This page collects some quotations that I like. It is updated from time to time. I try to clarify the source of each quotation. Before starting, remember that you never need quotations to justify your belief if you have one, and it is never a good idea to believe in something merely because some celebrity said so
(and it will be even worse if it turns out that she/he actually did not).
[中文] [English] [latest]
- Cum enim mundi universi fabrica sit perfectissima atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum, quin omnes mundi effectus ex causis finalibus ope methodi maximorum & minimorum aeque feliciter determinari queant, atque ex ipsis causis efficientibus. (For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves.)
— Leonhard Euler
“De Curvis Elasticis”, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Additamentum I (page 245), Apud Marcum Michaelem Bousquet & Socios, 1744
Translated from Latin to English by William A. Oldfather, C. A. Ellis, and Donald M. Brown, “Leonhard Euler's Elastic Curves”, Isis, 20(1):72–160, Nov. 1933
This statement (in German) is quoted by Professor Ya-xiang Yuan in “从瞎子爬山到优化方法”. It is also quoted by Martin Grötschel in Introduction (page 3) of Optimization Stories, an extra volume of Documenta Mathematica on the occasion of The 21st International Symposium on Mathematical Programming (ISMP 2012, Berlin, Aug. 19–24, 2012). The comment on this statement is
“Briefly and very freely translated: Nothing in the world takes place without optimization, and there is no doubt that all aspects of the world that have a rational basis can be explained by optimization methods. It is not so bad to hear such a statement from one of the greatest mathematicians of all time.”
- Why work on derivative-free optimization? Because the problems are important and cool.
— John E. Dennis, Jr.
In the beginning of Dennis’ talk
“Reasons to Study Derivative-Free Algorithms”,
given on July 24, 2013 (the last talk of morning session), at the conference Recent Advances on Optimization dedicated to Ph. L. Toint's 60th birthday, organized by CERFACS in Toulouse. Zaikun Zhang attended this talk.
- Wir müssen wissen, wir werden wissen! (We must know, we will know!)
— David Hilbert
This sentence is the concluding statement of the speech that Hilbert delivered during that annual meeting of the Society of German Natural Scientists and Physicians (Gesellschaft der Deutschen Naturforscher und Ärzte) on 8 September 1930.
Hilbert ended his speech by “Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und
überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus
gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die
Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heisse im Gegenteil
unsere Losung: Wir müssen wissen, Wir werden wissen!”
(We must not believe those, who today with philosophical bearing and a tone of
superiority prophesy the downfall of culture and accept the ignorabimus. For us there
is no ignorabimus, and in my opinion even none whatever in natural science. In place
of the foolish ignorabimus let stand our slogan: We must know, We will know!)
The radio broadcast and transcription (with English translation) of the speech are available online.
As a summary of Hilbert's beliefs on mathematics, this sentence was inscribed as the epitaph on his tomb in Göttingen.
In 1931, Kurt F. Gödel published his Incompleteness Theorems in the paper Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I (On Formally Undecidable Propositions of Principia Mathematica and Related Systems).
According to these theorems, roughly speaking, for any computable axiomatic system adequate to describe the arithmetic of the natural numbers, it holds that: If a axiomatic system is consistent, it cannot be complete; The consistency of axioms cannot be proved within their own system.
These theorems ended a half-century of attempts, beginning with the work of Friedrich Ludwig Gottlob Frege (see Die Grundlagen der Arithmetik, The Foundations of Arithmetic, 1884) and culminating in Principia Mathematica (by Alfred North Whitehead and Bertrand Arthur William Russell) and Hilbert's Formalism,
to find a set of axioms sufficient for all mathematics, concluding that some mathematical questions cannot be answered in the fashion that we would usually prefer.
- Die Wichtigkeit einer wissenschaftlichen Arbeit kann man daran messen, wieviele frühere Veröffentlichungen durch sie überflüssig werden. (One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.)
— David Hilbert
This statement is attributed to Hilbert without reliable source. See, for example, item 307° of [Howard W. Eves, Mathematical Circles Revisited: A Second Collection of Mathematical Stories and Anecdotes, Prindle, Weber & Schmidt, Inc., Boston, 1971] and [Markus Sigg, Humor in der Mathematik, in Lexikon der Mathematik: Band 2: Eig bis Inn, edited by Guido Walz, Springer Spektrum, 2017].
- L’analyse mathématique rapproche les phénomènes les plus divers, et découvre les analogies secrètes qui les unissent. (Mathematical analysis compares the most diverse phenomena and discovers the hidden analogies that unite them.)
— Jean-Baptiste Joseph Fourier
This statement comes from Fourier's book Théorie Analytique de la Chaleur (The Analytical Theory of Heat). In the Discours préliminaire (Preliminary discourse), Fourier wrote that “L’analyse mathématique est aussi étendue que la nature elle-même ; elle définit tous les rapports sensibles, mesure les temps, les espaces, les forces, les températures ; cette science difficile se forme avec lenteur, mais elle conserve tous les principes qu’elle a une fois acquis ; elle s’accroît et s’affermit sans cesse au milieu de tant de variations et d’erreurs de l’esprit humain. Son attribut principal est la clarté ; elle n’a point de signes pour exprimer les notions confuses. Elle rapproche les phénomènes les plus divers, et découvre les analogies secrètes qui les unissent”. (Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science
is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
Its chief attribute is clearness; it has no marks to express confused notions. It compares the most diverse phenomena and discovers the hidden analogies which unite them.)
Here, it seems that “mathematical analysis” means “analysis of mathematical kind” rather than the mathematical branch known as mathematical analysis. Therefore, one may paraphrase this statement as “La mathématique rapproche les phénomènes les plus divers, et découvre les analogies secrètes qui les unissent” (Mathematics compares the most diverse phenomena and discovers the hidden analogies that unite them).
- In many parts of mathematics a generalization is simpler and more incisive than its special parent.
— Paul Richard Halmos
In [How to talk Mathematics, Notices of the AMS (1974), 21:3, 155–158], Halmos suggested that “Be simple by being concrete. Listeners are prepared to accept unstated (but hinted) generalizations much more than they are able, on the spur of the moment, to decode a precisely stated abstraction and to re-invent the special cases that motivated it in the first place. Caution: being concrete should not lead to concentrate seeing the trees and not seeing the woods. In many parts of mathematics a generalization is simpler and more incisive than its special parent. (Examples: Artin's solution of Hilbert's 17th problem about definite forms via formally real fields; Gelfand's proof of Wiener's theorem about absolutely convergent Fourier series via Banach algebras). In such cases is always a concrete special case that is simpler than the seminal one and that illustrates the generalization with less fuss; the lecturer who knows his subject will explain the complicated special case, and the generalization, by discussing the simple cousin.”
- The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
— Paul Richard Halmos
This comment comes from Chapter 14 (“How to do almost everything”) of Halmos’ autobiography I Want to Be a Mathematician (Springer, 1985). In the “How to do research” part of this chapter, Halmos wrote “Where do the good questions, the research problems, come from? They probably come from the same hidden cave where authors find their plots and composers their tunes — and no one knows where that is or can even remember where it was after luckily stumbling into it once or twice. One thing is sure: they do not come from a vague desire to generalize. Almost the opposite is true: the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case. Usually it was the special case the suggested the generalization in the first place.”
- The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert
This statement is widely attributed to David Hilbert, yet apocryphally perhaps. For instances, see [Mark Kac, Wiener and integration in function spaces, Bull. Amer. Math. Soc., 72:52–68, 1966] and [Gizem Karaali, On Mathematical Ways of Knowing: Musings of a Humanistic Mathematician, in Interdisciplinary Perspectives on Math Cognition, edited by Marcel Danesi, 321–332. Springer, Cham, 2019].
- La mathématique est l’art de donner le même nom à des choses différentes. (Mathematics is the art of giving the same name to different things.)
— Henri Poincaré
This sentence comes from
Poincaré's book Science et Méthode (Science and Method) published in 1908. In Chapitre II (L'avenir des Mathématiques, i.e., The Future of Mathematics), Livre Premier (Le Savant et la Science, i.e., The Scientist and the Science) of the book (page 29), Poincaré wrote
“Je ne sais si je n'ai pas déjà dit quelque part que la mathématique est l'art de donner le même nom à des choses différentes. Il convient que ces choses, différentes par la matière, soient semblables par la forme, qu'elles puissent pour ainsi dire se couler dans le même moule. Quand le langage a été bien choisi, on est tout étonné de voir que toutes les démonstrations, faites pour un objet connu, s'appliquent immédiatement à beaucoup d'objets nouveaux ; on n'a rien à y changer, pas même les mots, puisque les noms sont devenus les mêmes. Un mot bien choisi suffit le plus souvent pour faire disparaître les exceptions que comportaient les règles énoncées dans l'ancien langage ; c'est pour cela qu'on a imaginé les quantités négatives, les quantités imaginaires, les points à l'infini, que saise encore ? Et les exceptions, ne l'oublions pas, sont pernicieuses, parce qu'elles cachent les lois. Eh bien, c'est l'un des caractères auxquels on reconnaît les faits à grand rendement, ce sont ceux qui permettent ces heureuses innovation» de langage Le fait brut est alors quelquefois sans grand intérêt, on a pu le signaler bien des fois sans avoir rendu grand service à la science ; il ne prend de valeur que le jour où un penseur mieux avisé aperçoit le rapprochement qu'il met en évidence et le symbolise par un mot.” (English translation by Francis Maitland:
I think I have already said somewhere that mathematics is the art of giving the same name to different things. It is enough that these things, though differing in matter, should be similar in form, to permit of their being, so to speak, run in the same mould. When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same. A well-chosen term is very often sufficient to remove the exceptions permitted by the rules as stated in the old phraseology. This accounts for the invention of negative quantities, imaginary quantities, decimals to infinity, and I know not what else. And we must never forget that exceptions are pernicious, because they conceal laws. This is one of the characteristics by which we recognize facts which give a great return: they are the facts which permit of these happy innovations of language. The bare fact, then, has sometimes no great interest: it may have been noted many times without rendering any great service to science; it only acquires a value when some more careful thinker perceives the connexion it brings out, and symbolizes it by a term.)
Cassius Jackson Keyser wrote in his book Mathematics as a Culture Clue (New York, Scripta Mathematica, 1947; see page 218) that
“Of the many pithy sayings of Henri Poincaré perhaps none is better than this:
‘Mathematics is the art of giving the same name to different things.’
I once quoted that mot to a poet, and got the quick response:
‘Poetry is the art of giving different names to the same thing.’”
- It is undeniable that some of the best inspirations in mathematics — in those parts of it which are as pure mathematics as one can imagine — have come from the natural sciences.
— John von Neumann
The Mathematician, Works of the Mind, by Robert B. Heywood, editor, vol. 1, no. 1, 180–196, University of Chicago Press, Chicago, 1947.
- As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers.
— John von Neumann
The Mathematician, Works of the Mind, by Robert B. Heywood, editor, vol. 1, no. 1, 180–196, University of Chicago Press, Chicago, 1947.
In the last two paragraphs of “The Mathematician”, von Neumann wrote “As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from ‘reality’ it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.” See also Donald E. Knuth's comment.
- A mathematician is a machine for turning coffee into theorems.
— Alfréd Rényi
This remark, often attributed to Paul Erdős (for example, see page 7 of the 1998 biography The Man Who Only Loved Numbers by Paul Hoffman, who did not give any source), was probably made by Alfréd Rényi.
In Bruce Schechter's biography of Erdős, My Brain is Open: The Mathematical Journeys of Paul Erdős (Simon and Schuster, 1998), page 155 reads:
“Rényi would become one of Erdős's most important collaborators … Their long collaborative sessions were often fueled by endless cups of strong coffee. Caffeine is the drug of choice for most of the world's mathematicians and coffee is the preferred delivery system. Rényi, undoubtedly wired on espresso, summed this up in a famous remark almost always attributed to Erdős: ‘A mathematician is a machine for turning coffee into theorems.’ … Turán (Pál Turán), after scornfully drinking a cup of American coffee, invented the corollary: ‘Weak coffee is only fit for lemmas.’” Similar attribution is given in MacTutor History of Mathematics Archive: Alfréd Rényi, where it is written that “Known to his many friends and colleagues by the nickname of 'Buba’, he (Rényi) often remembered as the author of the anecdote: … a mathematician is a machine for converting coffee into theorems. Turán developed the anecdote further by describing weak coffee as fit only for lemmas.” Additionally, in Section 6 (The Erdős graph) of the essay “Complex Networks” (Feature Column from the AMS, April 2004), Joseph Malkevitch writes: “Erdős has often been associated with the observation that ‘a mathematician is a machine for converting coffee into theorems’ but this ‘characterization’ appears to be due to his friend, Alfred Rényi. This thought was developed further by Erdős’ friend the Hungarian mathematician Paul Turán (1910–1976), who suggested that weak coffee was suitable ‘only for lemmas’.”
However, it is also suggested that this sentence was originally formulated in German: “Ein Mathematiker ist eine Maschine, die Kaffee in Sätze verwandelt”, where it can be interpreted as a wordplay on the double meaning of the word “Satz”, which can mean both “theorem” and “coffee grounds” (“Sätze” is its plural form).
See Math with Bad Drawings for interesting variants.
- The true Logic for this world is the Calculus of Probabilities, which takes account of the magnitude of the probability (which is, or which ought to be in a reasonable man's mind). This branch of Math., which is generally thought to favour gambling, dicing, and wagering, and therefore highly immoral, is the only ‘Mathematics for Practical Men’, as we ought to be.
— James C. Maxwell
This quotation comes from “Letter to Lewis Campbell, c. July 1850”, The Scientific Letters and Papers of James Clerk Maxwell: 1846–1862, edited by Peter M. Harman, CUP Archive, 1990 (page 197).
It was written in this letter that “As it is Saturday night I will not write very much more. I was thinking today of the duties of [the] cognitive faculty. It is universally admitted that duties are voluntary, and that the will governs understanding by giving or withholding Attention. They say that Understanding ought to work by the rules of right reason. These rules are, or ought to be, contained in Logic; but the actual science of Logic is conversant at present on with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true Logic for this world is the Calculus of Probabilities, which takes account of the magnitude of the probability (which is, or which ought to be in a reasonable man's mind). This branch of Math., which is generally thought to favour gambling, dicing, and wagering, and therefore highly immoral, is the only ‘Mathematics for Practical Men’, as we ought to be. Now, as human knowledge comes by the senses in such a way that the existence of things external is on inferred from the harmonious (not similar) testimony of the different senses, Understanding, acting by the laws of right reasoning, will assign to different truths (or facts, or testimonies, or what shall I call them) different degrees of probability.”
- There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense.
— Clifford A. Truesdell
In the very beginning of Chapter III (Thermodynamics of visco-elasticity, page 35) of Six Lectures on Modern Natural Philosophy (Springer-Verlag, 1966), Truesdell's comment on thermodynamics is: “Thermostatics, which even now is usually called thermodynamics, has an unfortunate history and a cancerous tradition. It arose in a chaos of metaphysical and indeed irrational controversy, the traces of which drip their poison even today … The logical standards acceptable in thermostatics fail to meet the criteria of other exact sciences; in books and papers concerning it the proportion of words if not prayers to equations is high — to proved theorems, almost infinite. There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense. When a physical writer expresses an assertion in words only, he is refusing to stand up to the test.”
- A large part of mathematics which becomes useful developed with
absolutely no desire to be useful, and in a situation where nobody could
possibly know in what area it would become useful; and there were no
general indications that it ever would be so.
John von Neumann
This quotation comes from “The Role of Mathematics in the Sciences and in Society”, Address at 4th Conference of Association of Princeton, Graduate Alumni, June 1954, pp. 16-29, [Vol. VI, No. 34],
Collected Works: Theory of Games, Astrophysics, Hydrodynamics and Meteorology, volume 6 of Collected Works of John von Neumann, edited by Abraham H. Taub, Pergamon Press, 1963. The quotation appears on page 489 of the book, where it reads: “But still a large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from thirty to a hundred years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.”
These sentences are quoted by Ted Hurley, “Algebraic structures for communications”, in Robert F. Morse, Daniela Nikolova-Popova, and Sarah Witherspoon, eds., Proceedings of the International Conference on Group Theory, Combinatorics and Computing (GTCC, Oct. 3–8, 2012, Boca Raton, Florida), pp. 59–77. In the last section (Section 9: “Finally”) of this paper, after quoting von Neumann, Hurley writes
“Indeed it took over 200 years for Field Theory to become ‘useful’ and this theory is now fundamental within the communications’ areas. Which areas are now ‘useful’ and which areas will become ‘useful’?”
- C’est par la logique qu’on démontre, c’est par l’intuition qu’on invente. (It is by logic that we prove, but by intuition that we invent.)
— Henri Poincaré
This sentence comes from
Poincaré's book Science et Méthode (Science and Method) published in 1908. In Chapitre II (Les définitions mathématiques et l’Enseignement, i.e., Mathematical definitions and Teaching), Livre II (Le Raisonnement Mathématique, i.e., Mathematical Reasoning) of the book (page 137), Poincaré wrote that “… c’est par la logique qu’on démontre, c’est par l’intuition qu’on invente. Savoir critiquer est bon, savoir créer est mieux.” (“… it is by logic that we prove, but by intuition that we invent. Knowing how to criticize is good, but knowing how to create is better.”)
- The moving power of mathematical invention is not reasoning, but imagination.
— Augustus De Morgan
This remark comes from the obituary of William Rowan Hamilton by De Morgan. The Obituary was published in [The Gentleman's Magazine, vol. 220 (January 1866), pages 128–134]. In this obituary, De Morgan wrote that
“Our notice of Hamilton's scientific character must be brief : and it is not in our power to dwell on those parts which are not in evidence before the public. The scholar, the poet, and the metaphysician must be set forth in some large and well studied memoir, or not at all. Hamilton himself often said, ‘I live by mathematics, but I am a poet.’ Such an aphorism may surprise our readers, but they should remember that the moving power of mathematical invention is not reasoning, but imagination. We no longer apply the homely term maker in literal translation of poet : but discoverers of all kinds, whatever may be their lines, are makers; or, as we now say, have the creative genius.” This sentence is quoted by Robert Perceval Graves in [The Life of Sir William Rowan Hamilton, vol. 3 (1889), page 219].
- Die Mathematik in gewissem Sinne von denen am meisten gefördert worden, die mehr durch Intuition als durch strenge Beweisführung sich auszeichneten. (Mathematics, in a sense, has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.)
— Christian Felix Klein
This sentence comes from page 271 of Klein's book Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert, Teil I (Lectures on the Development of Mathematics in the 19th Century, Part I) published in 1926. In Chapter Six, “Die allgemeine Funktionentheorie komplexer Veränderlicher bei Riemann und Weierstraß” (The general function theory of complex variables by Riemann and Weierstraß), commenting the work of Georg Friedrich Bernhard Riemann, Klein wrote that
“Ich mußte soviel erzählen, damit man einigermaßen schon hier erkennt, wie das imvergleichhche Genie von Riemann seiner Zeit vorauseilte und so die fernere Produktion weitgehend beeinflußte. Gewiß ist es der Schlußstein am Gebäude einer jeden mathematischen Theorie, den zwingenden Beweis für alle Behauptungen zu erbringen. Gewiß spricht sich die Mathematik selbst das Urteil, wenn sie auf zwingende Beweise verzichtet. Das Geheimnis genialer Produktivität wird es jedoch ewig bleiben, neue Fragestellungen zu finden, neue Theoreme zu ahnen, die wertvolle Resultate und Zusammenhänge erschheßen. Ohne die Schaffung neuer Gesichtspunkte, ohne die Aufstellung neuer Ziele, würde die Mathematik in der Strenge ihrer logischen Beweisführung sich bald erschöpfen und zu stagnieren beginnen, indem ihr der Stoff ausgehen möchte . — So ist die Mathematik in gewissem Sinne von denen am meisten gefördert worden, die mehr durch Intuition als durch strenge Beweisführung sich auszeichneten. Es ist kein Zweifel, daß Riemann derjenige Mathematiker der letzten Dezennien ist, der heute noch am lebendigsten nachwirkt.” (“I had to tell so much so that one can already see here to a certain extent how the comparative genius of Riemann preceded his time and thus largely influenced the further production. Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly, mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. — Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs. There is no doubt that Riemann is the mathematician of the last decade who is still the most alive today.”)
- Wir geben uns nicht gerne damit zufrieden, einer mathematischen Wahrheit überführt zu werden durch eine komplizierte Verkettung formeller Schlüsse und Rechnungen, an der wir uns sozusagen blind von Glied zu Glied entlang tasten müssen. Wir möchten vorher Ziel und Weg überblicken können, wir möchten den inneren Grund der Gedankenführung, die Idee des Beweises, den tieferen Zusammenhang verstehen. (We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper connection.)
— Hermann Klaus Hugo Weyl
Topologie und abstrakte Algebra als zwei Wege des mathematischen Verständnisses, Unterrichtsblatter fur Mathematik und Naturwissenschaften 38, 177-188 (1932), included in Gesammelte Abhandlungen: Band III as paper 95.
(English version published in The American Mathematical Monthly, Vol. 102, (1995), pages 453–460, entitled Topology and Abstract Algebra as Two Roads of Mathematical Comprehension)
Science and research
- Le savant n’étudie pas la nature parce que cela est utile; il l’étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle. (The scientist does not study nature because it is useful to do so; he studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.)
— Henri Poincaré
Poincaré made this remark in his book Science et Méthode (Science and Method) published in 1908. In Chapitre II (Le choix des faits, i.e., The selection of facts), Livre I (Le Savant et la Science, i.e., The Scientist and the Science) of the book (page 15), Poincaré wrote that “Le savant n’étudie pas la nature parce que cela est utile; il l’étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle. Si la nature n’était pas belle, elle ne vaudrait pas la peine d’être connue, la vie ne vaudrait pas la peine d’être vécue. Je ne parle pas ici, bien entendu, de cette beauté qui frappe les sens, de la beauté des qualités et des apparences; non que j'en fasse fi, loin de là, mais elle n'a rien à faire avec la science; je veux parler de cette beauté plus intime qui vient de l'ordre harmonieux des parties, et qu'une intelligence pure peut saisir.” (“The scientist does not study nature because it is useful to do so; he studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances; I am far from despising this, but it has nothing to do with science; what I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.”)
This quotation is similar to the remark that “The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.” Such a remark, often attributed to Poincaré, seems to be a misrepresentation of Poincaré's comment on nature rather than mathematics.
- The art of being wise is the art of knowing what to overlook.
— William James
The Principles of Psychology (1890), Chapter XXII. “Reasoning”
William James was an American philosopher and psychologist. He was one of the most influential philosophers of the United States, and considered to be the “father of American psychology”.
The original sentence that he wrote in his book was “As the art of reading (after a certain stage in one's education) is the art of skipping, so the art of being wise is the art of knowing what to overlook.”
This is similar to the remark made by
Hendrik W. Lenstra, Jr. that
“The art of doing mathematics is forgetting about the superfluous information.”
- There is nothing so practical as a good theory.
— Kurt Z. Lewin
Problems of research in social psychology, Field Theory in Social Science: Selected Theoretical Papers, edited by Dorwin Cartwright, Harper & Brothers Publishers, New York, 1951, pages 155–169 (see the last sentence of the penultimate paragraph, page 169)
Kurt Lewin was a German-American psychologist and widely recognized as one of the founders of modern social psychology (see “Kurt Lewin: The ‘Practical Theorist’ for the 21st Century”, D. Coghlan, T. Brannick, The Irish Journal of Management, 24:31–37, 2003). In the penultimate paragraph of his paper “Problems of research in social psychology” (written in 1943–44), Lewin wrote “The greatest handicap of applied psychology has been the fact that, without proper theoretical help, it had to follow the costly, inefficient, and limited method of trial and error. Many psychologists working today in an applied field are keenly aware of the need for close cooperation between theoretical and applied psychology. This can be accomplished in psychology as it has been accomplished in physics, if the theorist does not look toward applied problems with highbrow aversion or with fear of social problems, and if the applied psychologist realizes that there is nothing so practical as a good theory.”
- There is a pleasure in recognizing old things from a new point of view.
— Richard Phillips Feynman
Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics, 20:367–387, 1948
In the Introduction of this seminal paper, Feynman wrote
“It is a curious historical fact that modern quantum mechanics began with
two quite different mathematical formulations: the differential equation of
Schroedinger, and the matrix algebra of Heisenberg. The two, apparently
dissimilar approaches, were proved to be mathematically equivalent. These
two points of view were destined to complement one another and to be
ultimately synthesized in Dirac’s transformation theory.
This paper will describe what is essentially a third formulation of non-relativistic quantum theory.
This formulation was suggested by some of Dirac’s remarks concerning the relation of classical
action to quantum mechanics. A probability amplitude is associated with an entire motion of
a particle as a function of time, rather than simply with a position of the particle at a particular time.
The formulation is mathematically equivalent to the more usual formulations. There are, therefore,
no fundamentally new results. However, there is a pleasure in recognizing old things from a new point of view.
Also, there are problems for which the new point of view offers a distinct advantage. … In addition,
there is always the hope that the new point of view will
inspire an idea for the modification of present theories, a modification necessary to encompass present experiments.”
- If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.
— John von Neumann
In “Method in the physical sciences” (published in The Unity of Knowledge, ed. L. Leary (1955), pp. 157-164; and also
John von Neumann Collected Works, ed. A. Taub, Vol. VI, pp. 491-498.), von Neumann wrote:
“Thus whether one chooses to say that classical mechanics is causal or
teleological is purely a matter of literary inclination at the moment of
talking. This is very important, since it proves, that if one has really
technically penetrated a subject, things that previously seemed in
complete contrast, might be purely mathematical transformations of each
other. Things which appear to represent deep differences of principle and
of interpretation, in this way may turn out not to affect any significant
statements and any predictions. They mean nothing to the content of the
The sentence is quoted in Section 1.3 of the “Summary” part of the book Proportions, Prices and Planning: A Mathematical Restatement of the Labor Theory of Value (North-Holland Publishing Company, Amsterdam, 1970) by András Bródy (Hungarian economist).
- The person who discovers something rarely understands it …
If somebody else has created a field and he says this is the way to do it, you can entirely ignore it. On the other hand, when you create a field, remember to shut up. You don't really understand what you did.
— Richard W. Hamming
Hamming said these sentences during the “History of Computers — Software” part of the course The Art of Doing Science and Engineering: Learning to Learn, which was given to the graduate students at the Naval Postgraduate School (NPS) in Monterey, California on Apr. 4, 1995.
The video of this lecture is available on YouTube and also archived by NPS (as of July 22, 2015). See 15m42s–22m11s of the video.
Similar comments appear in Richard W. Hamming, The Art of Doing Science and Engineering: Learning to Learn, CRC Press, 2003 (Chapter 4, pages 27–28):
“History tends to be charitable in this matter. It gives credit for understanding what something means when we first to do it. But there is a wise saying, ‘Almost everyone who opens up a new field does not really understand it the way the followers do.’ The evidence for this is, unfortunately, all too good. It has been said in physics no creator of any significant thing ever understood what he had done. I never found Einstein on the special relativity theory as clear as some later commentators. And at least one friend of mine has said, behind my back, ‘Hamming doesn't seem to understand error correcting codes!’ He is probably right; I do not understand what I invented as clearly as he does. The reason this happens so often is the creators have to fight through so many dark difficulties, and wade through so much misunderstanding and confusion, they cannot see the light as others can, now the door is open and the path made easy. Please remember, the inventor often has a very limited view of what he invented, and some others (you?) can see much more. But also remember this when you are the author of some brilliant new thing; in time the same will probably be true of you. It has been said Newton was the last of the ancients and not the first of the moderns, though he was very significant in making our modern world.”
— 青原惟信禅师 (宋)
- 仅仅熟读了抽象的定义和方法而不知道它们具体来源的数学工作者是没有发展前途的，这样的人要搞深刻研究是可能会遇到无法克服的难关的。(One cannot go very far by merely learning bare definitions and methods from abstract notions without knowing the source of the definitions in the concrete situation. Indeed such an approach may lead to insurmountable difficulties later in research situations.)
— 华罗庚 (Hua, Loo Keng)
华罗庚, 《数论导引》, 北京: 北京科学出版社, 1957 年 (前言)
Hua, Loo Keng, Introduction to Number Theory (translated from the Chinese version by Peter Man-Kit Shiu (萧文杰)), Springer-Verlag, 1982 (page VIII)
- When theory becomes inbred — when it has grown several generations away from its roots, until it has completely lost touch with the real world — it degenerates and becomes sterile.
— Donald E. Knuth
Donald E. Knuth, Theory and practice, Theoretical Computer Science, 90 (1), 1–15, 1991 (paragraph eight)
This comment is similar to von Neumann's remark on mathematics.
- My experiences also strongly confirmed my previous opinion that the best theory is inspired by practice and the best practice is inspired by theory.
— Donald E. Knuth
Donald E. Knuth, Theory and practice, Theoretical Computer Science, 90 (1), 1–15, 1991 (paragraph 24)
- We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.
— Niels Bohr
Quoted by Freeman J. Dyson in “Innovation in Physics”, Scientific American, 199: 74–82, Sept. 1958, which reads: “A few months ago Werner Heisenberg and Wolfgang Pauli believed that they had made an essential step forward in the direction of a theory of elementary particles. Pauli happened to be passing through New York, and was prevailed upon to give a lecture explaining the new ideas to an audience which included Niels Bohr. Pauli spoke for an hour, and then there was a general discussion during which he was criticized rather sharply by the younger generation. Finally Bohr was called on to make a speech summing up the argument. ‘We are all agreed,’ he said, ‘that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.’ The objection that they are not crazy enough applies to all the attempts which have so far been launched at a radically new theory of elementary particles. It applies especially to crackpots. Most of the crackpot papers which are submitted to The Physical Review are rejected, not because it is impossible to understand them, but because it is possible. Those which are impossible to understand are usually published. When the great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer himself it will be only half-understood; to everybody else it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope.”
- The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.
— Niels Bohr
This quotation attributed to Bohr seems to be a popular but unsourced variant of the following sentences from N. Bohr, “Discussion with Einstein on epistemological problems in atomic physics”, in: P. A. Schilpp eds., Albert Einstein: Philosopher-Scientist, volume 7 in The Library of Living Philosophers, Northwestern University Press (Illinois), 1949 (the penultimate paragraph): “In the Institute in Copenhagen, where through those years a number of young physicists from various countries came together for discussions, we used, when in trouble, often to comfort ourselves with jokes, among them the old saying of the two kinds of truth. To the one kind belong statements so simple and clear that the opposite assertion obviously could not be defended. The other kind, the so-called ‘deep truths’, are statements in which the opposite also contains deep truth.”
- The task of the theorist is to bring order into the chaos of the phenomena of nature, to invent a language by which a class of these phenomena can be described efficiently and simply.
— Clifford A. Truesdell and Walter Noll
In Section 2 (Method and program of the non-linear theories, page 4), Chapter A of The Non-Linear Field Theories of Mechanics (second edition, Berlin: Springer-Verlag, 1992) by Truesdell and Noll, it reads: “While laymen and philosophers of science often believe, contend, or at least hope, that physical theories are directly inferred from experiments, anyone who has faced the problem of discovering a good constitutive equation or anyone who has sought and found the historical origin of the successful field theories knows how childish is such a prejudice. The task of the theorist is to bring order into the chaos of the phenomena of nature, to invent a language by which a class of these phenomena can be described efficiently and simply. Here is the place for ‘intuition’, and here the old preconception, common among natural philosophers, that natural is simple and elegant, has led to many great successes. Of course, physical theory must be based on experience, but experiment comes after, rather than before, theory. Without theoretical concepts one would neither know what experiments to perform nor be able to interpret their outcome.”
- This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new.
— Clifford A. Truesdell
This famous Mathematical Reviews quotation by Truesdell appears in Mathematical Reviews 12, page 561. The review of a 1950 paper titled “Equations of finite vibratory motions in isotropic elastic media” reads:
“This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new: if the reviewer has correctly understood the author’s undefined notations and misprints, the stress-strain relations used are those once proposed by St.-Venant [J. Math. Pures Appl. (2) 8, 257–295, 353–430 (1863); see §2], whose incorrect confusion of coordinates in the deformed and undeformed states of the body was pointed out by Brill and Boussinesq [cf. St. Venant, ibid. (2) 16, 275–307 (1871), see §7]. The falsity of the author’s results is obvious, since for the speed of propagation of finite waves in isotropic bodies he obtains expressions which are not scalars unless the displacement gradients are infinitesimal.” This quotation is recorded also in J. M. Ball and R. D. James, “The Scientific Life and Influence of Clifford Ambrose Truesdell III”, Archive for Rational Mechanics and Analysis, 161(1): 1-26, 2002 (Section 4: Influence, personality and style).
- You insist that there is something a machine cannot do. If you will tell
me precisely what it is that a machine cannot do, then I can always make
a machine which will do just that!
— John von Neumann
After giving a talk on computers at Princeton in 1948, John von Neumann
was met with an audience member who insisted that a “mere machine”
could never really think. Von Neumann's immortal reply was this, as quoted
in Section 1.3
of Probability Theory: The Logic of Science (Cambridge University Press, 2003)
Edwin Thompson Jaynes,
an attendee of the talk.
- Education is that which remains, if one has forgotten everything he learned in school.
— Albert Einstein
Einstein did write this quote in his essay “On Education” from 1936, which appeared as the 9th part of his book Out of My Later Years (Philosophical Library, New York, 1950). However, Einstein attributed it to an unnamed “wit”. He wrote that “If you have followed attentively my meditations up to this point, you will probably wonder about one thing. I have spoken fully about in what spirit, according to my opinion, youth should be instructed. But I have said nothing yet about the choice of subjects for instruction, nor about the method of teaching. Should language predominate or technical education in science? To this I answer: In my opinion all this is of secondary importance. If a young man has trained his muscles and physical endurance by gymnastics and walking, he will later be fitted for every physical work. This is also analogous to the training of the mind and the exercising of the mental and manual skill. Thus the wit was not wrong who defined education in this way: ‘Education is that which remains, if one has forgotten everything he learned in school.’ For this reason I am not at all anxious to take sides in the struggle between the followers of the classical philologic-historical education and the education more devoted to natural science.”
This sentence is similar to the French saying “La culture est ce qui reste lorsque l’on a tout oublié” (Culture is that which remains, if one has forgotten everything), attributed to Édouard Herriot.
Another French variant is “La culture est ce qui reste lorsqu'on a oublié toutes les choses apprises” (Culture is that which remains if one has forgotten everything one has learned), which appears in the 1912 book Propos Critiques by Georges Duhamel.
- There are three kinds of people in the world: There are those who cause things to happen, there are those who stand around and watch, and a vast majority who don't even know anything is happening. I want you to be one of the first; if not the first, surely the second.
— Richard W. Hamming
This quotation comes from Hamming's Superintendent's Guest Lecture (SGL) entitled “Learning to Learn: The Art of Doing Science and Engineering” given to all the students and faculty of the Naval Postgraduate School (NPS) in Monterey, California in on Apr. 30, 1990. The lecture provides a synopsis overview of many of the most fundamental ideas in Hamming's career.
The video of this lecture is available on YouTube and also archived by NPS (as of June 9, 2015), and the quotation appears at 33m56s–34m15s.
The first part of the quotation seems to be a common wisdom, also formulated as “There are three types of people in this world: those who make things happen, those who watch things happen, and those who wonder what happened.” It is attributed on the Internet to various people including Mary Kay Ash, Ann Landers, James A. Lovell, Jr., and Casey Stengel, but sources are barely given.
- O liberté, que de crimes on commet en ton nom ! (自由，自由，多少罪恶假汝之名而行！Oh liberty, what crimes are committed in thy name!)
— Marie-Jeanne Phlippon Roland
The last words of Roland (well known as “Madame Roland”) before she was executed on Nov. 8, 1793, during the French Revolution. She exclaimed the famous remark in front of the clay statue of Liberty in the Place de la Révolution shortly before the guillotine dropped on her.
See Helen M. Williams, Letters Containing a Sketch of the Politics of France, vol. 1, London: G. G. and J. Robinson, 1795 (page 201) and Wikiquote.
According to Jehiel K. Hoyt, Hoyt's New Cyclopedia of Practical Quotations (compiled by Kate L. Roberts), New York and London: Funk & Wagnalls Company, 1922,
the actual expression used is said to have been “O liberté, comme on t'a jouée !” (“Oh liberty, how thou hast been played with!”).
- The appeal of slavery is that it offers a life free of responsibilities, and this is always the appeal of slavery.
— Rousas John Rushdoony
“The fear of freedom”, California Farmer 242:3 (Feb. 1, 1975), pp. 55 (see Chalcedon)
- Sans la liberté de blâmer, il n'est point d’éloge flatteur. (若批评不自由，则赞美无意义。Unless there is liberty to criticize, praise has no value.)
— Pierre-Augustin Caron de Beaumarchais (博马舍)
Cette citation se trouve dans Le Mariage de Figaro, une comédie de Pierre-Augustin Caron de Beaumarchais écrite en 1778. Dans la troisième scène de l'Acte V, Figaro, le protagoniste, dit : « Je lui dirais … que les sottises imprimées n'ont d'importance qu'aux lieux où l'on en gêne le cours ; que, sans la liberté de blâmer, il n'est point d’éloge flatteur ; et qu'il n'y a que les petits hommes qui redoutent les petits écrits. » Cette citation est la devise du journal Le Figaro, qui a été nommé d'après Figaro, le personnage de Beaumarchais. Aujourd'hui (jusqu'au 15 mars 2016), on trouve cette phrase sous le titre du journal dans sa première page.
This quotation comes from The Marriage of Figaro, a comedy by Pierre-Augustin Caron de Beaumarchais written in 1778. In the third scene of Act V, the protagonist Figaro says: “I'd tell him that stupidities acquire importance only in so far as their circulation is restricted, that unless there is liberty to criticize, praise has no value, and that only trivial minds are apprehensive of trivial scribbling.” See Pierre-Augustin Beaumarchais and John Wood, The Barber of Seville and The Marriage of Figaro, Penguin Books, 2005. The quotation is the motto of the French newspaper Le Figaro, which was named after Figaro, the protagonist of Beaumarchais’ play. Nowadays (as of March 15, 2016) one can find this sentence on the first page of the newspaper right below its title.
语出陈寅恪为清华 “海宁王静安先生纪念碑” 所撰碑文《海宁王先生之碑铭》：“海宁王先生自沉后二年，清华研究院同人咸怀思不能自已。其弟子受先生之陶冶煦育者有年，尤思有以永其念，佥曰：宜铭之贞珉，以昭示于无竟。因以刻石之辞命寅恪，数辞不获已，谨举先生之志事，以普告天下后世。其词曰：士之读书治学，盖将以脱心志于俗谛之桎梏，真理因得以发扬。思想而不自由，毋宁死耳。斯古今仁圣所同殉之精义，夫岂庸鄙之敢望。先生以一死见其独立自由之意志，非所论于一人之恩怨，一姓之兴亡。呜呼！树兹石于讲舍，系哀思而不忘。表哲人之奇节，诉真宰之茫茫。来世不可知者也，先生之著述，或有时而不章，先生之学说，或有时而可商。惟此独立之精神，自由之思想，历千万祀，与天壤而同久，共三光而永光。”
- You only live once, but living once means living many times, as a series of similar but technically different people, who know each other, but only in one direction, and who can help each other, but only in the other direction.
— Michael Stevens
This sentence is an extension of “You only live once” (YOLO), an English motto that resembles the Latin aphorism “Carpe diem” (“seize the day”). It comes from Michael Stevens’ Vsauce video “Misnomers”, which covers various topics including linguistic discussions on nomenclature and philosophical discussions on identity. See 10m38s–11m03s of the video on YouTube.
- Mankind was born on earth. It was never meant to die here.
— Christopher Nolan and Jonathan Nolan
This quotation comes from the 2014 movie Interstellar, the director being Christopher Nolan, and the script writer being Christopher Nolan and Jonathan Nolan. It is similar to the sentence “Earth is the cradle of humanity, but one cannot live in a cradle forever.” (Планета есть колыбель разума, но нельзя вечно жить в колыбели, more precisely translated as “The planet is the cradle of the mind, but you cannot live forever in the cradle”), which was written by Konstantin E. Tsiolkovsky (Константин Эдуардович Циолковский, Russian and Soviet rocket scientist and pioneer of the astronautic theory) in a letter in 1911.
The quotation resembles a fact about mathematics: Mathematics starts from intuition, but it is never meant to stay there; ou: Good mathematicians start their journeys from intuition (and most likely return there in the end), while lousy ones stay there (without ever leaving). To see what it means, read Terence Tao's essay on the three stages of mathematical education (“There’s more to mathematics than rigour and proofs”) and the quotation on the three stages of Zen. Do not confuse the first stage with the third one.
《胡适文存》第三集第九卷《慈幼的问题》开篇：“我的一个朋友对我说过一句很深刻的话：‘你要看一个国家的文明，只消考察三件事：第一，看他们怎样待小孩子；第二，看他们怎样待女人；第三，看他们怎样利用闲暇的时间。’ ” 见远流出版公司 1986 年版《胡適作品集》第十四册 “三百年中的女作家:《胡適文存》第三集第七·八·九卷” 第二四三页。
子桑雽曰：“子独不闻假人之亡与？林回弃千金之璧，负赤子而趋。或曰：‘为其布与？赤子之布寡矣。为其累与？赤子之累多矣。弃千金之璧，负赤子而趋，何也？’ 林回曰：‘彼以利合，此以天属也。’ 夫以利合者，迫穷祸患害相弃也；以天属者，迫穷祸患害相收也。夫相收之与相弃亦远矣。且君子之交淡若水，小人之交甘若醴；君子淡以亲，小人甘以绝。彼无故以合者，则无故以离。”
《新中国未来记》(1902 年发表，录于 1989 年中华书局版《饮冰室合集》第十一本，目次 “饮冰室专集之八十九”) 第三回 “求新学三大洲环游 论时局两名士舌战”。
小说中黄克强 (虚构)、李去病二人游学海外，于癸卯 (1906) 年回国，途径山海关，触景生情，辩论时局。梁启超借黄克强之口写道：“兄弟，你是读过历史的。你看世界上那一国不是靠着国民再造一番，才能强盛吗？现在我和你两个，虽然是一介青年，无权无勇，但是我们十年来读些书是干甚么呢？(青年读书诸君想想。) 难道学几句爱皮西靠做将来的衣饭碗不成？(青年读书诸君想想。) 难道跟着那些江湖名士，讲几句激昂慷慨的口头话，拿着无可奈何四个字，就算个议论的结束吗？(青年读书诸君想想。) 我想一国的事业，原是一国人公同担荷的责任。(眉批：知责任者大，大夫之始也。任责任者大，大夫之终也。) 若使四万万人，各各把自己面分的担荷起来，这责任自然是不甚吃力的。但系一国的人，多半还在睡梦里头，他还不知道有这个责任，叫他怎么能够担荷它呢？既然如此，那些已经知道的人，少不免要把他们的担子，一齐都放在自己的肩膀上头了。(青年读书诸君想想。) 兄弟，我们两个虽算不得甚么人物，但已经受了国民的恩典，读了这点子书，得了这点子见识，这个责任，是平日知道熟了。今日回到本国，只要尽自己的力量去做，做得一分是一分。安见中国的前途，就一定不能挽救呢？” 文中批注为梁启超所作。另外，“爱皮西” 似为 “ABC” 之意。
- - Grandpa, were you a hero in the war?
- No… but I served in a company of heroes.
— Sergeant Myron N. Ranney (with his grandson), WWII veteran
This dialog between Ranney and his grandson was quoted by Major Richard D. Winters in the World War II miniseries Band of Brothers (HBO, 2011). During the interview segment at the very end of the miniseries, Winters quoted a passage from a letter that he received from Ranney: “I cherish the memories of a question my grandson asked me the other day, when he said, ‘Grandpa, were you a hero in the war?’ Grandpa said ‘No… but I served in a company of heroes.’” The dialog was also quoted in the book Band of Brothers: E Company, 506th Regiment, 101st Airborne from Normandy to Hitler's Eagle's Nest (Simon & Schuster, 2002) by Stephen E. Ambrose (on page 425) and in Beyond Band of Brothers: The War Memoirs of Major Dick Winters (Random House, 2011) by Richard D. Winters and Cole C. Kingseed (on page 262).
- When you realize you want to spend the rest of your life with somebody, you want the rest of your life to start as soon as possible.
— Nora Ephron
At the end of the movie When Harry Met Sally… (1989, written by Nora Ephron and directed by Rob Reiner), Harry's declaration of love to Sally on New Year's Eve: “I love that you get cold when it's 71 degrees out. I love that it takes you an hour and a half to order a sandwich. I love that you get a little crinkle above your nose when you're looking at me like I'm nuts. I love that after I spend the day with you, I can still smell your perfume on my clothes. And I love that you are the last person I want to talk to before I go to sleep at night. And it's not because I'm lonely, and it's not because it's New Year's Eve. I came here tonight because when you realize you want to spend the rest of your life with somebody, you want the rest of your life to start as soon as possible.”