...I am not sure that I shall still do geometry ten years from now. I also think that the mine is already almost too deep, and must sooner or later be abandoned. Today, Physics and Chemistry offer more brilliant discoveries and which are easier to exploit…
Lagrange, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1111
 
The theory of continued fractions is one of the most useful theories in Arithmetic… Since it is absent from most works on Arithmetic and Algebra, it may not be well known among geometers. I would be satisfied if I were able to contribute to make it slightly more familiar.
Lagrange, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1105
 
As long as Algebra and Geometry were separated, their progress was slow and their use limited; but once these sciences were united, they lent each other mutual support and advanced rapidly together towards perfection. We owe to Descartes the application of Algebra to Geometry; this has become the key to the greatest discoveries in all fields of mathematics.
Lagrange, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1097
 
Diophantus can be considered the inventor of Algebra…
Lagrange, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1098
 
Informal, quasi-empirical mathematics does not grow through a monotonous increase in the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.
Imre Lakatos, from Proofs and Refutations.
578
 
It has not yet been sufficiently realised that present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought. While in mathematics this authoritarianism follows the deductivist pattern just described, in science it operates through the inductivist pattern.
Imre Lakatos, from "The deductivist versus the heuristic approach," in Proofs and Refutations.
1152
 
The history of mathematics and the logic of mathematical discovery... cannot be developed without the criticism and ultimate rejection of formalism.
Imre Lakatos, from Proofs and Refutations.
577
 
In deductivist style, all propositions are true and all inferences valid. Mathematics is presented as an ever-increasing set of eternal, immutable truths. Counterexamples, refutations, criticism cannot possibly enter. An authoritarian air is secured for the subject by beginning with disguised monster-barring and proof-generated definitions and with the fully-fledged theorem, and by suppressing the primitive conjecture, the refutations, and the criticism of the proof. Deductivist style hides the struggle, hides the adventure. The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility.
Imre Lakatos, from "The deductivist versus the heuristic approach," in Proofs and Refutations.
1151
 
Most Bourbaki stories are unreliable. While the members of this cryptic organization have taken no blood oath of secrecy, most of them are so amused by their own joke that their stories about themselves are intentionally conflicting and apocryphal. They are happy to see themselves shrouded in mystery, and other mathematicians who were their intimate friends merrily went along.
Liew Hai Lam, from "Nicolas Bourbaki -- A non-existent mathematician," in Menumui Matematik (Discovering Mathematics; journal of the Malaysian Mathematical Society), vol. 2, no. 1, 1980, pp. 16-29.
1082
 
Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere trifle remains. But a careful analysis shows that in this seemingly trifle lies the crux of the matter; usually it contains either the proposition being proved or a postulate equivalent to it.
Lambert,
763
 
I have good reason to doubt that the present article will be read, or even understood, by those who should profit most by it, namely those who spend time and efforts in trying to square the circle. There will always be enough such persons...who understand very little of geometry…
Lambert, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1106
 
We preferred the comparatively simple but more intelligent life of Kansas to Washington. There are some intelligent people in Washington. More of 'em in Kansas.
Alfred M. Landon, in George Will, "Alf Landon's Little House on the Prairie," The Pursuit of Happiness and Other Sobering Thoughts, 1978.
1157
 
Read Euler, he is our master in all.
P. S. Laplace,
231
 
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
Marquis Pierre Simon de Laplace,
1400
 
We see... that the theory of probabilities is at bottom only common sense reduced to calculation.
P. S. Laplace,
230
 
Nature laughs at the difficulties of integration.
Laplace,
753
 
All the effects of nature are only the mathematical consequences of a small number of immutable laws.
P. S. Laplace,
229
 
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
Marquis Pierre Simon de Laplace,
1660
 
You see all of these statues of historical figures; there should be a monument to pi.
Justin Laramie, Westfield State College student
1130
 
What is tragedy? It is the failure to meet the challenge of the future; it is the failure to bring forth today, that which the small mind deems a seemingly "impractical action," but an action on which the possibility of existence of an acceptable tomorrow depends.
Lyndon LaRouche, from "On the Subject of Tariffs and Trade".
1047
 
The Cantorian infinite has been one of the main nutrients for the spectacular flowering of mathematics in the twentieth century, and yet it remains mysterious and ill understood.
Shaughan Lavine, from Understanding the Infinite.
541
 
The modern-day theory of the infinite did not begin with an effort to produce a theory of the infinite, and it did not build on a long history of attempts at mathematical theories of the infinite. It began instead with an attempt to clarify the foundations of analysis and specifically of the calculus -- that is, it grew out of the development of our theory of rates of change and of areas under curves.
Shaughan Lavine, from Understanding the Infinite.
542
 
Never give a man up until he has failed at something he likes.
Lewis E. Lawes,
1268
 
Born of man's primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art.
Lawrence University Catalogue, quoted in "Mathematics: an integral part of our culture" by Harald M. Ness, Jr., from Essays in Humanistic Mathematics.
506
 
The unread story is not a story; it is little black marks on wood pulp. The reader, reading it, makes it live: a live thing, a story.
Ursula K. Le Guin,
1706
 
What the pupils want to learn is as important as what the teachers want to teach.
Lois E. LeBar,
235
 
Unfortunately competitive examinations often encourage deception. The teachers must train their students to answer little fragmentary questions quite well, and they give them model answers that are often veritable masterpieces and that leave no room for criticism. To achieve this, the teachers isolate each question from the whole of mathematics and create for this question alone a perfect language without bothering about its relationships to other questions. Mathematics is no longer a monument but a heap
Henri Leon Lebesgue, quoted in Intersection: NCTM/Exxon Education Foundation Newsletter, May, 1998.
615
 
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.
Henri Lebesgue,
1544
 
The only instruction which a professor can give, in my opinion, is to think in front of his students.
Henri Lebesgue, quoted in Out of the Mouths of Mathematicians by Rosemary Schmalz
1545
 
Let the Math Love movement begin!!!
J. Todd Lee,
899
 
The combined opposition cannot prevent us from advancing so long as we have the road to books and schools open to us. Even the snub given to our political condition is as nothing compared with that it would be to shut the doors of the school against us.
Benjamin Lee, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
232
 
As I inched sluggishly along the treadmill of the Maycomb County school system, I could not help receiving the impression that I was being cheated out of something. Out of what I knew not, yet I did not believe that twelve years of unrelieved boredom was exactly what the state had in mind for me.
Harper Lee, from To Kill a Mockingbird.
233
 
Power is knowing your past.
Spike Lee,
234
 
The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another.
Gosta Mittag-Leffler,
260
 
What sane person could live in this world and not be crazy?
Ursula K. LeGuin,
978
 
If you see a whole thing - it seems that it's always beautiful. Planets, lives... But up close a world's all dirt and rocks. And day to day, life's a hard job, you get tired, you lose the pattern.
Ursula K. LeGuin,
979
 
It had never occurred to me before that music and thinking are so much alike. In fact you could say music is another way of thinking, or maybe thinking is another kind of music.
Ursula K. LeGuin,
980
 
The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic.
G. W. Leibniz, quoted in "The Twins" by Oliver Sacks.
238
 
Indeed in general I hold that there is nothing truer than happiness, and nothing happier and sweeter than truth.
Leibniz, from Philosophical Investigations.
1288
 
Taking mathematics from the beginning of the world to the time of Newton, what he has done is much the better half.
G. W. Leibniz,
237
 
The Lord loves odd numbers.
Leibniz, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1103
 
One must remember... that incomparably small quantities... are by no means constant and determined. On the contrary, since they may be made as small as we like, they play the same part in geometric reasoning as the infinitely small in the strict sense. For if an antagonist denies the correctness of our theorems, our calculations show that the error is smaller than any given quantity, since it is in our power to decrease the incomparably small... as much as is necessary for our purpose.
G.W. Leibniz, quoted in Calculus: A Liberal Art by W.M. Priestley.
723
 
It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used.
Gottfried Wilhelm von Leibnitz, quoted in The Magic of Mathematics, by Theoni Pappas.
493
 
[Complex numbers are] a fine and wonderful refuge of the divine spirit - almost an amphibian between being and non-being.
G. W. Leibniz, quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.
236
 
Schooling, instead of encouraging the asking of questions, too often discourages it.
Madeleine L'Engle, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
239
 
That is what learning is. You suddenly understand something you've understood all your life, but in a new way.
Doris Lessing, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
240
 
What you do should speak so loudly that no one will hear what you say.
Marv Levy,
609
 
I'm old enough to know my limitations and I'm young enough to exceed them.
Marv Levy, on becoming the General Manager of the Buffalo Bills at age 80.
1007
 
Adversity is an opportunity for heroism.
Marv Levy,
607
 
Where would you rather be than right here, right now.
Marv Levy,
608
 
Courage is not simply one of the virtues, but the form of every virtue at the testing point.
C.S. Lewis,
827
 
Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time.
Sophus Lie, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.
241
 
I am not, nor ever have been, in favor of bringing about in any way the social or political equality of the white and black races . . . I will say in addition that there is a physical difference between the white and black races which, I suppose, will forever forbid the two races living together upon terms of social and political equality; and in as much as they cannot so live, that while they do remain together there must be a position of the superiors and the inferiors; and that I, as much as any other man, am in favor of the superior being assigned to the white man.
Abraham Lincoln, 1858
1031
 
Life is like a landscape. You live in the midst of it, but can describe it only from the vantage point of distance.
Charles A. Lindbergh,
931
 
Only in growth, reform, and change, paradoxically enough, is true security to be found.
Anne Morrow Lindebergh, quoted in Wisdom for the New Millennium edited by Helen Exley.
1077
 
Only in growth, reform, and change, paradoxically enough, is true security to be found.
Anne Morrow Lindebergh, quoted in Wisdom for the New Millennium edited by Helen Exley.
1080
 
Uncertainty was hardly new to science in 1927... When scientists have resolved one level of disagreement, they move down to the next. Uncertainty, discrepancy, and inconsistency are the stock-in-trade of any lively scientific discipline... So Heisenberg didn't introduce uncertainty into science. What he changed, and profoundly so, was its very nature and meaning.
David Lindley, from Uncertainty: Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science.
1398
 
Points
Have no parts or joints.
How can they combine
To form a line?
J.A. Lindon, from A Dictionary of Quotations in Mathematics by Nowlan
1671
 
A heavy warning used to be given that pictures are not rigourous; this has never had its bluff called and has permanently frightened its victims.
Littlewood, quoted in "Pictures and Proofs" by Bill Casselman, Notices AMS, Vol. 47, No. 10, Nov. 2000.
1143
 
There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world.
Nicolai Lobachevsky,
242
 
Logic is the anatomy of thought.
John Locke,
243
 
How many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. We teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone needs is to be trained.
Paul Lockhart, From A Mathematician’s Lament (p. 54)
1648
 
So you would remove mathematics from the school curriculum? The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.
Paul Lockhart, From A Mathematician’s Lament
1642
 
We don’t need to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.
Paul Lockhart, From A Mathematician’s Lament (p. 39)
1643
 
What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources – beautiful works of art by some of the most creative minds in history – in favor of third-rate textbook bastardizations?
Paul Lockhart, From A Mathematician’s Lament (p. 40)
1644
 
School has never been about thinking and creating. School is about training children to perform so they can be sorted. It’s no shock to learn that math is ruined in school; everything is ruined in school!
Paul Lockhart, From A Mathematician’s Lament (p. 91)
1645
 
Mathematical Reality is… a landscape of wonderful, imaginary creatures who engage in all sorts of fascinating and curious behaviors… This place is so breathtakingly beautiful and entrancing that I actually spend a good part of my waking life there. I think about it all the time, as do most other mathematicians. We like it there, and we just can’t stay away from the place.
Paul Lockhart, From A Mathematician’s Lament (p. 92)
1647
 
Of course it is far easier to test someone’s knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning… How sad that fifth-graders are taught to say 'quadrilateral' instead of 'four-sided shape,' but are never given a reason to use words like 'conjecture' and 'counterexample.'
Paul Lockhart, From A Mathematician’s Lament (p. 58)
1649
 
Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus [high school geometry] attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way.
Paul Lockhart, From A Mathematician’s Lament (p. 67)
1650
 
Mathematics is not a language, it’s an adventure. Do musicians speak another language simply because they choose to abbreviate their ideas with little black dots? If so, it’s no obstacle to the toddler and her song.
Paul Lockhart, From A Mathematician’s Lament (p. 53)
1651
 
Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion – not because it makes no sense to you, but because you gave it sense and you still don’t have an idea what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive.
Paul Lockhart, From A Mathematician’s Lament (p. 37)
1652
 
You don’t need me to tell you that your math class was a boring, pointless waste of time – you went through it yourself, remember?
Paul Lockhart, From A Mathematician’s Lament (p. 91)
1646
 
Mathematics is the art of explanation. If you deny student the opportunity to engage in this activity – to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs – you deny them of mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
Paul Lockhart, From A Mathematician’s Lament
1641
 
This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way… On the other hand, once you have made your choices… then your new creations do what they do, whether you like it or not.
Paul Lockhart, From A Mathematician’s Lament
1640
 
That’s what math is – wondering, playing, amusing yourself with your imagination.
Paul Lockhart, From A Mathematician’s Lament
1639
 
There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
Paul Lockhart, From A Mathematician’s Lament
1638
 
The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.
Paul Lockhart, From A Mathematician’s Lament
1637
 
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house.
Lazarus Long, from Time Enough for Love by Robert A. Heinlein.
1454
 
Men of genius are often dull and inert in society; as the blazing meteor, when it descends to earth, is only a stone.
Henry Wadsworth Longfellow,
1437
 
Man is always more than he can know of himself; consequently, his accomplishments, time and again, will come as a surprise to him.
Henry Wadsworth Longfellow,
1436
 
If we could read the secret history of our enemies we should find in each man's life sorrow and suffering enough to disarm all hostility.
Henry Wadsworth Longfellow,
1434
 
There is a fountain of youth: it is your mind, your talents, the creativity you bring to your life.
Sophia Loren, quoted in "A Special Teacher."
244
 
I never am really satisfied that I understand anything; because, understand it well as I may, my comprehension can only be an infinitesimal fraction of all I want to understand about the many connections and relations which occur to me, how the matter in question was first thought of or arrived at, etc., etc.
Ada Lovelace, quoted in Modern Mathematicians, by Harry Henderson.
484
 
If we succeed in giving the love of learning, the learning itself is sure to follow.
John Lubbock,
613
 
Each definition is a piece of secret ripped from Nature by the human spirit. I insist on this: any complicated thing, being illumined by definitions, being laid out in them, being broken up into pieces, will be separated into pieces completely transparent even to a child, excluding foggy and dark parts that our intuition whispers to us while acting, separating into logical pieces, then only can we mover further, towards new successes due to definitions.
Nikolai Luzin, quoted in Naming Infinity by Loren Graham and Jean-Michel Kantor (p. 210)
1560
 

85 quotes found and displayed.