In Quotes

I've spent a lot of time recently immersed in books on spectral and pseudospectral methods and related topics (you can think of these methods as numerical tools based on generalizations of Fourier expansions -- and yes, that includes expansions in terms of Bessel functions!). There are technical reasons why I've been reading about -- and coding -- spectral methods recently, which I might write about at some other time. But right now, I want to mention a particular book, Chebyshev and Fourier Spectral Methods by J.P. Boyd.

Boyd's book is a gem. It's available as a PDF file from his web site; if you're interested in numerical ODE/PDE solvers or approximation theory, I recommend downloading it and skimming the table of contents. If you have only the vaguest interest in numerical mathematics, I still recommend that you download it and read the preface (to the first edition); I agree wholeheartedly with the sentiments expressed therein.

One of the things I like about Boyd's book is the quotes that appear at the beginning of each chapter. For whatever reason, such introductory quotes are common in books (and some papers!) on numerics. Several of the quotes looked familiar, and upon a little poking around, I discovered that many were also chapter quotes in Men of Mathematics -- one of my all-time favorite books, as I've mentioned a time or twelve.

So I've decided to collect a few favorite quotes of my own. I've recently spent some time re-reading Poincare's popular works, so the selection is biased; other quotes are taken from Chebyshev and Fourier Spectral Methods (Boyd), The Character of Physical Law (Feynman), Men of Mathematics (Bell), Galileo's Commandment (ed. Bolles), German Essays on Science in the 20th Century (ed. Schirmacher), and The Force of Symmetry (Icke). I'd thought about organizing these quotes under different headings, but that would destroy half the fun.

And here let me insert a parenthesis to insist on the importance of written exercises. Compositions in writing are perhaps not given sufficient prominence in certain examinations. In the Ecole Polytechnique, for instance, I am told that the insistence on such compositions would close the door to very good pupils who know their subject and understand it very well, and yet are incapable of applying it in the smallest degree. I said just above that the word understand has several meanings. Such pupils only understand in the first sense of the word, and we have just seen that this is not sufficient to make either an engineer or a geometrician. Well, since we have to make a choice, I prefer to choose those who understand thoroughly.

-- H. Poincare

A mathematician who is not also something of a poet will never be a complete mathematician.

-- K. Weierstrass

In my opinion a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy -- an opinion, moreover, which has been expressed by many philosophers.

-- H. Lebesgue

It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.

-- A. N. Whitehead

Six months in the lab can save you a day in the library.

-- A. Migliori

It is the increasingly pronounced tendency of modern analysis to substitute ideas for calculation; nevertheless there are certain branches of mathematics where calculation conserves its rights.

-- P.G.L. Dirichlet

Talk with M. Hermite: he never evokes a concrete image; yet you soon perceive that the most abstract entities are for him like living creatures.

-- H. Poincare

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.

-- H. Poincare

History shows that those heads of empires who have encouraged the cultivation of mathematics, the common source of all the exact sciences, are also those whose reigns have been the most brilliant and whose glory is the most durable.

-- M. Chasles

Nothing requires a rarer intellectual heroism than willingness to see one's equation written out.

-- Santayana

He studied and nearly mastered the six books of Euclid since he was a member of Congress.
   He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pollow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.

-- A. Lincoln (Short Autobiography)

In the terminology which you graciously ascribe to me, we might say that the atmosphere is a musical instrument on which one can play many tunes. High notes are sound waves, low notes are long inertial [Rossby] waves, and nature is a musician more of the Beethoven than of Chopin type. He much prefers the low notes and only occasionally plays arpeggios in the treble and then only with a light hand. The oceans and the continents are the elephants in Saint-Saens' animal suite, marching in a slow cumbrous rhythm, one step every day or so. Of course there are overtones: sound waves, billow clouds (gravity waves), inertial oscillations, etc., but these are unimportant and are heard only at NYU and MIT.

-- J. Charney

Physics is beautiful. It makes me sad beyond words to know that so many people think of the physical sciences as barren, boring, bone-dry. Not so: when you lie outside in the grass on a clear dark night and look up at the stars, what you see is splendid. It is also physics. Understanding can lift you off the Earth, safer and faster and further than any rocket. The mind can travel among the stars, even enter them to see what causes those fires inside. To the beauty of seeing, we can add the beauty of understanding. And there is another level of beauty beyond that: the beauty of discovery, of creation, of doing physics. This beauty I love the most.

-- V. Icke

To summarize, I would use the words of Jeans, who said that the Great Architect seems to be a mathematician. To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and peoplle who have not had this experience of understanding mathematics well enough to appreciate nature once.
   It is too bad that it has to be mathematics, and that mathematics is hard for some people. It is reputed -- I do not know if it is true -- that when one of the kings was trying to learn geometry from Euclid, he complained that it was difficult. And Euclid said, There is no royal road to geometry. And there is no royal road. Physicists cannot make a conversion to any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in. She offers her information only in the one form; we are not so unhumble as to demand that she change before we pay any attention.
   All the intellectual arguments that you can make will not communicate to deaf ears what the experience of music really is. In the same way all the intellectual arguments in the world will not convey an understanding of nature to those of the other culture. Philosophers may try to teach you by telling you qualitatively about nature. I am trying to describe her. But it is not getting across because it is impossible. Perhaps it is because their horizons are limited in this way that some people are able to imagine that the center of the universe is man.

-- R. Feynman

Play, art and science are the spheres of human activity where action and aim are not as a rule determined by the aims imposed by the necessities of life; and even in the exceptional instances where this is the case, the creative artist or the investigating scientist soon forgets this fact -- as indeed they must forget it if their work is to prosper.

-- E. Schrodinger

You have doubtless often been asked of what good are mathematics and whether these delicate constructions entirely mind-made are not artificial and born of our caprice.
   Among those who put this question I should make a distinction; practical people ask of us only the means of money-making. These merit no reply; rather would it be proper to ask of them what is the good of accumulating so much wealth and whether to get time to acquire it, we are to neglect art and science, which alone give us souls capable of enjoying it, and for life's sake to sacrifice all reasons for living.
   Besides, a science made solely in view of applications is impossible; truths are fecund only if bound together. If we devote ourselves solely to those truths whence we expect an immediate result, the intermediary links are wanting and there will no longer be a chain.

-- H. Poincare

Science knows only one commandment: contribute to science.

-- B. Brecht (The Life of Galileo)

It is only through science and art that civilization is of value. Some have wondered at the formula: science for its own sake, and yet it is as good as life for its own sake, if life is only misery; and even as happiness for its own sake, if we do not believe that all pleasures are of the same quality, if we do not wish to admit that the goal of civilization is to furnish alcohol to people who love to drink.

-- H. Poincare

Others will always ask themselves what use it is. They will not have understood, unless they find around them, in practice or in nature, the object of such and such a mathematical notion. Under each word they wish to put a sensible image; the definition must call up this image, and at each stage of the demonstration they must see it being transformed and evolved. On this condition only will they understand and retain what they have understood. These often deceive themselves: they do not listen to the reasoning, they look at the figures; they imagine that they have understood when they have only seen.

-- H. Poincare

This vain presumption of understanding everything can have no other basis than never understanding anything. For anyone who had experienced just once the perfect understanding of one single thing, and had truly tasted how knowledge is accomplished, would recognize that infinity of other truths of which he understands nothing.

-- Galileo (The Two Chief World Systems)

Would a naturalist imagine that he had an adequate knowledge of the elephant if he had never studied the animal except through a microscope?
   It is the same in mathematics. When a logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality; that indefinable something that constitutes the unity of the demonstration will still excape him completely.
   What good is it to admire the mason's work in the edifices erected by great architects, if we cannot understand the general plan of the master? Now pure logic cannot give us this view of the whole; it is to intuition we must look for it.

-- H. Poincare

What is it about nature that lets this happen, that it is possible to guess from one part what the rest is going to do? That is an unscientific question; I do not know how to answer it, and therefore I am going to give an unscientific answer. I think it is because nature has a simplicity and therefore a great beauty.

-- R. Feynman

Must we therefore say that science should be abandoned, and morality alone be studied? Does anyone suppose that moralists themselves are entirely above reproach when they have come down from the pulpit?

-- H. Poincare

I have heard myself accused of being an opponent, an enemy of mathematics, which no one can value more highly than I, for it accomplishes the very thing whose achievement has been denied me.

-- Goethe

I do not know.

-- J.L. Lagrange