Quotes about understanding
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The New Economics for Industry, Government, Education (1993)

“If children fail to understand one another, it is because they think they understand one another.”
The Language and Thought of the Child (1923) Tr. Marjorie and Ruth Gabain (1926)
Context: If children fail to understand one another, it is because they think they understand one another. The explainer believes from the start that the reproducer will grasp everything, will almost know beforehand all that should be known, and will interpret every subtlety. Children are perpetually surrounded by adults who not only know much more than they do, but who also do everything in their power to understand them, who even anticipate their thoughts and desires. Children, therefore... are perpetually under the impression that people can read their thoughts, and in extreme cases, can steal their thoughts away. It is obviously owing to this mentality that children do not take the trouble to express themselves clearly... This mentality does not contradict ego-centric mentality. Both arise from the belief of the child, the belief that he is the centre of the universe. These habits of thought account... for the remarkable lack of precision in the childish style.

Ruth in her letter to Robert
Ch. 19
I Am Legend (1954)
Context: I’m writing this note, though, because I want to save you if I can.
When I was first given the job of spying on you, I had no feelings about your life. Because I did have a husband, Robert. You killed him.
But now it’s different. I know now that you were just as much forced into your situation as we were forced into ours. We are infected. But you already know that. What you don’t understand yet is that we’re going to stay alive. We’ve found a way to do that and we’re going to set up society again slowly but surely. We’re going to do away with all those wretched creatures whom death has cheated. And, even though I pray otherwise, we may decide to kill you and those like you.

“It's not about money. It's about the people you have, how you're led, and how much you get it.”
As quoted in Fortune (9 November 1998); also quoted in "TIME digital 50" in TIME digital archive (1999) http://web.archive.org/web/20000612103032/http://www.time.com/time/digital/digital50/08.html
1990s
Context: Innovation has nothing to do with how many R&D dollars you have. When Apple came up with the Mac, IBM was spending at least 100 times more on R&D. It's not about money. It's about the people you have, how you're led, and how much you get it.

My Autobiography, p. 291
Context: I believe that faith is a precursor of all our ideas. Without faith, there never could have evolved hypothesis, theory, science or mathematics. I believe that faith is an extension of the mind. It is the key that negates the impossible. To deny faith is to refute oneself and the spirit that generates all our creative forces. My faith is in the unknown, in all that we do not understand by reason; I believe that what is beyond our comprehension is a simple fact in other dimensions, and that in the realm of the unknown there is an infinite power for good.
1983

“If you can't explain it simply, you don't understand it well enough.”

Source: Discovering Buddhism, 2004 https://www.youtube.com/watch?v=226w04QMPzQ

“You don't understand anything until you learn it more than one way.”
In Managing an Information Security and Privacy Awareness and Training Program (2005) by Rebecca Herold, p. 101

“You don't understand music: you hear it. So hear me with your whole body.”
Source: The Stream of Life

Source: Proudhon: What Is Property?

“Can you not understand that liberty is worth more than just ribbons?”
Source: Animal Farm

“Liberals can understand everything but people
who don't understand them.”

Physics and Philosophy (1958)
Source: Physics and Philosophy: The Revolution in Modern Science
Context: Whenever we proceed from the known into the unknown we may hope to understand, but we may have to learn at the same time a new meaning of the word "understanding."

“Oh, how hard it is to part with power! This one has to understand.”
Source: The Gulag Archipelago 1918-1956

“I am so clever that sometimes I don't understand a single word of what I am saying.”
Source: The Happy Prince and Other Stories

“If you love me, Henry, you don’t love me in a way I understand.”
Source: Crush

Source: On Becoming a Person: A Therapist's View of Psychotherapy

“A point of view can be a dangerous luxury when substituted for insight and
understanding.”
Source: The Gutenberg Galaxy: The Making of Typographic Man

“Faith must trample under foot all reason, sense, and understanding.”

“I understand HOW: I do not understand WHY”
1984
Variant: I understand HOW: I do not understand WHY.

Source: 1920s, "Picasso Speaks" (1923), p. 315.

As quoted in Stephen Hawking's Universe http://books.google.com/books?id=lkntNIwunAAC&pg=PA77&dq=hawking+%22my+goal+is+simple%22&ei=q5HtSvCOIoLklQTU_cWhDA#v=onepage&q=hawking%20%22my%20goal%20is%20simple%22&f=false (1985) by John Boslough, Ch. 7 : The Final Question, p. 77

“I can't understand why people are frightened of new ideas. I'm frightened of the old ones.”
Quoted in Richard Kostelanetz (1988) Conversing with Cage
1980s

Widely known as The Prayer of St. Francis, it is not found in Esser's authoritative collection of Francis's writings.
[Fr. Kajetan, Esser, OFM, ed., Opuscula Sancti Patris Francisci Assisiensis, Rome, Grottaferrata, 1978]. Additionally there is no record of this prayer before the twentieth century.
[Fr. Regis J., Armstrong, OFM, Francis and Clare: The Complete Works, New York, Paulist Press, 1982, 10, 0-8091-2446-7]. Dr. Christian Renoux of the University of Orleans in France traces the origin of the prayer to an anonymous 1912 contributor to La Clochette, a publication of the Holy Mass League in Paris. It was not until 1927 that it was attributed to St. Francis.
The Origin of the Peace Prayer of St. Francis, 2013-06-28, Renoux, Christian http://www.franciscan-archive.org/franciscana/peace.html,.
[Christian, Renoux, La prière pour la paix attribuée à saint François: une énigme à résoudre, Paris, Editions franciscaines, 2001, 2-85020-096-4].
Misattributed

Interview, quoted in "Words from the Master" http://www.co.uk.lspace.org/books/apf/words-from-the-master.html in The Annotated Pratchett File http://www.co.uk.lspace.org/books/apf/index.html
General sources
Context: As for The Mapp... I suspect it'll never get a US publication. It seemed to frighten US publishers. They don't seem to understand it.
That seems to point up a significant difference between Europeans and Americans:
A European says: I can't understand this, what's wrong with me? An American says: I can't understand this, what's wrong with him?
I make no suggestion that one side or the other is right, but observation over many years leads me to believe it is true.
“Aphrodite makes us understand why women have drowned their babies.”
Source: Untamed

“Most people do not listen with the intent to understand; they listen with the intent to reply.”
Source: The Seven Habits Of Highly Effective People (1989), p. 239
Source: The 7 Habits of Highly Effective People: Powerful Lessons in Personal Change

“Pain's not bad, it's good. It teaches you things. I understand that.”

“You always admire what you really don't understand.”

Hour of Gold, Hour of Lead: Diaries and Letters of Anne Morrow Lindbergh, 1929-1932 (1973), p. 3
Source: Gift from the Sea
Context: I do not believe that sheer suffering teaches. If suffering alone taught, all the world would be wise, since everyone suffers. To suffering must be added mourning, understanding, patience, love, openness and the willingness to remain vulnerable. All these and other factors combined, if the circumstances are right, can teach and can lead to rebirth.

“Never break a promise to an animal. They're like babies—they won't understand.”
Source: Wild Magic

“Religion is the impotence of the human mind to deal with occurrences it cannot understand.”

“I think that hate is a feeling that can only exist where there is no understanding.”
Source: Sweet Bird of Youth

“To understand is to perceive patterns.”

“Understanding is a kind of ecstasy”
Source: Broca's Brain: Reflections on the Romance of Science

“I don't pretend to understand the universe — it's much bigger than I am.”

“Can a man who's warm understand one who's freezing?”
Source: One Day in the Life of Ivan Denisovich (1962)

Si la coustume estoit de mettre les petites filles a l'escole, et que communement on les fist apprendre les sciences comme on fait aux filz, qu'elles apprendroient aussi parfaitement et entenderoient les subtilités de toutes les arz et sciences comme ils font.
Part I, ch. 27, p. 63.
Le Livre de la Cité des Dames (c. 1405)
Source: The Book of the City of Ladies

“No man understands a deep book until he has seen and lived at least part of its contents.”

As quoted in "Clemente Says He is Very Happy" https://news.google.com/newspapers?id=42oeAAAAIBAJ&sjid=QckEAAAAIBAJ&pg=1983%2C4221206 by the Associated Press, in The Daytona Beach Morning Journal (October 16, 1971), p. 1-C
Other, <big><big>1970s</big></big>, <big>1971</big>

General Theory of Law and State (1949), I. The Concept of Law, A. Law and Justice, a. Human Behavior as the Objects of Rules

“Man knows much more than he understands.”
As quoted in A Primer of Adlerian Psychology: The Analytic-Behavioural-Cognitive Psychology of Alfred Adler (1999) by Harold H. Mosak and Michael P. Maniacci

Source: Culture and Value (1980), p. 53e

2008, A More Perfect Union (March 2008)

“To understand a science it is necessary to know its history.”
A Course of Positive Philosophy (1832 - 1842) [Six volumes]
“We do have an organ for understanding and recognizing moral facts. It is called the brain.”
Paul Churchland. A Neurocomputational Perspective, 1989.

The Psychology of the Unconscious (1943)

In 1953; p. 21
before 1960, "Yves Klein, 1928 – 1962, Selected Writings"

Earliest published version found on Google Books with this phrasing is in the 1993 book The Internet Companion: A Beginner's Guide to Global Networking by Tracy L. LaQuey and Jeanne C. Ryer, p. 25 http://books.google.com/books?id=sP5SAAAAMAAJ&q=meowing#search_anchor. However, the quote seems to have been circulating on the internet earlier than this, appearing for example in this post from 1987 http://groups.google.com/group/comp.lang.c/msg/cc89abb5e065d23f?hl=en and this one from 1985 http://groups.google.com/group/net.sources.games/browse_thread/thread/846af15b5a38c35/3d6d5a639c24bba3. No reference has been found that cites a source in Einstein's original writings, and the quote appears to be a variation of an old joke that dates at least as far back as 1866, as discussed in this entry from the "Quote Investigator" blog http://quoteinvestigator.com/2012/02/24/telegraph-cat/#more-3387. A variant was told by Thomas Edison, appearing in The Diary and Sundry Observations of Thomas Alva Edison (1948), p. 216 http://books.google.com/books?id=NXtEAAAAIAAJ&q=edinburgh#search_anchor: "When I was a little boy, persistently trying to find out how the telegraph worked and why, the best explanation I ever got was from an old Scotch line repairer who said that if you had a dog like a dachshund long enough to reach from Edinburgh to London, if you pulled his tail in Edinburgh he would bark in London. I could understand that. But it was hard to get at what it was that went through the dog or over the wire." A variant of Edison's comment can be found in the 1910 book Edison, His Life and Inventions, Volume 1 by Frank Lewis Dyer and Thomas Commerford Martin, p. 53 http://books.google.com/books?id=qN83AAAAMAAJ&pg=PA53#v=onepage&q&f=false.
The wireless telegraph is not difficult to understand. The ordinary telegraph is like a very long cat. You pull the tail in New York, and it meows in Los Angles. The wireless is the same, only without the cat.
Variant, earliest known published version is How to Think Like Einstein by Scott Thorpe (2000), p. 61 http://books.google.com/books?id=9yrYQxBgIYEC&lpg=PP1&pg=PA61#v=onepage&q&f=false. Appeared on the internet before that, as in this archived page from 12 October 1999 http://web.archive.org/web/19991012152820/http://stripe.colorado.edu/%7Ejudy/einstein/advice.html
Misattributed

Letter to Dr. Theodore Canisius (17 May 1859)
1850s

Che cosa è fascismo? (What is fascism?), lecture delivered in Florence (March 8, 1925)

Connections (1979), 10 - Yesterday, Tomorrow and You
18
The Social Psychology of Organizations (1966)