.

I seem to have spent much more of my life not finding structures than finding them

Quoted in Georgina Ferry's Dorothy Hodgkin: A Life, Granta Books, 1999

Do not be afraid to skip equations (I do this frequently myself).

From the Preface to The Road to Reality, Vintage Books, London, 2005 (p. xix)

The subject of quantum gravity came up and Penrose and Feynman got into a heated argument. Penrose said, Feynman was so quick, he was usually about five steps ahead of me at any given point. Sometimes he didn't listen to what I was saying. The whole thing was mentally exhausting. I was completely drained at the end of the session. I have never encountered anyone so quick before. What Penrose and many other physicists didn't realize was the reason that accounted for Feynman's quickness on many matters in physics. Feynman thought about some of these areas in great depth and for long periods of time. A topic like quantum gravity would be one that Feynman had spent countless hours thinking about. It wasn't all off the cuff.

From Al Seckel on Feynman

In any thinking process there are moments when everything is going good and you've got wonderful ideas. Teaching is an interruption, and so it's the greatest pain in the neck in the world. And then there are the longer periods of time when not much is coming to you. You're not getting any ideas, and if you're doing nothing at all, it drives you nuts! You can't even say "I'm teaching my class."

From Feynman's "Surely You're Joking Mr. Feynman!" Adventures of a Curious Character, Bantam Books, New York, 1986. A slightly longer version can be read here.

Writing to his old friend Arthur Caley, Sylvester confessed, "I expect Poincarre [sic] tomorrow and he will have rooms in College. I rather dread the encounter as there is so little in the way of Mathematics upon which I can hope to talk to him!"

From J. Fauvel, R. Flood and R. Wilson, eds, Oxford Figures. Eight Hundred Years of the Mathematical Sciences, OUP, 2000

Being a great ditherer, I would frequently start an experiment far too late in the afternoon simply because I'd spent the whole day pondering exactly how to set it up.

In "Why won't the public put their faith in scientists?", Times Higher Education Supplement, 10/6/05

(he doesn't really belong here but I like this quote)

Ah, but a man's reach should exceed his grasp, Or what's a Heaven for?

From Andrea del Sarto, 1855 

This page is part of Robin Whitty's theoremoftheday site.

Robin Whitty, May 2021